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thesis entitled

ANALYSIS OF THE EFFECT OF AXIAL CONDUCTION ON THE
PERFORMANCE OF A COUNTERFLOW HEAT EXCHANGER

presented by

KEVIN J. DOWDING

has been accepted towards fulfillment
of the requirements for

MASTERS MECHANICAL ENGINEERING

degree in

 

 

 

Major professor

Date 3/3/ M3

 

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ANALYSIS OF THE EFFECT OF AXIAL
CONDUCTION ON THE PERFORMANCE OF A
COUNTERFLOW HEAT EXCHAN GER

By

Kevin J Dowding

A THESIS

Submitted to
Michigan State University

in partial fulﬁllment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1993

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ABSTRACT

ANALYSIS OF THE EFFECT OF AXIAL
CONDUCTION ON THE PERFORMANCE OF A
COUNTERFLOW HEAT EXCHAN GER

By
Kevin J Dowding

An investigation into the effect of axial (longitudinal) conduction
within the wall of a counterﬂow heat exchanger has been completed. The
exact solution was obtained for the temperature ﬁelds of the wall and ﬂuids
while including the possibility of heat ﬂow in the direction of the ﬂuid ﬂow
within the wall of the heat exchanger, so-called axial conduction.

The effect of axial conduction on the performance of the heat
exchanger was quantiﬁed by comparison to a solution that neglects this
effect. The results showed that the effectiveness could be over-estimated up
to 45% under certain conditions. These conditions were a small Biot number,
small ratio of the heat exchanger wall length to thickness, and large ﬂow heat
capacity ratios. The severity of the effect on the performance of the heat
exchanger depended on the magnitudes of these conditions. Predicting the
presence of axial conduction was shown to depend on a dimensionless
parameter called the Mondt number. For magnitudes less than .01 , axial con-
duction was negligible; while magnitudes greater displayed a nonzero axial
conduction, which grew larger as the Mondt number increased.

to my parents, Roger and Mary Ann Dowding

iii

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sions cone:

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ACKNOWLEDGMENTS

I am grateful for the opportunity to work with Professor Somerton; for
his guidance and encouragement conducting this research and the experience
gained in the undergraduate teaching laboratory. Also, his open, frank discus-
sions concerning professorial life were enlightening and appreciated.

The commitment of the remaining committee members, Professors
Beck and Schock, is also recognized. Their comments and input were instruc-
tive. Additional thanks to Professor Beck for his patience while this research
was completed.

The Department of Mechanical Engineering is acknowledged for the
continued generous support. Special thanks are extended to Professors Som-
erton and F oss for the involvement in their teaching laboratories.

Finally, I would like to thank my family, who have endured through
my education and given love and encouragement. My Mother and Dad are
appreciated for their endless unassuming support, without which this point
could have never been reached. While I am eternally grateful for the patience,
understanding, and support of my wife Tracey.

iv

list of Ta
List of Fig

Nomenclc

1 Introdt

1.0 I

2 Dacrit

2.0 1

TAB LE OF CONTENTS

List of Tables ............................................................................. ix

List of Figures ............................................................................ x

Nomenclature .......................................................................... xiv
Chapter

1 Introduction l

1.0 Heat Exchanger Analysis .......................................................... 4
1.0.1 Basic Analysis .............................................................................. 4
1.0.2 Dimensionless Analysis ............................................................... 6
1.1 Literature Review ..................................................................... 9

2 Describing Equations and Boundary Conditions 11

2.0 Dimensional Equations ........................................................... 11
2.0.1 Wall Conduction Equation ......................................................... 11
2.0.2 Fluid Energy Balance Equations ................................................ 12
2.1 Dimensionless Equations ........................................................ 14
2.1.1 Nondimensionalization of Lengths ............................................ l4

2.2

2.3

3 Metht

3.0

3.1

3.3

2.1.2 Nondimensionalization of Temperature ...... - - - -- ...15

 

2.2 Dimensionless Parameters ...................................................... 16

2.3 Heat Exchanger Performance Analysis .................................. 18

3 Method of Solution 20

3.0 General Method ...................................................................... 20

3.1 General Solution of the Wall Conduction Equation .............. 22

3.1.1 General Solution for Nonhomogeneous Boundary
Condition atx = L - - 23

 

3.1.2 General Solution for Nonhomogeneous Boundary

 

 

Condition on y .......................................... 27

3.2 Solution of the Fluid Energy Balance Equations .................... 28
3.2.1 Hot Fluid Formulation ............................................................... 28

3.2.2 Hot Fluid Solution - - - - -- - ..................................... 29

3.2.3 Cold Fluid Solution .................................................................... 31

3.3 Application of Nonhomogeneous Boundary

Conditions ........................................................................ 33

3.3.1 Application of Boundary Condition at x = L ........................... 34

3.3.2 Application of Boundary Condition at y = 0 ........................... 36

3.3.3 Application of Boundary Condition at y = 8 ........................... 38

3.3.4 Summary of Applying Orthogonality ........................................ 40

3.4 Computer Program AXCOND .................................................. 44

4 Raul

4.0

4.1 I

4.2

4 Results and Drscussron47

 

4.0 Identiﬁcation of Input Parameters .......................................... 47
4.0.1 Presentation of Results -- ..................................... 50
4.1 Inﬂuence of Axial Conduction at a Constant NTU ................. 52
4.1.1 Inﬂuence of the Wall Aspect Ratio ............................................ 52
4.1.2 Inﬂuence of the Wall End Biot Numbers ................................... 63
4.1.3 Inﬂuence of the Fluid Biot Numbers ......................................... 69
4.1.4 Summary of Investigation with a Constant NT U ..................... 74

4.2 Characterization of Axial Conduction as a Function

of NTU .............................................................................. 77
4.2.1 Ineffectiveness-NTU Results .................................................... 83
4.2.1.1 Heat Capacity Ratio of 1 ............................................. 83
4.2.1.2 Heat Capacity Ratio of .75 .......................................... 85
4.2.1.3 Heat Capacity Ratio of .50 .......................................... 87
4.2.1.4 Heat Capacity Ratio of .25 .......................................... 89
4.2.1.5 Discussion of Results .................................................. 91

4.3 Predicting the Presence of Axial Conduction with the

Mondt number .................................................................. 96
4.4 Application of Results .......................................................... 102
4.5 Comparison to Published Results ......................................... 108

5 Conclu
List of Re
Appendic

App

 

Appt

 

App<

Appt

Appc
App:

ADpe

Appe

5 Conclusions and Recommendations for Future Work .. 112

List of References ................................................................... 114
Appendices ............................................................................. 116

Appendix A.1: General Solution of the Wall

Conduction Equation ............................ 116
Appendix B. 1: Solution of Hot Fluid Energy

Balance ................................................ 125
Appendix B.2: Solution of Cold Fluid Energy

Balance ................................................ 130
Appendix B.3: Summary of Energy Balance

Solutions ............................................... 136
Appendix C. 1: Application of N onhomogeneous

Boundary Conditions ............................ 138
Appendix C.2: Summary of Applying

Nonhomogeneous Boundary

Conditions ............................................ 156
Appendix D.1: Solution for Constants ........................... 157
Appendix E.1: Derivation of Effectiveness-NTU

Relationship for a Counterﬂow

Heat Exchanger Neglecting Axial

Conduction ........................................... 163
Appendix E.2: Solution of Fluid Temperatures for

a Counterﬂow Heat Exchanger

Neglecting Axial Conduction ............... 167
Appendix F: Listing of Computer Program

AXCOND ............................................. 17 3

 

 

TABLE 4.3

TABLE 4.:

TABLE 4.3

TABLE 4.4

TABLE 4.5

LIST OF TABLES

TABLE 4.1: Input variables and dependencies for determining
heat exchanger performance 49

TABLE 4.2: Relationship between the slepe of the hot and cold

ﬂuids temperature proﬁle and heat capacity ratio 55
TABLE 4.3: Ineffectiveness as a function of the ﬂuid Biot

numbers from Figure 4.14 69
TABLE 4.4: Classiﬁcation of effects seen as the NTU varied 95

TABLE 4.5: Relationship among the Mondt number, wall aspect

ratio, NTU, and ﬂuid Biot numbers 96
TABLE 4.6: Magnitude of dividing Mondt number as a function

of the heat capacity ratio 99
TABLE 4.7: Operating conditions for axial conduction to exist 105

FIGURE

FIGURE
1 FIGURE

FIGURE

 

FIGURE .

FIGURE 1

 

A? '1 l noun-:2

t. i

FIGURE 4
FIGURE 4.
FIGURE 4..

FIGURE 4.4

FIGURE 4.

FIGURE 4.61

 

LIST OF FIGURES

FIGURE 1.1: Possible heat ﬂow paths for wall of a heat exchanger

FIGURE 1.2: Generic heat exchanger, nonspeciﬁc design exchang-
ing energy between two ﬂuids

FIGURE 1.3: Thermal circuit for a heat exchanger wall neglecting
any fouling

FIGURE 2.1: Heat exchanger geometry studied
FIGURE 2.2: Thermal circuit for heat exchanger wall
FIGURE 3.1: Flowchart of solution procedure

FIGURE 3.2: Flowchart of FORTRAN program AXCOND used to
compute solution

FIGURE 4.1: Ineffectiveness as a function of the wall aspect ratio
forNTU = 7 and CR =1

FIGURE 4.2: Ineffectiveness as a function of the wall aspect ratio
for NTU = 7 and CR = .75

FIGURE 4.3: Ineffectiveness as a function of the wall aspect ratio
for NTU = 7 and CR = .50

FIGURE 4.4: Ineffectiveness as a function of the wall aspect ratio
for NTU = 7 and CR = .25

FIGURE 4.5: Median dimensionless wall temperature as a function
of dimensionless position and Biot number for
NTU= 7,L =100,andCR =1

FIGURE 4.6: Ineffectiveness as a function of the wall aspect ratio
and heat capacity ratio for NTU = 7 and all Biot num-
bers equal .001

11
19
21

46

59

59

6O

61

61

 

FIGURE 4.

FIGURE 4.

FIGURE 4.

FIGURE 4.

FIGURE 4.

FIGURE 4.

FIGURE 4.-

 

FIGURE 4.7: Median dimensionless wall temperature as a function
of dimensionless position and heat capacity ratio for
NTU = 7, L = 100, and all Biot numbers .001

FIGURE 4.8: Median dimensionless wall and ﬂuid temperature as a
function of dimensionless position and heat capacity
ratio for NTU = 7, L = 100, and all Biot numbers
equal .001

FIGURE 4.9: Ineffectiveness as a function of the end Biot numbers
forNTU = 7, L =100,and CR =1

FIGURE 4.10: Ineffectiveness‘as a function of the end Biot numbers
for NTU = 7, L = 100, and CR = .75

FIGURE 4.11: Ineffectivenessps a function of the end Biot numbers
for NTU = 7, L = 100, and CR = .50

FIGURE 4.12: Ineffectiveness‘as a function of the end Biot numbers
for NTU = 7, L = 100, and CR = .25

FIGURE 4.13: Ineffectiveness as a function of the end Biot numbers
and heat capacity ratio for NTU = 7, L' = 100, and
ﬂuid Biot numbers equal .001

FIGURE 4.14: Ineffectiveness as a function of the ﬂuid Biot numbers
forNTU = 7,L =100,andCR =1

FIGURE 4.15: Ineffectiveness as a function of the ﬂuid Biot numbers
for NTU = 7, L = 100, and CR = .75

FIGURE 4.16: Ineffectiveness as a function of the ﬂuid Biot numbers
for NTU = 7, L = 100, and CR = .50

FIGURE 4.17: Ineffectiveness as a function of the ﬂuid Biot numbers
for NTU = 7, L = 100, and C, = .25

FIGURE 4.18: Ineffectiveness as a function of the hot ﬂuid Biot
number and heat capacity ratio for NTU = 7,
L = 100, and a cold ﬂuid Biot number equal .001

FIGURE 4.19: Ineffectiveness as a function of the NTU, wall aspect
ratio, and maximum ﬂuid Biot number for
Bil“! = 0.”)1 and CR =

62

62

66

67

67

68

68

72

72

73

73

74

78

 

 

 

 

 

FIGUI

FIGL'R

FIGL'R

 

FIGURE 4.20: Ineffectiveness as a function of the NTU and heat
capacity ratio for Bi”, = 0.0001, Bi”, = 0.001, and
L = 100

FIGURE 4.21: Ineffectiveness as a function of the NTU and wall
aspect ratio for Bin," = 0.0001 and CR = 1

FIGURE 4.22: Ineffectiveness as a function of the NTU and wall
aspect ratio for Bin“, = 0.001 and CR = 1

FIGURE 4.23: Ineffectiveness as a function of the NTU and wall
aspect ratio for Bin," = 0.01 and CR = 1

FIGURE 4.24: Ineffectiveness as a function of the NTU and wall
aspect ratio for Him,” = 0.1 and CR = 1

FIGURE 4.25: Ineffectiveness as a function of the NTU and wall
aspect ratio for Bin,” = 0.0001 and CR = .75

FIGURE 4.26: Ineffectiveness as a function of the NTU and wall
aspect ratio for Bin..." = 0.001 and CR = .75

FIGURE 4.27: Ineffectiveness as a function of the NTU and wall
aspect ratio for Bi,“ = 0.01 and CR = .75

FIGURE 4.28: Ineffectiveness as a function of the NTU and wall
aspect ratio for Bi = 0.1 and CR = .75

min

FIGURE 4.29: Ineffectiveness as a function of the NTU and wall
aspect ratio for Bin," = 0.0001 and CR = .50

FIGURE 4.30: Ineffectiveness as a function of the NTU and wall
aspect ratio for Bin..." = 0.001 and CR = .50

FIGURE 4.31: Ineffectiveness as a function of the NTU and wall
aspect ratio for Bi”, = 0.01 and CR = .50

FIGURE 4.32: Ineffectiveness as a function of the NTU and wall
aspect ratio for Rim." = 0.1 and CR = .50

FIGURE 4.33: Ineffectiveness as a function of the NTU and wall
aspect ratio for Bin,” = 0.0001 and CR = .25

81

83

83

84

84

85

85

86

86

87

87

88

88

89

 

 

 

F'IGUR

FIGUR'

FIGUR]

FIGURE

FIGURE

FIGURE 1

FIGURE 4

 

 

FIGURE 4.34:

FIGURE 4.35:

FIGURE 4.36:

FIGURE 4.37:

FIGURE 4.38:

FIGURE 4.39:

FIGURE 4.40:

FIGURE 4.41:

FIGURE 4.42:

Ineffectiveness as a function of the NTU and wall
aspect ratio for Bi,“ = 0.001 and CR = .25

Ineffectiveness as a function of the NTU and wall
aspect ratio for Bin,” = 0.01 and CR = .25

Ineffectiveness as a function of the NTU and wall
aspect ratio for Bin," = 0.1 and CR = .25

Predicted region affected by axial conduction in
terms of the NTU, Mondt number, wall aspect ratio
and minimum ﬂuid Biot number

Magnitude of pertinent heat exchanger operating
conditions as a function of wall thermal conductivity,
which is needed for axial conduction to exist

Comparison of present results to past result of Pan,
Welch, and Head [15] and Rohsenow [17]. Ineffec-
tiveness as a function of the NTU and wall aspect
ratio for Bin," = 0.0001 and CR = 1.

Comparison of present results to past result of Pan,
Welch, and Head [15] and Rohsenow [17]. Ineffec-
tiveness as a function of the NTU and wall aspect
ratio for Bin..." = 0.001 and CR = 1

Comparison of present results to past result of Pan,
Welch, and Head [15] and Rohsenow [l7]. Ineffec-
tiveness as a function of the NTU and wall aspect
ratio for Bin," = 0.01 and CR = 1

Comparison of present results to past result of Pan,
Welch, and Head [15] and Rohsenow [17]. Ineffec-
tiveness as a function of the NTU and wall aspect
ratio for Bin," = 0.1 and CR = 1

89

98

104

110

110

111

111

Arabic

 

 

0‘

Arabic
A
An’ Bn’ Cn

Bi

m- 63.9

. tau-a»

1~1-‘

23-

N TU

Q.~Q"u

a

Q‘l"

E

NOMENCLATURE

area [m2]

constants

Biot number

speciﬁc heat [Pi—IE]

heat capacity [WI K]

enthalpy [J]

speciﬁc enthalpy [— lg]
convective heat transfer coefﬁcient [n ]
thermal conductivity [in—K]
length of heat exchanger [m]
wall aspect ratio (L’ = 55')
mass ﬂow rate [s—ﬁc— 3]

number of termssec in series solutions
number of transfer units
perimeter [m]

heat [W]

heat ﬂux [—]

thermal resistance [— W]

time [sec ]

temperature [C or K]

thermal conductance [if—IE ]
heat exchanger wall windth [m]

xiv

 

 

 

 

 

 

Ly

Greek

min

”101

out

Superscr-

 

x,y

Greek

comm
1::

@@

Subscripts
C

H

in

L

min

max

out

Superscripts

+

Cartesian coordinates [m]

effectiveness

wall thickness [m]
kronecker delta function
scaled temperature [C or K]
dimensionless temperature

eigenvalues

cold ﬂuid wall side

hot ﬂuid wall side

inlet

1: = Lorx” =1wallend
minimum

maximum

x = 0 wall end

outlet

ratio

wall

dimensionless variable

XV

 

 

 

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Chapter 1

Introduction

Heat exchangers provide for the transfer of heat between two moving
ﬂuids. These devices are used in power generation, chemical and food pro-
cessing, heating and air conditioning, and motor vehicles. It is the most rec-
ognizable heat transfer device and one of the most widely used. Heat
exchangers are classiﬁed based on ﬂow arrangement and type of construc-
tion. There are many types of heat exchanger designs, each type with its own
characteristics that make it suitable for a particular application. The simplest
form of a heat exchanger would be a double pipe (or concentric tube) heat
exchanger. The construction of this type of heat exchanger consists of two
moving ﬂuids separated via a wall parallel to the ﬂuid motion; with the ﬂuids
moving in the same direction for a parallel ﬂow heat exchanger and in oppo-
site directions in a counterﬂow heat exchanger.

Design of heat exchangers in general, requires consideration of the
heat transfer occurring between the two ﬂuids in addition to the mechanical
energy needed to overcome the frictional forces to move the ﬂuids through
the heat exchanger. These two design criteria can be generally classiﬁed as
heat transfer and pressure drop. It is typically desired to achieve large heat
transfer yet maintain a small pressure drop. Large heat transfer rates can be
obtained by either having a larger heat transfer area or having a larger heat
transfer convection coefﬁcient. Unfortunately, both of these conditions cause
an increase in the pressure dr0p, since a larger area gives more frictional area
resulting in an increase in the pressure drop and the larger ﬂow rate to
increase the convection coefﬁcient would likewise increase the pressure drop.
There is a trade-off in these two design criteria; a beneﬁcial gain in one crite-

ria is usually at the expense of the other criteria and a compromise must be
established.

Focusing on the heat transfer, consider a simplistic heat exchanger that
has two ﬂuid streams separated by an inﬁnitesimally thin wall. The analysis

 

Cold Fluid

4c (1)

 
 
 

Heat Exchanger
Wall

9110‘)

Hot Fluid

 

 

 

Figure 1.1. Podble heat ﬂow paths for the wall on heat exchanger

of this heat exchanger could be accomplished by considering the energy bal-
ances of the two ﬂuids since the wall has negligible thickness. In reality, the
wall will be required to have some thickness to perform its function of sepa-
rating the two ﬂuids. As the thickness of the wall becomes nonnegligible,
some resistance to the ﬂow of heat between the two ﬂuids is added and now
the wall must be considered in the analysis. In addition, an alternate path for
the heat to ﬂow is provided. Figure 1.1 shows the heat ﬂow paths: q” is the
heat convected from the hot ﬂuid to the wall, q Ma, is the heat conducted
along the wall, and qc is the heat conducted to the cold ﬂuid from the wall.
When the wall is very thin, it is unlikely that heat transferred to the wall from
the ﬂuid will ﬂow parallel to the wall and the energy exchanged will balance
locally (qH (x) = qc (x) ). This may not be the case when the wall thickness is
increased; some of the heat may ﬂow along the wall. This effect is call axial
(longitudinal) conduction, and when axial conduction is nonzero(qAx,.a, = 0)
the ﬂuid streams may have a local energy imbalance (q H (x) at qc (x) ). The
energy is conserved, however.

Axial conduction of heat is typically neglected in the analysis of the
heat transfer in a heat exchanger. This assumption may be appropriate for
most cases, but under certain circumstances the effect of axial conduction can
become important. These circumstances depend on more than the wall thick-
ness, which was used to introduce the eﬁ‘ect. In general, axial conduction
depends on the material and physical dimensions of the wall and the proper-

 

 

ties and ﬁt
be the focr
quantiﬁed

ble identiﬁ
heat excha

The
heat exchai
tions and b
two, while
three. Chap
sponding d
in chapter f

ties and ﬂow rate of the ﬂuids. Assessing the effect of axial conduction will
be the focus of this thesis. Speciﬁcally, the effect of axial conduction will be
quantiﬁed and the conditions of the wall and ﬂuids for which it is nonnegligi-
ble identiﬁed. Also, the adverse effect on the performance of a counterﬂow
heat exchanger will be determined.

The remaining sections of this chapter will review the analysis of a
heat exchanger and conclude with a literature review. The describing equa-
tions and boundary conditions for the problem will be addressed in chapter
two, while the methods employed to solve the problem are shown in chapter
three. Chapter four will present the results of the investigation with corre-
sponding discussion. A summary and the resulting conclusions will be given
in chapter ﬁve, in addition to recommendations for future work

 

 

ci

 

 

116

1.0 Heat Exchanger Analysis

1.0.1 Basic Analysis

Figure 1.2 shows a heat exchanger that transfers energy between two
moving ﬂuids through a wall; the geometry and construction of this heat
exchanger may be considered arbitrary. A general thermal analysis of this
heat exchanger will be performed. First, taking the heat exchanger as a con-
trol volume and applying an overall energy balance, assuming no interactions
(work or heat) with the surroundings and steady state, results in an enthalpy
balance (it)

HI» = Hm (1.1)

which can be written in terms of the inlet and exiting conditions using spe—
ciﬁc enthalpy and mass ﬂow rate as

muﬁH|In+mcﬁclin = mﬂﬁulout+mcﬁclout (1'2)
and rearranged to group the ﬂuids
) = martial - incl“) (1.3)

Assuming the ﬂuid behaves as an incompressible liquid or an ideal gas and a
negligible pressure change, the change in enthalpy can be expressed as

"In (511", - 511'

0 I“ 0“!

din = deT (1.4)

 

 

  

Hot Fluid

«1

Cold I‘Iuld

   

 

    

 

 
 

 

ﬁgmlJ.Gmeﬂehuuchng«,noMpuEededgnuchngtngenugybemnthetwoﬂutds

 

5

which can be approximated by differences(dlr ~ 712 - 71, ). Using the approxi-
mate form of equation (1.4) in equation (1.3), the results for the overall
energy balance

are

q = mHCp, H (TH, in - TH, out) = mCCp, C (TC, out — TC, in) (1'5)
Introducing the heat capacity

Ca Inc, (1.6)

equation (1.5) can be rewritten as

q = CH (TH, in ' T1101“) = Cc (Team " Tc, in) (1:7)
In general, the heat transfer between the two ﬂuid streams is a function of the
following six parameters:

(1.8)

 

 

*

q = f(e my» "'0 TH,in’ Tarn, . Teen: TILoutl )

given unknown
where the ﬁrst four are typically given design parameters and the last two are
desired results. Thus, in order to obtain a solution more information is
needed. The additional information will come from the heat transfer analysis
of the wall separating the two ﬂuids.

A circuit describing the thermal communication of the two ﬂuids is
shown in Figure 1.3. This circuit shows the resistance that impedes the heat
ﬂow, neglecting any fouling on the heat exchanger wall. The total resistance
of the series circuit is the sum of the individual resistances

 

 

T T
” e—va w W—e C

 

 

 

 

Figure 1.3. Thermal circuit for a heat exchanger wall neglecting any fouling

R

Also, the overall conductance of the heat exchanger wall, that will give the
total amount of heat transferred if the temperature difference is known, is

m = Ru+RW+Rc (1.9)

UA = J— (1.10)

. tot

These terms are analogous to an electrical circuit, with the temperature differ-
ence as the voltage potential and the heat ﬂow as the current. The heat trans-
fer over a differential element of length (11: is

dq = UP [TH (x) - Tc (x) ] dx (1.11)

The heat transfer over the entire length of the heat exchanger is the sum of
these differentials elements over the total length

L L
q = jdq = UP] [7,,(x) -Tc(x)] dx (1.12)
0 0

The product UP was assumed constant and thus could be taken out of the
integral, but (TH - Tc) depends on x and cannot be removed from the inte-
gral. Because this integral is not known in general, we will deﬁne the mean
temperature difference as

L
(AT)... = H [1,,(x) -Tc(x)]dx (1.13)
0

Substituting equation (1.13) into equation (1.12), the total heat transfer can
be written as

q = UA (A1)" (1.14)

For a double pipe heat exchanger, equation (1.13) can be evaluated.
The result is the log mean temperature difference. However, for geometries
that are more complicated, evaluating equation (1.13) is difﬁcult if not
impossible. This suggests that another approach may be necessary to remain
a general analysis.

1.0.2 Dimensionless Analysis

To introduce another approach, consider the parameters that the mean
temperature difference is a function of

It depends on the inlet temperatures and heat capacities of the ﬂuids and the
wall conductance. The number of independent parameters can be reduced by
considering a second function times the temperature difference at the inlet

(A1)", = 3 (CC, C”, UA) (TH, .1.- TC, ,n) (1.16)

substituting equation (1.16) into equation (1.14) the functional dependence
for the heat transfer is given by

‘1 = UAg (Co CH: UA) (711.111 " Tarn) (1.17)

As typical in heat transfer, scaling will be introduced to provide dimension-
less parameters. Deﬁne the maximum possible heat transfer as

qrnax = Cmin (TH, in - TC, in) (1°18)
where Cm." = min (CH, CC) . Dividing equation (1.17) by equation (1.18)

_‘1 = ”A 3 (CC, C”, 0.4) (1.19)
anax Carin
and introducing another function that is in terms of the scaled independent

variables gives

 

 

= g (C , _) (1.20)
qrnax Cmin R Cmin
where
C .
C = "'"' (1.21)
R Curax

Finally, noting that equation (1.20) can be written for a ﬁnal function as

UA
C

ruin

This demonstrates that the performance of a heat exchanger can be expressed
in terms of three dimensionless variables. The ﬁrst was given previously in
equation (1.21) as the ratio of the heat capacities. The second is the effective-
ness

i=h(CR,

anax

) (1.22)

 

q q
a = —_ = ._ — — (1.23)
gm Carin (TH, in - TC, in)

 

 

 

,.
[:u.‘ r"~ - 3"?

 

which is the 1
maximum p0
last dimensio

which is a rat
mum ﬂuids 2

The pr:
moderately si
analysis of th.
sider a double
describing the
in Appendix I

for the case of

which is the ratio of the actual heat transfer given by equation (1.7) to the
maximum possible heat transfer. The number of transfer units (NTU) is the
last dimensionless parameter

UA
min
which is a ratio of the heat exchangers ability to transfer energy to the mini-
mum ﬂuid’s ability to retain energy.

N TU = (1.2/I)

 

The previously mentioned dimensionless parameters, at least for the
moderately simple heat exchanger geometry, can result during the analytical
analysis of the heat transfer occurring in a heat exchanger. For example, con-
sider a double pipe counterﬂow heat exchanger arrangement. The correlation
describing the performance of this heat exchanger, whose derivation is shown
in Appendix E, is

1_e-NTU(1-C.)

= 1 - Cge'muu - c,) (1.25)

 

for the case of CR :1: 1. If CR = 1, equation (1.25) reduces to

e ___ NTU

1+NTU (126)
This is the usual form used in evaluating or presenting the performance of a
heat exchanger, as a function describing the relationship between the three
dimensionless parameters. These functional relations are tabulated for many
different heat exchanger geometries and are the basis for easily predicting the
performance of a heat exchanger design.

The derivation of these functional relationships typically assumes neg-
ligible axial ﬂow of heat in the wall. The conditions for which this assump-
tion is true for a counterﬂow heat exchanger geometry is the focus of this
thesis. Furthermore, the amount that the performance is over-estimated by
assuming axial conduction is negligible will be addressed.

1.1 Literature Review.

The counterﬂow heat exchanger theory previously presented was for
the idealization of no axial conduction (in the ﬂow direction) of heat in the
wall or ﬂuids. It is not apparent under what conditions this assumption is not
valid, but the consequences can be an over-estimation of the effectiveness for
a given NTU by as much as 45%, as shown by Rosenhow [17].

Some of the earliest work performed in analyzing the effect of axial
conduction was done by Mondt [9]. By inspecting the ﬁnite difference equa-
tions for a solution that includes axial conduction, he proposed using a con-
duction parameter to correlate the results. This conduction parameter was

_ Kw...
" 1.me
and he showed that to accurately predict the convection from a surface at low
Reynolds numbers requires consideration of longitudinal conduction.

A (1.27)

In later works, the investigation was less speciﬁc and looked to predict
the inﬂuence of axial conduction on the performance of the heat exchanger.
Landau and Hlinka [8] and Pan and Welch [11] developed exact solutions to
predicting the performance of a counterﬂow heat exchanger while including
the effect of axial conduction. The solutions were in the form of ﬁnite series
and were algebraically involved. A general investigation was difﬁcult, if not
impossible, with the availability of the computing power at the time.

However, Pan, Welch and Head [10], were able to improve on their
earlier work [11] and predicted analytically the effect of axial conduction on
the performance of the heat exchanger as a function of the NTU and a “modi-
ﬁed-conduction- ﬂowrate-ratio”

N ____ KM... M

K mcp Lure“p
Using the exact solution, the existence of axial conduction was predicted for
a counterﬂow heat exchanger with balanced symmetric ﬂow, that is equal
mass ﬂow rates and convective heat transfer coefﬁcients for both ﬂuids.
Although more general than their past work, the balanced symmetric ﬂow
was still rather restrictive.

 

(1.28)

 

 

 

 

 

T
address:
incorp01
tion par:
Mondt 11

He then s
Mondt nr
worst and
mance of
large Mon
nant
Thi:
and solutic
ﬂow heat C
([8]. [10], l
90‘ needed
W 011'
tron to desc
diffePential
to solve the
eaSIIy cyalu
“has to obta
0f the heat e

10

The most recent work was performed by Rohsenow [17], who
addressed reasons that laminar ﬂow heat exchangers perform poorly. He
incorporated a similar formulation as [8], [10], and [11], but used the conduc-
tion parameter above in equation (1.27), from Mondt [9], which he called the
Mondt number

Kw...

M0 = mep
He then solved for the performance of the heat exchanger at the limits of the
Mondt number, zero and inﬁnity. The extremes of the Mondt number provide
worst and best case scenarios. A Mondt number of zero gives the perfor-
mance of the heat exchanger when axial conduction is negligible; while a
large Mondt number gives the performance when axial conduction is domi-
nant.

 

(l .29)

This investigation shall differ from previous work in the formulation
and solution of the problem. All previous investigations involving a counter-
ﬂow heat exchanger have neglected temperature gradients normal to the wall
([8], [10], [l l], [17]). Although this assumption does seem reasonable it is
not needed for this investigation to obtain a solution; whereas past studies
required this assumption. This diﬂerence results in a partial differential equa-
tion to describe the wall temperature for this study, instead of the ordinary
differential equation obtained for past studies. However, the techniques used
to solve the partial differential equation will allow for a solution that is more
easily evaluated; while past solution techniques required special ﬂow condi-
tions to obtain a solution. Therefore, by considering the two-dimensionality
of the heat exchanger wall the problem was more mathematically compli-
cated in comparison to past studies, but allowed for a more general investiga-
tion of the heat exchanger design parameters.

Chapter 2

Describing Equations and Boundary
Conditions

2.0 Dimensional Equations

2.0.1 Wall Conduction Equation

The geometry is shown in Figure 2.1, a two dimensional wall with hot
ﬂuid entering at one end and cold ﬂuid entering at the other. The problem was
formulated as a boundary value problem on the wall assuming steady state
and constant properties. The differential equation and boundary conditions
corresponding to the plane geometry with the given assumptions are

 

 

 

a’rw a’rw
a—XE +5?- = O (2.1)
or",
1&5; x=0 = halTWWJ) -TOI (2.2a)
at},
““53 “L = hLlTw(L, y) -TL] (2211)
Cold Fluid
Tom 4—
3 :2 WW
1: = 0 x = L
'—'> T3 (I)
Hot Fluid

 

 

 

 

Figure 21. Heat exchanger geometry sturﬂINote, the analysis was perErmed on a per
unit Width basis.

11

12

 

-K"3§wly=o = h” IT" 0‘) - T”(x’0)] (2.2c)
BTW
_K‘Q} ”5 = hen}, (x, 8) -Tc(x)] (2.211)

where boundary conditions of the third kind were used to represent the most
general case and still allow for boundary conditions of the ﬁrst and second
kind by adjusting the magnitudes of the constants (Kw, ha, 11,, I)”, he) .

2.0.2 Fluid Energy Balance Equations

The describing equations for the ﬂuids were formulated using an energy
balance on a diﬂ’erential length element of the heat exchanger. For the hot
ﬂuid the energy balance on a differential element of length Ax can be written

Hlx = HI” Ax+ q"w,,PHAx (2.3)
Substituting mass ﬂow rates and the speciﬁc enthalpy and rearranging gives
m£|x+ Ax "' litiil
Ax

Taking the limit as Ax -) 0 and assuming the ﬂuid behaves as an ideal gas or
an incompressible liquid (deT = din) gives

x +quwauPH = O (2.4)

 

. 47' H
(mHCp. Iﬂd—x '1’ ‘1"wauPH = 0 (25)
The heat ﬂux from the wall from Newton’s law of cooling is

4"...11 = hHITH (x) - T... (x. 0)] (2.6)
substituting into equation (2.5) and rearranging, results in the ﬁnal describing
equation for the hot ﬂuid

(IT 111’
”+111!

E mucp,[T"(x) -T,,,(x. 0)] = 0 (2.7)

Prescribing a constant temperature at the inlet gives the initial condition

1,,(0) .-.- TH’ 1.. (2.3)

Using a similar analysis, the describing equation for the cold ﬂuid and
corresponding initial condition are

l3

ch hcpc
,. — T = O (2.9)
T + .CCpc [Tw (x, 8) C (x)]

TC (L) = TC, 1.. (2.10)

14

2.1 Dimensionless Equations

2.1.1 Nondirnensionallzation of lengths

To remove any dimensional length dependence on the formulation of
the problem the length scales were nondimensionalized using the characteris-
tic lengths of the wall

3
L (2.11)
+ y
y =

8 (2.12)
Applying this nondimensionalization to the previously presented equations
for the wall and ﬂuid temperatures results in equations that only have units of
temperature, which were made dimensionless after solving (to be shown
later) for algebraic ease and for the physical insight available with units of
temperature. The wall conduction equation and boundary conditions are

371,, L2 3’1,

 

 

 

— — —- = 0 (2.13)
8x+2 52 ay+2
31,, +
"T +BioTw(0.y ) = BioT, (2.1421)
ax x’=0
or”,
— +BiLTw(19y+) = BiLTL (2.1411)
ax x‘ l
a W . + . +
”—I +B'nTw(x :0) = Br,,T,,(x ) (2.14c)
ay yO_o
an, . + . +
57 +B'cTw(x 11) = Bic Tc(x ) (2.141)
2’

 

y’=1

where dimensionless parameters introduced in the equations are deﬁned in
the next section.

Similarly, nondimensionalization of the equations for the hot and cold ﬂu-
ids gives

15

j{IEIHM(THO?) -T.,(x*.0)] = 0
dx“

711(0) = 771.1»
£0+NC[T,,(x+, 1) —Tc(x*)] = 0
dx+

Tc“) = Tc...

2.1.2 Nondimensionalization of Temperature

(2.15)

(2.16)

(2.17)

(2.18)

The problem was solved as presented in equations (2.13-2.18), giving the
solution with units of temperature only. The following equations were used to

present the results in a dimensionless form:

_ T..(x*.y*)_— Ta...

 

 

 

9 (x+9 +) ""
w y TH,in"-TC,in
T (x+)-T
9110‘” = I; -T C'in
H.111 C,in
T x+ -T
9C(x+) = C( ) C,in

TH, in - TC, In

(2.19)

(2.20)

(2.21)

 

 

 

2.2 Dim'

Th
alization .
madcal 11
wall cond
thickness

As this pa
tion must
must be is
likely the
duction is

The dir
the wall, e

16

2.2 Dimensionless Parameters

The dimensionless parameters that resulted during the nondimension-
alization of lengths in the equations will provide a link between the mathe-
matical model and the physical problem. The dimensionless parameter in the

wall conduction equation, equation (2.13), is a ratio of the wall length to
thickness

L‘ 3% (222)
As this parameter is increased, the change in heat conduction in the x-direc—
tion must also increase. Therefore axial conduction must increase; or the wall
must be isothermal in the y-direction (normal to the wall). The later is most
likely the dominant effect, but there may be values of L’ for which axial con-
duction is the dominant effect.

The dimensionless parameters that appear in the boundary conditions on
the wall, equations (2.14a-d), are Biot numbers

71,1.

Bin 1! 7?: (2.23)
Bi L I Elf—f (224)
Bi :1 a hr”; (225)
Bi C I hTC-s (2.26)

These parameters are expected to have the most inﬂuence on the solution and
axial conduction. The Biot numbers on the hot and cold sides of the wall,
equations (2.25-26), should be more inﬂuential than the Biot numbers at the
ends of the wall. Biot numbers, in general, give a ratio of the convection from
a surface of the body to the conduction through the body. This provides
insight into the temperature difference that will exist across the body com-
pared to the temperature difference between the surface and convecting ﬂuid.
For large Biot numbers the convection dominates, resulting in a large temper-
ature gradient across the body and a surface temperature that is close to the
convecting ﬂuid temperature. Conversely, for small Biot numbers the con—

 

 

duction v
and a larg

Th
of transfe
ters are

where the

Using equ
I61] In [em

which relal

17

duction will dominate, giving a small temperature gradient across the body
and a large difference between the surface and convecting ﬂuid temperatures.

The dimensionless parameters for the ﬂuids are similar to the number
of transfer units (NTU), but based on the pertinent ﬂuid only. These parame-
ters are

 

 

 

h h
N” = WA” -.- .55! (2.27)
mHCpn C”
h I:
NC = .cgc = 34C (2.28)
m Pc C
where the area in equations (2.27-28) is
A” = AC = Lw (2.29)

Using equation (2.29) with the heat capacity, equations (2.27-28) can be writ-
ten in terms of the corresponding Biot number

Bi K L’w

N1! = JG";— (2.30)
Bi K L'w

”c = _Cé_ (2.31)

which relates the wall parameters to the ﬂuid parameters.

 

23 Heat]

Solutior
ture ﬁelds 1
wall. But ti
heat excha]
mance ﬁ'on
lation neglc
terms of tht
These dime
eters from I
exchanger’:

Effectivl
considered
may be con
imbalance 1

energy bala

and the effe

18

2.3 Heat Exchanger Performance Analysis

Solution of the equations presented up to this point provides the tempera-
ture ﬁelds for the wall and ﬂuids, without neglecting axial conduction in the
wall. But the temperature as a function of the position along the length of the
heat exchanger is not the desired result. It is desired to compare the perfor-
mance from this solution with one that neglects axial conduction. The corre-
lation neglecting axial conduction was presented in equations (1.25-26) in
terms of the effectiveness, number of transfer unit, and heat capacity ratio.
These dimensionless parameters must be related to the dimensionless param-
eters from the mathematical solution to make the comparison in the heat
exchanger’s performance when axial conduction is included.

Effectiveness, deﬁned in equation (1.23), will depend upon which ﬂuid is
considered when axial conduction is included because some of the energy
may be conducted out the ends of the heat exchanger and result in an energy
imbalance between the ﬂuids. Thus, when including axial conduction the
energy balance on the ﬂuids are, from equation (1.5)

‘IH = CH (TH, in " TH, out) if 9c = Cc (Tc, out " Tc, in) (232)
and the effectiveness must be deﬁned as a function of the ﬂuid

qt!

8 = — (2.33)
” 9......
5c = _qC (2.34)
9
max

The deﬁnition of NTU is given in equation (1.24) and will not depend on
the inclusion of axial conduction, but must be related to the dimensionless
parameters used to model the physical problem. This was accomplished by
evaluating the thermal circuit that describes the heat exchanger wall. The
thermal circuit, neglecting any fouling, is shown in Figure 2.2, which is simi-
lar to the previously presented thermal circuit (except values for the individ-
ual resistances are shown for the geometry studied). The total resistance of
the circuit is evaluated using equation (1.9). The conductance for the heat
exchanger wall given by equation (1 . 10) is

l9

 

 

 

O—W 'ww «MA—o

1 6 1
hHAH Kwﬁw hcAc

 

 

 

 

Figure 2.2. Thermal circuit for heat exchanger wall

 

 

 

UA=-1—- = [_L.+_5 + 1 I1 (2.35)
Rm hHAH KM») ”MC
and NTU easily follows:
HA 1 1 8 1 '1
NTU = — = + + ] (2.36)
Cnu‘n Cmin[hHAH KHAN! hC‘AC

Relating equation (2.36) to the mathematical formulation was accomplished
through equations (2.27-28); substituting these equations into equation (2.36)
gives

1 1 8 1 '1

”T” ‘ c7; [~74 " V. * N—ccc]
Noting that A, = Lw, equation (2.37) can be rewritten as a function of dimen-
sionless parameters and the wall thermal conductivity using equations (2.30-

31), after some arranging it becomes

(2.37)

 

waL’ [ BiHBiC ]

7:," 31,,+ BiHBic+ Bic

N TU = (2.38)

The equations to be solved for the temperature ﬁelds of the wall and ﬂuids
are given in equations (2.13-2.18). Using the solution, the performance of the
heat exchanger can be presented with equations (2.33), (2.34), and (2.38) in
terms of heat exchanger dimensionless parameters, which can be compared
to solutions neglecting axial conduction to quantify the effect of axial con-
duction.

Chapter 3

Method of Solution

3.0 General Method

The problem presented in equations (2.13-18) is a system consisting of
a second order homogenous partial differential equation (PDE) with four
nonhomogeneous boundary conditions coupled with two ﬁrst order nonho-
mogeneous ordinary differential equations (ODE). The coupling occurs
through the boundary conditions of the PDE and in the nonhomogeneous
term for the ODEs. Thus, if a general solution can be obtained for the PDE
without evaluating all boundary conditions, that is in terms of a product of
unknown constants and functions, the solution can be substituted into the
ODEs and the ODEs solved as a function of the unknown constants. The
ODEs solution’s can then be substituted back into the boundary conditions of
the PDE and the unknown constants evaluated. With the constants speciﬁed,
the explicit solutions for all differential equations can be generated. This is
the general procedure used to solve the set of differential equations and is
shown schematically in Figure 3.1.

This section is intended to present the methods that were applied to
solve the problem. However, when the same methods were applied more than
once they are not shown in detail after the ﬁrst application. Also, some meth-
ods contain details that are not necessary in understanding a method, and
whose inclusion would overburden this section. For these reasons, the reader
is referred to the Appendices (A-D) for additional details about the solution.

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3.1 Genet

Sino
ferent matk
separation
of this met
ferential cc
lem has f0]
introducin;

one of the 1
matical for

22

3.1 General Solution of the Wall Conduction Equation

Since the problem is linear, the PDE could be solved using many dif-
ferent mathematical approaches; Fourier transforms, Green’s functions, or
separation of variables. The method chosen was separation of variables. Use
of this method requires that only one nonhomogeneous term exist in the dif-
ferential equation or boundary conditions. As presently formulated the prob-
lem has four nonhomogeneous boundary conditions equations (2.14a-d). By
introducing scaling on the temperature

9,,(x*.y*) = T,.(x*.y*) -T, (3.1)

one of the nonhomogeneous boundary conditions is eliminated. The mathe—
matical formulation with the scaled temperature becomes '

 

 

 

 

 

a’e , a’e
3+‘;’+L2 83:0 (3.2)
x y
30w . +
___; +8106“: (0’)) ) = O (3.3a)
x .
x =0
86w . + .
5;: +BzL6w(l,y ) = BtL(TL—T0) (33b)
x‘=1
33:" + Riﬂe“, (x+, 0) = Bi,,[T,,(x+) - To]
y y’=0 (33¢)
ml? 0 + _ . 4-
5y— +3100», (3: ,1) _ Blc[Tc(x )-T,] (33d)

 

y’ =1

Three nonhomogeneous boundary conditions, equations (3.3b-d), still exist.
Since the problem is linear the principle of superposition may be applied. The
problem is partitioned into a set of simpler problems that contain one nonho-
mogeneous boundary condition, and can be solved using separation of vari-
ables techniques. Then the solution to equations (3.2-3) is obtained by
application of superposition. The problem in equations (3.2-3) can be repre-
sented by three simpler problems, since there are three nonhomogeneous
boundary conditions. Using superposition the solutions are related as fol-
lows:

’thh‘v‘Tﬂ'Z‘i ‘ III-In).

 

 

I:

where 9‘. is

23

3
9,,(x".y+) = 20,.(x+.y+) (3-4)
i=1

where 0, is the solution to the problem given mathematically as

 

 

 

 

 

3’9, ,2 3’9,
+L — = 0 (3.5)
ax+2 ay+2
-'—: +Bioeg (0’)”) = O (3.63)
x .
x =0
5—} +BiL0‘.(O,y+) = SliBiL(TL—To) (3.6b)
x x’=l
803 . + . +
y y0=o
Bel
— +Bice‘.(x*, 1) = 53gBicch(x+) —T0] (3.6d)
8y y*=1
wherei = 1,2,3 and
8.. = kroneckerdelta = { 0 “‘1' (3.7)
1' 1 i=j

It can be shown by adding the three problems (i = 1, 2, 3) stated above that
the originally posed problem in equations (3.2-3) is obtained.

3.1.1 General Solution for Nonhomogeneous Boundary Condition at x = L

The separation of variables technique begins by assuming a product
solution for 91 of the form

91 = F(x*) G (y*) (3.8)

Substituting into the differential equation and boundary conditions, equations
(3 .5-6). yields

F, ,G+L"G , ,F = 0 (3.9)
H H

- GFx, +BiaGF(O) = o (3.103)

{=0

 

 

 

where the f(

Rearranging
tion (3.10) 1

Where a 001
(3.12). Sing
Only a func
StanL This
tial (limiter
6(Nation (3
Sign is ch01
Omogeno‘

24

 

 

 

CF", 2 =1+13i,,(;F(1) = BiL(TL— To) (3.101.)
—FG , +Bi,,m(0) = o (3.10c)
y y* =0
FGy, +BiHFG(1) = 0 (3.10d)
y’ =1

where the following convention for representing derivatives was used:

2
a F 3F . . (3.11)
8x+2 “‘ ‘

 

Rearranging equation (3.9) to group similar variables and simplifying equa-
tion (3.10) gives

 

 

 

 

 

l Fx‘x‘ = _ G”). = i 2 (3 12)
[1.2 F G u '
—F, +BioF(0) = o (3.13a)
x x0 = 0
017‘, . 1+BioGF(l) = BiL(TL-To) (3.131.)
x =
-G . +Bi,,o(0) = o (3.13c)
y y* = 0
Gy, +Bi,,G(1) = 0 (3.13.1)

 

y‘ = 1

where a constant, 1:112 (the eigenvalue), has been introduced in equation
(3.12). Since the left hand side is only a function of x+ and the right hand side
only a function of y+ , to obtain equality both sides must be equal to this con-
stant. This key result allows for separation of the variables, reducing the par-
tial differential equation to two ordinary differential equations. Before
equation (3.12) can be separated a sign must be chosen for the constant. This
sign is chosen to produce an eigenvalue problem for the function that has two
homogenous boundary conditions to evaluate. The eigenvalue problem will
have a general solution in terms of sine and cosine functions for this Carte-
sian coordinate system. After choosing the sign on the eigenvalue, two ordi-
nary differential equations can be written from equation (3.12) with the
appropriate boundary conditions from equation (3.13)

_ ‘2 2 _
Fr; L [1 F — 0 (3.14)

 

l'w- . :5

 

 

Note that eq
(3.16) has tv
tion (3.13b),
to either fun
tion Thus, t
complete so'

Equat
The general

Applying th
the two C01]:

WhiCh Can t
tiOn

25

 

 

-F . +BioF(0) = 0 (3.153)
x {=0
G , , +1120 = O (3.16)
y y
-G . +BiHG (0) = 0 (3.173)
y y0=o
G , +Bi”G(l) = 0 (3.17b)
y y*=1

 

Note that equation (3.14) only has one boundary condition, whereas equation
(3.16) has two boundary conditions. The reason for this can be seen in equa-
tion (3.13b), which depends on both functions F and G and cannot be applied
to either function singularly. This is the nonhomogeneous boundary condi-
tion. Thus, this boundary condition must be applied after assembling the
complete solution.

Equations (3.14) and (3.16) have easily obtainable general solutions.
The general solution for equation (3.14) is

F(x+) = Alcosth‘x”) +A,sinh(uL‘x+) (3.13)

Applying the boundary condition, equation (3. 15a), gives a relationship for
the two constants
Bio
A2 = uL' A1 (3.19)
which can be substituted into equation (3.18) and rearranged giving the solu-
tion

 

F(x”) = A2 [uL‘ cosh (uL’x‘) +Biosinh (uL‘x+)] (3.20)
The solution of equation (3. 16) is

G (3") = A3008 (11f) +A4sin (11)”) (3.21)

Applying the boundary conditions in equation (3.17a) solves for the relation-
ship between the constants

_ BiHA3
' u
Substituting into equation (3.21) and rearranging gives

 

A4 (3.22)

.‘V'1'I‘I‘I '

 

Lg

Applying C
any inform
equation. 1
equation 51

To meet [ht
satisﬁed fo:
it has an in.
solved to (11
is then

With 11, g1V«
Subm
Vide the sol;
‘0 equation
solution, A].
one constam
dition, aqua
rate and Was

26

60*) = A. [ucos my“) +Biysin (uy*)] (3.23)
Applying the ﬁnal boundary condition, equation (3.17b), does not provide
any information about the constant A 4 because the constant cancels from the
equation. However, it does provide information about the eigenvalue ()1). The
equation simpliﬁes to

(BiH+ Bic) u”
tan (tin) = 2 . .

To meet the boundary condition in equation (3. 17b), equation (3.24) must be
satisﬁed for all values of u". Equation (3.24) is a transcendental equation and
it has an inﬁnite number of solutions, hence the subscript on n. It will be
solved to determine the possible values of u". The solution for the function G
is then

(3 .24)

 

G(u,,.y*) = A2 [u,cos my) +Biusin (u,y*)] (3.25)
with u” given by the solution of equation (3.24).

Substituting equations (3.25) and (3.20) into equation (3.8) will pro-
vide the solution. However, since there exist an inﬁnite number of solutions
to equation (3.25) all possible solutions will be summed to obtain the ﬁnal
solution. Also, the undetermined constants, A2 and A 4, will be grouped into
one constant that will be determined later by applying the last boundary con-
dition, equation (3.6c), which was the boundary condition that did not sepa-
rate and was not evaluated. The solution for problem one is

91(x+,y+) = A" [an’ cosh (unL'f) +Biosinh (unL‘xtn *-
n = 0
[uncos (unf) + BiHsin (uny+)] (3.26)
and the eigenvalues are found from the solution of equation (3.24).

For ease in presenting these large solutions, the following functions are
deﬁned:

x1 (11» x") a put} cosh (unL‘xt) + Biosinh (nnL’xt) (3.27)

Y1 (u, y‘) a u..cos (u,y*) + Bigsin (u..y*) (3.28)
and the solution can be rewritten as

 

 

 

 

3.1.2 General:

The so,
ous section. "

repeated here
for problems

93(x

The eigenvalue

Deﬁning the fol

allows the 80111111

27

01(x*,y”) = A,x1(un,x*) Y1 (amy‘) (3.29)
=0

3.1.2 General Solution for Nonhomogeneous Boundary Conditions on y

The solution for i=2, 3 apply a similar procedure as used in the previ-
ous section. The details of which are shown in Appendix A and will not be
repeated here. Applying these procedures results in the following solutions
for problems two and three:

“ a a
02 (1:3)”) = Z B..[°°Sh (4f) - Cnsinh (4f N"
n=0 L L
[ancos (an?) + Biosin (anx+)] (3.30)
6H“+ “cm”shmH B'inha"+*
w»; new (9 )+ (a )1

[ancos (Gina?) + Biosin (anx+)] (3.31)

The eigenvalues are given by the positive roots of the transcendental equation

 

(Bio + Bi L) an
tan (an) = 2 . . (3.32)
(an—BrLBro)
Deﬁning the following variables:
X2 (“m x”) a: ancos (053+) 4» Bio sin (ans?) (3.33)
all an
Y2(an,y+) I cosh (—;y*)-§nsinh (7)”) (3.34)
L L
Y (a y“) sﬁcosh (25y+)+8i sinh (5f) (3 35)
3 n’ Lo Lo H Lo
allows the solutions to be presented in a compact form
02 (x*, y+) = 13”)!2 (an, y+) X2 (an, x”) (3.36)
n = 0

03 (x*, V) = CuY3 (an, y+) X2 (an, x*) (3.37)
- 0

 

 

 

 

 

LF

3.2 Soluti!

3.2.1 Hot F.

The
hot ﬂuid, p
equation in
r, (H, 0) 1
from equat

Try (1+: 0) =

Where the 12
(3.36) and (
(2-15) and a

dTH

dx‘ HT

With the initi;

1111's eqllation
131' 801111101]. T

28

3.2 Solution of the Fluid Energy Balances Equations

3.2.1 Hot Fluid Formulation

The ordinary differential equation describing the temperature of the
hot ﬂuid, presented in equations (2.15-16), is coupled to the wall conduction
equation in the differential equation itself. This is seen by the presence of
Tw (x+, 0) in the differential equation. The value of this term can be obtained
from equations (3.1), (3.4), (3.29), (3.36), and (3.37) as

Tw(x+,0) = 6w(x+,0) +70 = 70+ ZAnX1(un,x+)Y1(un,O)

n=0

+ 2‘, [B,Y2(a,,.0) +C,,Y3(an,0)]X2(ocn,x+) (3.33)

n = 0
where the last term in equation (3.38) is obtained from combining equations
(3.36) and (3.37). Substituting this into the diﬁ'erential equation in equation
(2.15) and arranging gives

d1” " ,
EF+NHTH = NH{T0+ 2A,,X1(un.x )Y1(ll,,.0)

n=0

+ 2 [BnY2(an,0) +CuY3(0t,,.0)]X2(a,,.x*) } (3.39)

n=0

with the initial condition from equation (2. 16a)

TH (0) = TH, ,, (3.40.1)

This equation is a linear nonhomogeneous ordinary differential equation. Its
solution may be written as the sum of a homogeneous solution and a particu-
lar solution. The homogenous problem is given as

dTH’ h
x‘.’

 

+NHT,“ = o (3.41)

and a particular problem is

 

 

 

41,,er

The particr
pler proble
eral solutio

where TH h

dTH. pi
d1+ + NH

29

 

dT a.
dSP+NHTH,p = NH{T0+ ZA”X1(un’x+)Yl(11n,O)

n=0

+ 2 [3..Yz(a,,.0) +CuY3(a,,,0)]X2(an,x+) } (3.42)
n=0

The particular problem in equation (3.42) was further divided into three sim-
pler problems for each term on the right hand side of the equation. The gen-
eral solution for equation (3.39) in terms of the simpler problem’s solutions is

3
TH(x+) = 1H,,(x+) + 2 TH,p,.(x*) (3.43)
i: l
where T”, n is the solution to equation (3.41) and T”, p, is the solution to

(IT
-—H-;p‘+NHTH.pr = NH{811T0 +821201!" X1 (umx )Y1(|.ln,0)

+ 53, 20 (Buy, (an, 0) + CRY3 (an, 0)]x2 (01”,?) } (3.44)

(i = 1, 2, 3) and 8], is the kronecker delta function as deﬁned in equation (3.7)

3.2.2 Hot Fluid Solution
The homogeneous problem in equation (3.41) is solved giving

T”, h (x*) = Dlexp (-N,,x*) (3.45)

where D1 is a constant that will be evaluated after assembling the complete
solution as shown in equation (3.43).

The particular solution was obtained by using standard variation of
parameter techniques. This method utilizes a general solution of functional
form similar to the nonhomogeneous term in the problem and then solves for
the unknown constants in this general solution. For the ﬁrst particular solu-
tion (i=1) the general solution would be a constant

7‘”, p1 (x‘) = k = constant (3.46)

Substituting equation (3.46) into equation (3.44) and solving for the constant
gives

Thesecon .
x, (1*) . sin
TEPZ (1

Substituting
constants, bu

 

TH, p2

 

 

L Finally. the th

TH,p3 (1

Which after su
stants can be ‘

TH,p3 (J

Assembling m.

(3'49)! and (3 _5
dance is

30

r”, p, (x*) = r, (3.47)

The second particular solution (i=2) would be of the same functional form as
X, (x‘) , since all other terms in this summation are constants

1,”, (x+) = for, (x+)) = 2 bucosh (11,131?) + casinh (unL‘r‘) (3.43)
n = 0
Substituting equation (3.48) into equation (3.44) and solving for the unknown
constants, b, and c", equation (3.48) can be written as

 

0- . (BioNH— ”ZL‘Z) t
T (x*) = A 2L 1 - ", cosh L Jr“)
H,p2 ago n {11, [ (Ni, " 11:1. 2) (”a

 

(BioNH - uﬁL‘z)
+
” " (Ni—93L”)
Finally, the third particular solution is of the functional form

:Isinh (unL’f) } (3.49)

T“, (x+) = f(X2 (an, in = 2 ancos (an?) + dusin (an?) (3.50)
n = 0
which after substituting into the differential equation and solving for the con-
stants can be written as

B' N 2
= 2(B.+1:c.){a.[1-‘ '0 W]

":0 (N;+a'2.)

 

BiN +042
+NH[ o H n

2 2
NH-t-an

]sin (aha?) } (3.51)
Assembling the solutions to the simpler problems, equations (3.45), (3.47),
(3.49), and (3.51), using equation (3.43) the solution for the hot ﬂuid energy

balance is

 

 

 

TH(X+) = 7

Applying m

31

T”(x*) = T0+D1exp (-N”x")

 

" , (BioNH-nﬁL‘z) , +
+ 2A,,{uﬁL [1- (N;_u:L,2) ]cosh(unL x )

n=0

 

(BioNH — uﬁL”)
+ H n

inh L‘ +
(Na-11:14.2) ]S (11,, x)}

 

.- B.” 2
+ 2 B +F—:C a 1-( 1° ”+a") cos(0t. x+)
n=0 n L n II II

(Ni, + (1%)

- 2

+N BroNH-Hxn
H N2“:2
H n

]sin (0%”) } (3.52)

Applying the boundary condition, equation (3.403), gives the constant

 

(BiaNH — ng‘z) ]

DI = TH,fn-To- zAnugL [1- (N2 _uzLez)
H u

n=0

_(BgNﬁ+ab

an
2w()[ 145»

 

3.2.3 Cold Fluid Solution

For the cold ﬂuid the ordinary differential equation describing the tem-
perature is slightly more involved due to the wall temperature being evalu-
ated at (y” = 1). This adds algebraic difﬁculty to the problem, but
procedurally the solution is obtained exactly as the hot ﬂuid. For this reason,
only the ﬁnal solution will be shown. All the details for this solution can be
found in Appendix B.

The solution for the cold ﬂuid energy balance equation is

ram
.
is.
V}
i

 

 

M

 

 

Tc(x+)

Where the i;

and the fOIIm

32

Tc(x+) = To+Elexp (ch+)

" . NcBio + ij’2] .
+ A ’ unL 1+ , cosh (unL 1:”)
ago I! { [ [Viv-[12L 2

N i+ ZL‘2 ..
+NC[ C3" 5"” :lsinhulnL m}
NC-p'uL

 

" NCBio—az
+ 2 (B ’+C ’) {a [l+——-—"]cos(a x+)
"=0 " " " Ni”: "
N Bi -a2
+NC[-C_2°__2_"]sin(anx*) } (3.54)
Nc+°‘,.

where the integration constant is

n=0

" _ NCBi +1121.” ,
151 = exp (-NC) [(TC’ m-To) — 2A":{un1, [1+ N2 0 “25.2 ]cosh(unL )
C- n

 

NcBiawﬁL’z ,,
+N , sinh(u L )
c[ Né-uib 2 "

" N B' - 2
.. z (Bn'+cn') {anl:1+_ci_§2

cos(a)
”=0 Nico-a: ] "

N i-oz2
+NC iii—J smut") (3.55)
”6"“:

and the following constants were introduced to simplify the expressions:

An’ = A,I [uncos (1.1") + Bi ”sin (un)] (3.56)

Bn’ = Bn[cosh (5—2) - Cusmh (:2 J] (3.57)

all all . all
Cn’ = C" [—. cosh (7) + Bi Hsmh (7 )] (3.58)
L L L

 

 

 

33 A991

Ha‘
boundary
condition:
condition:
to be appl
equation (
neous bou

The difﬂcu
an inﬁnite :
(3.37), are

33

3.3 Application of Nonhomogeneous Boundary Conditions

Having obtained the solution for the temperature of the ﬂuids, the ﬁnal
boundary conditions (nonhomogeneous) can now be applied. The boundary
conditions that need to be applied are the three nonhomogeneous boundary
conditions given in equations (3.6b-d). These boundary conditions only need
to be applied to the respective 6‘. solution, (i.e. equation (3.6b) applied to 6,,
equation (3.6c) to 92 and equation (3.6d) to 03 ). These three nonhomoge-
neous boundary conditions are

 

 

39
——1 +BiLe,(1.y*) = Bun-1,) (3.59)
ax x’ :1
392
--—, +3510; (x+.0) = BingHU“) - To] (3.60)
By y, ___ o
393
F +Bice3 (x+: 1) = Bic[Tc (x+) - To] (3.61)
y

 

y’ =1

The difﬁculty with equations (3.59-61) is that the relationships are in terms of
an inﬁnite series for 9." These series, from equations (3.29), (3.36), and
(3.37), are

61(x*,y+) = ZAnX1(“nvx+)Y1(lln:>’+) (3.62)
II

92(x*,y*) = 23nY2(°‘n’Y+)X2(°‘n:x+) (3.63)
n

63(x+,y*) = ECuY3(°‘.»Y+)X2(%x+) (3.64)
II

This means that theoretically an inﬁnite number of constants will need to be
determined. Of course, experience with series solutions has shown that less
than an inﬁnite number of terms (and constants) need to be considered since
the series converges after a ﬁnite number of terms; beyond which, the indi-
vidual terms are zero or negligible in comparison to the summation up to that
point and future terms decay further. However, the number of terms needed
for the series to converge is not known and must remain arbitrary and
assumed large.

34

The procedure used to determine the constants that will satisfy the
nonhomogeneous boundary conditions involves applying orthogonality.
Using the orthogonality of the eigenfunctions, the constants will be deter-
mined. An appropriate mathematical statement of this property is

l

gwny,(u,.y*)Y1(u,,.y*)dy* = { ”(ﬁm) '21:: (3.65)
01'

1 + + + 0 main

{wxyX2(an,x )X2(am,x )dx = { N(a,,,) m=n (3.66)

where wxy is a weighting constant, which for the Cartesian coordinate system
is equal to one. After substituting expressions for the temperatures and deriv-
atives of temperatures into the boundary condition in equations (3.59-61), the
equations will be multiplied by a second eigenfunction and integrated over
the boundary. Since the eigenfunctions have the property shown in equations
(3.65-66), the equations will be simpliﬁed and the unknown constants can be
determined.

3.3.1 Application of Boundary Condition at x = L

Beginning with the solution for (31 in equation (3.26) and taking the
derivative with respect to x+ and evaluating both expressions at the boundary
gives

91(1,y+) = A, [unL‘ cosh (unL') +Biasinh (unL')] *
=0

[u,,cos (my‘) + Biusin (11f) ] (3.67)

39 .-
5—:(l.y*) = ZAquiL’zsinhmuL') +BiouuL‘c6sh(unL‘)]*
x

n=0

tu,cos (u,y*) +Bigsin (u,y*)1 (3.68)

Substituting these expressions into the boundary condition, equation (3.59),
produces

35

A, [uﬁL‘zsinh (unL') + BiounL‘ cosh (uuL‘n *
- 0

[u,,cos my”) + BiHsin (u,y*)]

+ ML 2 A" [unL' cosh (unL’) +Biosinh (unL')] *

n=0
[uncos (Hf) + BiHsin (unﬂl

The orthogonality of the eigenfunctions is used to determine the constants.
The eigenfunction for this problem is

Y ("3' y+) = uncos (any) + BiHsin (unf) (3.70)
and these eigenfunctions have the property shown in equation (3.65). There-
fore, equation (3.69) will be multiplied by a second eigenfunction Y (um, y”)
and integrated over the boundary. Because this integral is only nonzero for
m=n, only this term will remain from the summations. The integrals of each
term in equation (3.69) after multiplying by a second eigenfunction are
shown in Appendix C.

After applying orthogonality, equation (3.69) reduces to
A," [uiL‘zsinh (uML') + BiaumL‘ cosh (umL‘) 1 N (pm) +

AmBiL [umL‘ cosh (umL') + Biosinh (umL') 1 Mum) =

_ . Bi H Bi H
BIL (TL- To) [srn (pm) - T008 (um) + T] (3.71)
where all summations that were present in equation (3.69) have been reduced
to a single term through the application of orthogonality. Equation (3.71) can

now be solved for the unknown constant

. . Bin Bill
3'1, (TL‘ To) [8111(um) "' T608 (11".) + T]
A - "' "' (3.72)

"' - [(uiL'2+BioBiL) sinhmML’) + (Bio+BiL)umL'cosh(umL')]N(um)

 

36

The functional relationship for N (um) is

 

N(u ) =1 (u2+Bi2) 1+ 8’" +Bi (373)
"' 2 "' ° ( (uﬁ,+BiL2)) ° '
This completes the solution of problem one, equation (3.62) could be evalu-
ated since A" is known from equation (3.72), noting that m is a dummy vari-

able.

3.3.2 Application of Boundary Condition at y = 0

The difﬁculty in applying the boundary condition in equation (3.60) is
recognized by noticing that the right hand side contains the temperature of
the hot ﬂuid. Using the solution for 62 from equation (3.30) and taking the
derivative with respect to y+ and evaluating both at the boundary gives

92 (x+, O) = 2 B" [ancos (an?) + Biosin (anx+)] (3J4)
n: 0
a62 0- an
——(x*, 0) = - z Bra-C, [ancos max”) +Biosin (anx*)] (3.75)
(3y+ "=0 L

In addition, the hot ﬂuid temperature is, from equation (3.52)
TH (x‘) = To + Dlexp (-NHx+)

- c (BioNH - “2L. 2) t
+ A (FL 1- '1 cosh (an x+)
"go n{ n [ (”z-“7231’ 2) ]

 

 

 

 

Putting these three expressions into equation (3.60) yields

37

" or
2 3,2,5; [ancos (anxt) + Biosin (63+)]
n = 0

+ Bi” 2 B" [ancos (0:31?) +Biosin (anx+)] =

n=0

BiH[Dl exp (—N,,x+)

 

" . (Rwy-uh”) .
+ A 2I. 1- '1 cosh( L x”)

n=0

 

 

.- B.” 2
+ 2 B +?—:C a l-( to ”+059 cos(a x‘”)
u=0 l L n n n

(N3, + 6:)

BiN +02
+1)!" "2” " sin(anx+) (3.77)
NH-t-ot:

 

for which orthogonality will be applied. Each term in equation (3.77) will be
multiplied by the eigenfunction

X (an, x*) = aucos (anal?) + Bio sin (anf) (3.78)
evaluated at term m, and integrated over the boundary. The eigenfunction has
the same property as discussed earlier, which is given in equation (3.66).
After applying orthogonality, equation (3.77) becomes

Bulk»: = -0)le - 2 An (pmn,1+pmn.2+pnm,3+pmn.4) -2(Bn+cni7)wmn(3'79)

n = 0 n
where the order of the terms in the two equations, equation (3.77) and equa-
tion (3.79), has been maintained, except the ﬁrst two terms in equation (3.77)
have been combined into a single term in equation (3.79). Additional vari-
ables were introduced to represent the results of the integration, the variable
deﬁnitions and integration details are shown in Appendix C.

The complexity added by the hot ﬂuid being the nonhomogeneous
term in the boundary condition is seen in equation (3.79), where summations

 

are pres:
cation 01
Because
perature .
term. Als
ally two 5
1 equation,
temperatu
the other I

 

3.3.3 Appli

The
very similz
exPression
both at the

 

 

aﬂd the temp

 

38

are present after applying orthogonality. This was not the case for the appli-
cation of orthogonality to 01, where all summations reduced to a single term.
Because the summations in equation (3.77) that represent the hot ﬂuid tem-
perature do not contain eigenfunctions, they were not reduced to a single
term. Also, equation (3.79) is a function of all three unknown constants (actu—
ally two since A” was just obtained). Nevertheless, the dependence of this
equation, which would only be a function of the constant 3,, if the hot ﬂuid
temperature were not present in the boundary condition, is now coupled to
the other boundary condition, equation (3.61), through the constant C”.

3.3.3 Application of Boundary Condition at y = 5

The evaluation of the boundary condition given in equation (3.61) is
very similar to the previously discussed method for 62. Starting with the
expression for 63 in equation (3.31) and taking the derivative and evaluating
both at the boundary gives

+ a all an . an
93 (I . 1) = C" —;cosh (—,)+BiHsmh (7) *
n= 0 L L L
[ancos (anf) + Biosin (anx+)] (3.80)

EBB . an an an an
—(x+, 1) = 2: Cn—,[—.sinh [.7)+ai,,cosh {—7)}-
ay“ n=0 L L L L

[ancos (an?) + Biosin (anx*)] (3.81)

and the temperature of the cold ﬂuid, from equation (3.54), is

 

 

 

 

TC (1+) =

39

TC (x+) = To + Elexp (ch+)

L, 1 + (BioNC + uﬁL‘z)
"l N2 _ 2L‘2
( c “n )

 

+ ZAn’

n80

J cosh (an' x+)

 

 

(Bi N +u2L‘2) ,
+NC 02 C 2 ".2 sinh (unL x")

(Nc‘unL )

“ (Bi N —62
+ 2 (Bn’+ Cn’) {an[1+ 02 C 2 ")Jcos (unit?)

":0 (Nc+orn)

Bi N —62

+NC[_°_2.C_2_'.'] sin (anx+) } (3.82)

Putting these temperatures into equation (3.61) gives

C n "sinh n B h n I B. . )1
"2 n ,I , ( ,) lHCOS ( J, ancos (aux ) rosrn (aux

“ or or or
+Bi C —:cosh(-—:)+Bi sinh(-—:')]‘
CE”, "[L L ” L
[ancos (uni?) + Biosin (anx+)] =
Bic [El exp (ch+)
“ , (Bi N + 2L”)
"I uﬂL 1+ 02 C “'12
(NC " 11,2,L )

 

] cosh (unL' x")

 

(Bi N + 2L”) .
+NC °2c ”'12 sinh(unL x“
(Ne-“zl‘ )
(BioNC-ag)

2
(NC + 63)

(BiONC- a?!) . +
+NC[ (Ni-+ai) Jsrn (crux ) } J (3.83)

 

] cos (ornf)

 

writ-:1- ‘

 

 

 

 

for which ’
multiplied ‘

evaluated a
the same pr
After applyir

where the or(
tion (3.83) ar
(3.83) have b
ilar procedure
to represent ti
details of this

Notice :
(3.61) had the
Summations e

t0 equation (3.

After ap
was obtained f

Where

No

NOE that the sub
s own since it is

40

for which orthogonality will be applied. Each term in equation (3.83) will be
multiplied by the eigenfunction

X(au, x‘) = ancos (uni?) + Biosin (an?) (3.84)
evaluated at term m and integrated over the boundary. The eigenfunction has
the same property as discussed earlier, which is given in equation (3.66).
After applying orthogonality, equation (3.83) becomes

Cmrbm = Quiz, 4» 2A,} (PM 1 + PM 2 + PM 3 + PM 4) + Z (Bn’ + Cn’) \PM (3.35)
n I!

where the order of the terms has been maintained for the two equations, equa-
tion (3.83) and equation (3.85), except that the ﬁrst two terms of equation
(3.83) have been combined into a single term in equation (3.85). Using a sim-
ilar procedure as the last boundary condition, new variables were introduced
to represent the results of the integration when orthogonality was applied. All
details of this step are shown in Appendix C.

Notice that the presence of the cold ﬂuid temperature in equation
(3.61) had the same effect as the hot ﬂuid in the previous boundary condition.
Summations exist after applying orthogonality and equation (3.85) is coupled
to equation (3.79).

3.3.4 Summary of Applying Orthogonality

After applying orthogonality to equation (3.59) a closed form solution
was obtained for the 01 constant, from equation (3.72)

 

. . Bin Bin
BrL(TL- To)[s1n(un) - u—cos (11") 4» ll ]

 

 

An = 2 .2 c c c (3'86)
[unL sinh(p.nL )+ (Bio-tBiLwnL cosh (unL )]N(un)
where
_ l 2 . 2 BiL .
Nut”) - §[(ll,,+3‘o ) (1+ (1134'3'12) )+Blo] (3.87)

Note that the subscript n was substituted for m in the form of the equations
shown since it is a dummy subscript.

 

 

 

Applica
the simple a1 gel
The diﬂerence
tion (3.59) this
a function (inﬁr
present two pro
the series will n
Secondly, they <
in equations (3.4
pled equations g
ranging, are

Burl. +2.: (11?,l +

 

Cmd’u'Z (gn'.

These equ
for the unknown
exist in the equai
number of terms
“V0 «muons; h
hence the subscr'
series (m = N), t
Stants.

The unkn
(" = 1. 7. -..N).
of B, and C", no
Also’ D1 and El
mbsﬁWIBd. Fro

 

41

Application of orthogonality to equations (3.60-61) did not result in

the simple algebraic equation with one unknown as it did for equation (3.59).
The difference is the terms on the right hand side of the equations. For equa-
tion (3.59) this term is a constant, whereas for equations (3.60-61) the term is
a function (inﬁnite series). These functions are TH (x”) and TC (x*) , and they
present two problems. First, they are not orthogonal functions; and therefore,
the series will not be reduced to a single term after orthogonality is applied.
Secondly, they contain both unknown constants, B” and C", which will result
in equations (3.60-61) being coupled after applying orthogonality. The cou-
pled equations given in equation (3.79) and equation (3.85), after some rear-
ranging, are

n+2 (313+ Cu (5;)wnm + 0) 1:301 = -2Aﬂ (pm, 1 + pm». 2 + pmn, 3 + prim. 4) (3‘88)
1:

C,,,<r>"l — 2 (B; + Cn’) ‘PM - one, = 2A,;(PM1 + Pm M 2 + PM 3 + PM 4) (3 39)
II II

These equations represent a set of simultaneous equations to be solved
for the unknown constants B" and C" (n = 1, 2, 3 ....... ). Because summations
exist in the equations it is now required to truncate the series after a ﬁnite
number of terms (n = N). This gives 2N constants to be determined from the
two equations; however, both equations can be written for each eigenvalue,
hence the subscript m. Since there are as many eigenvalues as terms in the
series (m = N), this totals 2N equations to solve for the same number of con-
stants.

The unknowns in equations (3.88) and (3.89) are 3,, and C"
(n = 1, 2, ....N). The primed constants in equation (3.89), which are functions
of B, and C", need to be substituted for, giving a new variable for that term.
Also, D1 and E, are function of the unknown constants and will need to be
substituted. From equations (3.53) and (3.55), D1 and E1 are respectively

.[1_ (BioNH- pita]
(Nir nil-’2)

 

D1 = TH,in—To— ZAnuyer

n=0

B'N 2
' 2 (B,+%C,)a,.[l-( 1" “69] (3.90)

":0 (Ni-Fai)

 

 

 

 

 

 

Introducing
(3.91) and (
reduced to

Xmas"- (\{1

“WP ( ‘N.

These CC
Own Const
[0 the left of [he
were emPle

42

 

B, = exp (-Nc) [(13, ,,-T,) - 2A,

n80

, NCBi,+n¢L‘2 ,
’ L 1+ ', cosh L )
{,5 l Né-uiL 2 l m"

NCBi,+uﬁL’2 ,
+N , sinh( L )
Cl Né-uib 2 u"

a. N ' _ 2
- 2 (B,'+c,') {6,[1+—C—B'°—ﬁ]cos(a,)

2 2
”=0 ”0+0."

N Bi -62
+NC —c—2i——" sin(a,) (3.91)
NC-t-or:

Introducing variables to represent the large terms in equations (3.90) and
(3.91) and converting B,’ and C,’ to B, and C, allows these equations to be
reduced to

 

D, = (THJu—T -2A, [3,,- 2 o,(B +C, 2"”) (3.92)
1130
E, = exp(-NC) (TQM-T0) - 2 A.’°,." Z (B,a,+c,c,) (3.93)
n=0 =0

where all new variables are deﬁned in Appendix D. Substituting for D1 and
El from equations (3.92) and (3.93) into equations (3.88) and (3.89) and
deﬁning new variables to convert the primed constants gives

a
3,3, + 2,119. N... - 7.0)..) + ZCnL—i' W... - 7.9)..) =
- mm (TH, in - To) - 2A7! (prim, 1 + prim, 2 + pm», 3 + prim, 4-mmBn) (3'94)
n

2 [9,6,- (‘l’B) M] B, + go, + 2 [9,5,- (‘PC) M] C, =

Qmexp (-NC) (Tc, ,, - T,) + 231; (PM 1 + PM +,,P 3 + P — 0,6,) (3.95)
n

mn,4

These equations are now in the form that they were solved for the
unknown constants. The equations are organized with functions of unknowns
to the left of the equal sign and known quantities to the right. Standard meth-
ods were employed to solve the set of simultaneous equations described by

43

equations (394-95) and these methods are addressed in Appendix D. In
short, the two equations were separated into coefﬁcient of B, and C, for each
eigenvalue and equation. These terms were put into a matrix and then solved
for the unknown constants using a partial pivoting scheme with the Gauss
method. Once the constants were determined, all solutions could be evalu-
ated.

 

 

 

 

ﬁsh)
of th
subr
tines
0U)

the s
were

44

3.4 Computer Program AXCOND

The development of the solution to this problem has been completely
analytical up to this point. A closed form solution is not possible since a set
of simultaneous equations must be solved for series constants, which depend
on a truncation of the general series solutions obtained for the temperature of
the wall and ﬂuids, as discussed in the previous section. With the solution
framework complete a computer program needed to be developed to perform
the evaluation of the analytical solution. The main functions of the program
were to:

1. Determine the eigenvalues from the transcendental equations.

2. Assemble the set of simultaneous equations and solve for B, and C,.

3. Evaluate the series to calculate the temperatures of the wall and ﬂuids.

4. Check that the boundary conditions are satisﬁed and that energy is conserved.

A FORTRAN program was developed to perform these functions and a
listing of the program is given in Appendix F. In addition, a general ﬂowchart
of the program is shown in Figure 3.2. It contains a main program that calls
subroutines to perform each of the individual tasks. There are nine subrou-
tines in the program: INPUTO, INPUT], ROOT, BUILD, GAUSS, PROFIL,
OUTO, OUTl, and OUPUT. Each of these will be brieﬂy discussed to sum-
marize its function. In the program, variables were named to coincide with
the solution shown herein, and deviations from the variables in the solution
were noted.

The ﬁrst two subroutines INPUTO and INPUTl, as there names imply,
import the data needed for a particular run. The former reads dimensional
data in and computes the dimensionless parameters needed for computation;
while the later reads in the dimensionless parameters, depending on the
option selected. In addition, INPUT] allowed for a particular input parameter
to be varied, with the ﬁrst call to the subroutine reading in the data and the
parameter to vary and subsequent calls updating the selected parameter for
another computation. This option allowed for assessing the eﬂect of various

parameters on the solution by observing the results as one parameter is var-
ied.

Subroutine ROOT is used to locate positive roots of the transcendental
equation. It employs a marching scheme that searches for a change in sign of
the function then backs up to locate the zero crossing, i.e. the root. A great

 

 

 

 

 

deal of tin:
the large c1
tion and m

ASSt
constants l
obtaining 2
partial pivc

Tern
solutions it
the derivat
lated. The .
ated in the
and X=L) t

Ewation (1
series. Usir
the ﬂuid en
ditions On 1
enabled a I!
two CheCks
Used ‘0 Selc
ity 0f the S(

45

deal of time was spent to make this subroutine sufﬁciently robust to handle
the large changes in magnitude of the parameters in the transcendental equa-
tion and not skip roots of the equation.

Assembling the matrix of simultaneous equation to be solved for the
constants B, and C, was done in subroutine BUILD. This was the key step in
obtaining a solution. After building the matrix, the subroutine GAUSS used a
partial pivoting scheme to solve for the constants.

Temperatures for the wall and ﬂuids were generated from the series
solutions in the subroutine PROFIL. In addition to computing temperatures
the derivatives of the temperatures with respect to the coordinates are calcu-
lated. The derivatives of the analytical series solutions were taken then evalu-
ated in the subroutine. Also, to calculate heat loss from the wall ends (x=0
and x=L) the integral of the heat ﬂux over the ends was taken

1
8T
q'end = IKw r
0

 

dy+ (3.96)

and

31:

Equation (3.96) was determined analytically and programmed to evaluate the
series. Using the derivatives of temperature it was possible to check whether
the ﬂuid energy balances equations were being met and if the boundary con-
ditions on the wall were being satisﬁed. The heat losses from the ends
enabled a total energy balance to be performed insuring conservation. These
two checks, conserving energy and satisfying the boundary conditions, were
used to select the number of terms to truncate the series and insure the valid-
ity of the solution.

The ﬁnal three subroutines generate the output ﬁle for a run. Subrou-
tines OU'FO and OUTl correspond to the option listed earlier for the input.
The main difference is that OUTO produces the results of one computed run
while OUTl varies a particular parameter giving the results as a function of
the varied parameter, but not listing the unchanged parameters again. Subrou-
tine OUTPUT can write the ﬂuid and wall temperatures as a function of posi-
tion if desired. These temperature are only written if a ﬂag is set in the input
ﬁle and there are separate ﬂags for the ﬂuids and wall. Likewise, another ﬂag
in the input ﬁle can check that the boundary conditions are satisﬁed and write
the results to the output ﬁle.

 

4.0 Ident'

In the
dimensionles
form a link
dimensionless

r— the wall aspec

 

and Biot numb

 

 

 

The Parameters

 

 

Chapter 4
Results and Discussion

4.0 Identiﬁcation of Input Parameters

In the formulation of the mathematical model for this problem certain
dimensionless groups appeared (Chapter 2). These dimensionless parameters
form a link between the mathematical model and the physical problem. The
dimensionless parameters needed to solve the wall conduction equation are
the wall aspect ratio

L
L - 5 (4.1)

and Biot numbers for each surface of the wall

3' h°L 42
'0 - 7?: ( . )
B' hLL 43
'L - T: ( - )
. M5
. has
BIC - —K: (4.5)
The parameters needed in the ﬂuid energy balance equations were
I: Bi K L w
N, = _"A” = _” W (4.6)
Ca Ca
h Bi K L w
NC = _CAC = ___c .. (4.7)

In addition to these dimensionless parameters the ambient temperatures at the
wall ends (T, and TL) and the initial or inlet condition for the ﬂuids (TH, 3, and
Tc, ,,) need to be speciﬁed. Mth these eleven parameters the mathematical

model can be solved for the wall and ﬂuid temperatures as functions of posi-

47

 

 

 

 

”I

tic

pd
Cl]

[16

de;
ass
ne:
enc

get

the
11101
the
deli

puti
part
do

inde
corn
tion
tains
solun
ccipa
mUm
tions.
Signir

48

tion. There are then eleven independent variables that may inﬂuence axial
conduction as the problem was formulated. The investigation of all eleven
parameters to determine their inﬂuence on axial conduction presents a difﬁ-
cult task. To reduce the number of independent variables, quantiﬁcation of
axial conduction will be shown by evaluating the solution using effective-
ness-NTU relationships. ‘

Presenting the solution in terms of the effectiveness eliminates the
dependence of the solution on any of the ambient or ﬂuid temperatures,
assuming a constant ambient temperature (T, = TL); because the effective-
ness is based on temperature differences and scaled by the temperature differ-
ence at the ﬂuid inlets. The NTU, which represents the physical and thermal
geometry of the heat exchanger, was earlier shown in Section 2.3 as

(4.8)

 

NTU K” L’
_ ...,L [

B 1' ”Bi c
Biu+ BiHBiC+ Bic]
This equation introduces the need for specifying the thermal conductivity of
the wall and the minimum heat capacity; whereas for the mathematical
model, the thermal conductivity was accounted for in the Biot numbers and
the heat capacity accounted for in NC and N H and did not need to be explicitly
deﬁned. It was decided that since the additional parameters of the wall ther-
mal conductivity and minimum heat capacity need to be speciﬁed for com-
puting NTU, they could also be used to compute NC and N,, from the other
parameters, as shown in equations (4.6) and (4.7). This allows for the more
physical parameter of heat capacity to be input.

The wall thermal conductivity and minimum heat capacity do not
independently affect the solution because they appear only as a ratio used to
compute the NTU in equation (4.8) and the parameters NH and NC in equa-
tions (4.6) and (4.7). It does not matter that the ratio in equations (4.6-7) con-
tains the individual heat capacities rather than the minimum. Since the
solution depends on the heat capacity ratio and the magnitudes of the heat
capacity will be set by the ratio of the wall thermal conductivity and mini-
mum heat capacity, the ratio will be accounted for in at least one of the equa-
tions. Therefore, the individual magnitude of these variables is not
signiﬁcant.

 

C.

 

 

 

The
mum heat c
ciﬁc pr0Pel'
dimension]:
adjusted to
these variat
group of ter
variable wil

where the w.
deﬁnition. 5

The Mondt 1
After Specif
number is see
.DUIIIber can ‘
‘3 important

 

49

The appearance of the ratio of the thermal conductivity to the mini-
mum heat capacity provides some additional ﬂexibility to this study. The spe-
ciﬁc properties of the wall and ﬂuids do not need to be speciﬁed, and a
dimensionless analysis can be performed. Furthermore, this ratio can be
adjusted to provide the desired magnitude of NTU. To present the ratio of
these variables, a new dimensionless variable will be deﬁned to represent the
group of terms that are not dimensionless in the NTU in equation (4.8). This
variable will be the Mondt number, from Rohsenow [1 7]

wa
M, s , (4.9)
CminL
where the wall aspect ratio was included to be consistent with Rohsenow’s
deﬁnition. Substituting the Mondt number into equation (4.8) gives

 

 

(4.10)

NTU = M L‘2[ BiHBiC ]
0

BiH-t- BiHBiC-t- Bic
The Mondt number will be used as the adjustable parameter in this study.
After specifying the ﬂuid Biot numbers and wall aspect ratio, the Mondt
number is set to provide the desired NTU. The magnitudes of the Mondt
number can be looked at to correlate the conditions in which axial conduction
is important

TABLE 4.1. Input variables and dependency for determining heat exchanger performance

 

 

 

Input Parameters Mathematical Model NTU Effectiveness

L' L' L' L‘
Bi, Bi, Bic Bi,
Bi L Bi L Bi H Bi L
Bic Bic M, Bic
Bi H Bi H Bi H
Kw N C CR
CC NH . Mo
CH TH, in

TH, in TC, in

TC, in To
To TL

1;

 

 

 

 

 

 

 

   
    

Presen f
number of ind

allows for the
with this soluti
tiveness, will (1
of Table 4.1.

An addit
terms needed b
vious discussio
overlooked in t
the series soluti
ary conditions (
ﬂed Since IllCSt
applying this c1
verge, Another
tion. This chec]
from the hot ﬂu
6°“ from the w
0f the TCSUlts, u
Validity of the s

The inﬂu
301116011 WOUId

50

Presenting the results in an effectiveness-NTU format will reduce the
number of independent variables. The input parameters that were speciﬁed to
obtain a solution are shown in the ﬁrst column of Table 4.1. These parameters
were used to calculate the information needed for the mathematical model
and NTU shown in columns two and three in Table 4. ]. This information
allows for the mathematical model to be solved and a NTU to be associated
with this solution. However, the solution, which will be in the form of effec-

tiveness, will depend only on the seven parameters shown in the last column
of Table 4. 1.

An additional input that is not listed in Table 4.1 was the number of
terms needed before the series is truncated, N. It was not included in the pre-
vious discussion because it lacks any physical signiﬁcance, but it was not
overlooked in the analysis. The value of N must be sufﬁciently large to allow
the series solutions to converge. Taking this criteria a step further, the bound-
ary conditions on the wall were checked to ensure that they were being satis-
ﬁed. Since these boundary conditions involve the wall and ﬂuid temperatures,
applying this criteria is equivalent to having all three series solutions con-
verge. Another check performed was an overall energy balance of the solu-
tion. This checked that the solution conserved energy, i.e. the energy lost
from the hot ﬂuid was accounted for in the cold ﬂuid and/or lost by convec-
tion from the wall ends. Although magnitudes of N will not be shown in any
of the results, using those two criteria the value of N was chosen and the
validity of the solution assured for all results reported.

The inﬂuence of N was not investigated in any detail. In general, the
solution would converge based on an energy balance within 10-20 terms.
However, the boundary conditions would not be met until more terms were
considered; but the solution (2 - NTU) did not change ﬁ'om the values seen at
a smaller number of terms. These results suggest it may be possible to obtain
simpliﬁed expressions for the solution, but this point was not pursued further.

4.0.] Presentation of Results

It was anticipated that the effect of axial conduction could be demon-
strated through the seven variables upon which the effectiveness depends
(shown in Table 4.1). These variables were to be investigated to determine
their inﬂuence on the performance of the heat exchanger. In order to present

51

these results it is useful to introduce a ﬁnal variable that will describe the deg-
radation in the performance of a heat exchanger. It is called the ineffective-
ness, and it represents the amount that a heat exchanger’s performance is
reduced due to the effect of axial conduction. It is deﬁned by

8 “-8
iC = M (4,11)
eNeg
8 "'6
i” = lit—l! (4.12)
8
Ne;

where cm, is the effectiveness calculated neglecting axial conduction using
equation (1 .25), which assumes one-dimensional heat ﬂow. The ineffective-
ness must be deﬁned for both hot and cold ﬂuids since the energy balances
may diﬂ'er due to heat lost from the wall ends. For the case that there is negli-
gible heat loss from the ends of the wall, the two values for ineﬂ'ectiveness
are equal

ic = iuai for 8c = e” (4.13)

Note that when the ineffectiveness is zero there is no degradation in the
performance of the heat exchanger due to axial conduction, indicating that
axial conduction is negligible. In addition to showing when axial conduction
is not negligible, the ineffectiveness gives the magnitude that the effective-
ness would be over-estimated or in some special cases under-estimated, using
a standard effectiveness-NTU relationship which neglects axial conduction. It
is the percentage of error introduced from assuming that axial conduction is
negligible.

52

4.1 Inﬂuence of Axial Conduction at a Constant N TU

To begin this parametric study the NTU will be held constant as other
parameters are varied. Through some initial computer runs it was seen that a
larger degradation in effectiveness was seen at higher values of NTU. Hence,
a NTU of seven was chosen for the study. This allows the identiﬁcation of the
more important parameters and will give insight into the conditions under
which axial conduction should not be neglected.

While maintaining a constant NTU, the Biot numbers are varied over a
range from .0001 to .1, the heat capacity ratios of l, .75, .50, and .25 are
investigated, and the wall aspect ratio is also varied. Variation of the Biot
numbers will be done by grouping all four or grouping the ends and the hot
and cold side. Although grouping all the Biot numbers may be unrealistic, it
will provide insight for future more realistic cases.

4.1.1 Inﬂuence of the Wall Aspect Ratio (L‘)

A natural choice to begin this parameter study is with the wall aspect
ratio, which will address the physical dimensions for which axial conduction
should not be assumed negligible and may need further investigation. Also,
by beginning with the wall aspect ratio some bounds can be established on
the other parameters that need to be investigated. This is achieved by elimi-
nating ranges of the other parameters for which axial conduction only occurs
under unrealistic physical dimensions. All four wall Biot numbers were set
equal, and the magnitudes were varied beginning with .1 and decreasing an
order of magnitude each run to the ﬁnal value of .0001. The results, showing
ineffectiveness as a function of the wall aspect ratio, are presented for the dif-
ferent Biot numbers in Figures 4.1-4.4 with each ﬁgure depicting a different
heat capacity ratio.

Examination of the data, prior to plotting ﬁgures, showed a negligible
difference in the effectiveness based on the cold ﬂuid and the effectiveness
based on the hot ﬂuid (2,. = e”). For this reason, only one ineffectiveness was
plotted in Figures 4.1-4.4. There was a slight deviation from the two effec-
tivenesses being equal for very low magnitudes of the wall aspect ratio
(L' < 40). Because this effect is not related to axial conduction and merely
demonstrates the effect of the wall aspect ratio becoming very small, it was
not included in Figures 4.1-4.4.

 

.1

N

.A

I» .3
I!

. 1'

v. . I“:

 

 

 

The 1
represents 2
nonzero. D
in the effec
The degrad
maximum 1
numbers th(
istic values

The t
can be expl.
exchanger 1

For a plane 1

53

The results indicate that as the aspect ratio decreases, which physically
represents a thick wall or short heat exchanger, the ineffectiveness becomes
nonzero. Depending on the heat capacity ratio (CR), one ﬁnds a degradation
in the effectiveness from 20% at small C, to nearly 50% at C, equal to one.
The degradation is further enhanced as the wall Biot numbers decrease. A
maximum Biot number of .1 is shown in the ﬁgures, since for larger Biot
numbers the ineffectiveness only departs ﬁ'om zero for very small and unreal-
istic values of the wall aspect ratio.

The trend of increasing ineffectiveness with decreasing aspect ratio
can be explained by considering the thermal resistances across the heat
exchanger wall in the two principal directions

L
RStreainwiu = K (4’14)
#‘x
R - 8 (4 15)
- — 0
normal K 4y

For a plane wall A, =-- 195 and A, = wL giving

L

RStreanrwiu = K W (4-16)
19

 

]
waL.
It can then be noted that as L' becomes small, the streamwise thermal resis-
tance becomes small while the normal thermal resistance becomes large. This
will lead to a greater energy ﬂow in the streamwise direction and ultimately a
larger degradation of the effectiveness.

 

Rnonnal = (4.17)

In considering the inﬂuence of the Biot number on the degradation of
the effectiveness, the increase in ineffectiveness with decreasing Biot number
is expected. Decreasing the Biot number results in a reduction in the convec-
tive heat transfer at the surface of wall and/or an enhancement of the conduc-
tive heat transfer within the wall, which is shown by considering the
deﬁnition of the Biot number

R 5/K
Bilxﬂ = __“fl = E]: (4.13)

convection y w

 

 

 

I

- I

I

‘t

1

, a

.3]

:--: :3

Cal

exc

54

Both these trends lessen the inﬂuence of axial conduction since energy moves
more freely from the surface while encountering more difﬁculty conducting
within the heat exchanger wall. This is further seen by observing the temper-
ature proﬁle along the wall. These results are shown in Frgure 4.5. Notice that
the temperature proﬁle along the wall ﬂattens as the Biot numbers decrease,
that is, the temperature at the ends of the heat exchanger becomes closer to
the temperature at the center of the heat exchanger as the Biot numbers
decrease. This is the effect of axial conduction moving energy along the
length of the heat exchanger wall. Coupling the effect of the Biot numbers to
the wall aspect ratio follows.

As the Biot numbers decrease, the alternate path of heat ﬂow along the
wall becomes more inﬂuential. At the smaller Biot numbers the wall aspect
ratio has less inﬂuence because the convective resistance is so high. There-
fore, the wall aspect ratio requires a larger magnitude to reduce the normal
resistance or increase the streamwise resistance to the point that axial con-
duction will not occur. Hence the smoothing of the curves in Figures 4.1-4.4
as the Biot numbers decreased.

The ﬁnal dependency that can be observed in Figures 4.1-4.4 is the
inﬂuence of the heat capacity ratio. To aid in this comparison, Figure 4.6 was
generated, which illustrates how the heat capacity affects the solution for a
Biot number of .001. This ﬁgure does not provide any new information, it
instead combines a curve from Figures 4.] through 4.4 into a single ﬁgure. In
Figure 4.6 it is seen that, at a particular Biot number, the ineffectiveness
increases as the heat capacity ratio increases.'lhis trend is not a physically
obvious result.

To better understand the effect, the dimensionless temperature proﬁle
along the heat exchanger at a wall aspect ratio of 100 was generated as a
function of heat capacity ratio and is shown in Figure 4.7. In this ﬁgure the
median wall temperature shows little or no variation in shape with heat
capacity ratio, the curves increase a constant amount as the heat capacity
ratio decreased. This can be explained through the existence of a greater dis-
parity in the amount of energy per degree of temperature between the two
ﬂuid streams as the heat capacity ratio decreases. The stream with the maxi-
mum heat capacity will undergo a smaller temperature change in the heat
exchanger than the ﬂuid with the minimum heat capacity. Thus, the ﬂuid with

 

 

the maximu
with the mi)
change, as I
energy to 1’3
perature as 1
wall does 111
examinatior

For n
sionless terr
dimensionlc
capacity rat
effect of ax
heat exchan
energy bala

However, i:
energy excl

for CH2 Cc

 

55

the maximum heat capacity can cause a larger temperature change in the ﬂuid
with the minimum heat capacity while experiencing less of a temperature
change, as the heat capacity ratio decreases. Therefore, there exists more
energy to raise the temperature of the wall which results in a larger wall tem-
perature as the heat capacity ratio decreases. The elevated temperature of the
wall does not explain the decreased ineffectiveness, however; and further
examination is needed.

For more insight into the effect of the heat capacity ratio the dimen-
sionless temperature proﬁles of the hot and cold ﬂuids were added to the
dimensionless wall temperatures and are shown in Figure 4.8, with a heat
capacity ratio of .75 omitted for clarity. This ﬁgure clearly demonstrates the
effect of axial conduction, which moves the energy along the length of the
heat exchanger within the wall. The magnitude of which can be gauged by an
energy balance on the ﬂuids. For negligible axial conduction

011:?! = Ccff (4.19)
However, if axial conduction is present there exist a local imbalance in the
energy exchange between the ﬂuids and equation (4.19) becomes

 

 

 

 

d9 d9
H c
( dx+ ' dx” )°‘ qAxial (4.20)

R
for CH2 CC, which can be rearranged to give

TABLE 4.2. Relationship between the slope of the hot and cold fluld’s temperature profile and the
heat capacity ratio

 

 

 

 

 

 

 

 

 

day/(1f
c, ((19,!de
CR x+=0 x+=0.5 x+=1
1.0 15.68 1.“) ”15.63
.75 15.53 1.28 1/11.58
.50 14.29 1.81 1n.75
.25 13.5 2.67 114.95

 

 

 

 

 

 

t4

Using
excha
the h<
show:
and a.-
for we
when

excha
shows

nituc'
Cold
on tl
the l
the \

56

 

d9 ”1 d3:+

(c, (dGC/dx“ )
Using equation (4.21), axial conduction at different locations along the heat
exchanger can be assessed. To help quantify the differences in the slope of
the hot and cold ﬂuid temperature proﬁles Table 4.2 was created, which
shows the ratio on the left hand side of equation (4.21) at various locations
and as a function of the heat capacity ratio. This ratio should be nearly one
for negligible axial conduction and considerably smaller or larger than one
when axial conduction is appreciable.

The data in Table 4.2 demonstrates that as the heat capacity ratio
decreases, axial conduction shows a local decrease at the ends of the heat
exchanger; while it increases near the center. Comparison of the magnitudes
shows the dominant effect at the x+ = 1 end of the heat exchanger, while the
x“ = 0 end and center show less signiﬁcant change. This can also be seen in
Figure 4.8, more qualitatively, by a comparison of the slopes of the tempera-
ture proﬁles. The net effect on the heat exchanger is that axially conducted
heath ineffectiveness decrease as the heat capacity ratio decreases, as dem-
onstrated in Figure 4.6 where ineffectiveness decreases with heat capacity
ratio.

-1)... 4M“, (4.21)

The difﬁculty with interpreting this case is that both solutions, includ-
ing and neglecting axial conduction, change with the heat capacity ratio.
Therefore, it is required to analyze why the solution more closely resembles
the case of neglecting axial conduction as the heat capacity ratio decreases.
There are two related reasons. The ﬁrst is the change in the available energy
on opposite sides of the wall. Second is the change in the driving potential on
the hot side of the wall, which is the difference between the ﬂuid and wall
temperatures.

The smaller heat capacity ratios were obtained by increasing the mag—
nitude of the hot ﬂuid heat capacity while maintaining the heat capacity of the
cold ﬂuid. Providing more energy per degree on the hot side of the wall than
on the cold side and this disparity in energy across the wall becomes larger as
the heat capacity ratio decreases. This situation was earlier shown to increase
the wall temperature.

j

' (“'91:

innv‘a 'u'

 

tion‘
obse

hast
pote:
ture.
incre
cons

eneq
toth
iBecz
rado
hste

ontl
ng

it im
03133
853;
thei
the 1

57

The increased driving potential on the hot ﬂuid side of the wall results
from the increase in the heat capacity of this ﬂuid, which can be shown by
observing the describing equations for the hot ﬂuid, from equation (2.15)

dT ,
CHE? +Bi,,K,,L w[T,,(x*) - T, (1:20)] = o (4.22)

The dimensionless parameter previously introduced in this equation (N H)
has been separated into components to isolate the heat capacity. The driving
potential balances the product of the heat capacity and slope of the tempera-
ture. Therefore, when the heat capacity increases, the driving potential must
increases to balance the equation since the slope of the temperature remains
constant.

The outcome of these two effects is that the cold ﬂuid acquires more
energy at or near its inlet, due to the hot ﬂuid having. more energy to transfer
to the cold ﬂuid and the increased driving potential to move more energy.
Because both these effects become more pronounced as the heat capacity
ratio decreases, the cold ﬂuid will depend less on axial conduction to provide
its temperature rise and the ineffectiveness will decreases.

Having an understanding of the overall effect of the heat capacity ratio
on the performance of the heat exchanger, it is now possible to refer back to
Figure 4.6 and the inﬂuence of the wall aspect ratio can be addressed. In this
ﬁgure it is seen that the wall aspect ratio reduces the ineﬂecﬁveness to zero as
it increases, and the convergence to zero is more pronounced at lower heat
capacity ratios. The increased convergence to zero is a consequence of the
effect of axial conduction already reduced by the heat capacity ratio, and then
the inﬂuence of the wall aspect ratio requires a smaller magnitude to decrease
the normal resistance and eliminate axial conduction.

The product of MaL‘2 is shown in the legend of Figures 4.1-4.4, from
which the Mondt number can be calculated. The use of the Mondt number in
this study was as a variable parameter, which could be adjusted to provide the
desired NTU as the other parameters were varied. In general, as the Biot num-
bers and wall aspect ratio increased; the Mondt number was decreased to
maintain a constant NTU, while the Mondt number was independent of the
heat capacity ratio because the minimum heat capacity remained constant.
Referring to equation (4.8) for NTU, as the wall aspect ratio and Biot num-

 

 

bers were inc1
by the ratio K
unit width.
The tre
reaction to ch
essence, the 1
properties to
the Mondt nu
increase sinc
be available.
by only the r

58

bers were increased, the noted trend in the Mondt number was accomplished
by the ratio Kw [Cum becoming smaller because the analysis was based on a
unit width.

The trends seen in the Mondt number to maintain a constant NTU are a
reaction to changes made in the wall aspect ratio and Biot numbers. In
essence, the Mondt number is compensating the wall (Kw) and ﬂuid (Cm)
properties to match the conditions (Bi H, Bic, L‘ ). Therefore, it is ﬁtting that
the Mondt number decreases as the wall aspect ratio and ﬂuid Biot numbers
increase since more energy can be transferred for these conditions and must
be available. This outcome suggest that axial conduction may be described
by only the magnitude of the Mondt number. A point to be investigated later.

 

”a ' Tue-u

 

 

 

0.8-1

0.6-1

/|

 

 

 

0.4-

0.2-

 

0.0‘ﬁ—

 

FIGURE 4.3»

0.8—

0.6-

0.4~

/l

0.2..

 

0.0J~,\

 

 

59

 

 

 

 

 

 

 

 

0.81 MOL“
—"" .0001 140007
. BiH=Bic=BiL=Bi°= —"" .001 14007
—" .01 1407
—- .1 147
0.6-
Ca=1
« NTU=7
.. tTa—T—‘W‘ ..
0.4 .\.. \.,\ \-..,
. \ \~. '°\ \-.
'\ \-- "\ \.
0.2- ' \
\. - \ \.
- ‘\, \.. \.
0.0 l I I I 1‘- r . I UT..“_1 r r 1
10 100 1000 10000
L.*

 

FIGURE4.1.IneffectivenessasafunctlonofthewallaspectratloforNTU = 7endCR = 1

 

 

 

 

 

 

 

 

0.8-1 M°Lu
----- .0001 140007
. BiH=Bic=BiL=Bio= —--~ -001 ”007
—-- .01 1407
_. .1 147
0.6-
C,=.75
. NTU=7
0.4J ___....____,.;--\——..\
. \. ... .n\
0.2« \ -\ \ .
. \ -- ‘\ ‘\.
\.\ .\.. \‘-. \"
0.0 u . Vv‘:"‘>" " \.
10 100 1000 10000
LII!

 

“CURE“.heﬂecﬂveneuasefuncﬂonofthewaﬂaspectratioforNTU = 7 and CR = .75

 

 

 

 

 

0.8'1
0.6“

0.4-1

/\

O.2~

 

 

0.0
10

 

 

FIGURE“ Inc

0.84

0.6-

0.4-

 

 

 

 

 

 

 

 

 

 

 

 

0.81 MoLn
—"" .0001 140007
- Bi =81 =31 =81 = —m .001 14007
" c L ° ——-- .01 1407
—' .1 147
(15-
C'=.50
.. NTU=7
(14‘
.- ...IIO""—‘--.\.. \H.
\\ '\.. \...
' \' a. .\
. \. .\ \..
\.~S-- .,\..—__'-\. .._-:.\--..__
0.0 r I I I V I I V I r‘ ' T I ' I
10 100 1000 10000
Lil!

 

FIGUREaneffectlveneuasafunctlon ofthewallaspectratlo for NTU = 7 and CR = .50

 

 

 

 

 

 

 

 

0.8- MOL”
—°"° .0001 140007
- 81H=Bic=BiL=Bio= —'" .001 14007
_.. .01 1407
—- .1 147
0.6-
c,=.25
- NTU=7
0.4-1
div
02‘ ‘;::"-..\
/e .. \
. -——-"-\,. \ \
\ \., \ \
0.0 I T \igr $iu—‘r . \n--_I‘ l r r 1
10 100 1000 10000
LII!

 

 

 

FIGUREMIneffectlveneuuefuneﬂonoﬂhewallupectntioforNTU = 7andCR = .25

 

l.(

0.E

0.6

=.5)

0.4

6w(y+

 

3‘“ numb;

 

004a

 

61

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I t I I —- [.0001r
:.\ Blﬂ=Blc=Bl°=BlL= : .361 1
0.8“ "\..\.\ __ .1 -.
\_ =
. .\:\ ET}! 7 ‘
.\"\ L:—100
{0‘ 0-6 h' ‘\ .\ \‘H = —J
n. 1 ' '\,_.\>‘
3; Q ~-.
’ - \. \' "~- -.~..
@ 0.4 ..\\ ﬁ
. ° \. i
0.2- 'Q-.\-
\.
« \<
0.0 . l , r j 1 r T
0.00 0.20 0.40 0.60 0.80 1.00
x+
FIGURE 4.5. Median dime wall temperature as a function of dimensionless position and
BiotnumhersforNTU = 7,L =100,ande =1
0.8-
—--- Cu-1
‘ — C.-=.75
--- c.=.50
----' C =.25
0.6- '
—— “..\ NTU=7
0-4‘ ——--—--\.j§_ Biu=Bic=Bi°=BiL=0.001
. .....———. \\. MoLn=14007
...—'- \. ..\
\. \.
0.2e ,,—-. .. .\ §
. \ \,.\..\
0.0 . . , .\-->‘—--+‘—’==1L—- . . ,
10 100 1000 10000
LII!

 

 

FIGURE 4.6. Ineﬂectiveness as a function of the wall aspect ratio and heat capacity ratio for

NTU = 7,L

= 100, and all Biot numbers equal .001

 

 

0.8

0.6-

O.(y*= .5)

0.4'

0.21

 

0.8

0.5

 

62

 

 

 

1-0 ' l ' l ' 1 F l
C.= .25 ..

 

 

A
‘0.
1|
...
>\
V
.
Q
" NTU=7 "
0.2- LI=100 ..

BiH=Bic=Bio=BiL=.OO1

 

 

 

0.00 0.20 0.14.0 0.60 0.80 1.00

 

 

FIGURE 4.7. Median dimensionless wpli temperature as a function of dimemionless position and
heatcapacity ratio for NTU = 7, L = 100, and allBiot numbers equal .001

 

 

 

1 o o ff‘? ...... 6 j 1 I I I I '

 

NTU=7

 

 

 

 

 

 

"\. -\\,
1 1.7-100 \‘ \ j
0.2~ 31H=Bic=Bio=BiL=,001 \.\. _
— W0" -.5 \...
‘ ------- Hot Fluid .
""'-' Cold Fluid
0.0 1 1— ' I ' 1 v I ' ﬂ
0.00 0.20 0.40 0.60 0.80 1.00

x41-

 

 

FIGUREMMedhndhnensimhuwaﬂandluidgempentunuafuncﬂonotdhnemionku
positionandheatcapadtyratioforNTU = 7,L = 100,andallBiotnumbersequal.001

 

 

 

 

4.1.1

C01]
end
0011

as 1
as t
nitt
Bio
hat
tic 1

hek
the
ity 1
fun:
tion
Not

two
and
alw

the
nun
incr
fOrx
bers
incr
CDC;

theh

63

4.1.2 Inﬂuence of the Wall End Biot Numbers (Bi 0 and Bi L)

During the investigation of the wall aspect ratio it was determined that
convective loses from the ends of the heat exchanger wall are negligible if the
end Biot numbers are less than or equal to the ﬂuid Biot numbers (hot and
cold side Biot numbers). This dependence will be further investigated for
cases that have end Biot numbers greater than the ﬂuid Biot numbers. These
cases are physically possible since the end Biot numbers have the wall length
as its characteristic length and the ﬂuid Biot numbers have the wall thickness
as the characteristic length, as shown in equations (4.2)-(4.5). Thus, the mag-
nitude of the end Biot numbers being greater than the magnitude of the ﬂuid
Biot numbers does not require the same dependence on the convective heat
transfer coefﬁcients, a case that would be unrealistic, because the characteris-
tic lengths proportionately increase the end Biot numbers.

Following a similar procedure as the previous section the NTU will be
held at seven while the end Biot numbers are varied. The same magnitudes of
the hot and cold side Biot number will be investigated at the four heat capac-
ity ratios for a prescribed wall aspect ratio of 100. The ineffectiveness as a
function of the end wall Biot numbers (Bio = Bi L) are shown for these condi-
tions in Figures 4.94.12, with a different heat capacity ratio in each ﬁgure.
Notice for these ﬁgures that each curve has a starting point at approximately a
zero end Biot number, then is double valued for larger end Biot numbers. The
two values at each end Biot number are the ineﬂ‘ectiveness based on the hot
and cold ﬂuids. For a speciﬁed end Biot number, the cold ineffectiveness is
always greater than or equal to the hot ineffectiveness in absolute magnitude.

Observing Figure 4.9, which shows the ineffectiveness as a function of
the end Biot number for a heat capacity of one, the effect of the end Biot
number is seen. As the end Biot numbers increase i H decreases while iC
increases, for all ﬂuid Biot numbers investigated. This divergence of the per-
formance based on the hot and cold ﬂuids is expected as the end Biot num-
bers increase because the heat lost by convection from the ends similarly
increases resulting in an energy imbalance between the ﬂuids and a differ-
ence in the ineffectiveness based on the hot or cold ﬂuid.

Recalling that an ineffectiveness of zero implies axial conduction can
be neglected, it is seen for smaller ﬂuid Biot numbers that the performance of
the heat exchanger is actually enhanced, as noted by the ineﬂ’ectiveness based

64

on the hot ﬂuid being negative. That is, the effectiveness calculated based on
the hot ﬂuid is greater that the effectiveness calculated neglecting axial con-
duction. However, this enhancement of the performance based on the hot
ﬂuid is at the cost of the performance based on the cold ﬂuid, as seen by the
hot ineffectiveness being negative while the cold ineffectiveness approaches
one. This outcome exists because the heat that is acquired from the hot ﬂuid
is lost from the wall ends and is not transferred to the cold ﬂuid. The appro-
priateness of this result depends on the use of the heat exchanger. For remov-
ing heat from the hot ﬂuid this outcome is beneﬁcial, but for adding energy to
the cold ﬂuid the results are discouraging. In systems which have one stream
as a waste stream, such as an automobile cooling system, this could be a use-
ful result.

The ﬂuid Biot numbers affect both the starting point of the curve and
the amount that the ineffectiveness calculated based on the hot ﬂuid diverges
from the ineffectiveness based on the cold ﬂuid. Figure 4.9 shows that the
curves begin at a higher ineffectiveness and diverge a greater amount
between the two ineffectivenesses as the ﬂuid Biot numbers decrease. Fur-
ther, the divergence of the two ineffectivenesses as the end Biot number
increase occurs at a faster rate at smaller magnitudes of the ﬂuid Biot number.
There are three relevant observations to be made as the ﬂuid Biot numbers
decrease 1) the starting ineffectiveness increases, 2) the magnitude of the
divergence of the two ineffectivenesses increases, and 3) the rate of the diver-
gence with respect to the end Biot number increases. These points will be
addressed in subsequent paragraphs.

The higher starting ineffectiveness of the curves at is a consequence of
the effect of axial conduction increasing as the ﬂuid Biot numbers decrease.
The basis of this outcome was addressed in the previous section.

The magnitude of the divergence of the ineffectivenesses increasing as
the ﬂuid Biot numbers decrease is also due to the effect of axial conduction
increasing as the ﬂuid Biot numbers decrease. Since energy is more likely to
proceed along the heat exchanger as the ﬂuid Biot numbers decrease, it fol-
lows that increasing end Biot numbers will result in more energy being
removed. The additional energy removed results in an increase in the magni-
tude of the divergence as the ﬂuid Biot numbers decrease, which correspond-
ingly grows with the end Biot numbers.

65

Finally, addressing the rate of the divergence with respect to the end
Biot numbers, which increases with the decreasing ﬂuid Biot numbers, is
again linked to the increased effect of axial conduction at lower ﬂuid Biot
numbers. With axial conduction being more prevalent at lower ﬂuid Biot
numbers it is expected that the sensitivity to the end Biot numbers will be
greater at the lower ﬂuid Biot numbers. With axial conduction more preva-
lent, a change in the end Biot number will have more effect than when axial
conduction is less and the effect of the end Biot number is diminished. This
sensitivity to the end Biot number is seen by the change in ineffectiveness as
the end Biot numbers change, which explains the change in rate of diver-
gence as the ﬂuid Biot numbers decrease.

The inﬂuence of the heat capacity ratio requires a comparison of infor-
mation from Figures 4.9-4.12. To aid in visualizing this information Figure
4.13 combines the curve for a ﬂuid Biot number of .001 from each of the four
ﬁgures, onto a single ﬁgure. In this ﬁgure the ineffectiveness is seen to
increase as the heat capacity ratio increases, which was covered in an initial
discussion of the heat capacity ratio, where this trend was also seen. How-
ever, as the heat capacity ratio varied; the end Biot number showed more
inﬂuence on the hot ineffectiveness.

At a heat capacity ratio of one the curve for ineffectiveness based on
the hot and cold ﬂuids is symmetric. As the heat capacity ratio decreased, the
curve for the ineffectiveness becomes asymmetric, sloping more on the lower
leg. This leg of the curve represents the ineffectiveness based on the hot ﬂuid.
The change with heat capacity ratio represents the performance of the heat
exchanger actually improving, as seen by the curving sloping more negative.
The effect is due to the increased driving force, or larger difference between
the hot ﬂuid and wall temperature, on the hot side resulting in more energy
available to the wall. This additional energy is then lost from the wall ends
through convection.

For this case the Mondt number depends only on the hot and cold Biot
numbers, and therefore was constant with respect to the end Biot numbers.
For this reason little can be learned about the Mondt number by varying the
end Biot numbers.

 

 

 

 

 

 

 

 

 

 

10 . a e T . . .
././".’._——-.——w_— J
08- /-/ -
0.61 /. . ’ .____ ....— -'"" "7" -1
0 -°
‘* 04- ,,/-'/ _ NTU=7
,g'z ' ../ C'=1
‘ .\ ‘ L‘=1OO
02- - '\-\. ...... -
\, ......... \ -'->-<:: """"""" 3159‘. MO
_I' _\.. ......... .— 1%.“... ...... 1
oo_— ...... :KV- oooooooo s—esIs-ssss-8218=::::=:;.=_:.3 -t —.... '1 '0147
. .\‘\_\. -""‘ .01 .1407
‘°“-~---..—-———— 4 —-- .001 1.4007
—° .0001 14.0007
—0.2 r l l I ' l
0.0 02 0.4 06 08
81° or BIL

 

22%“ ”i Ineffectiveness as a function ofthe end Biot numbers for NTU = 7, L' = 100,
R =

 

67

 

 

 

a———--—-'——°

.. 1 --"' . NTU=7
.‘r . c,=.7s

O .
0'2'\ \"\.. ‘ L*=1oo

 

. \ ...—...*OOO—.IC—I.I‘I..”'C
_ 8=::::..-J-<o.:—
O 0 ...—"'7..gﬁzﬁigﬁiia‘istgzur—n
e \ '\..\..\u" BI“=BIC Mo

 

‘\. ‘ -~ .1 .0147
-0.2- \. . - —--- .01 .1407
\\-.\ . —» .001 1.4007

o~o~.~.

l h’ —- .0001 14.0007

 

 

 

 

 

Bi or BiL

 

 

FIGURE 4.10. Ineffectiveness as a function ofthe end Biot numbers for NTU = 7, L’ = 100,
and c, = .75

 

 

 

1.07 I I 1 I , I 1 1

"ﬂ".-.
.

0.8 ,..——--

0.5 ./' ..

r l

 

0.4 ..——..--""-———-"“"’"" ......... _
0.2 °°"' _
‘ 0 .1 . ..\~ ........................... 4
7' O-O'#2357:;—<_m =m=mu=rtifffiff J NTU=7
Q: 1 ' '.\ ° .1 C.=. 0
\ \-
‘0-2‘ . \°-\..\ - L‘=1OO
-0.4- \ \ = N ..

~ '\. . 1315131¢ 11°
"0'61 \- ~— .1 .0147
081 \\\ 1 —--- .01 .1407

=~~—. . —-- .001 1.4007
‘ —- .0001 14.0007

 

J

 

 

 

 

 

 

-1 .0 I 1 r I y I 1 r
0.0 0.2 0.4 0.5 0.8

Bi 0r BiL

 

 

FIGURE 4.11. Ineffectiveness as a function ofthe end Biot numbers for NTU = 7, L‘ = 100,
and CR = .50

68

 

 

 

T j 1 T I ' I
I’I’C’O’.‘Iﬂ.ﬂ.
./.’ u-I
./
./ _

...- ”a." .— d
u _lllm'ﬁéﬁﬁ'ﬁﬁm:u—aa:;.—a;::—;;:a_.;
I.\ q
a. ..
\..
\..
\ \.O
\I -
a\..
\..
a \o.
\

_eo—co
00‘00‘0'“..
..g—aa—I—
00".

NTU=7
05.25

L’=100

 

 

 

 

 

 

-1 8.. \\ Biu=Bic MO
. ‘ .\. _"°‘ .1 .0147
_22_ \\.\ _ —--- .01 .1407
' ‘~-..- —-- .001 1.4007
—- .0001 14.0007
‘2-5 . ﬁ m V 1 1 ' I

or BIL

 

 

FIGURE 4.12. Ineffectiveness as a function of the end Biot numbers for N TU

andCR = .25

= 7,L‘

100,

 

 

 

 

 

 

 

 

100 U I I l I I I 1’
0.8- ..
0.6‘ ..'::——""".::;:::::::: -1
0.446, 5,:::?::-;j’__:,,___...,......———---—-—--.____.__ -
0-2-<3:‘"\5.1;:;-—«<:::.’ 7.. r
"‘ ° 6\. \°"\---\... """"" “ ---------- ‘ NTU=7
.- 0.0" \ ‘\ " 'H...‘ .1 . .
ea 3 - -\ "..\\. ""= ------- . Blu=Blc=.001
-0.2- '\. \".\\ '4 L'=100
o 44 \.\ \\\‘ ‘ Mo=1.4007
-0.6-i \.\. - —-- 0,-1.0
.. \.\ ' —° .=.75
“0.8‘ '\. ‘1 _. _._ 50
. ‘ J " ’
‘1-0 I 7 1 l l —- 08.25
0.0 0.2 0.4 0.6 0.8
BI0 or BIL

 

 

FIGURE 4.13. Ineffectiveness as a function of the end Biot numbers and heat capacity ratio for

NTU = 7,L‘ =

100, and ﬂuid Biot numbers equal .001

69

4.1.3 Inﬂuence of the Fluid Biot Numbers (Biﬂ and Bic)

To change the end Biot numbers requires the least effort without alter-
ing the operating conditions (ﬂow rates or inlet temperatures) of the heat
exchanger. Assuming that the end Biot numbers can be reduced to the mini-
mum Biot number by reducing the convective heat transfer coefﬁcient, the
ﬂuid Biot numbers will be investigated. These Biot numbers should represent
the core of this study because the end effects require some unusual circum-
stances to become appreciable; and it is these ﬂuid Biot numbers and the wall
aspect ratio that should characterize axial conduction.

Setting the NTU at seven with the end Biot numbers equal at a magni-
tude of .0001, the ﬂuid Biot numbers are varied. The cold Biot number was
set and the hot Biot number varied over the range of interest. The cold Biot
number was then increased, and the hot Biot number varied over the same
range. This procedure was repeated for the four heat capacity ratios. Figures
4.14—4.17 shows the ineffectiveness as a function of the hot and cold Biot
numbers at each heat capacity ratio. With such small end Biot numbers cho-
sen, negligible end effects were seen and the ineffectivenesses based on
either ﬂuid were equal (in = ic = i) .

Examining the results for a heat capacity ratio of one, Figure 4.14, the
insensitivity of the ineffectiveness to the hot Biot number is seen at larger
magnitudes of the hot Biot number. This result shows that beyond a certain
magnitude of the hot side Biot number the performance of the heat exchanger
depends minimally on this Biot number. This implies that the effect of axial
conduction depends on the minimum ﬂuid Biot number and increasing the
other ﬂuid Biot number does not improve matters signiﬁcantly. To determine
the magnitude of the hot Biot number beyond which the ineffectiveness does

TABLE 4.3. Ineffectiveness as a function of ﬂuid Biot numbers from Figure 4.14

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Bi )1
Bi C .0001 .0002 .0016 .0032 .0128 0256 .0512 .1024 .2048 .4096 .8192
.0001 .413 .408 .402 .402 .400 .402 .401 .401 .401 .401 .401
.001 .403 .383 .294 .276 .260 .257 .256 .255 .255 .255 .255
.01 .400 .378 .216 .155 .084 069 .062 .058 .056 .055 .055
.1 .401 .377 2% .136 .048 028 .017 .012 .009 .008 .007

 

 

70

not change, Table 4.3 was created. This table shows the ineffectiveness as a
function of the ﬂuid Biot numbers from Figure 4.14. In Table 4.3 it is seen
that for magnitudes of the hot Biot number less than the cold Biot number the
ineffectiveness shows a dependence on the hot Biot number. However, for
magnitudes of the hot Biot number greater than the cold Biot number the
ineffectiveness changes a maximum of 5% after the point of equal ﬂuid Biot
numbers. If more accuracy is desired, it is possible to be within 1%, for Biot
numbers separated by one order of magnitude or more. These results show
that axial conduction is weakly dependent on the larger ﬂuid Biot number.

This outcome is explained by realizing that an increase of one ﬂuid
Biot number will improve the heat transfer between the wall and that ﬂuid.
However, the increased thermal communication between the ﬂuid with the
higher Biot number and the wall will not signiﬁcantly affect the performance
of the heat exchanger because of conditions on the other side of the wall. 0n
the side where the Biot number is unchanged, any additional energy that
could be acquired by the increased Biot number cannot be handled on this
side. There can be some change in the amount of energy transferred between
the wall and the ﬂuid with the minimum Biot number; through an increase in
the wall temperature, but the magnitude of this increase will be much less
than the energy change due to the increased Biot number on the other side. To
summarize, the wall is unable to transfer more energy between the ﬂuids
when one Biot number is increased because it cannot increase the energy
transfer at the other side of the wall. The outcome is the wall not acquiring
any more energy even though one ﬂuid Biot number is increased.

To assess the inﬂuence of the heat capacity ratio, Figure 4.18 was cre-
ated, which combines a curve from Figures 4.14-17 on to a single ﬁgure for a
cold Biot number of .001. In Figure 4.18 the affect of the heat capacity ratio
is seen. The ineffectiveness increased as heat capacity ratio increased. How-
ever, little change in the shape of the curves is seen as the hot side Biot num-
ber varies. An interesting result, considering that the heat capacity was seen
to increase the driving temperature differential, as the heat capacity ratio
decreased, on the ﬂuid with the maximum heat capacity, which was the hot
ﬂuid for this case. Even with the increased driving force it is not possible to
move more energy while increasing only one ﬂuid Biot number.

71

For this case, each pair of hot and cold Biot numbers had a correspond-
ing Mondt number. Requiring a different Mondt number for each point on the
ﬁgures. Because the case of a constant NTU and L' = 100 is rather restrictive,
the inﬂuence of the Mondt number will be left for a more general case.

72

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Tl V T V VV'VYI I Y j ‘YUYFI r 1' U I VVWT' '1" V f
_j NTU=7 _
0'7 Bi°=BiL=.0001
. L’=100 .
c,=1
0.5~ -*
‘ _ Bic=.0001q
i.
0-3" .001 -
0.1- .01 ..
. .1 .
—0.1 0' I I I I 0000' ‘0 I h TTVI 0 I r7 010' I I I 1 ﬁII' 0 Vﬁ
0.0001 0.0010 0.0100 0.1000 1.0000
BIH
FIGURE4J4.IneffectivenessasafunctionoftheluidliiotnumhersforNTU = 7,L. = 100,
If I 'T0010' 1' T 0 0000'] I I 0 I 0000' 000‘
NTU=7
0'7 ' Bio=BiL=.0001 "
‘ L*=100 ,
c,=.75
0.5-1 -
1 _ Bic=.0001.
0.3-1 -
q .001 .
0.1- -
.01
. .1 .
-0.1 1' I I I IIIII' I I TTII'I r I TIIIII' I I IIIIII I r
0.0001 0.0010 0.0100 0.1000 1.0000
BIH
FIGURE4JS.IneifectivenessasafuncﬂonoftheﬂuidBiotnumbersforNTU = 7,L' = 100,

andCR = .75

 

 

 

73

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. . n"-.. . .....n, . . Tm... . .....n,
NTU=7
0'7“ 13158150001
. L*-1oo ,
c.=.50
0.5- —
i. .
0.3__ Blc=.0001_
0+1 .001 _
.01
1 .1 -
—O.1 T I IIIIII' I I I IIIII' 1 I ITIIII‘ I IIIIII' I
0.0001 0.0010 0.0100 0.1000 1.0000
Bi"
FIGURE4J6.IneﬂectivenessasafunctionoftheﬂuidBiotnumbersforNTU = 7,L‘ = 100,
CM CR = .50
. rm... . ......, . . ....n, . . ......T . ..
NTU=7
0'7 Bio=BiL=.0001
. L"‘=1OO
c,=.25 1
0.5- _
0.34 _
‘ Bic=.0001‘
0.14 .1
.001
1 .01. .1 -
_O.1 I I I I IITI" I I I IIIII' I I I I IIII' T I r IIIII' I I I
0.0001 0.0010 0.0100 0.1000 1.0000
BIH

 

 

FIGURE4.17.IneﬂectiveneasasafunctionoftheﬂuidBiotnumbersforNTU = 7, L’ = 100,
and C, = .25

 

 

 

 

 

 

 

 

 

NTU=7
0‘7 3158150001 7
LI=100
Bic=.001
0.5‘ -
5 \
0'3“ c,= 1.0 ‘
. .75 1
0.1- \ .50 -
. .25 1
_o.1 '1 I T fﬁIITI] I I IIIIII' I I IiIrIIl I I IIITII' I I I
0.0001 0.0010 0.0100 0.1000 1.0000

Bi

H

 

 

 

FIGURE 4.18. Ineffectiveness as a funcﬂon of the hot ﬂuid Biot number and heat capacity ratio for
NTU = 7, L =100,andacoldﬂuidBiotnumbersof.001

4.1.4 Summary of Investigation with a Constant N TU

Examination of the effect of axial conduction at a constant, relatively
high NTU has provided insight into the parameters that characterize condi—
tions that axial conduction will arise. There were also some possible simpliﬁ-
cations that arose. These issues are now discussed.

The wall aspect ratio is an important parameter in describing axial
conduction. This parameter represents the physical dimensions of the heat
exchanger; and it was seen that for large values of the wall aspect ratio axial
conduction could be neglected, almost regardless of the Biot numbers. It was
still possible for axial conduction to exist at the larger wall aspect ratios, but
only for very small Biot numbers. A case simulating a nearly insulated wall

75

(I: ~ 0) or an inﬁnite wall thermal conductivity (Kw » In), both physically trivial
cases.

Having investigated the physical dimensions that axial conduction was
important, a wall aspect ratio was set to exhibit axial conduction; and the
inﬂuence of the wall Biot numbers could be investigated. The Biot numbers
on the ends and for the hot and cold ﬂuids were grouped separately. This per-
mitted setting one group while varying the other, which reduces the possible
combinations to study.

The end Biot numbers cause energy loses from the wall ends through
convection and an energy imbalance between the ﬂuids. However, this effect
was not signiﬁcant for magnitudes of the end Biot numbers less than or equal
to the magnitude of the ﬂuid Biot numbers. When the end Biot numbers
exceeded the ﬂuid Biot numbers some interesting results were seen, with the
performance based on the hot ﬂuid actually larger than the performance cal-
culated neglecting axial conduction. This was at the cost of the performance
based on the cold ﬂuid, however.

Although the scaling used in the Biot numbers can result in the end
Biot numbers being larger than the ﬂuid Biot numbers, the proportionate
amount that the scaling changes the Biot numbers, in comparison to the con-
vective heat transfer coefﬁcients, is the wall aspect ratio. Making the end Biot
numbers greater than the ﬂuid Biot numbers, in most cases, would require a
very large wall aspect ratio, based on the expected magnitude difference in
the convection coefﬁcients; a case shown to result in negligible axial conduc-
tion. In addition, the effect of the end Biot numbers improves the effective-
ness in certain situations, and may not have any affect on axial conduction.
For these reasons, the end Biot numbers will be assumed negligible and not a
factor in further investigation.

With the end Biot numbers eliminated from further scrutiny, they were
set to the minimum magnitude of the Biot number, the focus turned to the
ﬂuid Biot numbers. Physically, the ﬂuid Biot number can be thought of as the
ﬂuid operating conditions. Describing the interface convection between the
ﬂuid and the wall. As the ﬂuid Biot numbers were varied, it was seen that the
results were fairly insensitive to the larger Biot number. This outcome simpli-
ﬁes the task of characterizing axial conduction, which now may be shown to
be a function of the wall aspect ratio and the minimum ﬂuid Biot number.

76

The heat capacity ratio plays a role in the effect of axial conduction on
the performance of the heat exchanger. The heat capacity ratio was varied
during the investigation of all other parameters. It can alter axial conduction
by inﬂuencing the interaction between the ﬂuid and wall. The eﬂ’ect of axial
conduction is smaller as the heat capacity decreases. Therefore, to reduce
axial conduction the heat capacity ratio can be decreased.

Although the Mondt number was not covered for all cases it was
shown to be inﬂuential. The Mondt number was adjusted as other parameters
were varied to maintain a constant NTU. As the wall aspect ratio and Biot
number became larger the Mondt number became smaller and the effect of
axial conduction decreases, which is interpreted as the Mondt number adjust-
ing the properties of the ﬂuids and wall to match the operating conditions.
This parameter may be useful to describe the condition when axial conduc-
tion will be seen.

In closing, this section has allowed the characterization of axial con-
duction to be reduced to a function of four variables; which includes the wall
aspect ratio, minimum ﬂuid Biot number, the heat capacity ratio, and Mondt
number. These variables will provide approximate estimates when axial con-
duction needs to be considered and to what extent the performance will be
altered if it is assumed negligible. Approximate, because only one variable is
considered for the ﬂuid Biot numbers. In the next section these results will be
used to produce ineffectiveness curves as a function of NTU for the parame-
ters of interest.

77

4.2 Characterization of Axial Conduction as a Function of NTU

Setting the NTU constant and varying the parameters revealed some
possible simpliﬁcations in the parameters needed to characterize the condi-
tions that axial conductions will arise. At the beginning of this chapter it was
shown that it requires eleven parameters to mathematically model this prob-
lem. 0f the number of parameters needed to mathematically represent the
problem the solution presented in dimensionless form would be a function of
seven of these parameters. Which includes four Biot numbers, a wall aspect
ratio, the ﬂuid heat capacity ratio, and Mondt number. Using the results of the
previous section the number of independent variables will be reduced.

The effect of the end Biot numbers was only seen for end Biot number
magnitudes greater than that of the ﬂuid Biot numbers. Although this is pos-
sible, it is not seen as a plausible operating condition. Therefore, the end Biot
numbers will be assumed to be the lowest possible magnitude and play no
role. This reduces the parameters needed to characterize axial conduction to
ﬁve.

Another simpliﬁcation that can be considered is the dependence of
axial conduction on the ﬂuid Biot numbers. For a constant NTU it was shown
that the ineffectiveness depends mainly on the magnitude of the smaller of
the two ﬂuid Biot numbers. However, these results were for a constant NTU
and wall aspect ratio of 100 and the dependence, or lack of, may differ with
these parameters variable. To explore this point, data were generated to pro-
duce ineffectiveness-NTU plots for a heat capacity ratio of one, which are
shown in Figure 4.19. The plots are shown as a function of the wall aspect
ratio and the maximum ﬂuid Biot number, while the minimum ﬂuid Biot
number remained constant. The minimum and maximum ﬂuid Biot numbers
are

Bi m
In Figure 4.19 the results of the previous section are supported for a wall
aspect ratio of 100 with less than a 1% change in the ineffectiveness as the
maximum ﬂuid Biot number increased three orders of magnitude. However,
for larger wall aspect ratios the ineffectiveness shows a dependence on the

maximum ﬂuid Biot number with as high as a 45% reduction in the ineffec-

ax = max (BiH,BiC) (423)

78

tiveness. This reduction was at a maximum Biot number one order of magni-
tude larger than the minimum Biot number, for further increases in the maxi-
mum Biot number the ineffectiveness was not changed signiﬁcantly.
Although Figure 4.19 shows a greater absolute change in the ineﬂ’ectiveness
at a wall aspect ratio of 500 than it does at a wall aspect ratio of 1000, the rel-
ative change in the ineffectiveness, which gauges the magnitude of axial con-
duction, is larger for a wall aspect ratio of 1000 with a 45% reduction in the
ineffectiveness.

 

 

 

 

 

 

 

 

 

0.8-
—-—- Bi”=.0001
Bim=.0001 ------- Bi_=.001
BiL=Bi°=.0001 "'- Bim=.01
--- BI__=
0.6~ Cu=1
L*= 100
e 0 4‘ /___”____...-———--
/‘
500
hays-vary” 1000
/ ##‘MW
0-0‘ 1 v 1 l I l
0 2 4 6 8 10
NTU

 

 

 

FIGURE 4.19. Ineffectiveness as a function of the NTU, wall aspect ratio, and maximum ﬂuid Biot
number for Him-n = 0.0001 and CR = l

79

It was shown earlier that axial conduction was greater for smaller wall
aspect ratios and ﬂuid Biot numbers. For this reason, at lower wall aspect
ratio, the ineffectiveness is less sensitive to the ﬂuid Biot numbers because
both parameters are at magnitudes that promote axial conduction. As the wall
aspect ratio increases, the solution becomes more sensitive to the ﬂuid Biot
numbers because this parameter is more inﬂuential in promoting axial con-
duction with the wall aspect ratio larger. This results in the ineﬁectiveness
decreasing signiﬁcantly as the maximum ﬂuid Biot number is increased at
larger wall aspect ratios. However, the maximum ﬂuid Biot number can only
improve the operating conditions so much, due to the limiting nature of the
conditions on the opposite side of the wall where the minimum ﬂuid Biot
number remains unchanged. The outcome is less degradation in the perfor-
mance of the heat exchanger as the maximum ﬂuid Biot number is initially
increased, but further change in the maximum ﬂuid Biot number produces no
improvement.

A ﬁnal observation that can be made concerning Figure 4.19 is the
increased ineffectiveness at lower values of NTU as the maximum ﬂuid Biot
number increased for a wall aspect ratio of 100, a result contrary to all other
cases, where ineffectiveness decreased as the maximum ﬂuid Biot number
was increased. This is only seen for values of NTU less than three, where
there is a crossing point and previous trends are seen, and only for this partic-
ular wall aspect ratio. This is due to the wall aspect ratio and minimum ﬂuid
Biot number both being at magnitudes that promote axial conduction.

Increasing the maximum ﬂuid Biot number is expected to decrease the
ineffectiveness because of the increased convection on one side of the wall.
In addition to increasing the maximum Biot number the Mondt number will
decrease to maintain the same NTU that existed when the ﬂuid Biot numbers
were equal. The lower Mondt number also would be expected to decrease
axial conduction. But it is the decrease in the Mondt number that actually
causes the increase in the ineffectiveness at low NTU.

To decrease the Mondt number the ratio (Kw! Cm) must be decreased.
At lower values of the NTU, the magnitude of the minimum heat capacity is
relatively larger and decreasing the ratio Kw! Cm." will make it even larger.
Therefore, the limiting factor on the energy transfer will be the magnitude of
the wall aspect ratio and ﬂuid Biot numbers. When the maximum ﬂuid Biot

80

number is increased, the corresponding increase in the Mondt number results
in the ineffectiveness increasing because the additional energy can not be
moved across the wall.

For larger NTU the heat capacity is relatively small and it becomes the
limiting factor on the energy transfer. For this case, the result is the ineffec-
tiveness decreases as the maximum ﬂuid Biot number is increased.

The results in Figure 4.19 show, contrary to the previous section, that
in addition to the minimum ﬂuid Biot number the maximum ﬂuid Biot num-
ber must also be considered for magnitudes of the maximum ﬂuid Biot num-
ber within one order of magnitude of the minimum ﬂuid Biot number.
Increasing the maximum ﬂuid Biot number further produces negligible
change in the results and does not need to be considered.

Because unequal ﬂuid Biot numbers must be considered the relative
magnitudes of the heat capacities on the streams will also play a role. For
example, at a particular heat capacity ratio and for unequal ﬂuid Biot num-
bers the results obtained when the ﬂuid with the minimum heat capacity has
the minimum ﬂuid Biot number will differ from the results obtained when the
minimum heat capacity is for the ﬂuid with the maximum ﬂuid Biot number.
To verify this point Figure 4.20 was created, which shows ineﬂ’ectiveness-
NTU curves at various heat capacity ratios with the minimum heat capacity
associated with the minimum ﬂuid Biot number and maximum ﬂuid Biot
number. At lower magnitudes of the NTU the two cases vary greatly, while at
larger values of the NTU the two cases converge toward a common result and
the convergence occurs at a lower NTU for larger heat capacity ratios.

The two noted dependencies are due to the magnitude of the heat
capacities at the referenced NTU. At lower values of NTU the minimum heat
capacity is relatively larger in comparison to the amount of energy that the
wall can transfer. For an opposite pairing of the extremes of heat capacity and
ﬂuid Biot numbers the amount of energy transferred is dictated mainly by the
ﬂuid Biot numbers. The larger heat capacity is limited by the smaller Biot
number on one side of the wall and similarly the larger Biot number is limited
by the smaller heat capacity on the other side of the wall. Producing an insen-
sitivity to the heat capacity ratio. However, if the pairing is changed to asso-
ciate extremes, the magnitudes of the heat capacity and ﬂuid Biot number

81

will not oppose each other and a dependence on the heat capacity ratio exists.

For larger values of NTU the minimum heat capacity is smaller in com-
parison to the possible energy transfer through the wall; and the energy trans-
ferred is dictated mainly by the heat capacity ratios. The insensitivity to the
Biot number is shown by the two cases approaching the same result at larger
values of the NTU. The smaller heat capacity ratios requires a larger magni-
tudes of the NTU to converge to the same result because the larger differences
in the heat capacity across the wall make the result more sensitive to the Biot
numbers, which requires a larger NTU to insure that heat capacity is the limit-
ing parameter on the heat transfer and not the ﬂuid Biot numbers.

 

 

 

 

 

 

 

 

 

 

 

0.8-
— C". on Bi”
Bim=.0001 Bim=.001 """"" C... 0" 31..
BiL=Bi°= .0001
05" L*=100
8:10
e 0 4- ----- - A“ _75
.............................................................. ._._. 50
0.2q ":5..." ............................................................................... .25
1
0.0- I 1 I I I
0 2 4 6 8 10
NTU

 

 

 

FIGURE 4.20. Ineffective-en as a function of the NTU and heat capacity ratio for
Bin," = 0.0001, mm = 0.001, and C, = 1

82

Between the limiting cases of a small NTU (where the heat transfer is
limited by the ﬂuid Biot numbers) and a large NTU (where heat transfer is
limited by the heat capacity ratio) both parameters are important. In this
region a combination of the outcomes is seen.

The results of Figures 4.19 and 4.20 have shown that axial conduction
can now be characterized for all ﬂuid Biot number combinations with mini-
mal error by two cases; the ﬂuid Biot numbers equal (Him: = Him) or the
maximum ﬂuid Biot number an order of magnitude larger that the minimum
(Him = 1083”,"). For the case of unequal ﬂuid Biot numbers, the minimum
heat capacity ratio must be considered to exist on the ﬂuid with the minimum
ﬂuid Biot number (C - Bind”) and with the maximum ﬂuid Biot number

(Cm, - Rim“) separately, since the results will differ.

Encompassing the previous results, the operating conditions that axial
conduction will arise can now be identiﬁed with ﬁve parameters. A minimum
and maximum ﬂuid Biot number, the wall aspect ratio, the heat capacity ratio,
and Mondt number. However, for the ﬂuid Biot numbers only one order of
magnitude larger than the minimum needs to be considered for the maximum
ﬂuid Biot number, greatly reducing the possibilities.

It is now possible to produce ﬁgures, which can be used to predict the
possibility of axial conduction affecting the performance of the heat
exchanger. To do this requires considering four minimum ﬂuid Biot numbers
at four heat capacity ratios, giving sixteen ﬁgures. On each ﬁgure, two cases
for the ﬂuid Biot number will be shown, ﬂuid Biot numbers equal and maxi-
mum ﬂuid Biot number ten times the minimum ﬂuid Biot number. Further-
more, for unequal ﬂuid Biot numbers both maximum and minimum Biot
numbers will be associated with minimum heat capacity. All of these cases
shall be shown at different wall aspect ratios.

83

4.2.1 Ineffectiveness-NT U Results
4.2.1.1 Heat Capacity Ratio of l

 

 

 

 

 

 

 

 

6.6—
— Bi__=ei,_
Bi“=.0001 ------- Bi_-1OBi,_ C... on Bin.
Bi =Bi =oooo1 —‘ Bl_=1OBl_h C‘ on Bin
L ° L‘= 1oo
CI=1 ”___
0.4“ /'/".
./'/
6 500
o.2~ / /./v’
/' ./'/ 1000
/ /
/,/ .ﬁf’.’“
./ ’.’,"’./
0.0“ L l l l l l
O 2 4 6 8 10
NTU

 

 

FIGURE 4.21. Ineffective-ace as a function of the NTU and wall aspect ratio for

Him." = 0.0001 and CR = 1

 

 

 

 

 

 

 

 

 

— Bi_=ai._.
Bim=.001 ....... Bina1oaiﬂ c... on Bil.
0,4: BiL=Bi°=.0001 —' Bi_=lOBi_,_ C_,_ on B|__
L = 100
Cl=1
‘ ./
/.//
b ./
./
0.2“ //
/'/
+ ./
// soo
' ___-___-————--——— — ““"1 000
0.0- “ """' - -_——.__r...._._.___.r
° 2 4 6 8 1o
NTU

 

FIGUREmIneffectivene-aaafunctionoftheNTUandwailaspectratiofor Rim-n = 0.001

andCR=l

 

84

 

 

 

 

 

 

 

 

 

 

 

 

0.20-a
— Bi_=3i__
Bi*=.01 ------ Bi_-1OBiﬂ_ C" on Bi ....
Bi=Bi = 0001 —- ai__=1oai_ Cion Bi__
L O '
0.1 - _
5 CH L*= 100
0.10-*
0.054
0.00- T K | ‘ 1
O 2 4 6 8 10
NTU

 

 

Elémuflmﬂecﬂvemuamoﬂhemandnﬂupednﬂofor Rim." = 0.01
R =

 

 

 

 

 

 

0.10‘
— Bi~=3iw
Bi*=.1 """" Bin—31033” C“. on Bid.
Bit=Bio=.0001 —' Bi_=103i_ CA on BL-
* Cl=1
0.05.,

5

 

 

 

 

HGUREmlneffecﬂveneuaaafunctionoﬂheNTUandwaﬂaspectratiofor Him-n = 0.1

 

85

4.2.1.2 Heat Capacity Ratio of .75

 

 

 

 

 

 

 

 

0.6-
-— BlusBiﬂ
Bi*=.0001 ....... Bin-1033* C.- on Bl"
‘ BiL=Bio='OOO1 -- Bi_=1OBi_,_ C_,_ on Bi_
C.=.75
1:.
0.4- 'm’bML’: 100
....ﬂ‘r'oﬁﬂrﬂ'ﬁsmﬂ"
o . /,°,. ......
/ 500
o 27 /
. / ./-/,...’ --------- 1000
./ ..........
- .. 4 -------- ,___._————---'""——'-' ..........
0.o_ ----.:' ....l. ................ I I I I
O 2 4 6 8 10
NTU

 

 

FIGUREdJSJneﬂecﬂveaaaaaaafunctionoftheNTUandwaﬂaapectraﬂofor
31m," = 0.0001 and CR = .75

 

 

 

 

 

— ai__=ai_
ai...=.oo1 ------- Big-1035.. 0.. 0" Bin»
0.4- BiL=Bi°=.OOO1 —' B'-=1°B"‘ C“ °n 8"
C.=.75
L*= 100

 

 

 

 

 

 

 

 

FIGUREmIneffectivencaaaaai’unctionoftheNTUandwallaspectratiofor Him-n = 0.001
and CR = .75

86

 

 

 

 

 

 

0.20q
— Bi_=Bi__
Bi*=.o1 ....... Bin-=1OBL. C.h on Bi”
BiL=Bi°=.0001 —- BI_=lOBI_,_ C_ on Bum
0.15u C'___.75
0.10-

0.05%

 

 

 

 

 

NTU

 

FIGUREUIInaffecﬂveneaaaaafnnctionofthaNTUandwaﬂupactntiofor Bimin = 0.01
M CR = .75

 

 

 

 

 

 

0.10
1 — Bi_=ai__
Bim='1 ------- Bi_=1OBiﬂ_ C... on Bi”
BiL=Bi°=.OOO1 —' B|_=1OBI... C... on 8|...
q C.=.75
& 0.05~

 

 

 

 

 

 

 

FIGUREMIneﬂectivenca-aaafunctionoﬂhemandwailaspectraiiofor Bimin = 0.1

 

 

87

4.2.1.3 Heat Capacity Ratio of .50

 

 

 

 

 

0.6-
— Bi_=Bi*
Bi_h=.0001 ------- Bim=1OBim C“. on Bi".
Bi =Bi = 0001 —- Bi__=1OBi___ C_,_ on Bi__
L O '
C'=.5O
0.4-

 

 

 

0.2-

 

 

 

 

0.0d

 

 

 

 

 

FIGURE429JneﬂecﬂvenuaaaafunctionoftheNTUandwanaspectratiofor

 

 

 

 

 

mm," = 0.0001 and CR = .50
0.41
— Bi_=ai__
_‘ Biw=.001 ....... Bi~31oaiﬂ c“ on Bi...
BiL=Bi°=,0001 -° Bi_=1OBi_,_ C‘_ on Bi__
0'3“ C'=.50

 

 

 

 

 

 

FzgléRE 4.3%01neffectiveneas as a function ofthe NTU and wall aspect ratio for Him." = 0.001
g =
R a

88

 

 

 

 

 

0.20q .
— Bi_=ei__

Bi*=.o1 ....... Bingt'OBi'. Cut on Bi”
BiL=Bi°=.OOO1 —- BI..=1OB|.. e... on an

0.15" C'=.50

0.10“

0. -

05 L*= 100

 

 

 

 

 

NTU

 

FIGUREulJneffectiveneasaaafnnctionoftheNTUandwailaspedratiofor Rim-n = 0.01
3!!! CR = .50

 

 

 

 

 

 

 

 

 

0.10-
— Baum-mim
Bi*=.1 ------- Bin-2108i” c__ on Bi“
BiL=Bi°=.0001 —' Bi_=1oau.. 0.. on a...
1 C.=.SO
i 0.05-
L‘= 100
0.00—W ................. I. .................. - .i..................%...................,"
0 2 4 6 8 10

NTU

 

mlémugolmﬂecﬂveneuuafmdondthemandwaﬂaspednﬂofor Rim-n = 0.1
R = -

 

 

89

4.2.1.4 Heat Capacity Ratio of .25

 

 

 

 

 

0.4m
— ai_-=Bi_,_
Binh=’0001 ....... Bin-3103i... C.- on Bi".
Bi :3: = 0001 —- Bi_=1OBi__ C‘ on 83
L o ‘
.1 C.=.25

 

 

 

 

 

 

 

 

 

FIGUREdJJJneffectiveneuasafhncﬂonoftheNTUandwaﬂaspectnﬂofor
Bum,” = 0.0001 and CR = .25

 

 

 

 

 

0.2!
— Bi~=aim
Bi*=.001 ------- Bi_=1OBi__ c“, on Bi”
BiL=Bi°=.0001 _' Bi_=1OBI__ CA on Bl-
C.=.25

 

 

 

 

 

 

 

 

E'IIEUCRE 4.33.51neffectiveneu as a function ofthe NTU and wall aspect ratio for Rim-n = 0.001
R = -

 

 

6

0.10-

0.05“

 

Biw=.01
BiL=Bi°=.OOO1
CR: .25

 

 

—- Bi__=Bi__
------- Bi_=1OBiw C... on Bi“.
-- Bil-5108i C on Bi

L"= 100

 

 

 

 

 

 

 

 

 

NTU
FIGURE 4.35. Ineffectiveness as a function ofthe NTU and wall aspect ratio for Rim-n = 0.01
0.10q
— Bi_=3i,_,
Bim=.1 ------- Bi__=1oai_ c... on Bi,“
Bi =Bi = 0001 —' Bi_=1OBi_ C_,_ on Bi—
L o '
Cl=.25
a 0.05-

 

 

 

0.00_______-ﬁ==‘—

NTU

L*= 100

 

 

FIGUREmlneffectivenessasafnnctionoftheNTUandwaﬂaspectraﬂofor Bimin = 0.1
andCR = .25

 

 

91

4.2.1.5 Discusdon of Results

In Sections (4.2.1.1-4.2.1.4) results were presented to give an indica-
tion of the operating conditions when axial conduction will exist and the
extent to which the performance of the heat exchanger will be affected. Much
of the information pertaining to the underlying physics was covered when the
NTU was held constant and therefore will not be covered in detail in this sec-
tion. Instead, the general trends in Figures 4.21-4.36 will be discussed.
Because the scope of this study was intended to be general, all magnitudes of
the describing parameters were considered for which axial conduction was
nonzero. However, only the ﬁgures that best demonstrate the trends will be
discussed.

The Mondt number is the only describing parameter that is not shown
on the ﬁgures because it was the adjustable parameter in this study. Equation
(4.10) shows that NTU is a function of the ﬂuid Biot numbers, wall aspect
ratio, and the Mondt number. To vary the NTU the ﬂuid Biot numbers and
wall aspect ratio was set and an appropriate magnitude of the Mondt number
selected to provide the desired NTU, which is analogous to adjusting the ﬂow
rate of the ﬂuids since the remaining terms in the Mondt number depend on
the properties of the wall and ﬂuid. Thus, the NTU is indicative of the Mondt
number, and the two variables are linearly related for speciﬁed a wall aspect
ratio and ﬂuid Biot numbers. Although the inﬂuence of the Mondt number
will be discussed in this section the real power of this variable is to predict
the presence of axial conduction, which is covered in the next section.

The wall aspect ratio and ﬂuid Biot numbers display a similar inﬂu-
ence on axial conduction. At smaller magnitudes of these parameters the
effect of axial conduction is more prominent. This is demonstrated in Figures
4.21 and 4.24. In Figure 4.21 for L‘ = 100 and Rim." = Him = 0.0001 the
ineffectiveness is over 40%, maintaining the same ﬂuid Biot numbers the
ineffectiveness decreases to a maximum of 10% at a wall aspect ratio of
1000. In Figure 4.24 the ﬂuid Biot numbers are increased to .1 while main-
taining the wall aspect ratio at 100 and the ineffectiveness decreases to 2%.
These trends are seen for the wall aspect ratio regardless of the ﬂuid Biot
number and for equal ﬂuid Biot numbers (Rim, = Bin“), but are not as uni-
versally true for unequal ﬂuid Biot numbers. Since the maximum ﬂuid Biot

92

number introduces secondary effects as it is increased, a point to be discussed
later.

On a physical basis the smaller wall aspect ratios and ﬂuid Biot num-
bers represent less thermal communication between the ﬂuids because the
smaller wall aspect ratios increase the wall thickness or decreases the wall
length while the smaller Biot numbers decrease the convection between the
wall and the ﬂuid. These changes result in a greater amount of energy ﬂow
along the wall because the resistance to the energy ﬂow in this direction is
less than the resistance to the ﬂow across the wall, which increased due to the
changes in the wall aspect ratio and ﬂuid Biot numbers.

In general, increasing the maximum ﬂuid Biot number reduces the
effect of axial conduction. This is shown by the ineffectiveness decreasing by
as much as 40% in Figure 4.21 when the maximum ﬂuid Biot number was
increased at a wall aspect ratio of 1000, which is explained by the resistance
to convection decreasing at the larger maximum ﬂuid Biot number and axial
conduction having less inﬂuence with the lower resistance. An exception to
this outcome is seen for a combination of small magnitudes of the NTU, wall
aspect ratio, and ﬂuid Biot numbers, where the ineffectiveness increased at
the larger maximum ﬂuid Biot number, as shown in Figure 4.21 for L' = 100
and NTU<2.5. This is a result of the energy transfer switching from being
limited by the thermodynamic parameter (Cm) to the heat transfer parame-
ters (Bimx, Him, L' ). To show this, the effect of the Mondt number must ﬁrst
be considered.

The Mondt number varies with NTU, ﬂuid Biot numbers, and the wall
aspect ratio. To maintain a constant NTU at the larger maximum ﬂuid Biot
numbers the Mondt number is smaller than it was for equal ﬂuid Biot num-
bers, which means the minimum heat capacity must become larger with the
maximum ﬂuid Biot number since all other parameters are constant in the
Mondt number for a speciﬁed wall and ﬂuids. In addition, at small NTU the
Mondt number is relatively small in comparison to its magnitude at larger
values of the NTU. Therefore, the minimum heat capacity is larger for small
magnitudes of the NTU, and larger yet at the increased maximum ﬂuid Biot
number; due to the reciprocal relationship between the Mondt number and
minimum heat capacity.

93

With these arguments, at the small values of the NTU the energy trans-
fer is limited mainly by the minimum Biot number and wall aspect ratio since
the minimum heat capacity is large. Then increasing the maximum ﬂuid Biot
number increases the ineﬂ'ectiveness because the additional energy cannot be
moved across the wall. While at larger NTU the energy transfer is limited
mainly by the magnitude of the minimum heat capacity and when the maxi-
mum ﬂuid Biot number is increased the additional energy can be moved
across the wall ineffectiveness decreases. Note that at larger minimum ﬂuid
Biot numbers and wall aspect ratios this effect is not seen because the condi-
tions are not as limiting on the energy transfer, as shown in Figure 4.21 at
L' =500 and 1000 and for all wall aspect ratios at Bin," = 0.001 in Figure
4.22.

Because unequal ﬂuid Biot numbers needed to be investigated, the
association between the magnitude of the heat capacity and ﬂuid Biot num-
bers became an issue. Both possibilities are shown on all ﬁgures: minimum
heat capacity with maximum ﬂuid Biot number and with minimum ﬂuid Biot
numbers. The outcome of these two cases is straightforward. When like
extremes of the heat capacity and ﬂuid Biot numbers are associated an
increase in the maximum ﬂuid Biot number will improve conditions more so
than when opposite extremes are associated Since similar capabilities of
energy transfer are being matched in the former case, largest heat transfer
parameter with largest thermodynamic parameter. While for the later case the
opposite extremes limit the energy transfer because the heat transfer parame-
ter restricts the thermodynamic parameter or the thermodynamic parameter
limits the heat transfer parameter. The outcome is, associating like extremes
of heat capacity and ﬂuid Biot numbers always has an equal or lower ineﬂ‘ec-
tiveness compared to the case of opposite extremes associated.

The heat capacity ratio reduces the effect of axial conduction as it
decreases. Considering a wall aspect ratio of 100 and equal ﬂuid Biot num-
bers of magnitude .0001, at CR = 0.25 the ineﬁ’ectiveness if less than one-half
the ineffectiveness at CR = 1.0, in Figures 4.33 and 4.21 respectively. The
heat capacity ratio decreases the effect of axial conduction because there is
more energy available on one side of the wall and this increased amount of
energy causes a larger driving temperature potential between the wall and the
ﬂuid. The improved energy transfer and larger amount of available energy

94

reduces the effect of axial conduction and hence decreases the ineffective-
ness.

The heat capacity ratio also has some secondary effects that are related
to the maximum ﬂuid Biot number and the association between the magni-
tudes of the heat capacity and ﬂuid Biot numbers. For the minimum ﬂuid heat
capacity associated with the maximum ﬂuid Biot number the ineffectiveness
was shown to increase as the maximum ﬂuid Biot number increased for small
magnitudes of the NTU, wall aspect ratio, and minimum ﬂuid Biot number.
Comparing the results of Figures 4.21, 4.25, 4.29, and 4.33 at L' = 100
shows that the heat capacity ratio magniﬁes the amount that ineffectiveness
increases at the larger maximum ﬂuid Biot number in comparison to the inef-
fectiveness at equal ﬂuid Biot numbers. Also, the magnitude of NTU at which
the crossing point is seen and the ineffectiveness decreases for larger maxi-

mum ﬂuid Biot numbers moves to larger NT U as the heat capacity ratio
decreases.

Recalling that the larger ineﬂ’ectiveness seen for small NTU as the
maximum ﬂuid Biot number increases was due to the energy transfer being
limited by the magnitude of the wall aspect ratio and minimum ﬂuid Biot
number. The larger maximum ﬂuid Biot number resulted in additional energy
available, which could not be moved across the wall and hence the ineffec-
tiveness increased. This was seen until at larger NTU the minimum heat
capacity was smaller and the energy transfer limited by the available energy.
In which case the ineffectiveness decreased at the larger maximum ﬂuid Biot
number.

As the heat capacity ratio decreases, the energy disparity across the
wall grows. Mth opposite extremes of the heat capacity ratio and ﬂuid Biot
numbers paired, the growing energy disparity results in a larger increase in
the ineffectiveness at the larger maximum ﬂuid Biot number in comparison to
the ineffectiveness at equal ﬂuid Biot numbers. In Figure 4.29 for L’ = 100
the ineffectiveness increases nearly 10% for CR = 0.5 at the larger maximum
ﬂuid Biot number and small NTU, while similar conditions produce an
increase of only 5% for CR = 0.75 in Figure 4.25. Note that the magnitude of
the ineffectiveness is greater at CR = 0.75 than it is at CR = 0.5 for similar
conditions, but the change seen in the ineffectiveness as the maximum ﬂuid
Biot number increased is larger for CR = 0.5.

95

To summarize the results, the effects seen can be separated into two
groups: primary effects and secondary effects. The primary effects occur due
to changes in the wall aspect ratio, ﬂuid Biot numbers, or heat capacity ratio.
Secondary effects occur due to changes in the Mondt number and minimum
heat capacity to maintain a constant NTU after varying the wall aspect ratio
or ﬂuid Biot numbers. These effects are shown in Table 4.4.

The primary effects occur as a direct result of a change in the amount
of axial conduction. Table 4.4 shows that these effects decrease the amount of
axial conduction and the ineffectiveness. However, when the conditions
reach certain magnitudes; the secondary effects occur.

The secondary effects are the result of varying the NTU and show the
status of the interaction between the heat transfer and thermodynamics. For
small magnitudes of the NTU the minimum heat capacity is relatively large,
and the amount of energy transferred is dictated mainly by the heat transfer.
Whereas for larger magnitudes of the NTU, the minimum heat capacity is rel-
atively small; and the amount of energy transfer dictated by the thermody-
namics or available energy. Therefore, at small NTU the larger
ineffectiveness seen as the parameters were varied is not due to an increase in
axial conduction, instead it reﬂects that the limit of the energy transfer based
on the heat transfer parameters has been reached or exceeded. This produces
a larger ineffectiveness because the available energy increased as the parame-
ters were varied but no additional energy could be transferred; due to the
magnitude of heat transfer parameters(Bimw Rim, L' ). The secondary effects
demonstrate interactions between the heat transfer and thermodynamics that
are quite complex to analyze.

TABLE“ CiassiﬂcationofeffectsaeenastheNTUwasvaried

 

 

 

 

PrimaryEffect SecondaryEffect
Ineﬂ'ectivenessdecreasesatlarger L', Rim, Ineffectigeness ina'easesforsmallmagnitudes of
andBimu. NTU,L ,andBimmwhenBimisincreased.
IneffectivenessdecreasesatsrnallerCR. Ineffectiminaeasesforsmallmagnimdesof
NTU, L , Rim-n, when Bin“ isincrcasedand
theamountot‘theinereasegrowsas CRis
decreased.

 

 

 

96

4.3 Predicting the Presence of Axial Conduction with the Mondt number

Having shown that axial conduction can indeed adversely affect the
performance of the heat exchanger, the conditions at which this will occur
will now be analyzed. These operating conditions are represented in dimen-
sionless form by the minimum and maximum ﬂuid Biot numbers, wall aspect
ratio, heat capacity ratio, NTU, and the Mondt number. Note that not all of
these condition can be independently speciﬁed, as seen in equation (4. 10).
The ﬁrst ﬁve parameters are deﬁned on all ﬁgures in Section 4.2.1. The
Mondt number however, is not shown on these ﬁgures because its magnitude
was calculated to provide the desired NTU. Hence, as the minimum and max-
imum ﬂuid Biot numbers and wall aspect ratio were varied the Mondt num-
ber was changed. However, using the equation for NTU the Mondt number
can be calculated from the information on the ﬁgures

 

_ NTU[Bimax+BimaxBimin+Bimin:l (424)

M, - LU2 BiminBimax
Using equation (4. 24) the information in Table 4.5 was created allowing the
Mondt number to be easily calculated as a function of the ﬂuid Biot numbers,
wall aspect ratio and NTU. Note that the Mondt number, which was deﬁned
earlier, is
wa
M0 = , (4.25)
CminL
and equation (4.24) is used to calculate the magnitude of the Mondt number
for a particular operating condition, not deﬁne it.

 

TABLE 4.5. Relationship among Mondt number, wall aspect ratio, NTU, and Fluid Biot numbers

 

 

 

 

 

 

M01."
W
Bing
Bin...” .0001 .001 .or .1 r
.0001 20001 11001 10101 lWll 1W2
.001 2(01 1101 1011 1002
.01 201 111 102
l 21 12

 

 

 

 

 

 

 

 

 

97

Now the magnitude of the Mondt number as a function of the other
parameters can be investigated. This was done for equal minimum and maxi-
mum ﬂuid Biot numbers.The results are shown in Figure 4.37 for the Mondt
number as a function of the NTU while varying the minimum ﬂuid Biot num-
ber and wall aspect ratio. The legend identiﬁes the correspondence between
the minimum ﬂuid Biot number and line pattern and to the right of the legend
the wall aspect ratio for each curve is shown. Following upwards from a
curve the corresponding wall aspect ratio for that curve can be found then
moving horizontally from the wall aspect ratio the associated minimum ﬂuid
Biot number can identiﬁed. The legend also allows the minimum Biot num-
ber to be identiﬁed directly from the curve, but when the curves overlap this
is difﬁcult and the alternate method may be used. Also, the trends in the mag-
nitude of the wall aspect ratio can be visualized at the various minimum ﬂuid
Biot numbers.

The information concerning the relationship among the Mondt num-
ber, NTU, ﬂuid Biot number, and wall aspect ratio by itself is not very useful.
Using the information from Section 4.2.1 it was possible to draw the line
depicting the dividing line between axial conduction being present and negli-
gible. Ineffectiveness is negligible for parameters to the left of the line and
nonzero for parameters to the right. The signiﬁcance of this line is that it
identiﬁes the regions in terms of all the inﬂuencing parameters when axial
conduction needs to be considered, albeit for a heat capacity ratio of one and
equal ﬂuid Biot numbers. After addressing the limitations on Figure 4.37 the
basis for axial conduction depending mainly on the Mondt number is dis-
cussed.

As noted, the line identifying where axial conduction becomes nonzero
is restricted to equal minimum and maximum ﬂuid Biot numbers. However,
as shown earlier, only a maximum ﬂuid Biot number one order of magnitude
larger than the minimum ﬂuid Biot number needs to be considered since fur-
ther increases in the maximum Biot number do not result in signiﬁcant
changes. Because the Mondt number changes with the maximum ﬂuid Biot
number, this requires a shifting of the curve for a particular wall aspect ratio
and minimum ﬂuid Biot number. The individual curve on Figure 4.37 needs
to be shifted an amount equal to the ratio of the values in Table 4.5.

98

 

 

 

 

 

 

 

 

 

02
PO _. oO _. TO —. «IO _. nIO —. TO —. oIO r oIO _. 70 r
p ....p: . . b.3PP. _ ___..p. . c .2: . . . FI-p _ . ___..prp p 7:: . c . VECPc b O
..\ ..\..\ .\.. -
.\ .\ .\ \ ..\ ..N
\.. .\ \.. .\ -
.\ .\ ....
\ \.. .\ \ ....
\ .\ .\ \ x. [m
.2 N .\ \ x .
cop . can 000' . FOOD."d_ﬂ_m ..| [Or
8. con 08. POO..|Ic_E_m ..l I
oo— eon coo. "and FO.H£E_m I
co. con 08. F.Hc_e_m I IN_.

 

 

 

 

oLoNcoz A
P5303950 _o_x<

L_V 299.32
COBODUCOO _o_x<

 

ﬂlN

FIGURE 4.37. Predicted region affected by axial conduction in terms of the NTU, Mondt number,

wailaspectratio,andminimumiluidBiotnumber

 

 

 

99

To use the curves for equal ﬂuid Biot numbers in Figure 4.37 when
ﬂuid Biot numbers are not equal requires the Mondt number or NTU read
from the ﬁgure to be corrected. The Mondt number for unequal ﬂuid Biot
numbers is

 

 

 

 

{MoL . . \
NTU (Brmx>Brmin)
Mawimwim) = M0(Bim=mm,n) .2 (4.26a)
MoL . .
k NTU (Blmax=Blmin) J
while the NTU is
w." . . \
W (Blmax=Blmin)
NTU(Bimu>Bim-n) = NTU(Bim=Bi,,,,.,,) 2 (4.26b)
MOL. . .
K-N—TIT (Blmax>Blm‘-n))

 

 

where M0 (Bimx=Bim.n) in equation (4.26a) and NTU (Bimeimin) in equa-
tion (4.26b) are read from Figure 4.37 and the ratio on the right hand side is
obtained from Table 4.5 at the corresponding maximum and minimum ﬂuid
Biot numbers. Equation (4.26a) corrects the Mondt number to correlate to the
NTU in Figure 4.37 for unequal ﬂuid Biot number and equation (4.26b) cor-
rects the NTU to correlate to the Mondt number in Figure 4.37 for unequal
ﬂuid Biot numbers. Only one of the variables (NTU or M) needs to be cor-
rected; the other can be read from the ﬁgure directly.

A ﬁnal restriction to the dividing line in Figure 4.37 is that it applies
for a heat capacity ratio of one. Because this heat capacity ratio is the most
adversely affected, these results will be correct in predicting when axial con-
duction is negligible for smaller heat capacity ratios. However, the results
may indicate axial conduction is nonzero when in fact it is negligible at
smaller heat capacity ratios. To alleviate this restriction, the dividing line can

TDABIJE4J£.thpﬂhmkaofdhﬂdhmghimndtnunﬁuusurlhuunhniofﬂue

 

 

 

 

heateapaeﬂyraﬂo
C M
1.0 .010
75 .015
.50 .030
25 .070

 

 

 

 

100

be shifted for smaller heat capacity ratios, where the effect of axial conduc-
tion is less, if all other conditions are the same. Table 4.6 gives the approxi-
mate Mondt numbers that divide axial conduction as a function of the heat
capacity ratio. Because the line does not shift very much, the additional lines
were not added to Figure 4.37.

The Mondt number varies with the wall aspect ratio, NTU and ﬂuid
Biot numbers, as seen in equation (4.24). The Mondt number can be inter-
preted as the parameter that matches the properties of the wall and ﬂuids (Kw
and Cm) to the operating conditions (Binm, Him, L', NTU). Figure 4.37 in
conjunction with the results of Section 4.2.1 shows that as the Mondt number
approaches zero the effect of axial conduction is negligible, while at larger

values axial conduction is present.

The smaller Mondt number results in less axial conduction because to
become smaller the product of the streamwise resistance and minimum heat
capacity must get larger, which is shown using equations (4.25) and (4.16)

Kw 1
M0=CWL'=C R
min

min streamwise

 

 

(4.27)

A larger minimum heat capacity means the ﬂuid retains its energy better, and
the energy that is transferred to the wall is hindered from conducting axially
by the large resistance in that direction. However, for larger Mondt numbers
the heat capacity is small, and the ﬂuid gives up its energy easily. This energy
can then be conducted down the wall because of the smaller streamwise resis-
tance.

Physically the Mondt number represents the capabilities of the ﬂuids
and wall, the ﬂuid’s ability to carry energy and the walls ability to move
energy in the axial direction. Because the Mondt number varies with ﬂuid
Biot number, wall aspect ratio and NTU it will reﬂect any changes seen in
these variables, which explains why it is possible to base the presence of
axial conduction on this parameter

The results in Figure 4.37 are intended to serve as a design guide to
predict if axial conduction will exist. A certain amount of caution must be
exercised in applying the results in Figure 4.37 and Table 4.6, however. The
line presented on the ﬁgure for a heat capacity ratio of one was estimated by
observing the ineffectiveness-NTU plots presented earlier, as were the magni-

101

tudes listed in Table 4.6. The criteria was an ineffectiveness less than 1% was
negligible. For the regions near the dividing line, it is recommended to check

the magnitudes of the ineffectiveness on the appropriate ﬁgure in Section
4.2.1.

The corrections allowing Figure 4.37 to be applied for unequal ﬂuid
Biot numbers in equations (4.26a) and (4.26b) will be in error for small val-
ues of the NTU, wall aspect ratio, and ﬂuid Biot numbers if the relationship
between the heat capacity and ﬂuid Biot numbers associates the opposite
extremes (Cm, - Bin“). This is due to the inﬂuence of the secondary effects
when the thermodynamic energy becomes large. Even though the Mondt
number is getting smaller the ineffectiveness increases, and this outcome
reaches larger values of the NTU as the heat capacity ratio decreases. It is rec-
ommended not to use the corrections for this association of the ﬂuid Biot
numbers and heat capacity ratio.

102

4.4 Application of the Results

Up to this point, all operating conditions of the heat exchanger have
been in terms of the Biot numbers, wall aspect ratio, heat capacity ratio, and
Mondt number. Also the material properties of the wall or ﬂuids have not
been an issue since they were absorbed into dimensionless parameters. In this
section the dimensional variables will be presented to show speciﬁc operating
conditions (,i.e. ﬂow rates, ﬂuid types, and wall material) when the previ-
ously presented results will apply. The results of this section could be used to

establish experiments to verify the analytical solution.

The parameters that will be addressed in dimensional terms are the

wall aspect ratio

L
L=8

minimum and maximum ﬂuid Biot numbers

where

heat capacity ratio

where

and the Mondt number

Bi in = min (Bic, Bi”)

Bi a = max (Bio Bi”)

 

Cm." = min (Co CH)

C

max = max(Co CH)

 

min

(4.28)

(4.29)

(4.30)

(4.31)

(4.32)

(4.33)

(4.34)

(4.35)

(4.36)

103

which were all deﬁned earlier, but are shown again for discussion purposes.
The NTU is then

 

NTU = M L‘2[ “mm“ ] (4.37)
0 Bimx + BimmBim‘ + Bin,“

In the previous section it was shown for a Mondt number less than .01
axial conduction was negligible. Consequently, the Mondt number must be
larger to see axial conduction, and this is the parameter to begin the investiga-
tion of the dimensional quantities. Figure 4.38 gives an indication of the con-
ditions required on the wall and ﬂuids (CminL‘ Iw) as a function of the wall
thermal conductivity, that are needed to provide the required Mondt number.
Also, for the given wall thermal conductivity the product of the convective
heat transfer coefﬁcient and wall thickness, calculated from equations (4.29-
32), are given as a function of the minimum ﬂuid Biot number by the addi-

tional labelled axes.

The results in Figure 4.38 give insight into the magnitudes and trends
of the basic heat exchanger parameters required for axial conduction to exist.
The thermal conductivity that intuitively would improve the performance if it
were larger actually increases the possibility of axial conduction because the
value of (CmmL'lw) becomes larger as the thermal conductivity increases, as
does the product 1:5. The larger these two groups of parameters are the more
likely the chance they will be seen in a typical application. Noting that typical
magnitudes of the wall aspect ratio and minimum heat capacity are at mini-
mum 102 - 103, the required Mondt number will surely not be reached unless
a moderate thermal conductivity exist. This result gives an indication of the
conditions necessary for axial conduction to exist; it is not an ordinary occur-
rence.

To experience axial conduction when the wall aspect ratio is relatively
large (L’ > 1000), the heat capacity must be small or the wall width must be
large. These conditions require a ﬂuid with a low speciﬁc heat (such as a gas),
a creeping ﬂow to provide the small mass ﬂow rate, or a large heat transfer
area with a correspondingly large wall width.

The wall thickness is present on both axis and produces opposite out-
comes for each. As the wall thickness becomes smaller, the wall aspect ratio
gets larger and the possibility of axial conduction decreases because the

104

 

 

10‘

10’

10‘

10’

CmL/dw

102

10‘

 

 

 

 

 

 

10°
1o-t i
10-' 10° 10' 102 10
KW
164 . ....qu . ....qu ....qu . '11‘73-
h6(Bim=.OOO1)
1(5-4 . --.-.wd . ”Hid-2 . .84 . ””"tba
h6(Bim=.001)
154 . H.164 . ””164 . .T....1.bo . ......{10‘
h6(Bim=.01)
16-2 - ””164 . ”Hub . ......{b‘ . ......PO
hd(Bim=.1)

 

FIGURE 4.38. Magnitude of pertinent heat exchanger operating conditions as a function of the
wallthermalconductivity,whichisneededforaxialconductiontoexist

105

required minimum heat capacity is very snmll. However, the convective heat
transfer coefﬁcient on the horizontal axis becomes larger and more likely,
which is due to the scaling scheme on the ﬂuid Biot numbers while, the
former effect of the wall thickness is due to its physical role.

From the previous discussion it is clear that to obtain the Mondt num-
ber necessary for axial conduction to exist requires one or more of the follow-

ing conditions:
1. High wall thermal conductivity
2. Small ratio of wall length to thickness

3. Larger heat transfer area and corresponding wall width
4. Low speciﬁc heat ﬂuids

5. Small mass ﬂow rates

TABLE 4.7. Operating conditions for axial conduction to exist

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

niL/w (kg/sec) h (W/mzK)
“it B' ' Wall
.01 .1 1 .0001 .001 .01 .1 Material
Air 1.5E—3 1.5E-4 1.5E-5 1.5 15 150 1500 Swel
Water 3.75E-4 3.7555 3.75E-6
Air 4.&-2 4.&-3 4.&-4 40 400 4000 40000 Copper
Water l.&-2 LOB-3 LOB-4
= .01
Air 1515-2 1513-3 1513-4 .15 1.5 15 150 Steel
Water 3.75E-3 3.75134 3.7SE-5
Air 4.&-1 4013-2 4.&-3 4 40 400 4000 Copper
Water 1.&-l 1.&-2 l.&-3
= .1
Air 1.5E-1 1.5E-2 1.55-3 .015 .15 1.5 15 Steel
Water 35755-2 3.75E-3 3.75B-4
Air 4.0E+0 4.&-1 408-2 .4 4 40 400 Copper
Water 1.0500 1.&-1 1.&-2

 

 

 

 

 

 

 

 

106

Consider the conditions needed for axial conduction in a wall material of
steel (Kw ~ 15) or copper (Kw ~ 400) and ﬂuids of water (Cp ~ 4000) or air

(Cp ~ 1000). The possible combinations of wall material and ﬂuid type are
shown in Table 4.7 at various Mondt numbers and wall thicknesses. The
results shown in Table 4.7 are the mass ﬂow rate combined with the wall
length and width and the convective heat transfer coefﬁcient as a function of
the ﬂuid Biot number. The ﬂuids, wall materials, and wall thicknesses were
chosen to give a perspective over the range of the parameters that may be
encounter for typical applications. Even though a wall thickness of .1 is quite
extreme, it gives an indication of the magnitudes required for other parame-
ters to become reasonable.

It was not possible to reduce to only the mass ﬂow rate because the
case would be too specialized. But noting that Llw is a minimum of one, and
most likely larger than one; the magnitude of the mass ﬂow rate is even
smaller than the values shown for niL/w in Table 4.7.

It is apparent from the data presented in Table 4.7 that small mass ﬂow
rates and corresponding convection coefﬁcients are required for axial con-
duction to exist. The required mass ﬂow rate decreases as the Mondt number
increases while the convection coefﬁcient decreases as the minimum ﬂuid
Biot number decreases. Both trends demonstrate the requirement for axial
conduction to become larger since large Mondt numbers and small ﬂuid Biot
numbers increase the effect of axial conduction on the performance of the
heat exchanger.

Recalling that the results predicting axial conduction were based solely
on the Mondt number, the results presented in Figure 4.38 and Table 4.7 do
not particularly depend on the magnitude of the convective heat transfer coef-
ﬁcient. In essence, the magnitude of the convective heat transfer coefﬁcient is
arbitrary as long as the requirement on the Mondt number is met. But for a
required Mondt number the magnitude of the convective heat transfer coefﬁ-
cient and wall dimensions will set the NTU. To obtain a particular NTU with
a prescribed Mondt number and convective heat transfer coefﬁcient may
require unrealistic wall dimensions. Likewise, if the Mondt number and wall
dimensions are prescribed, unrealistic convective heat transfer coefﬁcients
may be required. The magnitudes of the convective heat transfer coefﬁcients
are intended to represent the range that axial conduction was shown to exist

107

in physical situations. The range of the convective heat transfer coefﬁcient,
however, may not be physically realistic for the entire range of Mondt num-
bers shown.

108

4.5 Comparison to Published Results

All previous studies of the effect of axial conduction have been consid-
erable less general than the present work. For this reason, there is only one
speciﬁc case that can be compared to previously published results. This is the
case of balanced symmetric ﬂow in which the mass ﬂow rates, heat capaci-
ties, and convective heat transfer coefﬁcient between the ﬂuid and wall are all
equal for the two ﬂuids. Therefore, the heat capacity ratio is one (CR = 1) and
the ﬂuid Biot numbers will be equal (Bi = Rim“).

min

For this case, Pan, Welch, and Head [15] derived a closed form expres-
sion for the effectiveness as a function of three variables

NL = ”—611 = NTU (4.33)
Kw“ hA WK 2 .
”K = (—'E':) (a) = (_C—w) 3' (4.39)

1

2 2
p =(1+IVK) (4.40)

which are related to the variables of the present study as shown. Note that the
subscripts on heat capacity and Biot numbers have been dropped since the
conditions are the same on both sides of the wall. The effectiveness is then
given ([15] equation (34)) as

_ NL+ (NR/2p) tanh (pNL)

- NL+ (NK/Zp) tanh (pNL) + (NR/2) +1
Using equation (4.41) the effectiveness (and ineffectiveness) was cal-

culated and is shown in Figures 4.39-4.42 as a function of NTU at various

minimum Biot numbers and wall aspect ratios. The results show excellent

agreement with the present work. The results (i - N TU ) differ only beyond the

third decimal place, which can be attributed to computational round-oﬂ’.

8

 

(4.41)

The same restrictive case was investigated by Rohsenow [17]. How-
ever, he proposed the effectiveness as a function of NTU and the Mondt num-
ber (Mo), which was earlier shown to be

KVAW wa
C

M: = .
CL

0 L

 

 

(4.42)

109

and solved for the effectiveness at the extremes of the Mondt number, ([17]
equations (32) and (34))

M0 = o
NTU
8 " TIN??? “'4'”
Mo = on
e = %(1-e'm”) (4.44)

Obviously, with M, = 0 the ineffectiveness is zero, since this is the case of
neglecting axial conduction by allowing the wall thermal conductivity to
shrink to zero; whereas, the wall conductivity becomes inﬁnite for M0 = co.
Both of these cases are shown also in Figures 4.39-4.42, and they provide
bounds on the present results.

110

 

 

 

 

 

 

 

— Bi_-ai__
J Bi*=.0001 X Pan, Welch, Head [15]
0'6 81:81 = 0001 + Rohsenow [17] MO-..
‘ ° ' x Rohsenow L17] tin-=0
CI=1
L*= 100
+ +
0.4“ +
. +
b
+ 500
4-
0.2-
+ 1000
0.0 ‘ ‘
0 2 4 6 8 10
NTU

 

 

FIGURE 4.39. Comparison of present results to past result of Pan, Welch, and Head [15] and
Rohsenowblaxn ffectivene: as a function ofthe NTU and wall aspect ratio for

 

 

 

 

 

 

— Bin-Bi“
Bi*=.001 3! Pan, Welch. Head [15]
0-5" Bi =3; =.0001 + Rohsenow [17] Mos—o-
‘ ° X Rohsenow [lzLMQ=0
4 C.-.-1
...
+ + + +
0.4" + L*= 100

 

 

 

 

 

 

FIGURE 4.40. C of present results to past result of Pan, Welch, and Head [.15] and
Rohsenow [17]. Inc ectiveness as a function of the NTU and wall aspect ratio for B t = O. 001

111

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.
— Bi_=Bi_.
Bi_.=.01 3r Pan. Welch, Head [15]
0'60-4 BlL=Bio=.0001 "l’ Rohsenow [17] M0309
X Rohsenow [17] MO=0
q Cn=1
+ 1’ + +
0.40-« + 1
, +
a
+
‘I’
0.20q
+ L*= 100
0.00
NTU
FIGURE4.41. ComparisonofpresentresnltstopastresultofPan,Welch,andHead [15]and
mmwuplnsffectivenessmafunctlonoftheNTUandwallaspectratiofor Bin“ =0.01
R — a
0.701
— ai_=ai__
Bug-7.1 * Pon, Welch, Head [15]
Bi =5; =.0001 + Rohsenow [17] M08-
" ° X Rohsenow [17] Mn=0
050-] C.-1
+ "' + +
+ +
+
a 0.30- 1’
9’
+
0.10- +
+ 'I- 100
W— #4 x x
’0-10 I l l l l
0 2 4 6 8 10

NTU

 

FIGURE“2.C0mparbonofpresentruultstopastresultofPan,Welch,and Head [15] and
{Isabenowﬂ‘ln .IneffectivenessasafunctionoftheN’I‘Uandwallaspectratiofor Bi

R-

ruin"

 

Chapter 5

Conclusions and Recommendations for
Future Work

An exact solution for the analysis of the heat transfer occurring in a
counterﬂow heat exchanger including the eﬁect of axial conduction has been
presented. In addition to the solution methodology a computer program to
evaluate the numerical aspects of the exact solution was given. The program
allowed for the investigation of the inﬂuential parameters in the solution and
veriﬁcation of the solution’s validity. The role of all describing parameters
was investigated to determine their importance in affecting axial conduction.
The effect of axial conduction was quantiﬁed by the amount that the perfor-
mance of the heat exchanger was degraded in comparison to the case of
neglecting axial conduction.

The following results and conclusions are drawn from this investiga-
tion:
1. Axial conduction has less effect on the performance of the heat
exchanger at larger wall aspect ratios.

2. Axial conduction has less effect on the performance of the heat
exchanger at larger ﬂuid Biot numbers.

3. Axial conduction is reduced at smaller heat capacity ratios.

4. The effect of the end Biot numbers on the performance of the heat
exchanger is negligible for magnitudes less than or equal to the magnitude
of the ﬂuid Biot numbers.

5. Axial conduction is negligible for magnitudes of the Mondt number
less than .01.

6. High wall thermal conductivity promotes axial conduction.
7. Axial conduction is more likely for large heat transfer areas.
8. Applications with low heat capacity ﬂuids promote axial conduction.

The most obvious continuation of this work would be to experimen-
tally verify the results presented herein. Although the eﬂ’ect requires some

112

113

atypical conditions, it was shown possible for common ﬂuids. The applicabil-
ity of this solution to an experimental set-up, which would typically be con-
centric tubes, will require the set-up to be approximated by the model, a plane
dividing wall. Therefore, it must be possible to “cut and unroll” the concen-
tric tubes and represent them as parallel plane walls without incurring an
appreciable difference in the areas on opposite sides of the wall. In addition,
the issue of identifying the type of boundary condition to model the physical
problem at the ends of the wall will need to be addressed.

The computer program AXCOND could be coupled with existing ther-
mal design software to predict the performance of a counterﬂow heat
exchanger. Use of this program will be computationally expensive but will
only need to be used when certain conditions exist, otherwise the traditional
effectiveness-NTU relationships can be used. Incorporating the program into
existing software will require conversion of the main program into a subrou-
tine and consideration of the number of terms needed to produce a converged
solution. Also, if the required analysis is a sizing problem, solving for the
heat exchanger dimensions needed for a particular operating condition, an
iterative scheme needs to be implemented.

The issue of the number of terms needed to obtain an accurate solu-
tions could also be further investigated. The outcome of this additional work
may lead to possible simpliﬁcations if only a small number of terms are
needed. Furthermore, simpliﬁed equations may result under certain condi-
tions, which may allow for approximate solutions. In general, the complex
solution presented may be studied for possible simpan cases.

The solution presented could also be modiﬁed rather easily to deter-
mine the performance of a parallel ﬂow heat exchanger. The formulation of
the problem for the heat exchanger wall is exactly the same. The formulation
of the ﬂuid energy balance equations will differ by a sign and the location of
the speciﬁed initial condition. The ﬁrst difference will inﬁltrate the applica-
tion of orthogonality to apply the nonhomogeneous boundary conditions on
the wall and change one of the ﬂuids temperature solution, but only by sign
change(s). Whereas the initial condition will change the form of the integra-
tion constant on one of the ﬂuid temperature solution. The conversion may be
tedious, yet would require minimal computational eﬂ’ort or programming
changes since the outline of the solution and program are given.

LIST OF REFERENCES

[1]

[2]

[3]

[4]

[5]

[6]

[7]
[8]

[9]

[10]

[11]

[12]

[13]

114

Beyer. W. H. Wallabies. New York: CRC
Press Inc., 1987

Campo, A., and J. C. Auguste, “Axial Conduction in Laminar Pipe
Flows with Nonlinear Wall Heat Fluxes.”11;te;n_ational_193gal_gf_lieat_

MW; 1987, volume 21, p. 1957-1607

Chiou, J. P., “The Advancement of Compact Heat Exchanger Theory
considering the Effects of Longitudinal Heat Conduction and Flow
Nonunifonnity.” Compag Heat Exghangers,

Chiou, J. P., “Thermal Perforrnanoe Deterioration in Crossﬂow Heat

Exchangers Due to Longitudinal Heat Conduction in the Wall.” ASME
paper no. 76-Wa/HT-8

Chiou, J. P., “The Effect of Longitudinal Heat Conduction on Cross-
ﬂow Heat Exchangers.” f H Tr f r May 1978, volume
100, p.346-351

Edwards D. K., V. E. Denny, and A. F. Mills, W, New
York: Hemisphere Publishing Corporation, 1979

Eisensmith, S. P., W, September 1990

Hewitt. G. F.. W. New
York: Hemisphere Publishing Corporation, 1990

Incropera. F. P.. and D. P. Dertt. W New
York: John Wiley and Sons, 1985

Kays W. M., and A. L. London, W, New York:
McGraw Hill Book Company, 1984

Landau, H. G., and J. W. Hlinka, “Steady State Temperature Distribu-
tion in a Counterﬂow Heat Exchanger Including Longitudinal Conduc-
tion in the Wall.” ASME paper no 60-Wa/HT-236

Mondt, I. R., Effects of Longitudinal Thermal Conduction in the Solid
on Apparent Convection Behavior, with Data for Plate-Fin Surfaces.”

WM; 1960-61. P 614-621

Nyhoff. L.. and S. Leestma. W
New York: Macmillan Publishing Company, 1988

[14]

[15]

[16]

[17]

[18]

[19]

[20]
[21]

115

Ozisik M. N., W90, New York: John Wiley and Sons,
1980

Pan, C. L., N. E. Welch, and R. R. Head, “A New Concept to the Gen-
eral Understanding of the Effect of Longitudinal Conduction for Multi-
stream Counter Flow Heat Exchanger.” Proceedings of the 3rd
International Heat Transfer Conference, 1966, volume I, p159-169

Pan, C. L., and N. E. Welch, “Exact Analytical Wall and Fluid Temper-
ature Field for a Counterﬂow Heat Exchanger with the Effect of Longi-
tudinal Heat Conduction.” ASME paper no. 65-HT-63, 8th National
Heat Transfer Conference, August 1965

Rohsenow, W. M ., “Why Laminar Flow Heat Exchangers Can Perform
Poorly.” W, New York: McGraw Hill Publishing Corpo-
ration, 1981, p. 1057-1072

Shah, R. K., “A Correlation for Longitudinal Heat Conduction Effects
in Periodic-Flow Heat Exchangers.” Journal of Engineering For
Power, 1875, volume 97, p. 453-454

Somerton, C. W., Notes on Heat Exchanger Experiment, ME413
Course Notes, Michigan State University, 1990

Somerton, C. W., Personal Communication on Axial Conduction, 1990

Wark, K., Thermodynamics, New York: McGraw-Hill Book Company,
1983

APPENDICES

APPENDIX A

A.1 General Solution of the Wall Conduction Equation

Beginning with the partial differential equation for the wall and bound-

ary conditions that were derived in Chapter 2

 

ﬂq-L'l'z 326” = O
ax+2 ay+2
80
”—7 +Bi00w(0,y+) = O
x e
x =0

 

W

+BiL0w(l,y+) = BiL(TL—To)

 

 

 

3):“ x. =1
30w
--—+' +3139... (1+a0) = BiHITH(x+) -T0]
8y y’=0
aew , + . .
5? ”teem .1) = B‘CITCO‘ >-Tol
y‘=1

(A.l)

(A.2a)

(A.2b)

(A.2c)

(A.2d)

The problem is split into three simpler problems each with only one nonho-

mogeneous boundary condition.

 

Problem 1
329‘ +L"‘2 9:31- = 0
ax+2 ay+2
301 '
-—+ +Bi091(0,y+) = 0
81 {=0

 

1

$7 +BiLe,(1,y*) = Brim-T.)

x’=l
801 . +
73—; +Bz,,el(x ,0) = 0
y ,.=0

 

 

116

(A3)

(A.4a)

(A.4b)

(A.4c)

117

+3.09, (x+, 1) = o

 

Problem 2
3’9, 3’9,

2 ..
ax+2+Lit ay+2 - O

 

+ BiaG2 (0, y+) = 0

x’=0

 

+BiL02(l,y+) = o

 

x’=1

892

ay"

 

4495102 (x*,0) = BiH[TH(x+) —To]
y‘ =0

892

 

+3369, (x", 1) = 0
y'=l

Problem 3

3’9, ”*2 air, = o
ax+2 ay+2

303

 

 

+Bi003(0,y+) = o

 

+BiL03(1,y+) = o

 

 

+Bi303 (x+,0) = 0
y’=0

__1
a)!“

 

«t-Bice3 (x‘, 1) = Bicch(x+) -To]
y‘ =1

(A.4d)

(A5)

(A.6a)

(A.6b)

(A.6c)

(A.6d)

(A.7)

(A.8a)

(A.8b)

(A.8c)

(A.8d)

It can be shown by adding the three problems in Eq. (A.3-8) that the original

problem is obtained in Eq. (A.1-2). The solutions are related as follows:

118

0“, (x1, y”) = 01(x+, y“) + 02 (x‘, y”) + 03 (x+, y+) (A.9)

A.l.l Solution for Nonhomogeneous Boundary Condition at x“ = 1
(Problem 1)

Assuming a product solution for 01 of the form

01 = F(x”) G(y*) (A.10)

Substituting into the partial differential equation and boundary conditions,
Eq. (A.3-4), gives

 

 

 

 

F , .G+L’ZG . .F = 0 (A.11)
x X y y
-GF . +BioGF(0) = 0 (11.123)
x x1 =0
orx. e =1+BioGF(l) = BiL(TL- To) (A.12b)
-FG . +Bi,,FG(0) = 0 (A.12c)
y ’9 = o
my. +Bi,,ro(1) = 0 (1112.1)
r’ = l

with the following convention for representing derivatives used:

air
ax” - Fm. (A.13)

 

Rearranging Eq. (A. 1 l) to group similar variables and simplifying Eq. (A. 12)
gives

 

 

 

 

 

_L Ff” ._ _GY'Y _ :l: 2 (A14)
Leg 1; - G - 11 °
—F, +Bi,r(0) = o (A.15a)
x x‘ =0
3 =
-G . +Bi,,G(0) = 0 (A.15c)
y ’930
G . +Bi,,G(l) = 0 (A.15d)
1' f =1

119

where a constant, 1:111, was introduced in Eq. (A. 14). With the left hand side
only a function of x“ and the right hand side only a function of y+ for equality
both sides must be equal to a constant, hence the introduction of the constant.
This key result allows for separation of the variables, reducing the single par-
tial diﬂ‘erential equation to two ordinary differential equations. Before Eq.
(A. 14) can be separated however, a sign must be chosen for the constant. This
sign is chosen to produce an eigenvalue problem for the function that has two
homogenous boundary conditions to evaluate. The eigenvalue problem will
have a general solution in terms of sine and cosine functions for this cartesian
coordinate system. After choosing the sign, two ordinary differential equa-
tions can be written from Eq. (A.14) with the appropriate boundary condi-
tions from Eq. (A.15)

 

 

F . . -L'2u2F = 0 (A.16)
x I
-r. +Bi,r(0) = 0 (11m)
x x’ 80
G . . +1126 = 0 (A.18)
Y y
-G, +BiHG(0) = o (A.19a)
y y9=o
Gy. . +Bi,,G(1) = o (A.19b)
y =1

 

Note that Eq. (A.16) only has one boundary condition, whereas Eq. (A.18)
has two boundary conditions. The reason for this can be seen in Eq. (A.15b),
which depends on both functions F and G and cannot be applied to either
function singularly. Thus, this boundary condition must be applied after
assembling the complete solution.

Eq. (A.16) and Eq. (A.18) have easily obtainable general solutions.
The general solution for Eq. (A. 16) is

r(x+) = Alcosth’xt) +A2sinh(uL‘x+) (.4120)

Applying the boundary condition, Eq. (A.17a), gives a relationship for the
two constants

120

A2=

 

"HA, (A.21)

which can be substituted into Eq. (A20) and rearranged giving the solution

F(x”) = A, [pL‘ cosh (uL‘xt) +Biosinh (pL‘x*)] (A22)
The solution of Eq. (A.18) is

G (yl) = A3608 my") +448in (11f) (A.2a)

Applying the boundary conditions in Eq. (A.19a) solves for the relationship
between the constants

 

BiHA3
A = .24
4 u (A )
Substituting into Eq. (A23) and rearranging gives
60*) = A. [ucos (11)”) +Biusin (uy“)] (A25)

Applying the ﬁnal boundary condition, Eq. (A.19b), does not provide any
information about the constant A 4 because the constant cancels, it does pro-
vide information about the constant )1 though, the equation simpliﬁes to

 

(11" + -BtHBrC)
To meet this boundary condition Eq. (A.26) must be satisﬁed for all values of
u". Eq. (A.26) is called a transcendental equation, it has an inﬁnite number of
solutions, hence the subscript on u, and will be solved to determine the possi-
ble values of u". The solution for the function G is then

(A.26)

601,. f) = 112111.003 01,3”) + Biusin (u,y*)] (11.27)
and u" is given by the solution of Eq. (A.26).

Putting Eq. (A27) and Eq. (A.22) into Eq. (A.10) will give the solu-
tion. However, since there exist many solutions to Eq. (A.27) all possible
solutions will be summed to obtain the ﬁnal solution. Also, the undetermined
constants, A2 and A ‘, will be grouped into one constant that will be deter-
mined later by evaluating the last boundary condition, Eq. (A.4b), which was
the boundary condition that did not separate and was not evaluated. The solu-
tion for problem one is

121

(11 (x+, m = .1,,[unL’ cosh (unz‘xt) + Biosinh (an‘x+)] *
n = 0
[mos My“) + Bigsin (u,y*)] (A28)
and the eigenvalues are found from the solution of Eq. (A.26).

A.1.2 Solution for Nonhomogeneous Boundary Condition at y+ = 0
(Problem 2)

Using similar methods problem two can be solved. Deﬁning an
assumed product solution

02 = F(x*)G(y*) (1129)

and substituting into the partial differential equation and boundary conditions
and choosing a sign to make the proper eigenvalue problem allows the partial
differential equation to be separated into two ordinary diﬁerential equations

 

 

 

F . . +0121r = 0 (A30)
x x
"Fe +BioF(0) = 0 (A.3la)
x’ = 0
F. +BiLF(1) = o (A.31b)
x x‘ =1
G . . -a20 = 0 (A32)
I y
G . +Bi,,o(1) = 0 (11.33...)
Y y. =1

The solutions are of the form, respectively
F(x*) = Blcos (ax‘) +stin (of) (11.34)
+ (I + - (l +
G(y ) = B3cosh(-Ey )+B4s1nh (Ey ) (A.35)
Applying the boundary conditions in Eq. (A.3la-b) to Eq. (A.33) gives

F (16‘) = Br [ancos (an?) + Biosin (anx*)] (A36)

where the values of an are obtained from the solution of the transcendental
equation

122

(Bi, + 31,) an
(ah-aim»

Applying the boundary condition given in Eq. (A.33a) to Eq. (A.35) gives

tan (01") = (A.37)

 

0 (y‘) = B.[cosh (1%)“) -Csinh (fl-ﬂ] (1.33)
where C is a constant, given by

ﬁmﬂb +Biccosh (3;)
c = L L L (A39)

(1 a a
-;cosh (-—;) +Bi sinh (-;-)
L L C L

 

As before, putting the two solutions together and noting that the solution to
Eq. (A36) is the sum of all possible solution, and lumping undetermined
constants together allows the solution to problem two to be written as

" or or
02 (x+. y“) = 2 Bn[cosh (4f) - 9.8m (_fy‘t )] e
n = o L L
[ancos (of) + Biosin (anx*)] (A.40)
and the values of an are obtained ﬁ'om the roots of Eq. (A.37).

A.1.3 Solution for Nonhomogeneous Boundary Condition at y+ = 8
(Problem 3)

Finally, applying the method again to solve problem three. Deﬁning an
assumed product solution

03 = F(x") G (y”) (A.41)

and substituting into the differential equation and boundary conditions and
choosing a sign to make the pr0per eigenvalue problem allows the partial dif-
ferential equation to be separated into two ordinary differential equations

 

F . .+a2F = 0 (A42)
1 x
- r1. +Bi,r(0) = o (A.43a)
x’ = 0
17‘. + BiLF(1) = o (A.43b)
x’ = l

 

123

G . . -—azG = 0 (A44)
H

G . +Bi,,o (0) = 0 (11.45.)
y y. g 0

Realizing that Eq. (A.42-43) is exactly the same problem solved in the previ-
ous section, Eq. (A.30-3l), the solution is

 

F(x+) = Cl [ancos (aan) + Biosin (anx+)] (A46)

where the values of an are obtained from the solution of the transcendental
equation in Eq. (A.37). The solution to Eq. (A4445) is

G(y ) = C3[:—cosh(— —y )-1-Bil,,sirrh(i1 y )] (A47)

Putting the two solutions together and noting that theL solution to Eq. (A.46) is
the sum of all possible solution, and lumping undetermined constants
together allows the solution to problem two to be written as

or
e (x+,y CnLl:-§cos 1):!“ )+Bi sinh(—,y+)]*
3 )=n§0n Ly H L
[ancos (an?) +Biosin (anx*)] (A.48)

and the eigenvalues are given by the roots of Eq. (A.37)

A.1.4 Summary of the Solutions

The solutions for the three problems without applying the ﬁnal (nonho-
mogeneous) boundary conditions are:

01(x+,y+) = 2An[ttnL'cosh(unL.x+) +Biasinh(unL‘x+)]*
n=0

[u,cos (11,.y“) + Bigsin (u,y*)] (A.28)

92 (1+, ”.20" a "[cosh (L, y )- ; "WGWH'

[ancos max“) +Biosin (anx*)] (A39)

124

" or or a
0 x1. + = Cl:—1-'cosh(-';' *)+Bi sinh(—§ +)]*
3( y ) "go a L L y H L y
[ancos (up?) + Bio sin (anx*)] (A47)

where the eigenvalues u" and a" are given by the positive roots of the tran-
scendental equations

 

(Bi +Bi )u

taunt.) = 2" , C, " (A.26)
(nu-BrHBrc)
(Bi +Bi )(X

mum") = ° L " (A.36)

 

(a: + —BiLBi0)

The remaining boundary conditions to be applied to determine the con-
stants A", B", and C", are Eq. (A.4b), Eq. (A.6c), and Eq. (A.8d) and they
depend on the ﬂuid temperatures for B, and C", therefore all will be applied
after the ﬂuid temperatures are determined in Appendix B.

The solution for the wall conduction equation is given by Eq. (A.9) as
the sum of the three solutions and the unsealed temperature (needed to solve
for the ﬂuid temperatures) is

T,.(X*.y*) = 9,,(x*.y+) +T, = 0101.)”) +92(x*.y+) +93(x+.y*) +7}. (448)

APPENDIX B

B.l Solution of Hot Fluid Energy Balance

Taking the describing differential equation for the hot ﬂuid that was
derived in Chapter 2 and rearranging gives

d1” +

1,, (0) = 1,“, (B.2a)
The expression for the unsealed wall temperature (from Appendix A) is

1,, (x*.0) = 10+ 2.1.11" [unL'coshan’xU +Biosinh (an36)]

71:0

" or
+ 2 (Bn+L—:Cn)[ancos(anx*) +Biosin (anx*)] (B3)
n-O
Putting this into the differential, Eq. (B1), and arranging gives
dTH - a s + . . t 4'
Ewing, = NH{T0+ 211,41”an cosh (an x )+Brosrnh(|.tnL x )1

n=0

" or
+ 2 (Bn+i-:C.)[ancos(anx*) +Biasin(anx+)] } (3,4)
n=0

To solve this problem it was ﬁrst split into two problems, a homogeneous
problem

dTH, h
dx+

 

“var“ = o (13.5)

and a particular problem

(1TH, p

d1:+

 

+~,,Tu.. = N..{T.+ 2'. .u.tu,.L‘cosh(u,L’x‘) +Bi.sinh(u.L'x*)l

n=0

.. an + - - +
+ 20(Bn+17cn)[ancos (crux )+Brosm(ornx )] } (8.6)

125

126

The particular problem in Eq. (B6) was further divided into three problems
for each term on the right hand side of the equation. The general solution for
Eq. (B. l) in terms of the simpler problem’s solutions is

3

7“,,(x+) = Tﬂ’h(x+) + z Tmp‘(x+) (3.7)
i=1

Where 7'”, p, is the solution to the problem

dTH, p,

+

 

+NHTH,pi = NH{5ttT,

+ 52, 2 Ann” [u'L' cosh (unL'f) + Biosinh (11,5151

u=0

+53, 20(Bn+;§cn)[ancos(aux+) +Biosin(anx+)] } (13.3)

(i = l, 2, 3) and 5], is the kronecker delta function as deﬁned in Eq. (3.7).
Superposition of the problems in Eq. (BS) and Eq. (B.5) will give the origi-
nally posed problem in Eq. (B.4).

B.1.1 Hot ﬂuid Homogenous Solution
The homogenous problem in Eq. (B5) is solved giving

TH, ,, (x+) = chxp (N”x+) (13.9)

where D1 is a constant to determined after assembling the complete solution
as shown in Eq. (B.7).

B.1.2 Hot ﬂuid Particular Solution 1

Solution of the particular problem in Eq. (B8) was obtained using
standard variation of parameter techniques. This method substitutes a general
solution of functional form similar to the nonhomogeneous term in the prob-
lem then solves for the unknown constants in this general solution. For the
ﬁrst particular solution (i=1 ) the general solution would be a constants,

TH,” (x1) = f(To) = k = constant (B.10)

127

Substituting this into the diﬁerential equation Eq. (B.8) yields
NH]: = NHTO (3.11)

and the solution is

T”, p, = To (3.12)

B.1.3 Hot ﬂuid Particular Solution 2
The functional form of the second particular solution (i =2) is

T“, = 2 bacoshan'x") +cnsinh(|.tnL.x+) (B.13)

n=0

Substituting into the differential equation, Eq. (B.8), (note the summations
are not shown) gives

buan‘ sinh (nave) + cnunL. cosh (anti)
NH [bucosh (unL‘xt) + cnsmh (unL'f) 1

= NHAnun [unL' cosh (uuL‘xt) +Biasinh (an’x+)] (3.14)

For this particular solution to be correct equality must hold, which is accom-
plished through the constants b" and en. Grouping all constants for the sinh
and cash terms separately into two equations produces

bnunL' +1)!ch = NHAnttnBio (3.15)
for the sinh terms and

cnunL‘ 1'”an = NﬂAnuﬁL’ (B.16)
for the cash terms. Solving these two equations for the unknown constants

 

 

. (BiN - 2L‘z)
bn=AnuﬁL 1- "2” 5”,: (3.17)
(NH-11314 )
(33,117,, - 33L”) ]
c =ANHu , (3.18)
" " "[ (Ni-uiLz)

and substituting into Eq. (B.13) gives the second particular solution

128

" . (3.- N wit”) . .
Tmnw) = 2.1,{332 [1- (1:219:32) :Icosh (11,,L x )
H— n

 

(31,111,, - uﬁL‘z)
(Ni-uﬁL")

 

+Nulln[ ]sinh (unL’f) } (3.19)

B.1.4 Hot ﬂuid Particular Solution 3
The functional form of the third and ﬁnal particular solution (i=3) is

1,,” (x+) = 2 aucos (anxt) + dnsin (a,x*) (3.20)
n = 0

Substituting into the differential equation

—a,,ct,sin (0.3”) +dnctucos (an?) +NH[a,cos (our?) +d,,sin (anx+)]

or
= NH[B,+ i—an) [ancos (uni?) + Bi,sin (anx+)] (B21)

then separating into two equations, the coefﬁcients are

 

 

a
dud" + NHa, = NH (B, + Egg)», (3.22)
for the cos terms and
all .
- and" + N ”d" = N H (B, + Z;- C")Br, (B23)
for the sin terms. Solving for the unknown constants
0. (Bi N + a2)
a" = (B,+-£C,)an[1- "2” 2" :l (3.24)
L (N ”-1- or")
or Bi ”3* a2
d = (B +—-f-C )N ° " (3.25)
" " L " ”[ N§,+ a},

and substituting into Eq. (B.20) gives the particular solution

129

 

" ct (Bi NH-t-a’)
711.33 0+) = 2 (Bn+i-§Cn){au[l- (13244;; ]cos (anx+)
H n
+N 3i,N,,+a3,
H N2+a2
H n

:lsin (any) } (3.26)

Assembling the solutions to the simpler problems, Eq. (B.9), Eq. (B.12), Eq.

(B.19), and Eq. (B.26), using the relationship given in Eq. (B.7), the solution
for the hot ﬂuid energy balance is

TH (15*) = T, + Dlexp (-NHx+)

“ , (Bi,NH-|.t:L'2) , +
+ 2A,,{nﬁL [1- (Nil-113132) :lcosh (3,1. x )

 

(BLNH-uiL’z) .
N inh L +
+ ”u"[ (Na-33L”) ls ('1’ “l

“ B'N 2
+ 2 (Bn+a—2Cn){au[l-( r, H+a")]cos(anx+)

(N3, + 01:)

3i,1v,,+ a:
+ N” 2 sin (our) (3.27)
NH-t- or:

Applying the boundary condition, Eq. (B.2a), the constant is

 

 

 

 

°' , (Bi,NH-112L‘2)
D=T.—T- AuzL 1- ",
l H.173 o 2 n n [ (Ni-11:1! 2)

n=0

- 0‘. (3i,N,,+ a3)
- 3 —c 1 - .23
n§0( "+1: a)0‘n[ (Niﬁcri) ] (B )

 

130

B.2 Solution of Cold ﬂuid Energy Balance

The differential equation that was derived in chapter 2 for the cold
ﬂuid, after rearranging is

are ,
-;: +NCTC = NCTW (x ,1) (B29)

TC(1) = TC, , (3.30.1)
The wall temperature solution (from appendix A) is

7,06, 1) = 70+ 2 A,[n,L’cosh(n,L‘x+) +3i,sinh(nnL‘x+)] *

n=0

[uncos (11,.) + Biysin 05.)]

+2:~[cosn(:—r)-c~m(%)]=~

[ancos (mg?) + Bi,sin (anx+)]
”Canha" B’sinha"*
+ — -— + —
.2. "[L‘ M (L) '” (L' )]
[ancos (an?) +Bi,sin (0131)] (B31)

Deﬁning the following constants:

A,’ = A, [uncos (11») + BiHsin (11") ] (B32)
0: or
3,; = 3,,[cosh {—2) — gen). (4')] (3.33)
L L
or a or.
C,’ = Cn[—: cosh (4) + Bi ”sinh (—f )] (B34)
L L L

simpliﬁes the expression for T, (x”, 1) to

Tw(x*, 1) = 70+ 2A,’[ttnL.cosh(ttnL‘x*) +3iosinh(n,L“x+)]

n=0

+ 2 (Bn'+C,,') [ancos (crux‘) +Bi,sin (anx+)] (3,35)

n=0

131

Putting this into the differential Eq. (B29) and arranging gives

dTC . , O 3 + o . . +
_quvcrc = NC{T0+ 211nm}. cosh(|.tnL x )+Bzosmh (unL x )1

n=0

+ 2 (3,; + C,’) [ancos (any) + Biosin ((13+)) } (3.36)
n = 0
To solve this problem it was ﬁrst split into two simpler problems, a homoge-
neous problem

ch

(hf +NCTC’,, = 0 (3.37)

 

and a particular problem

dTC’ p
+

+ NCTC’ p = NC {To + Z A.’ [unL' cosh (11.13:?) + BioSinh muff“

n=0

 

+ 2 (Bn’+ Cn’) [ancos (anf) +Biosin (anx+)] } (B38)

3 = 0
The particular problem in Eq. (B3 8) was further divided into three problems
for each term on the right hand side of the equation. The general solution for

Eq. (B29) in terms of the simpler problem’s solutions is

3

raw) = Twat) + 2 Tam-(f) (3.39)
i=1

Where Tc, p, is the solution to the problem given as

”Q“ NT -N 51‘
17+ cc,p:- c{uo

+ 52: X A,’ [unL' cosh (unL'f) + Biosinh (an‘r‘n

n=0

+5”; (3,,’+c,,') [ancos(anx+) +Biosin(ocnx+)] } (3.40)

n=0

(i = 1, 2, 3) and 5;.- is the kronecker delta function as deﬁned in Eq. (3.7)

132

32.] Cold ﬂuid Homogenous Solution
The solution to the homogeneous problem in Eq. (B.37) is

T0,). (x+) = Elexp (NC?) (B.41)

and E1 is a constant to be determined, after assembling the complete solution.
Superposition of the problems in Eq. (B40) and Eq. (B.37) will give the orig-
inally posed problem in Eq. (B.36).

B.2.2 Cold ﬂuid Particular Solution 1

Assuming the ﬁrst particular solution ( i=1 ) is a constant and substitut-
ing into the differential equation Eq. (B.40)and solving for the magnitude of
the constant gives

Tm,1 (x*) = To (13.42)

B.2.3 Cold ﬂuid Particular Solution 2
The form of the second particular solution (i =2) is

Tamer) = 2 bucoshan'f) +cnsinh(unL’x+) (3.43)

n=0

substituting into the differential equation Eq. (B.40) (with summations not
shown) gives

bnan‘ sinh (uuL'JF) + enunL‘ cosh (puL‘xt)
- Nc[bncosh (ﬂuff) + cnsinh (unL'x+)]

= —NCA,,'[unL‘cosh (unL'f) +Biasinh (unL.x+)] (3.44)
and the coefﬁcients of the sinh terms and cash terms are, respectively

bnunL’ -Nccu = -NCA,’Bia (13.45)

canal: - NCbn = 'NCAn’unL. (13°46)
Solving for the unknown constants

133

 

 

. N 1' MPH2
b" =31;an [1+ Cf" 2".2 ] (3.47)
NC—unL
N Bi +1121."
c" = An’NC[ C2 ° 2",2 J (B.48)
NC-uﬂL

and substituting into Eq. (B.43), the second particular solution is

 

" , NCBi +1125" ,
Tcﬂoc”) = 2A,;{an [1+ N2:112£‘2 ]cosh(|.1nL x+)+
C

n=0 1!

 

[NCBio + 33L”
C

big—11%” ]sinh(unL x*) } (3.49)
II

B.2.4 Cold ﬂuid Particular Solution 3
The last particular solution is of functional form

Tc’p3(x+) = Z ancos(anx*) +dnsin(anx+) (3.50)

n = 0
The resulting equation after substituting into the differential equation (with-
out summations shown) is

(-anan) sin (Olaf) + dnancos (uni?) - NC [ancos (an?) + dnsin (anx+)]
= —NC (Bn’ + Cn’) [ancos (an?) + Biosin (anx+)] (B31)
Separating into coefﬁcient equations for the sin and cos terms
- an“, - Ned, = -NC (Bn’ + Cn’) Bio (352)

dud" - Ncan = -NC (Bn’ 4» Cn’) an (3.53)
and solving for the unknown constants

 

(13.54)

N '- 2
an: (Bn’+Cn’)an[l+ CBIO an]

2 2
”CHI,

134

 

d (B ’ C ’)N "emf“: (355)
= + —— .
n n u C N2- + a:
and substituting into Eq. (B50) gives the ﬁnal particular solution
+ ” I I NOEL-0 - a: +
TC,p3 (x ) = "go (8,. +C,. ) {an[l + $441: ]cos (aux )
NCBio - (xi
+ NC ‘7‘“?- sin (any) (B.56)
NC + an

Assembling the simpler problem’s solutions, Eq. (B.41) Eq. (B .42),
Eq. (B.49), and Eq. (B56), using Eq. (B.39) the'solution for the temperature
of the cold ﬂuid can be written as

TC(x+) = T0+Elexp (New?)

 

 

'° , NCBio-t-p’L'z] ,
+ A’ uL 1+ '1 cosh(uLx+)
ago 8{ n I: [Viv-[1:14 2 II
NCBio+p§L‘2 ,
+N . sinh( Lx")
C[ ”is-nil- ” p”

r N Bi -a2
+ 2 (3,;+C,,’) an l+—CE°—-2—" cos(anx+)
"=0 Newt-an

 

NCBio - a: . +
+NC[ Ni. +01: :lsmuxnx ) (3.57)
By applying the boundary condition in Eq. (B.30a) the unknown constant E1

can be determined

135

 

" . NCBio-t-uiL'z .
E, = exp(-NC) (Tm—To) - 2A,; uuL 1+ 2 2 .2 cosh(|.tnL )

n=0 :3

N Bi +11%” ,
+Nc[ 02° 2",2 ]sinh(unL)}
NC-uﬂL

 

" N ' -a2
- 2 (Bn’+C,,’){an[l+ 2131:“: ":lcosmn)
C n

It: 0
Nam-a3] . H
+ NC[ sm (an) (3.58)

2 2
Nc+an

 

 

136

B.3 Summary Energy Balance Solutions

This section does not provide any new information it instead summa-
rizes the solutions obtained in this appendix. The equation numbers will aid
in locating the details of these solutions.

B.3.1 Hot Fluid Energy Balance Solution
Solution for the hot ﬂuid temperature is

TH(x+) = To-t-Dlexp (-NHx+)

0- . (BioNH - ”3L. 2) :l e
+ A 32L 1- , cosh(|.le+)
.20 "{ " [ (33,—ng 2) "

 

(BioNH - uﬁL‘Z)
(33,-1133‘2)

.- B. N 2
+ 2(Bn+g;cn){an[1-( I" "+a")]cosm,,x*)

(N3, + 01%)

 

+NHP..[ ]Sinh (lint?) }

 

BioNH + (:2

"Jsin (anf) } (3.27)

 

+N
"[ ”31+“:

where

 

(BioNH - 33L”) ]

D1=T,“-T— ApZL‘[1 ,
"' ° .21. " " (Na-3:1. 2)

 

" a. (BioNH+ aﬁ)
- 3 —c 1 - .
Ell " + L‘ ")a”[ (N§,+ 0:) ] (B 28)

137

B.3.2 Cold Fluid Energy Balance Solution
The solution for the cold ﬂuid temperature is

Tc(x+) = T0+Elexp (NC-x”)

" , N i+ ZL‘2 ,
+ZA,'{IJ,,L [1+ CB” l1" Jcoshut'L x‘)

N26 - 113,1."

 

n=0

 

NcBio + 1131.”
+ NC N2 _ 21:2
C ”I!

Jsinh (H133) }

 

" NCBio-aﬁ
+ (B ’+C ’) a 1+ cos (a x+)
E." ~{~[ ~33] ~

New), - a:
+ No 2 sin (of) (3.57)
NC + a:

 

where

New, + 33L”

N3 - #3132

 

E1 = exp (-NC) [(TC,in - To) - Z An’{uul‘. [1 + :ICOSh ﬂint.)
n=0

 

NCBio+11§L‘2
+NC 2 2 '2
NC-unL

]sinh(u,L‘) }

 

- E B +C, a + COS a
11:0 II n n ”Cl “)2; n

N Bi -a2
+NC C2" 2" sin(0tn) (3.53)
Nc+an

 

APPENDIX C

C.1 Application of Nonhomogeneous Boundary Conditions

In Appendix A the solution of the wall conduction equation describing
the temperature distribution in the wall was solved to the point of applying
the nonhomogeneous boundary condition for the three simpler problems that
the original problem was split into, because two of the three boundary condi-
tions depended on the temperature of the ﬂuids. With the solutions for the
ﬂuid temperatures complete (Appendix B) the ﬁnal boundary conditions can
be applied.

C.1.1 Application of Orthogonality at x” = 1

The solution for 61 up to the point of applying the nonhomogeneous
boundary condition, and the boundary condition (Appendix A Eq. (A28) and
Eq. (A.4b)) are

91(x+,y*) = 2An[uuL'cosh(uuL'x+) +Biosinh(|.tnL'x+)]*

n80
[u,.cos day“) + Bil-(Sin (u,y”)] (c1)

36
57: ”31.91 (1, y+) = BiL(TL- TO) (0.22))
x’ =1

Taking the derivative of Eq. (Cl) and evaluating the expression and it deriv-
ative at the boundary gives

91(1,y+) = ZAuan'coshwnL‘)+3iosinh(nnL‘)]*
u=0

[1130s (11,?) + Biysin (u,y*)] (C3)

391 -+ _ 2.2. t . t c *
—(1..V ) - A, [11,14 smh(unL )+Bloll,,L 00511 (11,1! )]

...
ax n=0

[uncos My") + Bigsin (u,y*)l (CA)

138

139

Substituting these expressions into the boundary condition, Eq. (C23), pro—
duces

A, [uﬁL‘zsinh (unL') + BiouuL' cosh (unL’) 1 *
- 0

[woos (11.3”) + Biysin (unyin

+ 31L 2 A, [unL' cosh (unL') + Biosinh (unL')] *

n=0
[11,608 (113*) + Biysin (11M) ]

= BiL(TL- To) (05)

The orthogonality of the eigenfunctions was used to solve for the constants,
A". The eigenfunction for this problem is

Y (nu. y*) = new (113*) + Bigsin (unyi) (as)
and these eigenfunctions have the property that

0 natm

l
{wwi’muwhY(u,,..y*)dy+ = { N01,.) n=m ((2.7)

where wxy is a weighting constant and for the Cartesian coordinate system it
is equal to one. Therefore, Eq. (C.5) will be multiplied by a second eigen-
function Y (11”, y*) and integrated over the boundary. Because this integral is
only nonzero for m=n that is the only term that will remain from the summa-
tion. The integrals for each of the three terms in Eq. (C5) are

1 a.

I { 2 A. [uﬁL’zsinh (unL’ ) + 31011,!) cosh (unL’ )] *

0 n=0

[(1,608 (ﬂy) + Bigsin (u,y*)] [uncos (umf) + BiHsin (umy+)] }dy+ __-

A,” [uiL’zsinh (me‘) + BioumL' cosh (11mm) No.1,") ((2.3)

140

l -
j {31L 2 A, [unL’ cosh (nut) 4.3103311) (11_L‘)] *
0 n=0

[uncos (11f) +BiHsin (11.3”)1 [umcoswmyh +191},,sin(tl,,.y*)]aly+ } =

AmBiL [me‘ cosh (umL‘) + Biosinh (umL') 1 N (11,") (C9)

1
Biz. (TL- T,)j[u,,,cos (umf) +Biusin (twin ally+ =
0

_ . Bi H Bi H
31L (TL - To) [3m (um) - E—cos (um) + T] (C.10)
After applying orthogonality Eq. (C5) is
A," [piL'zsinh (umL') + BioumL' cosh (umL' ) 1 N01") +
AmBiL [umL' cosh (umL' ) + Biosinh (me‘) ] N (ll...) =
. Bin 31“,,
Eli (TL- To) [sin (11,") - Tcos (um) + r] (C.11)
which can be solved for the unknown constant Am
B ' B'
BiL(TL— To) [sin (11 )- ﬁcos (u )+ it]
III ”In In “In
A _ (C.12)

 

- [ (11:1.‘2 + 31,311) sinh (umL') + (310 + 35) umL‘ cosh (me‘ ) ] N (11,")

The functional relationship for N01,") is

 

N(|.1) = l[(uerBiz) 1+ Bi" +Bi (€13)
"‘ 2 "' ° ( (1133832)) °] '
Having this information everything is completely known for evaluat-
ing 91. It would be possible to substitute Eq. (C.13) into Eq. (C.12) and this
result into Eq. (CI) to obtain a closed form solution for 01, but because the
expression would be algebraically cumbersome this will not be done.

141
C.l.2 Application of Orthogonality at f = 0

The equation and boundary condition to be applied for determination
of the unknown constants (Appendix A Eq. (A.40) and Eq. (A.6c)) are

" a a
02(x+.y+) = ZBn[cosh(-7"y+)-Cnsinh (4y+]]*
n=0 L L

[ancos (an?) +Biosin (anx+)] (C.14)

 

“Riﬂe2 (x‘, 0) = BiH[T,,(x*) - To] (c.1531)
y‘ = 0
The difﬁculty in applying this boundary condition is recognized by noting the
right hand side contains the temperature of the hot ﬂuid. The needed temper-
ature expressions for evaluating Eq. (C. 15a) are:

92 (x+, 0) = B,I [ancos (anf) + Biosin (anx+)] (C.16)
11:0

392 " an
— (x”, O) = - 2 337C. [ancos (anf) +Biosin (anx+)] (C.17)
3)” n = o L

TH 0:“) = To + Dlexp (-N,,x+)

~ . (aw-3.1L”) ..
+ 24,,{113L [1- (N; #13132) ]cosh(p.an)

 

 

 

 

Putting these expressions into Eq. (C.15a)

142

" a
Z 3,,L—fgn [aﬂcos (mg?) + Biosin (up?) 1
n=0

+ Bi” 2 B" [ancos (an?) + Biosin (anx+)] =
n=0

" , (BiN - 2L”) ,
+ 2A,,{113L [1— ”2” 5",: cosh(ttuL x+)
"=0 (NH-1131' )

 

 

:lsinh (11,113+) }

a. B. N 2
+ 2 (Bn+a_:cn){an[l-( 1° "+a")]c08(a,,x1)

2 2
(NH+°‘..)
Bi N +012
+NH[ 1:2: 2"]sin(anx+) }] (C.19)
H a).

for which orthogonality will be applied. Each term in Eq. (C. 19) will be mul-
tiplied by the eigenfunction

 

 

X (an,x*) = ances (an?) +Biosin (czar) (C20)
evaluated at term m and integrated over the boundary. The eigenfunction has
the same property as discussed earlier, which is given in Eq. (C.7). Each of

the integral will be listed separately and evaluated starting with the ﬁrst term
in Eq. (C.19) and proceeding to the last term.

1 a.

a
J { 2 Bn-écnIancosmnf) +Biosin(aux+)] *
0 n=0 L

[amcos(amx*) +Biasin(amx*)] }ch+ = Bm%CmN(am) (C21)

143

1 to
j (31,, 2 3,, [ancos (an?) +Biosin (anxtn *
0 n=0

[amcos (an?) +Biosin(amx*)] }dx+ = BiHBmNmm) (C22)

1
31,4), 1' {exp (-1~/,,,x+ ) [amcos (amxt) +Biosin (amx+)] }dx* =
0

 

 

BiHDl {(N +B' )a + -N”[ (N +B° )a cos(a )+
l e "' l
(”31+a'2n) H a m H 0 m m
(Uzi-Nam.) sin (am) ] } (0.23)
l on - 2 .2
t (BIONH-unl‘ ) t
{Bi 3,1131. 1- , coshutnL x+)*

1 ”E. 1 (Nib-nil- ”) 1

[amcos (an?) +Biasin(amx*)] }dx* =

. . t2
2 An31ﬂam 2 , _ Name - 1131.
(113132 + cf") " N2, - 11:1.”

 

 

] {sinh (unL’) unL' cos (am)

n=0
+ cosh (unL‘) amsin (am) }

+ i 11,133,131, “2 , 1_NHBio-u:L
(nﬁt‘zwg) " Ng-uﬁL‘z

n=0

 

 

2
] {sinh (11,13) u,L‘ sin (a...)

-cosh (unL‘) amcos (am) + am } (C34)

144

 

1 c- . 2 02
(BloNH-Ll'L )
B ° A N

]sinh(uuL'x+)*
0 13:0

[aacos (an?) +Biosin (anxtn }dx+ =

 

 

i AnBiHam NHBio - 33L”
(11,2,L’2 4» 01,2") l4, H N}, - uﬁL"

n30

] {cosh (unL’) unL' cos (am)

+ sinh (uuL')amsin (am) ‘11,}; }

 

 

" AnBiHBio NHBio- p 2‘1.
) n H

+ 2 N},-:

"=0 (uiL‘2+a: 22,,]{008h(u L) p. L sin(am )

-sinh (uuL’) amcos (am) } (C25)
The remaining integrals depend on whether n=m.

(BiaNH-I- 01:) +
I“; iunzo (B +_ L' 1C")a"[l - (N},+ 01%) Jcos (“"x )*

[amcos (an?) +Biosin (amx+)] }dx” =

 

For n=m

" 3' N 2
Biﬂz (B"+E;C")anl:l- ( '02H+:n):l*
"=0 L (NH+an)

(1m 1 2
[Tm+43in(amff)+

 

Bi
sin (a )] (C26)

Fornstm

Bi” i (Bu-(- %Cn)an[l _ (BiaNu+ (1%)] {am|:8in(a’"-a”) + sin (“n+“n)]

n=0 (Nift- (1:) 2 (am-0%,.) 2 (am+an)

 

 

 

 

+ Bio[l - COS (am— an) 1- cos (am+ an)

2(am-an) + 2(am+an) ]} (€27)

145

 

(31°,N,,+01:) . +
j{31,,2(3,, +—c )NH[ (Nfﬁaﬁ) ]sm(anx )2

n=0
[amcos (an?) +Biosin (amx+)] }dx+ =

For n=m

 

. u an (BioNH+atzr) Bio Bio . 1 .
Bl” 2 (Bu '1' PC")NH[ (Nil... (13) ]|:T - 33:83] (2am) '1' 581112 (am)] ((3.28)

n=0
Fornaem

. " on,l (BioNH-t-ai) . sin(am-an) sin(am+an)
B'”.§.(B"*PC")"”[ mama) 1W1 21.1,-..) " 2(a..+a,.)]
1- cos (“111 - an) 1- cos (“111 + at")

+“[ mam-01") + 2(am+an)

 

 

 

 

 

] (C29)
Instead of substituting these large expression into the Eq. (C.19) some new
variables are deﬁned. For Eq. (C23) let

“{(N +31H)01 +eN”[- (N,,+31°)01m cos(a )

 

(OMB N2:

+ (ag-NHBio) sin (01,") ] } ((2.30)
In Eq. (C.24-25) deﬁne:

 

 

3' N 3' - 2L’2
'21“... ) 2L'[l- H 10 ll, ]{sinh (unL') unL'cos (am)

 

 

5
p "'"1‘ (ngL‘2+o1; N§,-11,2.L“2
+cosh(unL')amsin(am) } (c.31)
BiHBio 2L! 1 NHBio_u:L.2 {. ( L.) L. . ( )
p a . - , smh 11 [l sm 0:
"'2 (nthag) 33-1192 " " "'

-cosh (unL') amcos (“111) + am } (C32)

146

 

 

31,,01," [NHBio -11§L‘2

p "‘2" (1121.32.32) 11,, " Ni-uiL" 1km (""L ) ”"1“ com")

+ sinh (unL' ) a," sin (01m)-11nL‘ } (C33)

 

 

3533'} ”3310 " "3L” { sh ( L.) L" - (a )
a co sm
pmn,4 (“£12.2‘1'az.) n H [Vii-11:14.2 “I! "a m
-sinh (nut) amcos (am) } (C34)
and for Eq. (C.26-29), for m=n

 

 

 

 

 

 

_ . (BioNH+a:) an: 1 . Bio . 2
wmn-Btuau 1" (N§+a:) [7+zm(2am)+2am81n (am)]
(Bi.N11+a§> 31, 31,, . 1 . 2
+NH (N;l+a:) [T ”Ia—MSU! (2am) +5811! (am):| (C35)
and for man:
-3° 1 (31',N,,+01§) sin(am-au) sin (amwn)
“'"m’ ”“1 ’ (was) "'1 2111",-..) “ 2(a..+a.>]

 

 

. 1-cos(am-an) 1-cos(am+an)
+B‘°[ 2(a111'0‘11) + 2(“111+°‘11) ]

 

 

 

(BioNH + 01%) . sin (“11 - on") sin (am + “11)
+NH[ (N21411:) '0': 2(am-an) - 2(am+an) :|

 

 

+am[1-cos (“111-“11) l-cos (“111+a11)

nan-01,.) + 2(am-t-an) ] (c.2115)

Using the newly deﬁned variables given in Eq. (C.30-36) in the integrals of
Eq. (C.21-29); Eq. (C.19) after applying orthogonality, can be written as

147

“111 . ”
BMFCmNﬂlm) + BmB'HNuxm) = (0le + Z A” (plan. 1 + pm”. 2 + pm": 3 + pm", 4)

n=0

" a
+ 2 (Egg-[gum (C37)

n=0

where the orthogonality relationship is

N01,, ) 1 ((114-32) +——-Bic +B° (C38)
= _ 1” l .
[ ( .2...) .1
Deﬁning one last variable
a
A,” = —[ZT-CM+BiH]N (“111) ((3.39)

and rearranging the equation gives

3317‘". +1.20"? +C,, gm)wm+o) D1: “zAu(P11111,1+P11111,2+P11111,3+Pm11,4)
n=0

(C.40)
where the sign changes were introduced (in A," and Eq. (C.40)) so that the
variables are exactly as programmed in the computer program used to evalu-
ate the problem (in Appendix F).

148

C.l.3 Application of Orthogonality at y” = 1

The equation and boundary condition to be applied for determination
of the unknown constants (Appendix A Eq. (A.48) and Eq. (A.8d)) are

93(X+.)’+) = ZCn[%cosh(%gy*)+3iﬂsinh(%y+)]*

n=0

[ancos (anf) + Biosin (anx+)] (C41)

803
a— +BiC93 (x+, 1) = BiC[TC(x+) - To] ((2.423)
y .
y = l
The evaluation of this boundary condition is very similar to the previously
discussed method for 62, except the algebra is a little more involved because
the boundary condition is evaluated at y“ = 1 instead of O. The needed tem-

perature expressions for evaluating Eq. (C.42a) are

 

u a" all . an
9303.1) = 2 Cn[—,cosh (7) +BiHsmh (7)] *
n g 0 L L L
[ancos (anf) + Bio sin (anx*)] (C43)

89 " a a a a
—3-(x+.l) = ZCn-§[—fsinh(—:)+Biucosh(—§)]*
ay+ ":0 L L L L

[ancos (anf) + Biosin (anx+)] (C.44)

 

149

Tc (x‘) = To + Elexp (NC?)

 

(Ni-nib”)

n=0

.0 B. N 2Lt2
+ 2A.’{uuL‘[l+( 1" Cw" )]cosh(unL'x+)

 

(BiN + 2L”) .
+NC ”2C 5"” :|sinh(unL m}
(NC-“"1! )

" B'N — 2

n = 0
BioNC — a:
+ ”c 2 sin (uni) (c.45)
NC + a:

 

 

Putting these expressions into Eq. (C.42a) gives

“ an an a" an
2 Cr; [—, sinh (7) + BiHcosh (—, II [ancos (anx+) + Biosin (anx*)]
n: o L L L L

+Bic i C,[%cosh (;§)+Biusinh (2.3)]:-

n=0

[ancos (Qua?) +Biosin (a,x")] =

Bic [El exp (ch)

" , (BioNc+ ng‘z) .
+ A ’ L l + . cosh L x+

 

 

(BioNC+u:L'2) .
+N , sinh( L 1:”)
C[ (NE-uiL 2) ""

“ (BiONC-ag)
B ’ C ’ l
+ n§0( n+ n){an[ + (Né+a:)

+N (Riﬂe-ab ' ( +) (C46)
(1 .
C[ (Ni-mi) is” "x ]

 

:l cos (053+)

 

150

for which orthogonality will be applied. Each term in Eq. (C.46) will be mul-
tiplied by the eigenfunction

X(an,x*) = aucos (ans?) +Biosin(anx*) (C.47)

evaluated at term m and integrated over the boundary. The eigenfunction has
the same property as discussed earlier, which is given in Eq. (C.7). Each of
the integral will be listed separately and evaluated starting with the ﬁrst term
in Eq. (C46) and proceeding to the last term.

{Ca"a"inha” 3 ha I B. ( )1...
I 2 nL.[L.S (U) tycos (21)] ann”c08(ax) lsin ax
[amcos (an?) +Biosin (amx+)] }dx+ =

am am am an
Cm—. [—;— sinh(— T)+ BIHCOSh(-— ?)]N (am ) (C.48)
L L L L

an an
3,131.0 20 C "E 0031‘ (L—f )+B‘33inh (f: [I [ancos (an?) +Biosin (anx+)] *
[amcos (amf) +Biasin (amx+)] }dx" =

am am am
31'ch [— cosh ( ) + BiHsinh(— )ﬂN(aM ) (C.49)
L L L

1
area, 1 {exp (ch+) [amcos(amx+) +Biosin(0tmx+)] }dx+ =
0

BicEi

N2 :{wi -Nc)am +eN‘[(NC-Bi )am cos(am )+

 

(0:,+Nc3i,)sin(am) ] } (C.50)

151

 

l 0- ° 2 ‘2
. (B'oNc'l'llnL )

{Bi An’u L 1+ ,

i C 2 " [ (Ni-nil- 2)

n=0

]cosh (unL'f) *

[amcos (amf) + Biosin (amx*)] }dx* =

 

 

" Au’BiCamttnL' (BioNcwﬁL'z)
2 -2 1+ 2 .2
(nib mi.) (NC-nil. )

n=0

] {sinh (nnL‘) an" cos (am)

+ cosh (unL') amsin (am) }

 

 

+ i An’BicBiounL’ 1+ (BioNC+u:L'2)
with 0:3,.) (N3 wil-'2)

n=0

] {sinh <u,L‘) unL’ sin (am)

-cosh (uuL') amcos (am) + am } (C.51)

 

1 .. - 2 ‘2
(BtoNc+tt L ) .
Bi A ’N '1 sinh L 15* *
1‘ C20 " C[ (NE-uiL’) ] m" )

[amcos (amf) + Biosin (amx*)] }dx+ =

 

i An’BicamNC (BioNCwﬁL’z)
(nil-”wﬁ.) (Ni-uiL")

n=0

] {cosh (unL’) unL' cos (am)

+ sinh (unL’) amsin (am) —u,,L" }

 

- An’BicBioNC [ (Bi.Nc+ uﬁL")
"’ 2 2 ’2 2 2 2 ‘2
(uni. +a,,,) (Ne-uni. )

n=0

] {cosh (an‘) unL' sin (0cm)

-sinh (an') atmcos (am) } (C52)

152

The remaining integrals depend on whether n=m

(BioNc - (2:)

(Ni.+ (1:)

 

l .-
jw ic 2 (B,'+ C,’)a,[1 +
0

:lcos (an?) *
n80

[amcos (uni) + Biosin (amx*)] }dx+ =

for n=m

 

 

. on I I (BioNC-a'z') a". 1 . Bio . 2
BIC 2 (Ba + Ca )anl:1+ (Né+a2) [—2— + Zsm (2am) + 2“ sm (am)](C.53)
ﬂ

n=0 m

f0rn¢m

 

 

 

" B'N — 2 - _ -
Bicz (Bn’+cn’)an[l+ ( to C (1')] ”[8111(am an) sm(am+an)]

 

 

 

"-0 (Né+a:) “am-“..) + 2(am-I-an)
, l-cosmm-an) l-cos(am+0tn)
+314: 2(am-an) + 2(am+an) ]} (€54)
1 ' ~ I I (BioNC-ai) . + at
{{Blcn=o(33 +C,)Nc[ (Né-l-az) ]sm(anx)
[amcos (amf) +Biasin (amx*)] }dx+ =
forn=m

 

 

Bic 2 (B; + (2;) NC

sin (2“...) + 1 sin2 (0%)] (C55)

74a 2

[ (BioNc - a3.) ] [Bio Bin

2 2 -
(NC + an)

fornaem

 

 

 

(BioNC-a:)] .[sinmm-an) sin(am+an)]

Bic-2 (Bn’+Cn')Nc[ avg-”12) 2(am-an) - 2(am+an)

u=0

 

 

441,,[1- cos (am-an) 1- cos (am+0tn)

Nam-an) + 2(“m+°‘n) ] (C56)

153

Instead of substituting these large expression into the Eq. (C.46) some new
variables are deﬁned. For Eq. (C50) let

Bic
‘

ﬂ 2 2
NC-I-am

{ (Bio - NC) an + eNCI (NC - Bio) amcos (am)

 

+ (a; —NHBi0) sin (am) ] } (C.57)
In Eq. (051-52) deﬁne:

 

 

. t . .2
P __ BicamunL [ +(BroNC+u:L )

= sinh L‘ L‘
mn,l (uiL‘2+a:) (Né-Ilzla.2) :|{ (11,, ) l1" COS (am)

+cosh(unL‘)amsin(am) } (C58)

P

 

 

BicBiaunL’ (BioNC + ngL‘z)
a 2 '2 2 1+ 2 2 '2
(u..L +a,,.) (NC-11,1. )

mu, 2

] {sinh (unL') an’ sin (am)

—cosh (unL‘) amcos (am) + am } (C59)

P

 

a BicamNc [(BioNcng‘z)
mn,3

(11.2.1324'0‘2) (NZC-uﬁL‘z) ]{°°‘°’h (”nl‘ ) ”nL C05 (0‘...)

+ sinh ( unL‘) am sin (am) -an‘ } (cm)

 

BiCBiaNC [ (BioNC+ ngL")

h 0 l .
”“2 (uiL‘2+a;) (Né-uﬁL”) Jms (“"LN'JJ 8mm")

—sinh (unL') amcos (am) } (C.6l)

 

154

and for Eq. (C.53-56) deﬁne

for m=n

 

 

[ (BioNC—a§)] an 1 Bio 2
‘1'” 'Bicau l + [— + -sin (2am) + sin (am)]

 

 

 

 

 

(N%+T%) 2 4 2am
(BioNc- a3.) Bi, Bio _ 1 ,
+NCI: (”3“. (1:) “:7 - 4°". sm (2“...) + -2- sm2 (am)] (C.62)
for m it n
. (BioNC — of.) sin (an — an) sin (am + an)
ngmca" 1+ (N§+a§) "'[ 2(“m'“n) + 2(“m+°‘n) ]

 

 

. l—cos(am-an) l—cos(am+an)
+B'°[ Mam-an) + Nam-ion") ]

 

 

 

(BiaNC-az) . sin(am-an) sin(am+an)
+"C[ wing) ] B‘°[2<a..-a.> ‘ 2(am+a,)]

 

 

+ (1,,[1 - cos (am-0t") 1— cos (am+an)

2(am-au) + 2(am+an) ] (C63)

Using the newly deﬁned variables in the expressions for the integrals, Eq.
(C.46) after applying orthogonality is
C — [—,sinh (1)4- BiHcosh—;-]N(am)

L L L

ML.

am am am
+ BiCCm[—;- cosh (T) + BiHsinh—;:|N (am) =
L L L

QME1+ Z An’ (PM 1 + PM 2 + PM 3 + PM .) + 2 (B; + 0,5) ‘1’” (C64)
n=0 n=0

Deﬁning a ﬁnal variable

 

052.. . . . a... . . a...
(but 2 [(14.2 +BICBIHJSlnh (F)... (BIH+BIC) COSh (FHNUIH') (C55)

and substituting into Eq. (C64) and rearranging gives

155

Cmom - 2 (B; + Cn’) ‘l'm — QmEl = 221,319”, 1 + PM 2 + PM 3 + P ) (C66)
I! l

mu, 4

156

C.2 Summary of Applying Nonhomogen. Boundary Conditions

The ﬁnal boundary conditions were applied and simpliﬁed using the
orthogonality of the eigenfunctions. This procedure produced a close form
solution for the constant A, in 61 since the nonhomogeneous term in the
boundary condition was a constant

, 3 Bi” Bi”
BIL(TL- To) [sm (“111) - -u—cos (“111) + T]

 

 

Am = 2 t2 . o . o a ((112)
[(umL +Bi0BiL)s1nh(umL )+ (B10+BiL)umL cosh(ttmL )]N(um)
where
_ _1_ 2 - 2 Bi!— -

However, for the constants B3, in 63 and Cu in 03 the boundary condi-
tions to be applied have nonhomogeneous terms that are functions containing
the unknown constants B, and C3,. These functions are contained in the solu-
tions for the ﬂuid temperatures. Thus, in applying the boundary conditions
and orthogonality the resulting equation is simpliﬁed, yet still contains sum-
mations and is a function of both unknown constants B" and C3,. The equa-
tions to be solved to determine these constants are

B", 2 “+203? +C,—2:1,”)wm+m 1),:

" 2 A”(91.111.12“ Pm, 2 + PM 3 + pm, 4) (C40)

n=0

Cmtbm - Z (B3'+ Cn’)‘1’,..,.

13:0

(25,:

2A,’(PM3+PM +Pm m3+P3M3) (C.41)

These two equations represent a set of simultaneous equations that need to be
solved for the unknown constants B" and C,I (n= 1,2 ....... N) . The terms on the
left hand side contain unknown constants and the right hand side contains all
known information for two equations. The solution of these equations is
given and discussed in Appendix D.

APPENDIX D

D.l Solution for Constants B, and c,

After evaluating the ﬁnal nonhomogeneous boundary conditions and
applying orthogonality two equations were derived to solve for B, and C,,
(Appendix C Eq. (C40) and Eq. (C.66), which are

B 1,, + 2 0(8, +C, :Z-MJVMR'i-O) DI: -2A,(pm1+pm,,2+pm,,3+p,,,4XD.l)

u=0

C,,o, - 2 (B,’+ c,') ‘l’m- 9,51 = 2 A,’(P,,,,1+P,,,,2+Pm,’3 +P,,4) (D2)
n=0 n=0

where all the greek variables are deﬁned in Appendix C and the expressions

for E1 and D1 are (from Appendix B Eq. (B.25) and Eq. (B.52))

o 1 (BiaNH-ﬂila.2)
" N2_ 2L'2
H "a

 

D1: Tam-To- 2"»sz
n=0

.- B. N 2
-2 (B,+%C,)a,[l- ( t°2H+a")] (D.3)
n=0

2
NH+a,

 

 

" , , (Bi NC+ pZL’Z) ,
E1 = exp (-NC) [(Tmn" T,)- 2 A, {unL [1+ 02 2 ",2 cosh (unL )
n=0 Nc‘llnL

(BiN +p2L’2) . ,
+Nc[ ”2C 2",, ]smh(u,L)}
NC-u'nL

 

 

“ B'N — 2
-2 (B,'+C,'){a,[1+ 1" c a"]cos(a,)+

2 2
,go Neal-oz,

N (BiaNC—aﬁ) . ( ) 4
[ (Pew: ism" i“

 

Deﬁning the following variables:

157

158

 

 

(Bi N” - ﬂ")
2 t - 0 n
B, a (u,L ) [I N}, __ ”it, (D.5)
(BioNH + (23,)
Y..'°‘..[1' ”3”“; (13.6)
ll

_ . (BiN wil-‘2) .
Date Ncﬂ‘lnl‘ [1+ £2.qu .2 ]c08h(}.lnL)

 

 

(BioNC+ uﬁL‘z)
+NC 2 2 ‘2
NC- ”EL

4 a, a, (BiaNC - aﬁ) ‘
an?" C[cosh (—,)-Cnsinh (7)] {ual} + 2 2 cos (an)
L L NC+ a, _

(BiONC - a3) ‘
+ NC 2 2
NC + a,

Jsinh (u,L‘) } (v.7)

 

 

 

sin ((1,) } (D8)

 

_N a, a, a, (BiaNc- uni)
ruse C[_.cosh (—;)+Biusinh (7)] a,[l + ]cos (an)
L L L

 

N3: + a:
(BioNC- a3) .
+ NC 2 2 sm (a,) (D9)
NC + a,
and substituting into Eq. (D.3-4) gives

a Q a
D1 = (TH, ,., - To) - 2 A,p, — 2 (B, + c,—f)y,) (D10)

1: = o n = o L
E, = exp (-NC) (TC, ,,— :20 A, ',u - 20 (B, a, + C,1:(D.11)

where Eq. (D.8-9) contain the necessary constants to convert B,’ and C,’ to
B, and C,. Deﬁning two last variables, to convert all unknown constants to a
form without a superscript prime

(VB) M, = ‘Pm [cosh (g) — Cnsinh (E; )] (D.12)

159

a a a
($10,, = W,,[—3 cosh (4)+Bi,sinh (4)] (0.13)
L L L

Introducing Eq. (D.10-13) into Eq. (D.1-2) and rearranging gives

" " a
2 W.,-m.) + 2 C,,_ W.,-m.) =

n=0 n=0

-mm(THJﬂ—T0) - 2Al(pmn,l+pmn,2+pmn,3+pmn,4-mmﬂn) (1114)

n=0

2 [annou- (YB) mu] 3,, + Cm¢m + 2 [ﬂm1n_ (WC) mu] C, ___

n=0

Qm‘xp(‘ c) (TQM-To) + zAn’u’nm.1+Pmm2+Pmm3+Pmm4-Qmun) (13°15)

n = 0

These two equation represent a set of simultaneous equations to be solved for
the constants B,and C,. To accomplish this the series must be truncated at a
value n=N giving 2N constants to solve for (B, and Ci 2' = 1, 2 ..... N).There are
only two equations, however these equations can be written for each eigen—
value; hence the subscript m denotes the eigenvalue considered. Therefore,
since there are m=N eigenvalues and two equations for each eigenvalue the
total number of equations are 2N to solve for the same amount of constants.

For illustrative purposes consider a matrix representation of the equa-
dons

[ME] = [f] (D.16)

where [K] is the coefﬁcient matrix of order (2Nx2N) and [f] is the forcing
vector (2le) and the unknown constant vectors given as

 

It‘xli‘et :11:- 'FDFJ-

 

160

 

 

 

 

81 Cl
B2 C2
LB] = LC] = - (D.17)
_BN. -CNJ
Further, the coefﬁcients matrix can be quartered and the forcing vector
halved
K K
[K] = " 1 " 2 (D.18)
L. .J
F)
[El = (D.19)
[2]

Each quarter of the coefﬁcient matrix is a square matrix of order (MW) and
half the forcing vector is of length (NxI). Referring to Eq. (D. 16) the simulta-
neous equations can be represented as

[§:§:]E’;~]=[§;]

This shows where the terms that need to be put into the coefﬁcient matrix and
forcing vector will come from. The rows in the coefﬁcient mauix in Eq.
(D20) represent the coefﬁcients of B, and C, for each equation. Row one
corresponds to Eq. (D.14) and row two to Eq. (D.15). Similarly, the entries in
the forcing vector correspond to Eq. (D. 14) for row one and Eq. (D.15) for
row two. Presenting the equations in this form, Eq. (D20), makes it possible
to write out some of the terms of the matrices to show the developing pattern
and indicate where these terms originate. The individual matrices and forcing
vectors that form Eq. (D.18) and Eq. (D.19) are listed below.

161

 

 

(A, My" -11co,) (Wu-72ml) .. (Wm-won
(V21 "' 7102) (A2 + V22 " 7202) - - (WZN — YNmz)
[K1] = (0.21)
(VNI - 710)”) (VNZ — 720)”) . . (2%” + VNN -' yNtoN)
" T

(anl a2 a
((V11'71m1)27) (le’yzml)f) '°((W1N-7le)zg)

 

 

((v vana‘) ((v vanaz) (( )a")
21‘12'7 22‘227 -- W'Y‘D'T
[K2] = L L 2N N 2 L
(( - (0)11) (( - (0)93) a”

WM 71 N L‘ Wm 72 N L' -- ((WNN'YNlezr)

(D22)
(alol-(WB)11) (Gig—(1%)”) . . (91°N‘(‘FB)1N)
(Q,a,-(\PP)21) (Q,a,-(\PB)22) . . (azoN-(WB) 2N)
[.Kg] =

(Owl-(WM) (ﬂNaz-(WB),2) .. (Q,a,-(\PB)NN)

 

 

(D23)

162

P

 

 

 

 

(<bl+nltl-(‘FC)11) 0112-0?le . . QItN-(WC) 1N
(lz‘cl-(‘I’Q21 (¢2+0212—(‘I‘C)22) . . “25/" PVC)”,
[K4] =
ﬂNtl-(‘I’Om QNIZ—(‘I’O N2 . . (¢N+QNtN-(‘I’C)NN)
(D24)
"‘91 (THJn-To) "2’41. (pln,1+pln,2+pln,3+pln,4)
ﬂ
’ ‘02 (Tutu " To) " 2A» (92», 1 + Pzn, 2 + Pzn,3 + 92:“)
[£1] = n ([125)
L-mN(TH,in-'To) "'zAu(p1§Iu,1"I.pNn,2+pNn,3'*-pNn,4)_i
" W
...N ’
n
.N ’
‘ CQZ(TC,in-To) +2An (1,211.1+P2m2+P2m3+P2n,4—02Dn)
[F2] = "
.N ’

 

(13.26)

The indices, row-i and column-j, of the coefﬁcient matrix can be inter-

preted as the coefﬁcient of the constant B] (or C!) for the it” eigenvalue. That
is, a row corresponds to a single eigenvalue and a column to a single con-

stant.

 

d

APPENDIX E

El Derivation of Effectiveness-NTU Relationship for a
Counterﬂow Heat Exchanger Neglecting Axial Conduction

Consider a double pipe heat exchanger geometry as shown in Figure
E], with the energy balances on the ﬂuids as shown. For the ﬂuids, the
energy balances give

(me) CTCL = (me) CTCIH Ax+dq (E.1)

(ch) ”Tulx = (mCP) Hrnlxqu (132)

In addition to the ﬂuid energy balances over the differential length a thermal
circuit can be analyzed for the heat exchanger wall. The thermal circuit for
this heat exchanger is shown in Figure B.2, neglecting any fouling. The heat
transfer over a differential element of length Ax is

dq = aura-rem): (13.3)
where

 

insulatio .

  

 

 

 

 

      
   

\\\\\\\\\\\\\\\\\\\\\\\\\\\V

 

 

 

 

Figure E.1. Double i heat exch er come and the cue balance for a
dill’er'engage element ofﬁnggﬂrg Ax try 1'8!

163

164

 

 

 

1,, —/wv~ w W— Tc
1 In (do/d1.) 1
nap” szu hCPC

 

 

 

Figure B.2. Thermal circuit for heat exchanger wall

1 In (do/d1) 1
up - [_hﬂpu.» 2M... + thc] (5.4)

Substituting Eq(E.3) into Eq(E.2) and Eq. (El) gives

[ (mop) Job...- (mcp) cTclx]

 

 

 

- Ax = UP (TH- TC) (E5)
[(mC) TI -(mC) Tl]
_ p H ”“2; p H ”x = UP(TH-TC) (E.6)
Introducing the heat capacity
C a me (12.7)
and taking the limit as Ax -> 0, Eq. (E5) and Eq. (E6) can be written as
are
_ 7x. = _ (Ii-(TH Tc) (E8)
dry
.. 7x. : _ U:-(T” Tc) (E9)

Subtracting Eq. (E9) from Eq. (E.8) gives

d l 1
dx H C CH Cc ” C

This equation can be separated and integrated over the length of the heat
exchanger

idﬂ'ﬂ- Tc) 1

L
1
(Tu- Tc) =- UP( )J‘dx (12.11)

ETC—c o

 

giving

165

zn(r,,- Tc)|o=-UP(C—H --C—C) 1113 (12.12)

Evaluating the terms and rearranging Eq. (E.12), provides a solution in terms
of the operating conditions of the ﬂuids and the heat exchanger wall design

(TH, out - TC, in)_1
(1.11.313 _ TC, out): exp-[ -UA (ClH -CC)] (B.13)
The transformation of Eq. (B.13) into a dimensionless form will begin
by subtracting and adding the inlet temperatures of the ﬂuids to the numerator
and denominator on the left hand side of the equation

 

 

(TH, in - TC, in) - (TH, in - TH, out) 1

(TH, in — TC, in) - (TC, out _' TC, in) = €xp[" -UA (El-H- CC ] (13.14)
Next, each term of on the left hand side will be divided by the temperature
difference at the ﬂuid inlets, (711.1» — TC. M), and the numerator multiplied by
(C,,! C,,) and denominator by (CC/ CC)

 

 

 

 

 

 

-1 _ 311(THJn — TH,out)1
C (T - T ) 1
L H H in C in .1
_ ' ' = ex -UA(-—— —— ] (5.15)
1 _ CC (TC, out - Tails)1 P[ CH CC
CC (TH, in — TC, in) ..
Deﬁning the effectiveness as
e = ‘1 (E.16)

Carin (TH, in — TC, in)
where

C

min = min (Co CH) (13.17)
and q is the actual heat transfer that occurs over the length of the heat
exchanger, which was derived in chapter 1 using an energy balance on the

ﬂuid, and is written as

q = CC (TC, out - TC, in) = CH (TH, in — TH, out) (5’18)
The effectiveness could be substituted into Eq. (EJS) if it was know which
ﬂuid had the minimum heat capacity, for the sake of generality all possibili-
ties will be considered. Two of three possible expressions for Eq. (E.15) can
be written as follows, the third will be addressed later:

166

For C,,, = C,,
[ ”‘( C”)]
—— = exp -— l-— (E19)
1- E’ e C” CC
Cc
For Cm." = CC
1- (J3):
CH UA Cc
—.-. «wet-.7)]

Solving Eq. (E.19) and (E20) for the effectiveness, after introducing the
number of transfer units

 

 

N TU - UA (E21)
len
and the heat capacity ratio
(3 .
CR = mm (E22)
CM“

into the equations gives the same solution for both Eq. (B.19) and Eq. (E20),
which is

_ 1- exp [-NTU(1 - CR”

‘ 1- 5Rexp [-NTU (1 - 511)]
This provides results for C,, at CC, however, if the heat capacity ratios are
equal, which 1s the third possibility that was mentioned previously, Eq. (E. 23)

is indeﬁnite (% ). For this case L’ Hosp1tal s rule can be applied to determine
the limit as the heat capacity ratio goes to 1 giving

(E23)

 

1 + NTU
e - W (E24)

167

E2 Solution of Fluid Temperatures for a Counterﬂow Heat
Exchanger Neglecting Axial Conduction

E2.1 Unequal Heat Capacities (C,, at CC)

Solving for the general solution of the differential equation in Eq.
(E10) for the temperature difference between the hot and cold ﬂuid gives

TH-TC = Clexp[- (Ina-I - C811] (5.25)

where Cl is a constant. Substituting Eq. (E25) into the individual differential
equations for the ﬂuids, in Eq. (E8) and (E9), gives the equations to be
solved for the temperatures of the ﬂuid, with appropriate initial conditions

H” C U-P

K = -C-_: 18Xp[- (El; " 61"?)X] (13.26)
TH(0) = 711,111 (E2621)

ch UP 1

—d—x— - ~55 Clexp[- UP(C1CH —E;)x] (E27)
TC(L) = TC,“ (E2721)

These two equations, Eq. (E26) and (E27), are easily solved for the temper-
atures of the hot and cold ﬂuids

 

 

TH = 1C1 1 exp[- -UP (C—IH - C—C) X] + C2 (13.28)
“‘61; ‘ c—c’

TC = 1C1 l exp[-UP1 (— C "' -Cl—)X] + C3 (E29)
CC‘Q'E’ ”' c

where C2 and C3 are constants, but noting that Eq. (E25) must holds gives

C4: C2 = C3 (530)
Incorporating Eq. (E30) and rearranging Eq. (E28) and (E29) gives

 

exp[- UP(—— 1--——)11] (E31)

CHCC

C
T" (x): (1- C,,/CC)

168

 

Tc(x) = C,- —UP(-1— 1 )x] (5.32)

l
(1— can.) “”[ CH " E";
which can be solved for the unknown constants using the initial conditions in

Eq. (E26a) and (E27a). Applying these conditions to the equations for the
hot and cold ﬂuids produces

 

 

Cl
TH.in = C4+ (l-CHICC) (E33)
C
Tc: =C‘- 1 exp -UP(—l---L)L (E34)

a set of two equations with two unknowns. Solving, the two unknown con-
stants are

(TH, in ‘ TC, in) (CC ‘ CH)

 

C1 = C C UP( 1 1 )L
c' H“? [" — ' — ]
CH CC (5.35)
l 1
C4 = H C (E36)

 

1
Cc- CHCXP[-UP(Z‘1; - EQL]

If C,, > CC and the heat capacity ratio is used C,, = CC/ C,,, Eq (E35) and Eq.
(E36) become

(7.11.17: ' Tc, in) (CR '1)

C1 = ER-exp [-UP/CC(CR-1)L]

 

(E37)

_ CRTC’ 1.. - 73,1119” [-UP (CR - 1) L]
‘ - CR-exp[-UPICC(CR-1)L]
Substituting Eq (E35) and (E36) into the solutions for the hot and cold ﬂu-
ids, Eq. (E31) and (E32), gives

(E38)

 

169

CCTC,iu “Tu,tnCH¢xP[’UP(Z~l—"C—)L]
1,,(x) = C +
C -C x -UP(— —)L]
C- He pl: C1” -CC

 

(TH, in - TC, in) (CC - CH)

(1 - C,,/CC) {Cc— C,,exp[-UP(EI; - —1— L]}

exp[- UP(-— -2}- )x](E (13.39)
C ) CH C
C

CCTC ,n— TH inCHexp[- UP(— - —)L]
Tc(x) = CH CC —
CC- Cuexp[-UP(— -—)L]
CH Cc
”min" Tarn) (CC' CH)

exp ”PFC-l- --)x (13.40)
(l-CCJCH){CC-Cuexp[-UP(£;—EIE)L]} [ Cu Cc ]

For the case of CH> CC Eq. (E37) and (E38) can be used in place of Eq.

(E35) and (E36) giving the following expression for the hot and cold ﬂuid
temperatures:

 

 

+ - - .—
THO‘ ) ' CR-exp[-UP/CC(CR-1)L]
(TH,in-TC,in) CR
CR- exp [—UPICC(CR- 1)L] ”‘1’ [‘UP (CR ’1)“ “‘14”
T (x+) = CRTC,“in-TH,in:xp[-UPI_CC(CR-1)L]_
C

 

(TH, in - TC, in)
CR - exp I-UIVCE (CR ‘1)L] exp l-UP (CR -1)” 05.42)

Nondirnensionalizing the length scales by the total length of the heat
exchanger

 

+-§
x -L (13.43)

and introducing the number of transfer units (NTU)

170

UPL
C

min

 

N TU = (E44)

the ﬁnal equations for the hot and cold temperatures become

- CRTC, in "' TH, mexp ["NTU(CR - 1)]
H- CR-exp[—NTU(CR-1)L]

_ (TH, in "' TC, in) CR
CR-exp [-NTU(CR- 1)]

 

 

“P [‘NTU (CR - 1)x*] (545)
T = CRTC, in " Tn, inexp [-NTU (CR - 1)]
C CR-expl-NTU(CR— 1)]

(Tu, m' TQM)
CR - exp [-NTU(CR -1)]

 

exp [-NTU(CR - l)x+] (E46)

 

Note that Eq. (E45) and (E46) apply for C,, > CC, which could easily be con-
verted if the opposite were true.

E.2.2 Equal Heat Capacities (C,, = 0,)

If the heat capacity ratio of the two ﬂuids are equal the solution of the
differential equation, in Eq. (E10), for the temperature difference between
the ﬂuids is a constants.

Which upon substituting into Eq. (E8) and (E9) give the following differen-
tial equations and initial conditions for the temperatures of the hot and cold
ﬂuids, respectively.

dTH -UP
7; — WC, (5.48)
T” (0) = T", in (E.483)
dTC -UP
TC (L) = To». (E49a)

The general solution of these differential equations is

171

-UP

TH(x) = C —Clx-t-C2 (E50)
1!

Tc(x) = Z-Cl—J-I—iClx-t-C3 (13.51)
c

where C2 and C3 are constants, giving a total of three constants to be deter-
mined. The initial conditions in Eq. (E48a) and (E49a) provide two condi-
tions and the third condition comes from Eq. (E25). Applying these three
conditions gives the following set of equations to be solved for the unknown
constants.

T”, M = C2 (5.52)
TC, in = -g—:+:C_ L+C3 (5.53)

The solution for C2 is obvious, solving fore the other constants gives

 

 

 

 

(Ta, 1.. - Tc 1..)
C1 " (1 +_ UPL (355)
Cc
UPL
TC,’ in + TH, in CC
C3 = (1+___UPL (E56)
CC
which can be rewritten using Eq. (E44) as
(TH, in - TC, in)
CI " 1+ NTU (EN)
Tc, in + T”. “N TU
C3 ' 1 + NTU (E'SB)

Substituting Eq. (E57) and (E58) into the general solutions, Eq. (E50) and
(E51), the expressions for temperature of the hot and cold ﬂuids are

_ UP (TH, in - TC, in)
CH(1+NTU)

 

711(1) = x + T”, in ([559)

1‘72

UP(THiu-TCIn) (TCin+TH,inNTU)
Tc(x) = — ’ ' x+ ’
CC(1+NTU) 1+NTU

If length dimensions are scaled according to Eq. (B.43), expressions for tem-
perature become

(E60)

 

 

NTU (Tu, :. - Tarn)
(1+ NTU)

NTU (TH, in ' Tc, in) x” + (Tc, in "' TH, inNTU)
(1+NTU) 1+NTU

 

TH (x) = - x” + T”, in (E.61)

TC (1:) = (5.62)

 

 

 

APPENDIX F

Program AXCOND

BOUNDRY CONDITIONS OF THE THIRD KIND
ORTHORGONALITY APPLIED WITH COSINE AND SINE FUNCTION

ASSUMING To = Tl (CONSTANT AMB TEMP)

LAST MODIFIED 4127/92

************

implicit double precision (a-h,o-z)
character para*10,runﬁl*60
integer size,size2
double precision len,Nc,Nh,Kw,Lstr,mu,NTU
parameter (size=200,size2=50)
dimension coeff(size,size+l),BC(size),A(size),
+ Tcold(size2),Thot(size2),Twall(size2,size2),
+ mu(size),alph(size),
+ chold(size2),dThot(size),dexxO(size2),dexx1(size2),
+ deyy0(size2),deyy1(size2)

common delta,len,Lstr,Nc,Nh,Bic,Bih,Bio,Bil,Tcin,Thin,To,'l‘l

it

** INPUT DATA

*1-

” delta - wall thickness (m)

** len - total lenth (m)

""" Cc - heat capacity cold stream (WIK)

** Ch - heat capacity hot stream (WIK)

*" hc - convection coeff cold side (W/m**2 K)

** hh - convection coeﬂ“ hot side (Wlm**2 K)

*"' Kw - thermal conductivity of wall (W/m K)

** ho - convection coeﬂ‘ to ambient at x=0 (W/m* *2 K)
** hl - convection coeﬂ‘ to ambient at x=L (Wlm**2 K)
** To - Temp of ambient at x=0 (C)

** Tl - Temp of ambient at x=L (C)

*"' Tcin - temp of cold ﬂuid at x=L

** Thin - temp of hot ﬂuid at x=0

** Lx(L) - scale length for x-direction

173

I”
13*
it
it
it
it
it
*4!
*II
it
II"!
it
*1!
I”
**
iii
IMF
ill
**
It'll
It'll

C

*I-I'i-‘I-‘I'i‘I-‘I-‘I-‘I-‘I‘I'i-‘I'I-M

*1-

174

Ly(delta)- scale length for y-direction

CALCULATED DATA

Lstr -Lley

Nc -thch/Cc

Nh -thth/Ch

Bio - ho Lx I Kw all variables dimensionless
Bil -h1Lx/Kw

Bic -thy/Kw

Bih -thy/Kw

read the input data in and deﬁne looping method
iopt - 0 -input raw data (dimensional)
1 -input Non-dimensional data

iloop = 0
irun = 1
continue

READ IN NEXT RUN FILE NAME AND BEGIN CALCULATIONS

if(irun.eq.l)runﬁl='[dowding.thesis.rundata]Bih.one'
if(irun.eq.2)runﬂl='[dowding.thesis.rundata]Bih.low'
if(irun.eq.3)runﬁl='[dowding.thesis.rundata]Bio.one'
if(irun.eq.4)runﬂl='[dowding.thesis.rundata]Bil.one'
if(irun.eq.5)runﬁl='[dowding.thesis.rundata]BihBic.one'
if(irun.eq.6)runﬂl='[dowding.thesis.rundata]BioBil.one'
if(irun.eq.7)runﬂl=’[dowding.thesis.rundata]Lstr.one'
if(irun.eq.l)runﬁl='[dowding.thesis.rundata]crl.low'
if(irun.eq.2)runﬂl='[dowding.thesis.rundata]cr75.low'
if(irun.eq.3)runﬁl=‘[dowding.thesis.rundata]cr50.low'
if(irun.eq.4)runﬁl='[dowding.thesis.rundata]cr25.low’
if(irun.eq.l2)runﬂl=’[dowding.thesis.rundata]nterm.one'
if(irun.eq.5)go to 300

print *,'running data set ',irun

open(20,status=‘unknown',ﬁle=runﬁl)
read(20,'(i10)')iopt
print *,'enter ioption O-single run l-vary parameter 2-stop'
I'ead("'.'(il)')iOPt

175

if(iopt.eq.2)go to 300

100 continue

*‘I'i'l‘l'

{ii-1"

**'I-*

if (iopteq.0)then
call input0(Cc,Ch,Kw,wid,nterrn,ibound,iﬂuid,iwall,ider,
deltax,deltay,istop)
elseif(iopt.eq. l)then
iloop=iloop + 1
call inputl (Cc,Ch,Kw,wid,ntenn,ibound,iﬂuid,iwall,ider,
deltax,deltay,istop,iloop,para,ipara,del,parmax)
endif
Nc = Bic*Kw*I.su*widICc
Nh = Bih*Kw*Lstr*wid/Ch
print *,' read in input'
if(istop.eq.l)go to 200

SOLVE FOR THE EIGENVALUES

print *,'going to calculate eigenvalues‘
SOLVE FOR ALPHA AND MU
call root(alph,size,Bio,Bil,nterm)
call root(mu,size,Bic,Bih,nterm)
if(alph(l).gt.3.l41592654)then
write(10,"')alph(l),' eigen troubles'
go to 300
endif
if(mu(l).gt.3.l41592654)then
write(10,*)mu(l),' eigen troubles'
go to 200
endif

print *,' calculated eigenvalues'
BUILD MATRDK OF UNKNOWN COEFFCIENTS

call build(mu,alph,coeﬂ',nterm,size,A)
print *,' built matrix'

SOLVE THE COEFF MATRIX FOR THE NEEDED CON STANT S

call gauss(coeﬂ‘,size,nterrn,BC)
print *,' solved matrix '

CALCULATE THE FLUID AND WALL TEMPERATURES

call proﬂl(BC.A,size,size2,nterrn,deltax,deltay,Thot,

176

+ Tcold.Twall,chold,dThot,mu.alph,d'I‘dxx0,dexx1,
+ deyyO,deyyl,qx0,qxl)
print ‘3' solved for temperatures '

 

CALCULATE THE EFFECI'IVNESS AND NUMBER OF TRANSFER UNITS
WITH AND WITHOUT AXIAL CONDUCTION INCLUDED

*§**§

n = nint(lldeltax) + 1
m = nint(lldeltay) + 1

if(Cc.le.Ch)then
Cmin = Cc
Cr = Cc/Ch
else
Cmin = Ch
Cr = Cthc
endif

 

NTU = (Bic*Bih"Kw*Lsu'*wid)/(Cmin*(Bih+Bic*Bih+Bic))
effc = Cc*(Tcold(1) - Tcold(n))/(Cmin*(’l'hot(l)-Tcold(n)))
effh = Ch*(Thot(l) - 'Ihot(n))l(Cmin*('Ihot(1)-Tcold(n)))

* write(*,*)'NTU ',N'IU

if(Cr.eq.1)then
eff = NTU/(1.0 + NTU)
else
eff = ( 1-exp(-NTU*(l-Cr)))/(1-Cr*exp(-NTU*(l-Cr)))
endif
print *,' calculated Effect and NTU '

CALCULATE THE HEAT FLUXED AND INSURE THAT ENERGY IS CONSERVED

*****

qhot = Ch "' (Thot(l) - Thot(n))
qcold = Cc * (Tcold(l) - Tcold(n))

qu = -wid"‘Kw/Lstr‘qx0
qxl = -wid*Kw/Lstr'"qxl

*write(10,*)'heat ﬂuxes'
*write(10,*)'qx0 =',qx0
*write(10,*)'qxl =',qxl
*write(10,*)'qhot =',qhot
I"write(10,"')'qcold =',qcold

**§***

4:

177

qtot = (qhot - qcold + qu - qxl)/(rnin(qhot,qcold))
print ‘, ' calculated heat ﬂuxes '

GENERATE THE OUTPUT FILE OF THE TEMPERATURE DISTRIBUTIONS OF THE
FLUIDS AND WALL AND CHECK THAT THE BOUNDRY CONDITIONS ARE MET

if(iopt.eq.0)then

call output(deltax,deltay,size2,dexxO,dexx1,deyyO,
+ deyyl,Thot,Tcold,Twall,iwall,iﬂuid,ibound,
+ idenponplxpoymly)

call out0(NTU,eﬂ',eﬂ'c,eﬂh,nterm,Kw,Cc,Ch,qtot,p0x,p1x,
+ P0y.ply)

else

call output(deltax,deltay,size2,dexxO,dexx1,deyy0,
+ deyyl,ThoLTcold,Twall,iwall,iﬂuid,ibound,
+ ider,p0x,p1x,p0y,ply)

call out1(NTU ,eﬂ'pﬂ‘c,eﬂh,nterm,Kw,Cc,Ch,para,qtot,p0x,p1x,
+ p0y,ply,iloop,ipara)

endif

print *, ' Generated output ﬁles '

if(iopt.eq.0)close(10)

goto 100

200 continue

if (iopLeq. l)then
close(10)
close(40)
irun=irun+l
ismp=0
iloop=0

go to 50

endif

300 continue

a:
t
a.

Stop
end

DATE 3105/92

SUBROUTINE TO GENERATE THE OUTPUT FILE FOR OPTION 0

178

‘I'

- SINGLE DATA FILE OPTION VARYING SOME PARAMETER

*

subroutine out0(NTU,eﬂ,effc,eﬂh,nterm,Kw,Cc,Ch,qtot,
+ Pokplwomly)

implicit double precision(a-h,o-z)
double Precision NTU,Nc,Nh,len,Lstr,Kw

common deltaJen,I..str,Nc,Nh,Bic,Bih,Bio,Bil,Tcin,'I‘hin,To,Tl

1"}

WRITE THE NONDIM DATA AND INLET PARAMETERS USED IN CALCULATION

write(10,"')"
write(10,*)' Nondimensional Data'
write(lO,'(a,f12.5)')' L“ =',Lstr

 

write(lO,'(a,f12.5)')' Hot side(Bih) =',Bih
write(10,'(a,f12.5)')' Cold side(Bic) =',Bic
write(10,'(a,f 12.5)')' Wall End(Bio) =',Bio
write(10,'(a.f12.5)')' Wall End(Bil) =',Bil
write(10,‘(a,f12.5)')' 'ntu" hot (Nh) =',Nh
write(10,'(a,f12.5)')' 'nm" cold(Nc) =',Nc
write(10.*)' '
write(10,*)' Fluid and Wall conditions '
write(10,'(a,fl 2.5,a)')' Wall Conductivity(Kw) =',Kw,’ W/mAZK'
write(10,'(a,f12.5,a)')' Heat Cap cold (Cc) =',Ch,‘ W/K'
write(10,'(a,fl 2.5,a)')' Heat Cap hot (Ch) =',Cc,‘ W/K'
write(10,'(a,f12.5,a)')' Inlet Temp cold (Tcin) =',Tcin,‘ C'
write(10,'(a,f12.5,a)')' Inlet Temp hot (Thin) =',Thin,’ C'
write(10,*)' '
write(10,*)'Ambient Temperatures '
write(10,'(a.f 12.5)')' At x=L (Tl) =',Tl
write(10,'(a,f12.5)')' At x=0 (To) =',To
write(10,"‘)"
write(10,'(a,iZ,a)')' Summations terminated at ',nterm,' terms'
write(10,"')"
write(lO,'(l7x,a,5x,a,4x,a,4x,a)')'I-lot',’Cold',
+ 'Negl', 'Eng Boundry Condition err Bi'
write(l0,'(5x,a,8x,a,6x,a,5x,a,5x,a)')'NTU','Eﬂ‘,'Eﬂ‘,'Eﬂ‘,
+ 'Bal Bio Bil Bih Bic'
write(10,'(2x,t7.4,2x,3f9.4,lx,e8.2,4f7.4)')NTU,effh,
+ eﬁc,eﬂ’,qtot,p0x,plx,p0y,ply

return
end

* DATE 3/05/92

i-I-‘I'ﬁi

*

+

4..

+

179

SUBROUTINE TO GENERATE THE OUTPUT FILE FOR OPTION 1
- OPTION VARYING SOME PARAMETER

subroutine outl(NTU,eﬂ‘,eﬂ'c,eﬂ‘h,nterm,Kw,Cc,Ch,para,qtot,
pOx.p1x.pOy.ply.iloop.ipara)

implicit double precision(a-h,o-z)

double Precision NTU,N'IUlam.Nc,Nh,len,Lstr,Kw

character para" 10

common delta,len,Lstr,Nc,Nh,Bic,Bih,Bio,Bil,Tcin,Thin,To,Tl
WRITE THE NONDIM DATA AND INLET PARAMETERS USED IN CALCULATION
if(iloop.eq.1)then

write(10,*)' '
write(lO,"')' Nondimensional Data'
write(10,'(a,f12.5)')' L“ =',Lstr
write(10,'(a,f12.5)')' Hot side(Bih) =',Bih
write(10,'(a,f12.5)')' Cold side(Bic) =',Bic
write(10,'(a,f 12.5)')' Wall End(Bio) =',Bio
write(10,'(a,f12.5)')' Wall End(Bil) =',Bil
write( lO,'(a.f 12.5)')' "ntu" hot (Nh) =',Nh
write(10,'(a,f12.5)')' ”ntu" cold(Nc) =',Nc
write(10.*)' '
write(10,*)' Fluid and Wall conditions '
write(10,'(a,f12.5,a)')' Wall Conductivity(Kw) =‘,Kw,‘ W/m"2K'
write(10,'(a,f12.5,a)')' Heat Cap cold (Cc) =',Cc,’ W/K'
write(10,'(a,fl 2.5,a)')' Heat Cap hot (Ch) =',Ch,‘ W/K'
write(10,'(a,fl 2.5,a)')' Inlet Temp cold (Tcin) =',Tcin,’ C'
write(10,'(a,f12.5,a)')' Inlet Temp hot (Thin) =',Thin,‘ C‘
write(lO,‘)' '
write(lO,‘)’Ambient Temperatures '
write(10,'(a,f12.5)')' At x=L (Tl) =',Tl
write(10,'(a,f 12.5)')' At x=O (To) =',To
write(lO,*)' '
write(10,'(a,i3,a)')' Summations terminated at ',nterm,' terms'
write(10,*)' '
write(10.'(a.a)')' varying parameter 29m
write(10,*)' '
write(lO,'(3x,a9,l6x,a,5x,a,4x,a,4x,a)')para,'Hot','Cold',

'Negl', 'Eng Boundry Condition'
write(lO,'(l9x,a,6x,a,5x,a,5x,a,5x,a)')'NTU',

'Eﬂ‘, 'Eﬂ‘,'Eﬁ‘,'Bal max error'

endif

 

180

BCerr = max(p0x,plx,p0y,p1y)

uneff = (eﬂ‘ - (eﬂ'h+eft‘c)l2)leff
Cmin = min(Ch,Cc)
Cmax = max(Ch,Cc)
NTUlam a Kw*Lstr/Cmin
if(iparaeq.l)then
write(10,'(3x,f7.2,5x,f8.3,1x,3f8.5,lx,e8.2,3x.f7.5)')
+ Isu',NTU,eﬂh,eﬂ'c,eff,qtot,BCerr
write(40,'(lx,f10.3,lx,f4.2,2f10.6,2x,4f9.5)')Lstr,Cmin/Cmax,
+ Cmin,NTU,eﬂh,eﬂ'c,eff,uneﬂ‘
return

elseif(ipara.eq.2)then

write(10,'(3x,f7.4,5x,f8.3,lx,3f8.5,1x,eB.2,3x,f7.5)')Bih,NTU,
+ eﬂh,effc,eﬂ',qtot,BCerr

write(40,'(1x,1f 10.6,lx,f4.2,2f10.6,2x,4f9.5)')Bih,Cmin/Cmax,
+ Cmin,NTU,eﬂh,effc,eff,uneff

return

elseif(ipara.eq.3)then

write(10,'(3x,f7.4,5x.f8.3,lx,3f8.5,1x38.2,3x.f7.5)')Bic,NTU,
+ eﬂ'h,eﬂ'c,eff,qtot,BCerr

 

write(40,'(lx,f 10.6,1x,f4.2,2f10.6,2x,4f9.5)')Bic,Cmin/Cmax,
+ Cmin,NTU,effh,eﬂ'c,eﬂ‘,uneff
return
elseif(ipara.eq.4)then
write(10,'(3x,f7.4,5x,f8.3,lx,3f8.5,lx,e8.2,3x,f7.5)')Bio,NTU,
+ effh,effc,eﬂ‘,qtot,BCerr
write(40,'(lx,f10.6,lx.f4.2,2f10.6,2x,4f9.5)')Bio,Cmin/Cmax,
+ Cmin,NTU,eﬂh,eﬂ'c,eff,uneff
return
elseif(ipara.eq.5)then
write(10,'(3x,f7.4,5x,f8.3,1x,3f8.5,1x,e8.2,3x,t7.5)')Bil,NTU,
+ eﬂh,effc,eﬂ',qtot,BCcrr
write(40,‘(1x,f10.6,1x,f4.2,2fl0.6,2x,4f9.5)')Bil,Cmm/Cmax,
+ Cmin,NTU,eﬂh,eﬂ’c,eff,uneff
return
elseif(ipara.eq.6.and.Ch.lt.Cc)then
write(10,'(3x,f7.4,5x,f8.3,lx,3f8.4,lx,e8.2,3x.t7.5)')Cthc,NTU,
+ eﬂ‘h,eﬂ'c,eff,qtot,BCerr
write(40,'(1x,f4.2,1x,2fl0.6,2x,4f9.5)')Cmin/Cmax,Cmin,NTU,eﬂh,
+ eﬂ'c,eff,uneﬂ‘
return
elseif(ipara.eq.6.and.Ccle.Ch)then
write(10,'(3x,f7.4,5x,f&3,1x,3f8.5,lx,e8.2,3x.f7.5)')Cc/Ch,NTU,

****

+

+

+

+

+

+

+

+

+

+

+

181

eﬂh,eﬂ'c,eff,qtot,BCerr

write(40,'(1x,f4.2,1x,2flO.6,2x,4f9.5)')Cmin/Cmax,Cmin,NTU,effh,
eﬂ'c,eff,uneﬂ‘

return

elseif(ipara.eq.16.and.Cc.le.Ch)then

write(10,'(f7.4.f8.2,f8.3,lx,3f8.5,lx,e8.2,3x,f7.5)')Cc/Ch,LsU,NTU,

eﬂh,eﬂ‘c,eff,qtot,BCerr

return

elseif(ipara.eq.l6.and.Ch.lt.Cc)then

write(10,'(t7.4,fB.2,f8.3,lx,3f8.5,lx,e8.2,3x,ﬂ.5)')Ch/Cc,lsu,NTU.
eﬂ‘h,effc,eﬂ‘,qtot,BCerr

return

elseif(ipara.eq.23)then

write(10,'(3x,t7.5,5x,f&3,lx,3f8.5,1x,e8.2,3x,f7.5)')Bih,NTU,
eﬂh,effc,eﬂ',qtot,BCerr

write(40,'(lx,2f8.5,lx,f4.2,lx,f8.4,lx,f$.2,4f8.5)')Bic,Bih,
Cmin/Cmax,Cmin,NTU,eﬂh,effc,eﬂ‘,uneﬂ‘

return

elseif(ipara.eq.45)then

write(10,'(lxﬂ.5,lx,f7.5,f8.3,1x,3f8.5,1x,e8.2,3x,t7.5)')Bio,
Bil,NTU,eﬂ'h,eﬂ'c,eﬂ',qtot,BCerr

write(40,'(1x,2f8.5,lx,f4.2,1x,f8.4,lx,f5.2,4f8.5)')Bio,Bil,
Cmin/Cmax,Cmin,NTU,eﬂh,eﬂ‘c,eﬂ',uneﬂ’

return

elseif(ipara.eq.8)then
write(lO,’(3x,iS,7x,f8.3,lx,3f8.5,1x,e8.2,3x,f7.4)')nterm,NTU,eﬂh,
eﬂ‘c,eff,qtot,BCerr
write(40,'(lx,3i,f10.6,2x,4f9.5)')N,NTU,eﬂ'h,
eﬂ'c,eff,uneff
return
endif
end

DATE 2/17/92
SUBROUTINE TO OPEN FILE AND READ IN RAW DATA
subroutine input0(Cc,Ch,Kw,wid,nterm ,ibound,iﬂuid,iwall,ider,

deltax,deltay,istop)

implicit double precision (a-h,o-z)
double precision len,Lx,Ly,Kw,Nc,Nh,Lsu'

 

 

182

character outﬂl‘60.inﬁl*60

common deltalenlsu',Nc.Nh,Bic,Bih,Bio,Bil,Tcin,Thin,To,Tl
print ‘,Enter the input ﬁle'

read(*.'(a)')inﬁl

print ‘,Enter the output ﬁle'

read(*.'(a)')outﬁl

CHECK IF EOF FLAG TO TERMINATE THE PROGRAM HAS BEEN READ IN

if(outﬂl.eq.'eof')then
close(20)

istop = 1

return

else

istop=0

endif

READ IN THE RAW DATA

open(10,status='new',ﬁle=outﬁl)
0pen(40,status='new',ﬂle=outﬁl)
open(50,status='new',ﬁlwoutﬁl)
open(30,status='unlrnown',ﬁle=inﬂl)

read(30,'(l7x,i20.0)')l..str
read(30,'(l7x,f20.0)')Bih
read(30,'(l7x,f20.0)')Bic
read(30,'(l7x,f20.0)')Bio
read(30,'(l7x,i20.0)')Bil
read(30,'(l7x,f20.0)')Kw
read(30,’(l7x,i20.0)')Ch
read(30,'(l7x,f20.0)')Cc
read(30,'(l7x,f20.0)')'lhin
read(30,'(l7x,f20.0)')Tcin
read(30,'(l7x,f20.0)')To
read(30,'(l7x,f20.0)')'1'l
read(30,'(l7xf20.0)')wid
read(30,'(l7x,i20)')nterm
read(30,'(l7x,i20)')ibound
read(30,'(l7x,iZO)')iﬂuid
mad(30,'(l7x520)')iwall
read(30,'(l7x,iZO)')ider
read(30,‘(l7x,i20.0)')deltax
read(30,'(l7x,i20.0)')deltay
close(30)

 

*****§

1.
at
a:

183

return
end

DATA 3105/92

SUBROUTINE TO OPEN FILE AND READ IN RAW DATA

subroutine inputl(Cc,Ch,Kw,wid,nterm,ibound,iﬂuid,iwall,ider,
+ deltax,deltay,istop,iloop,para,ipara,del,parmax)

implicit double precision (a-h,o-z)
double precision len,Lstr,Nc,Nh,Kw
character outﬁl*60,para*10,inﬁl*60

common deltaJeanu;Nc,Nh,Bic,Bih,Bio,Bil,Tcin,Thin,To,Tl

if(iloop.eq. l)then

READINPARAMETERTOVARYJNCREMENTANDMAXVALUETHENTHERAW
DATA FILE NAME AND PERTINENT DATA

print *,'enter the para to vary'
l'ead("‘.'(i10)')il>ﬁm
print *,'enter the del'
read(*,'(f20.0)')del
print*,'enter the max value'
read(*,'(f20.0)')parmax
print ‘,‘enter the input ﬁle name'
read(*.'(a)')inﬁl
print *,'enter the output ﬁle name'
read(*.'(a)')outﬁl

open(40,status='new',ﬁle=outﬂl)
Open(10,status='new',ﬁle=outﬁl)
open(30,status='unhiown',ﬁle=inﬁl)

read(30,'(l7x,f20.0)')Lstr
read(30,'(l7x,f20.0)')Bih
read(30,'(l7x,i20.0)')Bic
read(30,'(l7x,i20.0)')Bio
read(30,'(l7x,f20.0)')Bil
read(30,'(l7x,f20.0)')Kw
read(30,'(l7xﬂ0.0)')Ch
read(30,'(l7x,i20.0)')Cc

*i't‘l'

184

read(30,'(l7x,i20.0)')Thin
read(30,’(l7x,i20.0)')Tcin
read(30,'(l7x,i20.0)')To
read(30,'(l7x,i20.0)')Tl
read(30,'(l7x,i20.0)')wid
read(30,'(l7x,iZO)’)ntenn
read(30,'(l7x,i20)')ibound
read(30,'(l7x,i20)')iﬁuid
read(30,'(l7x,im)')iwall
read(30,'(l7x,i20)')ider
read(30,'(l7x,f20.0)')deltax
read(30,'(l7x,f20.0)')deltay

close(20)
close(30)

 

OTHERWISE INCREMENT THE PARAMETER THAT IS BEING VARIED AND SEND
DATA BACK TO PROGRAM

endif
if(iparaeq.l)then
para = 'Lstr'
if (iloop.eq. l)then
Cmin = min(Cc,Ch)
Cr = Cchh
ratio = Lstr/Cmin
else
Lsu' = Lstr + del
write(*,"')'Lstr ',Lstr
Cmin = Lstr/ratio
if(Cc.le.Ch)then
Cc = Cmin
Ch = Cc/Cr
else
Ch = Cmin
Cc = Cr " Ch
endif
endif
istop = 0
if(I.str:gt.parmax)istop=l
return
elseif(ipara.eq.2)then
para = 'Bih'
istop = 0
if(iloop.eq. l)then
Cmin = min(Cc,Ch)
Cr = Cchh

185

ratioi = Lstr/Cmin“(Bih"Bic*Kw*wid/
(Bih + Bic‘Bih + Bic))
else
Bih = Bih * del
ratio = Lstr *(Bih*Bic*Kw*widl
(Bih + Bic*Bih + Bic))
Cmin = ratio/ratioi
if(Cc.le.Ch)then
Cc = Cmin
Ch = Cc/Cr
else
Ch = Cmin
Cc = Cr " Ch
endif
endif
if(Bih.gt.parmax)istop=l
return
elseif(ipara.eq.3)then
para = 'Bic'
istop = 0
if(iloop.eq. l)then
Cmin = min(Cc,Ch)
Cr = Cc/Ch
ratioi = Lstr/Cmin*(Bih*Bic*Kw*widl
(Bih + Bic*Bih + Bic))
else
Bic = Bic * del
ratio = Lstr *(Bih*Bic*Kw*widl
(Bih + Bic‘Bih + Bic))
Cmin = ratio/ratioi
if(Cc.le.Ch)then
Cc = Cmin
Ch = Cc/Cr
else
Ch = Cmin
Cc = Cr * Ch
endif
endif
if(Bic.gt.parmax)istop=l
return
elseif(ipara.eq.4)then
para = 'Bio'
kmp=0
if(iloop.gt.1)Bio = Bio * del
if(Bio.gt.parmax)istop=l
return
elseif(ipara.eq.5)then

 

 

*********************

186

para = 'Bil'
istop = 0
if(iloop.gt.l)Bil = Bil * del
if(Bil.gt.parmax)istop=l
return
elseif(ipara.eq.6)then
para = ‘Cr'
istop = 0
Cr = Cthc
if(iloop.gt.l)Ch = Ch 4» del
if(iloop.gt.l)Cc = Cthr
if(Cc.gt.parmax)istop=l
return
elseif(ipara.eq. l6)then
para = ‘Cr & L*'
istop = 0
Cmax = max(Cc,Ch)
Cmin = min(Cc,Ch)
if(iloop.eq.1)then
ratio = Lsu'lCmin
deLsu' = del * (parmax-Lstr)
delCr = del "' ratio/(Cmax - Cmin)
write(*,*)'delCr delLstr =',delCr,deLstr
else
if(Ch.le.Cc)then
Lsu' = Lsu' + deLstr
Ch = Ch + delCr
else
Lstr = Lsu' + deLstr
Cc = Cc + delCr
endif
endif
if(Lsu'.gt.parmax)istop = 1
return
elseif(ipara.eq.23)then
para = 'Bih & Bic'
istop = 0
if(iloop.eq. l)then
Cmin = min(Cc,Ch)
Cr = Cc/Ch
ratioi = Lstr/Cmin*(Bih*Bic*Kw*wid/
(Bih + Bic‘Bih + Bic))

else
Bih = Bih + del
Bic = Bic + del
ratio = Lstr *(Bih*Bic*Kw*wid/

187

+ (Bih + Bic‘Bih + Bic))
Cmin = ratio/ratioi
if(Cc.le.Ch)then

Cc = Cmin
Ch = Cc/Cr
else
Ch = Cmin
Cc = Cr "' Ch
endif
endif
if(Bih.gt.parmax)istop = 1
return
elseif(ipara.eq.45)then
para = 'Bio & Bil'
istop = 0
if (iloop.eq. l)then
Bioi=Bi0
Bili=Bil
else
Bio = Bioi "‘ del
Bil = Bili "‘ del
Bioi = Bio
Bili = Bil
endif
if(Bio.gt.parmax)istop = 1
return
elseif(ipara.eq.8)then
para = 'N'
kmp=0
if(iloop.gt.l)nterm = nterm "' del
if(nterrn.gt.parmax)istop=l
return
endif

end

date 2109/92

SUBROUTINE TO BUILD THE MATRIX OF UNKNOWN COEFF

*i-ii-I'l’l-

subroutine build(mu,alph,coeﬂ',nterm,size,A)

implicit double precision (a-h,o-z)
integer size

 

{ii-I-

188

double precision 1en,Nc,Nh,mu,mulsq,rnul,Lstr,Ncsq,Nhsq,lam,musq
common deltalen,Lsu',Nc,Nh,Bic,Bih,Bio.Bil,Tcin,Thin,To,T1

dimension coeﬂ‘(size,size+l),mu(size),alph(size)
dimension A(size)

GENERATION OF THE MATRIX, ORDER WILL BE (2*NTERM X 2*NTERM +1)
SET CONSTANTS AND INITIALIZE THE PARAMETERS

pi = 1141592654

Ncsq = Nc * Nc
Nhsq = Nh "' Nh
rhsl = 0.0
rhs2 = 0.0
rhsB = 0.0
rhs4 = 0.0

exch = dexp(-Nc)

BEGIN COMPUTTIN MATRIX -i CORRESPONDS TO ROW NUMBER
do 10 i=1,nterm

mul = mu(i)*Lstr

mulsq = mul "' mul

musq = mu(i) * mu(i)

alpl = alph(i) I Lstr

alphsq = 81111110)“ alph(i)

alplsq = alpl * alpl

hsin = sinh(alpl)

hcos = cosh(alpl)

sine = sin(alph(i))

cosin = cos(alph(i))

zeta = (alpl*hsin + Bic‘hcos)l(Bic*hsin + alpl*hcos)

orth =((alphsq+Bio"'*2.0)*(1.+Bill(alphsq+Bil**2.))+Bio)l2.0
orthmu =((musq+Bih"2.0)*(l.0+Bicl(musq+Bic**2.0))+Bih)/2.0
lam = -(alpl*zeta + Bih) "' orth

if(Nh.lt.100)then
0mg = Bih/(Nhsq+alphsq)*((Nh+Bio)*alph(i)+dexp(-Nh)*
((alphsq-Nh‘BioY’sine - (Bio+Nh)*alph(i)*cosin))

else

omg = Bil1l(Nhsq+alphsq)*(Nh+Bio)*alph(i)
endif

 

 

189

cphi =((alplsq+Bic*Bi11)*hsin+(Bic+Bih)"'alpl*hcos)*orth

1}

ABSORBED EXP(-NC) INTO COMG FROM LOADING OF THE MATRIX

if(Nc.lt.100)then

comg = Bic/(Ncsq+alphsq)‘(exch*(Bio-Nc)*alph(i)+
+ ((alphsq+Bio*Nc)*sine + (Nc-Bio)*alph(i)*cosin))

else

comg = Bic/(Ncsq+alphsq)*((alphsq+Bio*Nc)*sine +
+ (Nc-Bio)*alph(i)"'cosin)

endif

I-

j CORRESPONDS TO COLUMN NUMBER

 

do 20 j=1,nterm

alphi = alph(i)
alpl = alph(i)/Lstr
alphsq = alph(i) " alph(i)
musq = mu(i) "' mUG)
mul = mu(j)*Lstr
mulsq = mul*mul
* hsinmu = sinh(mul)
"' hcosmu = cosh(mul)
hsinal = sinh(alpl)
hcosal = cosh(alpl)
sinal = sin(alph(i))
cosal = cos(alph(i))
sinmu = sin(mu(i))
cosmu = cos(mu(i))
chl = (Nh*Bio-mulsq)/(Nhsq-mulsq)
ch2 = (Nh*Bio+alphsq)/(Nhsq+alphsq)
ccl = (Nc‘Bio+mulsq)/(Ncsq-mulsq)
* cc2 = (Nc*Bio-alphsq)l(Ncsq+alphsq)

* CONSTANTS 01.62.03.04 AND cps HAVE A COSH FACTORED OUT OF THEN WHICH
"' CANCELS WITH THE COSH FACTORED OUT OF THE CONST AG) WHEN MULT

:- TOGET'HER WITH rho AND crho IN RHS CONSTANTS AS COMPARED TO SOLUTION

c1 = tanh(mul)*mul*cos(alphi)
+ + alphi*sin(alphi)
c2 = tanh(mul)*mul*sin(alphi)
+ - alphi‘cos(alphi) + alphi

 

 

++

190

c3 = mul*cos(alphi)

+ tanh(mul)*alphi*sin(alphi) - mul
c4 = mul‘sin(alphi)

- tanh(mul)*alphi*cos(alphi)

denoml = mulsq + alphi*alphi
zeta = (alpl‘hsinal + Bic*hcosal)l
(Bic‘hsinal + alpl*hcosal)

rhol = Bih*alphildenom1*musq*Lstr*(l-chl)*cl
rh02 = Bih‘Bio/denoml*musq*Lsu'*(1-chl)*c2
rh03 = Bih‘Nh/denoml*mu(i)*alphi*chl*c3
rho4 = Bih*Nh*Bio/denom l *mu(i)*chl *c4

crhol =Bic‘alphildenoml*mul*(l+ccl)*cl
crh02 =Bic*Bio/denom1*mul*(l+ccl)*c2
crho3 =Bic*Nc/denoml*alphi*ccl*c3
crho4 = Bic*Nc*Bio/denom1*ccl*c4

beta = (l-ch1)*musq*Lstr

8am = (l-ch2)‘alph(i)

eps = (l+ccl)*mul+ Nc*ccl*tanh(mul)

sig = (hcosal-zeta*hsinal)*((1+cc2)*alph(j)*cosal
+ Nc*cc2*sinal)

tau =(alpl*hcosal+Bih*hsinal)*((1+cc2)*alph(j)*cosal
+ Nc*cc2*sinal)

ammn = alphi - alph(i)
ampn = alphi + alph(i)
anmm = alph(i) - alphi
anpm = alph(i) + alphi

if(i.eq.j)then

psi = (.5+1/(4*alphi)*sin(2*alphi))*(l-ch2)*alphsq
+ ((Bio*(l-ch2) + Nh*ch2)l2) *(sin(alphi))**2
+ (.5-ll(4"'alphi)*sin(2*alphi))*Nh*Bio*ch2

cpsi = (.54-ll(4*alphi)*sin(2*alphi))*(1+cc2)*alphsq
+ ((Bio*(1+cc2) + Nc*cc2y2) *(sin(alphi))*"'2
+ (.5-1/(4‘alphi)*sin(2*alphi))*Nc‘Bio‘ch

 

*‘I-i-I'i-

191

else

sinlm = sin(ammn)/(2*ammn)
sinlp = sin(ampn)l(2"'ampn)
cos2m = cos(ammn)l(2*ammn)
cos2p = cos(ampn)/(2*ampn)
cos3m = cos(anmm)l(2*anmm)
cos3p = cos(anpm)l(2*anpm)

argl = sinlm + sinlp

arg2 = -cos2m - cosZp + l/(2*ammn) + 1/(2*ampn)
arg3 = -cos3m - cos3p + 1/(2*anmm) + ll(2*anpm)
arg4 = sinlm - sinlp

psi =arg1‘alph(j)*alphi*(1-ch2)
+ arg2‘alph(i)*(l-ch2)*Bio
+ arg3*ch2*Nh*alphi + arg4’ch2*Nh*Bio

cpsi =argl’alph(i)*alphi*(l+cc2)
+ arg2*a1ph(i)"'(l+cc2)"‘Bio
+ arg3‘cc2‘Nc‘alphi + arg4*cc2*Nc*Bio

endif

psiB = psi * Bih
psiC = psi * Bih * alpl

cpsiB = cpsi "' Bic * (hcosal - zeta "‘ hsinal)
cpsic = cpsi * Bic * (alpl‘hcosal + Bih*hsinal)

LOAD THE MATRIX, LOADING IS DONE IN QUARTERS(i.e.TOP LEFT,
top right, bottom left. ..... ) but concurrently

lST EQUATION LOADING - TOP LEFT AND TOP RIGHT
coeﬂ‘(i,j) = -omg*ga1n + psiB
coeff(iJ+nterm) = -omg*gam"'alpl + psiC

2ND EQUATION LOADING - BOTTOM LEFT AND BOTTOM RIGHT
coeff(i+nter1n,j) = comg‘sig - cpsiB
coeﬂ'(i+nterrn,j+ntenn) = comg*tau - cpsiC

if(i.eq.j)then
coeff(i,j) = coeﬂ'(i,i) + lam
coeff(i+nterrn,j+ntenn) = coeﬂ‘(i+nterm,j+ntenn) + cphi

192

endif

CONSTANT AG) HAS A COSH FACTURED OUT OF THE DENOMINATOR THAT
IS CANCELLED BY THE COSH FROM C1..C4 WIEN MULTIPLIED BY rho AND
crho IN TIE RHS CONSTANTS CALCULATIONS

*‘I'O‘I'ﬁ

if((Tl-To).ne.0)then

A0) = (Bil"(Tl-To)*(sin(mu(i))—Bih/mu(i)*cos(mu(i))
+ +Bihlmu(j))) l(orthmu"'
+ ((mulsq+Bil*Bio)*tanh(mul)+(Bil+Bio)*mul))

Aprm = A(j)*(mu(i)*cosmu + Bih’sinmu)

else

A0) = 0.0

Aprm = 0.0

endif

rhsl = (rhol+rho2+rho3+rho4)*A(i) + rhsl

rth = (crhol-+crho2+crh03+crho4)*Aprm + rhs2
rhs3 beta*A(j) + rhs3

rhs4 eps‘Aprm + rhs4

LOADING THE RHS CONST ANTS - DONE FOR EQUATION 1 THEN EQ 2

**'I-*

if(i.eq.nterm)then
coeﬂ‘(i,2"'nterm+1) = -rhsl + omg’rhs3 - omg*(Thin-To)

coeff(i+ntenn,2*ntenn+l)=rhs2 - comg*rhs4

+ + comg'CTcin-To)
endif

20 continue
rhsl = 0.0
rhs2 = 0.0
rhs3 = 0.0
rhs4 = 0.0

10 continue

return
end

DATE 51 12192

*I'i'i'

*I'I'i

Mia-{area-

193

SUBROUTINE TO CALCULATE EIGENVALUES FROM THE TRANSCENDENTAL
EQUATIONS. USING A SCHEME THAT MARCHES OUT UNTIL TIE FUNCTION
CHANGES SIGN THEN BACKSTEPS TO TIE ROOT

subroutine root(eigen,size,B1,B2,ntenn)

implicit double precision(a-h,o-z)

integer size

double precision Nc,Nh,len,Lsu'

common deltalenJAu',Nc,Nh,Bic,Bih,Bio,Bil,Tcin,Thin,To,Tl

dimension eigen(size)

func(x) = tan(x) - (x*(B l+B2))/(x**2-B1*B2)

irep = 0

delin = .5d+0

ndel = 2

pi = 4.0d+0*atan(1.0d+0)
eps = 1d-8

ep = 1d-10

del = delin/ndel

K = 0

arg =sqrt(Bl*B2)

idenom = 0

CHECK IF FUNCTION CHANGES SIGN DUE TO CHANGE OF SIGN IN DENOM-
INATOR OF THE FUNCTION. (X<B 132 ON TIE INTEVAL ) IDENOM
SIGNALS LEG WIERE DENOM CHANGES SIGN HAS BEEN PASSED

continue
if(arg.lt.(k+l)*pi .and. idenom.eq.0)then
if(arth.(2*(k+l).l)*pi/2)then
= 3’8 + 61)
del = ((2d+0"'(k+ld+0)-ld+0)*pi12d+0 - x)lnde1
ileg = 1
else
= (2d+0*(k+1d+0)—1d+0)*pi/2d+0 + ep
del = (arg -(2d+0*(k+1d+0)-1d+0)*pi/2d+0)/ndel
ileg = 1
endif
idenom = 1

else

194

IF DENOMINATDR DOESN'T CHANGE SIGN OD INTERVAL [K*P1,(K+1)*PI]
SEARCH FOR ROOT ON TIE PROPER LEG OF FUNCTION

**§*

ileg = 0
if(idenom.eq.0)then
x = (2d+0*(k+1d+0)—1d+0)*pil2d+0 4» ep
del = delin/ndel
else
x = k "' pi + ep
del = delin/ndel
endif
endif

10 continue
if(irep.gt.10)go to 100
fx = func(x)

 

* CIECK SIGN OF TIE FUNCTION

12 if(fx)13,13,15
l3 n=0

i=n

goto20
15 n=1

i=n

* INDEX X AND MARCH FORWARD

20 x=x+del
hx=func(x)

* CHECK SIGN AT NEXT STEP

if(hx)23.23.25
23 n=0
go to 30
25 n=l
*
"' CHECK IF SIGN CHANGED. IF NO SIGN CHANGE CONTINUE MARCHING
* IF CHANGED GO BACK AND TAKE A SMALLER STEP FORWARD.
1.:
30 if(i-n)40,20,40
ﬂ

* CIECK FOR CONVERGENCE

195

a:

40 if(eps-del)45,51,51
45 x = x - del
del = del/2d+0
go to 20
51 k = k + l
eigen(k)=x

* CIECK E EIGENVALUE WAS MISSED _ E 80 DECREASE STEP SIZE AND REPEAT

if(lr.gt.l)then
if((eigen(k)—eigen(k-l)).gt.pi)then
ndel = 2*ndel
ep = ep/2.0d+0
k = k—l
if(ileg.eq.l)idenom=0
irep = irep + 1
go to 5
endif
else
if(eigen(k).gt.pi)then
ndel = 2*ndel
ep = ep 1 2.0d+0
k = k-l
idenom = 0
irep = irep + 1
go to 5
endif
endif
up
* CHECK IF NEED TO COMPUTE MORE ROOTS
4.
if(k-nterm)5,60,60
100 continue
write(10,*)'eigentroubles; irep =',irep
write(10,*)'cigen funct =',x,fx

60 continue
return
end
*
* date 2109/92
:1:
* SOLVES THE MATRIX USING A PARTIAL PIVOTING SCHEME
*

SUBROUTINE GAUSS(A,SIZE,N,X)

-I-

"' A - SQUARE MATRIX TO SOLVE WIHT ADDITIONAL RHS CONSTANT S

196

N - NUMBER OF TERMS IN THE SERIES (A IS ORDER (2N X 2N+l)
SIZE - DECLARED MAX SIZE OF A IN MAIN PROGRAM
X - SOLUTION

{*I'f'l-

implicit double precision (a—h,o-z)
double precision MULT

INTEGER SIZEPIVOT

DIMENSION A(SIZE,SIZE+1),X(SIZE)

N=2*N

I-

GAUSSIAN ELIMINATION USING PARTIAL PIVOTING
DO 701=l.N

IF(A(I,I).EQ.0) THEN
PIVOT=0
J=J+l
30 IF((PIVOT.EQ.0). AND .(J.LE.N)) THEN

IF(A(J,I).NEO) PIVOT = J
J=J+l
GO TO 30

ENDIF

IF(PIVO‘T.EQ.0) THEN
STOP 'SINGULAR MATRIX'

ELSE

t

* INTERCHANGE ROWS
a:
DO 40 J=1,N+1
TEMP = A0,!)
A0,!) = A(PIVOTJ)
A(PIVOT‘J) = TEMP
40 CONTINUE
ENDIF
ENDIF

up

* ELIMINATE TIE ITH UNKNOWN

*
DO 60 I: I + l, N
MULT = -A(I,I)IA(I,I)

DO 50 K=I,N+l
A(I,K) = A(I,K) + MULT " A(I,K)

50

60

70

up
a:
:1.

80

*‘lr‘I‘i‘I-‘I'

197

CONTINUE
CONTINUE

CONTINUE

FIND SOLUTION

X(N) = A(N.N+l)/A(N.N)

DO 90 J=N-1,1,-1
X(J) = A(J,N+1)
DO 80 K = J+l,N
X0) = X0) - A(I,K) * X(K)
CONTINUE
X0) = XU) / A(JJ)
CONTINUE
N=N12
RETURN
END

date 2116/92

SUB ROUTINE TO CALCULATE TI-IE FLUID TEMPERATURES
AND WALL TEMPERATURES ALONG WITH TIEIR DERIVATIVES

subroutine proﬁl(BC,A,size,size2,nterm,deltax,deltay,
Thot,Tcold,T\vall,chold,dThot,mu,alph,dexx0,

dexx 1 ,deyy0,deyy 1 ,qx0,qx l)

implicit double precision (a-h,o-z)

double precision len,Nc,Nh,mu,musq,mulsq,mulx,Ncsq,Nhsq,Lstr,

muy,mul
integer size,size2

common delta,len,Lstr,Nc,Nh,Bic,Bih,Bio,Bil,Tcin,Thin,To,Tl

dimension BC(size)A(size),Tcold(size2),Thot(size2),

+ d'Tcold(size2),dThot(size2),TWall(size2,size2),

+ mu(size).alph(size),dT dxx0(size2),dT dxx1(size2),deyy0(size2),
+ deyy1(size2)

pi = 11415926535

*‘I-‘I-‘I-

198

n = nint( lldeltax) + 1
m = nint( lldeltay) + l
Nhsq = Nh * Nh
Ncsq = Nc * Nc

CALCULATING FLUID TEMPERATURES
NOTE: ONLY A FUNCTION OF X
do 10 i=l,n

sumhl =0.0
sumh2 = 0.0
sumcl =0.0
sumc2 = 0.0
sumdhl = 0.0
sumdh2 = 0.0
sumdcl =0.0
sumdc2 = 0.0

if(i.eq.l)sumD1=0.0
if(i.eq.l)sumEl=0.0

do 20 j=1,nterrn

mul = mu(i)*I.str

musq = mu(i)*mu(i)

mulsq = mul*mul

mulx = mu(i)*Lstr*(i-1)*deltax
alphl = alph(i)/Lsu'

alphx = alph(j)*(i-1)"‘deltax
alphsq = alph(i)*alph(i)

chl = (Nh*Bio-mulsq)l(Nhsq-mulsq)
0112 = (Nh*Bi0+ﬂPhsq)/(Nhsq+alph8®
col = (Nc*Bio+mulsq)/(Ncsq-mulsq)
cc2 = (Nc*Bio-alphsq)/(Ncsq+alphsq)

zeta = (alphl‘sinh(alphl)+Bic*cosh(alphl))/
(Bic*sinh(alphl) + alphl‘cosh(alphl))

if(AG).ne.0)then

Apr = A(i)lcosh(mul)*(mu(i)*cos(mu(i))+Bih*sin(mu(i)))

else

Apr = 0.0

endif

Bpr = BC(i)*(cosh(a1phl)-zeta*sinh(alphl))

Cpr = BC(i+nterm)*(alphl‘cosh(a1phl)+Bih*sinh(alph1))

 

199

if(i.eq.l)then

if(A(i).ne.0)then

sule = A(i)/cosh(mu1)*musq*Lsu'*(l—chl)+(BC(i)+
BC(j+nterrn)"'alphl)"(1-ch2)*alph(i) + st1

else

st1 = (BC(i)+
BC(i+nterrn)*alphl)*(l-ch2)"'alph(i) + st1

endif

if(AG).ne.0)then

sumEl = Apr*(mul*(1+cc1)*cosh(mul)+Nc*cc1*sinh(mul))
+ (Bpr+Cpr)*((l+cc2)*alph(i)*cos(alph(i))
+ Nc‘cc2‘sin(alph(i))) + sumEl

else

sumEl = (Bpr+Cpr)*((l+c02)"‘a1ph(i)*cos(alph(j))
+ Nc‘cc2*sin(alph(i))) + surnEl

endif

endif
if(A(i).ne.0)then
sumhl = A(i)lcosh(mul)*((l-ch1)*musq*Lsu'*cosh(mulx)
+Nh*mu(i)"'chl"sinh(mulx)) + sumhl

else

sumhl = 0.0

endif

hot2 = (l-ch2)*alph(i)*cos(alphx)

+ Nh‘ch2‘sin(alphx)
sumh2 = (BCG)+BC(i+nte1m)*alphl)*hot2 + sumh2

if(A(j).ne.0)then

sumcl =Apr‘((l+ccl)*1nul*cosh(mulx)
+ Nc*cc1*sinh(mulx)) + sumcl

else

sumcl = 0.0

endif

sumc2 = ((1+cc2)*alph(i)*cos(alphx)+Nc‘cc2*sin(alphx))
*(Bpr + Cpr) + sumc2

if(AG).ne.0)then

sumdhl = A(j)/cosh(rnul)*((1-ch1)*musq*mul*Lstr
*sinh(mulx) + Nh*musq"'Lsu'*chl*cosh(mulx)) +sumdhl

else

sumdhl = 0.0

endif

dhot2 =(-1+ch2)*alphsq*sin(alphx)

20

*******

200

+ Nh’ch2‘alph(j)*cos(alphx)
sumdh2 = (BC(i)+BC(i+nterrn)*alphl)*dhot2 + sumdh2

if(AG).ne.0)then

sumdcl = Apr"((l+cc l)*mulsq*sinh(mulx)
+Nc*ccl*mul*cosh(mulx)) + sumdcl

else

sumdcl = 0.0

endif

sumdc2 = (-(cc2+l)*alphsq*sin(alphx)
+Nc*cc2*alph(i)*cos(alphx))*(Bpr+Cpr)+sumdc2

continue

if(i.eq.l)then

D1 = (Thin-To) - sule
E1 = (Tcin-To-sumEl)
endif

if((Nh*(i-l)*de1tax).lt.100)then

Thot(i) = D1*Dexp(-Nh*(i-1)*deltax) + To + sumhl + sumh2
dThot(i)= -D1*Nh*dexp(-Nh*(i-1)*deltax) +sumdh1 +sumdh2
else

Thot(i) = To + sumhl + sumh2

dThot(i)= sumdhl + sumdh2

endif

if((-Nc*(l-(i-l)*deltax)).lt.100)then

Tcold(i)= E1A*Dexp(-Nc*(l-(i-1)*deltax)) + To + sumcl + sumc2
chold(i) = E1*Nc*Dexp(-Nc*(1-(i-1)*deltax)) + sumdcl + sumdc2
else

Tcold(i)= To + sumcl + sumc2

chold(i) = sumdcl + sumdc2

endif

continue

COMPUTE TEMPERATURES IN THE WALL
SUMWl-THETAl SUMW2-THETA2 SUMW3-THETA3
TWALL=THETAI +THETA2 +THETA3 +To

if(iwall.eq.0)go to 100

201

dex 1=o.o
dex2=0.0
dex3=0.0
dey 1=o.o
dey2=0.0
dey3=0.0
Q1 =o.o
Q2 =o.o
Q3 =o.o

i X—DIRECTION MARCHING

do 30 i=1,n

j Y-DIRECTION MARCHING
do 40 j=1,m

sumwl = 0.0
sumw2 = 0.0
sumw3 = 0.0

k SUMMATION OVER THE N-TERMS

do 50 k=l,nterrn

alphl = alph(k)/Lsu'

mul = mu(k)*Lstr

mulx = mul‘(i-l)*deltax

alphx = alph(k)*(i-l)*deltax

muy = mu(k)*(i-1)*deltay

alphy = alphl*(i-l)"'deltay

zeta =(alphl*sinh(alphl)+Bic*cosh(alphl))

/(Bic*sinh(alphl)+alphl*cosh(alphl))

if(A(j).ne.0)then

sumwl = A(k)/cosh(mul)*(mul‘cosh(mulx)+Bio

*sinh(mulx))*(mu(k)*cos(muy)+Bih*sin(muy))+ sumwl

else
sumwl = 0.0
endif

sumw2 = BC(k)*(cosh(alphy)-zeta*sinh(alphy))*
(alph(k) *cos(alphx)+Bio*sin(alphx)) + sumw2

sumw3 = BC(k+nter1n)"'(alphl*cosh(alphy)+Bih
*sinh(alphy))*(alph(k)*cos(alphx)+Bio"'
sin(alphx)) + sumw3

202

if(i.eq.l.or.i.eq.n)then
if(A(j).ne.0)then
dex1 = A(k)lcosh(mul)*mul*(mul*sinh(rnulx)+Bio*
cosh(mulx))*(mu(k)*cos(rnuy)+Bih*sin(muy)H dexl
else
dex1 = 0.0
endif

dex2 = BC(k)*(cosh(alphy)—zeta*sinh(alphy))*
(~alph(k)*sin(alphx)+Bio*cos(alphx))*alph(k)
+ dex2

dex3 = BC(lr+nterm)*(alphl*cosh(alphy)1-Bih
*sinh(11191110)*(-alph(k)*sin(alphX)+Bi0*
ms(alphx))*alph(k) + dex3
if(A(j).ne.0)then
Q1 = A(k)/cosh(mul)*mul*(mul"‘sinh(mulx)+Bio*
cosh(mulx))*(sin(mu(k))-Bih/mu(k)*cos(mu(k))+
Bih/mu(k» «1» Q1
else
Q1 = 0.0
endif

Q2 = BC(k)*Lsu"(sinh(alphl)-zeta*cosh(alphl)+zeta)*
(~a1ph(k)"'sin(a1phx)+Bio*cos(alphx))
+ Q2

Q3 = BC(k+nterrn)*Lstr*(alphl*sinh(alphl)+Bih
*cosh(a1phl) -Bih)*(-alph(k)*sin(alphx)+Bio*
cos(alphx)) + Q3

endif

if(i.eq.l.or.j.eq.m)then

if(A(i).ne.0)then

dey1 = A(k)lcosh(mul)*(mul‘cosh(mulx)+Bio*
sinh(mulx))*(-mu(k)*sin(muy)+Bih*cos(muy))*mu(k) + deyl
else

dey1 = 0.0

endif

dey2 = BC(k)"‘alphl"(sinh(alphy)-zeta*cosh(alphy))*
(alph(k)*cos(alphx)+Bio*sin(alphx)) + dey2

dey3 = BC(k+nterm)*(alphl*sinh(alphy)+Bih
*cosh(alphy))*alphl*(alph(lr)"cos(alphx)+Bio

 

50

40

203

*sin(alphx)) + dey3

endif

continue
Twall(i,j) = sumwl + sumw2 + sumw3 + To

if(i.eq.l)then

dexx0(i)= dex1+dex2+dT dx3
dex l=0.0

dex2=0.0

dex3=0.0

qx0 =Ql +Q2+Q3

Q1 = 0.0

Q2 = 0.0

Q3 = 0.0

endif

if(i.eq.n)then

dexle)= dexl+dex2+dex3
dex1=0.0

dex2=0.0

dex3=0.0

qxl =Q1+Q2+Q3

Q1 = 0.0

Q2 = 0.0

Q3 = 0.0

endif

if(i.eq.l)then
deyy0(i)=dey1+dey2+dey3
dey 1 =0.0

dey2=0.0

dey3=0.0

endif

if(i.eq.m)then

deyyl(i)= deyl+dey2+dey3
deyl=0.0

dey2=0.0

dey3=0.0

endif

continue

204

30 continue
100 continue
return

end

date 3105/92

SUBROUTINE TO GENERATE TIE OUTPUT OF TIE RUN

****‘I-*

subroutine output(deltax,deltay,size2,dexx0,dT‘dxx1,deyyO,
+ deyyl,Thot,Tcold,T\vall,iwall,iﬂuid,ibound,
+ ider,p0x,plx,p0y,ply)
implicit double precision(a-h,o-z)
integer size2
double precision Nc ,Nh,len,Lstr
common deltaJen,Lsu',Nc,Nh,Bic.Bih,Bio,Bil,Tcin,Thin,To,Tl
dimension dexx0(size2),dexxl(size2),deyy0(size2),
+ deyyl(size2),Thot(size2),Tcold(size2),TWall(size2,size2)

n = nint( lldeltax) + 1
m = nint( lldeltay) + l
k = m15

if(iﬂuid.eq.0) go to 100

WRITE OUT THE FLUID TEMPERATURES - E IFLUID=1

*i-‘I'I-

write(10,*)' '
write(lO,‘)' '
do 10 i=1,n
x=(i-1)*deltax

if(i.gt.1) go to 15
write(10,"')' x"‘ Tc" Th"l '
15 write(10,'(f4.2,3x,2f15.8)')x,Tcold(i),
+ Thot(i)
write(40,'(f4.2,3x,2fl5.8)')x,(Tcold(i)-Tcin)/(Thin-Tcin),
+ (Thot(i)-Tcinyahin-Tcin)

10 continue

write(10,*)' '
write(10,*)' '

205

100 continue

if(iwall.eq.0)go to 200
* WRITEOUTT‘HEWALLTEMPERATURE- IFIWALL=1
*

write(10,‘)'Wall temperatutes '

do 20 1:1,]:

y=(i-l)*deltay*5

memory y*'

write(10,'(a,5f12.3)')' x“,y,y+deltay,y+2"‘deltay,y+3*deltay,

+ y+4*deltay

do 25 ii=1,n
x=(ii-l)*deltax
write(10,'(lx.fS.2,5fl2.7)')x,((Twall(ii,i+(k-1)*5)-Tcin)
+ lClIrin-Tcin),j=l,5)
write(50,'(1x55.2,5f12.7)')x,(('1\vall(ii,j+(k-1)*5)-Tcin)
+ I(Thin-Tcin),i=1,5)

25 continue
write(10,*)"

20 continue

200 continue

* PRINT THE DERIVATIVES AT THE WALL ENDS. IF IDER=1
if(ider.eq.0)go to 250

write(10,*)' '

write(10,*)'dTw1dx*(x"‘=0)'

write(10,*)' '

x=0.0

write(10,'(1x,f4.2,3x,6f1 1.6)‘)x,(dexx0(i),i=1,1n)
write(lO,‘)' '

write(10,*)' '

write(10,*)'dT\v/dx*(x*=l) '

write(10,*)' '

x=(n-l)*deltax

write(10,'( 1x,f4.2,3x,6f1 l.6)')x,(dexle),j=1,m)
write(10,‘)' '

250 continue

if(ibound.eq.0)go to 300

:11
*

 

iii-{*‘Ir'l’

30

206

CIECK BOUNDRY CONDITIONS- E IBOUND= 1 OR 2

dT/dx"(0.y"') = Bio ('I‘W(0.y"') - T0)
-dl‘ldx"'(l.y”) = BiL (“(1.1") - TL)
-dT/dy"'(x"',0) = Bih (11101“) - TW(X"'.0))
-dT/dy"'(x"'.1) = Bic ('I\v(x*,l) - TC(X"'))

dif 1m = 0.0
dif2m = 0.0
dif3rn = 0.0
dif4m = 0.0

if(ibound.eq.2)then

write(10,*) ' '

write(10,*) ' '

write(10,*)‘ Check of Boundry Conditions'

write(10,*)' In the Y-diretion BCy0 - y*=0 BCyl - y*=1'
write(10,‘)' x" BCy0 BCyl '

endif

do 30 i=2,n-1

x=(i-l)‘deltax

difl = -deyy0(i)/('1'hot(i)-Twall(i,l))

difZ = -deyyl(i)/('Nall(i,m)—Tcold(i))

if(abs(dif1-Bih).ge.ahs(dif1m))then
dif1m=dif1-Bih
iml = 1

endif

if(abs(dif2-Bic).ge.abs(dit2m))then
dif2m=dif2-Bic
i1n2 = i

endif

if(ibound.eq.2)write(10,'(1x,f4.2,3x,2f1 8.9)')x,dif 1 ,dit2

continue

if(ibound.eq.2)then

write(10,*) ' '

write(10,*) ' '

write(10,*)' In the X-diretion BCx0 - x*=0 BCxl - x*=l'
write(10,*)' y"' BCx0 BCxl '

endif

do 40 j=2,m-1
FG-ll‘deltay
difl = d'dex0(i)l('TWall(l,i)-To)
W = -dexx1(i)/Wall(nJ)-Tl)

207

if(abs(dif1-Bio).ge.abs(dif31n))then
dif3m=difl-Bio
im3 =1'
endif
if(abs(dif2-Bil).ge.abs(dif4m))then
dif4m=dif2~Bil
im4 =J'
endif
if(ibound.eq.2)write(10,'(1x,f4.2,3x,2f18.9)')y.dif1,difZ
40 continue

p0y = diflm/Bih
ply = dif2mlBic
p0x = dif3m/Bio
plx = dif4m/Bil

if(ibound.eq. l)then

write(10,*)"

write(10,*)"

write(10,*)'Max Errors in the boundry conditions and locations '
write(10,'(a,f5.2,f 12.6)')'x* = 0 y" = ',(jm3-l)*de1tay,p0x
write(10,'(a,1'5.2,f 12.6)')'x* = 1 y" = ',(im4-l)*deltay,plx
write(10,'(a,f5.2,f 12.6)')'y* = 0 x" = ',(iml-l)*deltax,p0y
write(10,'(a,f5.2,f12.6)')'y“' = 1 x"' = ',(im2-l)*deltax,ply

endif

*********

300 continue
retum
end

"17111111111111111115