HEAT TRANSFER DURING PNEUMATIC
TRANSPORT 0F SPRAY- DRIED
FOOD PRODUCTS

Thesis for the Degree of Ph; D.
MiCHéGAN STATE UNEVERSETY.
PlE-YI WANG ‘
1970

LIBRAR Y
MlChlgan S i, 5.;
University I

    

This is to certify that the

thesis entitled

HEAT TRANSFER DURING PNEUMATIC TRANSPORT
OF SPRAY-DRIED FOOD PRODUCTS

presented by

Pie-yi ‘Wang

has been accepted towards fulfillment
of the requirements for

Ph.D. Agr. Eng.

degree in

:91 K. HJLLMM

Major professor

Date February 26, 1970

 

O~169

Y BINDING BY ‘
‘ "DAB & SUNS'
800K BINDERY INC.

s IWJI I

 

 

 

 

ABSTRACT

HEAT TRANSFER DURING PNEUMATIC TRANSPORT
OF SPRAY-DRIED FOOD PRODUCTS

By

Pie-yi Wang

Since spray drying improves the storage of products, reduces
the cost of transportation, and provides a product which can be served
as a convenience-type food, it is used extensively to manufacture
several food products. Dry milk probably is the most widely known
spray-dried food product; non-fat dry milk was therefore used as the
representative test material. Most of the spray-dried products are
subjected to cooling after leaving the drying chamber. In order to
effectively control the product temperature, the investigation of
heat transfer rate during the processing was needed.

In order to predict the temperatures of air and particles, the
equations describing the temperatures of air and product particles
have been developed based on the enthalpy balance of the system with
three different wall conditions; uniform wall temperature, linear
wall temperature, and uniform heat flux. Since the mixture heat trans-
fer coefficient was employed to develop the governing equations, the
mixture Nusselt number was experimentally determined in order to
evaluate the derived equations.

Only the uniform wall temperature experiments were conducted
in this investigation. In the food industry, the spray-dried food

products should be cooled immediately after entrance into the air

Pie-yi Wang

stream. The average mixutre Nusselt number, therefore, was experimen-
tally determined at the entrance region. In addition, the mixture
Nusselt number was determined for two situations; equal and different
inlet temperatures of product particles and air. The experimental
results illustrated that there is no significant difference in heat
transfer rate between equal and different inlet temperatures. For
this reason, the experimental results of both cases were combined

to establish the general correlation. The experimental results were
correlated to be:

G

c
= 0.61 0.75
(1mm 0.17 (Re)a (1 + -P-—BGaca) (i 77.)

This empirical equation is valid for convective capacity parameter

greater than unity and within the entrance region with a uniform

wall temperature. With the aid of this empirical equation, the temper-

atures of air and particles can be predicted from the derived equations.
There was no direct method to verify the derived equations of

temperature history. The mixture temperature was, however, predicted

from the calculated air and particle temperatures and measured by

the experiment. Since the predicted and measured mixture temperatures

were in agreement, the predicted air and particle temperatures were

assumed to predict the true temperature of both phases. This indirect

method was used to verify the derived equations. By comparison

of the experimental and predicted results, the temperatures of air and

product particles were well predicted by the derived equations. Tem-

perature histories of product particles were then plotted in charts.

Pie-yi Wang

Approved

Q. R “&W\ 4/99/70

Major Professor
iv” M

Department Chairman
4/14/70
Date

HEAT TRANSFER DURING PNEUMATIC TRANSPORT

OF SPRAY-DRIED FOOD PRODUCTS

By

Pie-yi Wang

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
' for the degree of

DOCTOR OF PHILOSOPHY

Department of Agricultural Engineering

1970

€22qu
7'17 70

For their personal sacrifice, understanding and encourage-

ment, this thesis is dedicated

To my wife, Chun Rwang,
My newborn daughter, Karin H. Wang, and
My mother, Mrs. Lai-hou Wang

ii

ACKNOWLEDGMENTS

The author wishes to express his sincere appreciation and gratitude
to:

Dr. D. R. Heldman, Associate Professor, Agricultural Engineering
Department and Food Science Department, for his constant inspiration,
encouragement, interest and guidance.

Dr. A. W. Farrall, Professor and Chairman, emeritus, Agricultural
Engineering Department, for his encouragement and opportunity to work
on thermal conductivity measurement of dry milk and field performance of
dry milk cooling and storage systems.

Dr. A. M. Dhanak, Professor, Mechanical Engineering Department, for
his depth and thoroughness in teaching heat transfer at the graduate
level and serving as a guidance committee member. Consultations were
freely given and most rewarding.

Dr. F. W. Bakker-Arkema, Associate Professor, Agricultural Engin-
eering Department, for his depth and thoroughness in teaching bio-
processing engineering and serving as a guidance committee member.

Dr. T. I. Hedrick, Professor, Food Science Department, for
consultations and serving as a guidance committee member.

Dr. C. W. Hall, Chairman and Professor, Agricultural Engineering
Department, for the graduate assistantship that enabled the author to

undertake this investigation.

iii

ACKNOWLEDGMENTS. . .

LIST OF

LIST OF

LIST OF

FIGURES. . .

TABLES . . .

APPENDICES. .

NOMENCLATURE. . . .

Chapter
I.
II.

III.

VI.

INTRODUCTION .

OBJECTIVES. .

LITERATURE REVIEW AND JUSTIFICATION

TABLE OF

CONTENTS

Particle to Fluid Heat Transfer.

Relative Velocity between Particle and Fluid

Heat Transfer from Wall to Gas-Solids Mixtures.
Fluid Mechanics of Gas-Solids System .

DEVELOPMENT OF THEORY .

Assumption and Justification.

l. Fluid and Particle Properties are Independent
of Temperature

2. The Particles are Sufficiently Small to Allow
the Assumption of no Temperature Gradient

Within the Particle.

3. Heat Transfer from the Pipe Wall to the Parti-

cles is Negligible .

4. Particles are Uniformly Dispersed and Reach
Their Terminal Velocity at the Entrance

0

Position of the Heat Exchanger .

‘ 5. Velocity Profile is Plug-like
6. Nb'Mass Transfer Occurs

Mathematical Models.

Case 1.
Case 2.
Case 3.

Uniform Wall Temperature
Linear Wall Temperature.
Uniform Heat Flux.

iv

Page
iii
vii
viii

ix

ll
15

16
16

l6

l7

l7

17
18
18

18
19
24
26

Chapter Page
v I mpERmEmAL O O O O O O O O O O O O O O O O 29

Experimental System. . . . . . . . . . . . . 29
Particle Feed and Recovery System . . . . . . . 29
Air Metering! . . . . . . . . . . . . . . 31
Heat Transfer System. . . . . . . . . . . . 31

Temperature Measurement . . . . . . . . . . . 33

Experimental Procedures . . . . . . . . . . . 37

Product Characteristics . . . . . . . . . . . 39

VI. RESULTS AND DISCUSSIONS . . . . . . . . . . . . 42

Heat Transfer Results . . . . . . . . . . . . 42
Temperature Histories . . . . . . . . . . . . 51

VII. ANALYSIS. .. . . . . . . . . . . . . . . . . 62
VIII. CONCLUSIONS . . . . . . . . . . . . . . . . 78
IX. RECOMMENDATIONS FOR FUTURE WORK. . . . . . . . . . 79
BIBLIOGRAPHY. . . . . . . . . . . . . . . . . . . 81

APPENDICES...................86

Figure

5.1

5.2

5.3

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

7.1

LIST OF FIGURES

The System Investigated. . . . . . . . . . .
Experimental System . . . . . . . . . . . . .

Experimental Nusselt Number of Air Compared with Kreith
Proposed Equation for Constant wall Temperature. .

Locations of Thermocouples. . . . . . . . .

Mixture Nusselt NUmber vs. Reynolds for Various Convective
Capacity Parameters . . . . . . . . . . . . .

Mixture NUSselt Number vs. Convective Capacity Parameter
for Various Reynolds Number With Equal Inlet Temperature.

Mixture Nusselt Number vs. Convective Capacity Parameter
for Various Reynolds Number With Different Inlet Temper-
ature. O O O O I O O O O O O O O O O O O

The Combined Nusselt Number vs. Convective Capacity Para-
meter. 0 C C O O O O l. O O . O O I O C 0

Comparing the Measured Outlet Temperatures of Mixture with
the Predicted Temperatures of Particles, Mixture, and Air
for (Re)a = 5.6 x 104 . . . . . . . . . . . .

Comparing the Measured Outlet Temperatures of Mixture with
the Predicted Temperatures of Particles, Mixture, and Air
for (Re)a = 5.6 x 104 I I O C O O O O O O O O

Thermocouples Locations at the Exit Position With Its
Represented Areas. . . . . . . . . . . . . .

Comparing the Measured Outlet Temperature of Particles
and Mixtures with the Predicted Temperatures of Particles,
Mixture and Air for (Re)a = 5.6 x 104 . . . . . . .

Comparing the Measured Outlet Temperature and the Measured
Mixtures Temperature With the Predicted Temperatures of
Particles, Mixture, and Air for (Re)a = 4.7 x 10 . .

Temperature Histories vs. Axial Distance under the System
of Adiabatic Condition and Uniform Wall Temperature for
Loading Ratio of 0.56 and Reynolds NUmber of 5.6 x 104 .

vi

Page
20

3O

34

38

45

47

48

50

54

55

58

59

60

65

i.—

I—“. l.l

Figure Page

7.2 Dimensionless Temperature History of Particles Along the
Axial Distance during Pneumatic Transport of Spray-Dried
FOOd PrOduc ts O O I O O O O O O O O O O O O O 74

7.3 Dimensionless Temperature History of Particles Along the
Axial Distance during Pneumatic Transport of Spray-Dried
Food Products . . . . . . . . . . . . . . . . 75

7.4 Dimensionless Temperature History of Particles Along the
Axial Distance during Pneumatic Transport of Spray-Dried
Food Products . . . . . . . . . . . . . . . . 76

7.5 Dimensionless Temperature History of Particles Along the

Axial Distance during Pneumatic Transport of Spray-Dried
Food Products . . . . . . . . . . . . . . . . 77

vii

Table

6.1

7.1

LIST OF TABLES

page

Characteristics of Heat Exchanger. . . . . . . . . . 35

Comparison of the Predicted and Measured Mixture Temperature
(Different Inlet Temperature) . . . . . . . . . . . 56

The Outlet Temperatures of Particles at the Exit Position of
a 10 ft. Conveying Tube under Various Loading Ratios and Dif-
ferent wall Conditions, (°F) . . . . . . . . . . . 69

viii

LIST OF APPENDICES

Appendix Page
A. Solutions for the Case of Uniform Wall Temperature. . . . 87
B. Solutions for the Case of Linear Wall Temperature . . . . 92
C. Solutions for the Case of Uniform Heat Flux . . . . . . 97
D. Experimental Results (Equal Inlet Temperature) . . .. . . 100

E. Experimental Results (Different Inlet Temperature . . . . 102

F. Measured and Predicted Temperatures (Equal Inlet Temper-
attire. o o o o o o o o o o o o o o o o o o 1.03

ix

4. ni»

NOMENCLATURE

Experimental constant.

Particle surface area as defined in equation (5-6), ft2.ft3.

Individual particle surface area, ftz.

Tube wall thickness, ft.

n = l, 2, 3, 4, 5, 6, parameters as defined in Appendix A.

Experimental constant.

n = l, 2, 3, 4, integration constant.
Specific heat of air, Btu/lb-°F.

Drag coefficient.

Specific heat of particle, Btu/lb-°F.
Particle diameter, ft.

Small or large tube diameter, ft.

Tube diameter, ft.

(D1+D2)/2, equivalent diameter, ft.

Experimental constant.
2
Gravitational constant, ft/sec .

Vapa, Mass flow rate of air, lb/hr-ftz.
Vppp, Mass flow rate of particles, lb/hr-ftz.
Heat transfer coefficient of air alone, Btu/hr-ft2-°F.

Heat transfer coefficient of air at the entrance region,

BtUIhr-ft2-°F o

(NU)m

(mop
(Nu)w

Pr

Heat transfer coefficient of gas-solids mixture, Btu/hr-ft2—°F.
Water side heat transfer coefficient, Btu/hr-ft2—°F.
Thermal conductivity of air, Btu/hr-ft-°F.

Thermal conductivity of particle, Btu/hr-ft-°F.
Thermal conductivity of water, Btu/hr-ft-°F.

Tube length, ft.

ELL)
, operator.

dx
Experimental constant
Mixture Nusselt number as defined in Appendix A.

hpdp/ka, particle Nusselt number.
the/kw’ water side Nusselt number.
Capa/ka, Prandtl number

Radius from center of tube at the subscribed locations (showed
in Figure 11), ft.

Mixture side resistance, hr-ft2-°F/Btu.
. 2 .

Overall resrstance, hr-ft - F/Btu.

Tube wall resistance, hr-ft2-°F/Btu.

Water side resistance, hr-ft2-°F/Btu.

Reynolds number for air alone.

Modified particle Reynolds number.

Vd/

r P a’ particle Reynolds number.

V D / , water side Reynolds number.
w e a

xi

(St)a

(St)m

(St)d

ai

30

mi

mo

pi

p0

. 2
Cross-sectional area of tube, ft .

n = l, 2, 3, 4, sub-divided cross-sectional area according to

Figure 11, ftz.

Stanton number for air alone.

Mixture Stanton number as defined in Appendix A.
Particle Stanton number as defined in Appendix A.
Air temperature, °F.

Air inlet temperature, °F.

Air outlet temperature, °F.

Mixture temperature, °F.

Mixture inlet temperature, °F.

Mixture outlet temperature, °F.

Particle temperature, °F.

Particle inlet temperature, °F.

Particle outlet temperature, °F.

n = l, 2. ......... 12, temperatures with respect to the
thermocouples locations as indicated in Figure 4, °F.

Wall (or water) temperature, °F.
Logarithmic mean temperature difference, °F.

Overall heat transfer coefficient, Btu/hr-ft-°F.

Air velocity, ft/sec.
Individual particle volume, ft3/1b.

xii

Particle velocity, ft/sec.
Relative velocity between particle and air, ft/sec.
Free fall terminal velocity, ft/sec.

Convective capacity ratio as defined in Appendix A.
Axial distance, ft.

Particular solution.
Complementary solution.

Parameter as defined in equation (7-3).

Constant, axial temperature gradient, °F.
Parameter as defined in Appendix A.

Dynamic velocity of air, lb/sec«ft.

(T .)/(Tpi-Tw)

pi-Tai
Kinematic viscosity of air, ft2/sec.

Air density, lb/ft3.

Particle density per unit volume of mixture as defined in
equation (5-7), lb/ft3.

True particle density, lb/ft3.

Mixture density, 1b/ft3.

Shape factor.

xiii

CHAPTER I
INTRODUCTION

Spray drying plays an important role in food processing, since
it improves the keeping quality of products, reduces the cost of
transportation, and provides a product which can be served as a
convenience-type food. Spray drying is used extensively to manufac-
ture several products such as tomato powder, orange powder, bean pow-
der, etc. Probably the most widely known spray-dried product is dry
milk. Non-fat dry milk, therefore, was used as the test product in
this investigation.

The quality of spray-dried food products may be decreased
considerably by prolonged storage at high temperatures. In most cases,
the prolonged heating causes heat damage. Hall and Hedrick (1966)
indicated that the continued heating causes staleness in non-fat dry
milk flavor and causes the melted fat to move to the surface of whole
milk powder. Heat damage such as clumping, off-flavor, and off-color
may develop very rapidly. Hanrahan gt El. (1967) illustrated that
slow cooling results in a cooked or scorched flavor in spray-dried
milk. Most dry milk plants prefer the milk powder to be cooled below
80° F as indicated by Farrall _£ 31. (1968 b). tha _5 a1. (1966),
Farrell 25 11° (1968 a) and Chen (1969) showed that the dry food pro-
ducts possess a very low thermal conductivity. The cooling rate in

the package, therefore, will require several days to reach the ambient

temperature. It would appear that cooling rate during storage is not

 

suffi

produ
to ca
the p
dmer
mix h
pneum
rate .
ature
order
conve;
illus
heate.
is on
effec
temps

aPPro.

sufficiently rapid to prevent heat damage.

Since spray drying utilizes hot air as drying medium, the dried
product leaving the drying chamber is at a sufficient high temperature
to cause heat damage. For the purpose of maintaining high food quality,
the products should be cooled as rapidly as possible after leaving the
dryer. As indicated by Farrall _£,§l° (1968 b) most dairy plants
mix hot powder with room temperature air and the mixture is then
pneumatically conveyed through a tube to the collectors. The cooling
rate of this type of system depends mainly on the ambient air temper-
ature. Slow cooling rates can be expected during summer season. In
order to obtain low temperature products, drying plants may utilize
conveying tubes which may be longer than 100 ft. Heldman _£ 31. (1969)
illustrated that the rapid cooling rate can be achieved by mixing
heated powder with cooled air. However, the direct cold air injection
is only effective for low loading ratios. In order to obtain a more
effective cooling system for high loading ratios, reduction of surface

temperature of the conveying tube seems to offer the most satisfactory

approach.

CHAPTER II

OBJECTIVES

The application of gas-solids mixture as a heat transfer and
transport media was first used in petroleum-cracking plants. The
evidence that solids in gaseous heat transfer systems increased heat
transfer rate was found experimentally by the petroleum industry.

In this decade, the technique has been utilized in the field of solid
propellent combustion, cooling nuclear power plants, food processing,
etc. The increasing interest has emphasized the study of heat trans-

fer and fluid mechanics of the gas-solids mixture systems. Farbar

and Morley (1957) were the first investigators in this field. Due to

the lhnited number of investigations and the complexity of the flow
mechanism, most available information is based on experimental results.
Heat transfer data are limited to: (1) within the fully developed region
(aerodynamically) and (2) the equal inlet temperature of gas and particles.
In the food industry, powder cooling will begin immediately after
entrance to the air stream and a temperature gradient should be main-
tained between the two phases. It is necessary to investigate the
heat transfer mechanisms within this entrance region and with the
different inlet temperatures of the two phases.

The objectives of this research were to (1) determine heat trans-
fer coefficients between mixture and tube wall within the entrance
region, (2) investigate whether the different inlet temperatures in

the two phases influence heat transfer coefficients, and (3) develop

the mathematical models to predict the temperatures of both air and
product particles. The final objective of this investigation was

to use the experhmental heat transfer coefficients and the derived
equations to analyze different cooling systems for spray-dried food

products.

CHAPTER III
LITERATURE REVIEW AND JUSTIFICATION

Zenz and Othmer (1960) presented an extensive literature review
of both fluid and heat transfer mechanisms in pneumatic conveying systems.
Before 1960, most investigations were experimental in nature. The book,
"Fluid Dynamics of Multiphase Systems," by $00 (1967) describes
both fluid dynamics and heat transfer mechanisms in the multiphase
systems. In this book the continuous mechanics of a single-phase
fluid is extended to the multiphase systems. The general equations
of motion and energy have been developed. Due to the complexity of
fluid dynamics of multiphase, the theoretical solutions are limited
to a few simple cases. The following literature review discusses

each parameter involved.

Particle to Fluid Heat Transfer

When a rigid sphere at a given temperature is submerged in a
quiescent fluid at another temperature, heat transfer takes place
only by conduction through the boundary layer surrounding the particle.
For very small particles under free convection, it is known that the
boundary layer is appreciably larger than the particle radius.
Based on this assumption, Themelis t_§l. (1963) derived the theoreti-

cal solution in which the particle Nusselt number is described as:

h d
(Nu)p - {-2 = 2 ---------------------------------------- (3-1)
a

Equation (3-1) is appliable only when dp is greater than the mean

free path of molecules of the surrounding medium. Since relative

motion of the fluid to the particle exists, the heat transfer rate
depends on the relative velocity. If the motion of particle falls
within Stokes regime, i.e. Reynolds number less than one; Breiman

(1952) theoretically derived Nusselt number as a series expansion

form:

_ l _1_ 2 2
(Nu)p - 2 + 2 (Re)pPr + 6(Re)p Pr + ------------------ (3-2)

Equation (3-2) asymptotically approaches the value of 2, corresponding
to the conduction in quiescent fluid as described by equation (3-1).

For the high particle Reynolds number, theoretical analysis is very
complex. Only experimental correlations were conducted within the
intermediate and Newton's Law regimes. From literature reviews of

Rowe 35 31. (1965), $00 (1967) and Rubnek £5 21- (1968), most of forced-
convection heat transfer data for single particle have been correlated

in the following forms:

(Nu)? = 2 + f(Re): (Pr)c -------------------------------- (3-3)
(Nu)p = f(Re): (Pr)C ------------------------------------ (3-4)

where a, f, and c are experimental constants.

For the low particle Reynolds number, Nusselt number takes the
form of equation (3-3). The advantage of using equation (3-3) is to
allow description of the heat transfer rate for the particle in
quiescent fluid. Various correlations were made by Zenz t 1. (1960)

and $00 (1967). The latter suggested that the best fit was proposed

by Drake (1961),

0.55 0.33

(Nu)P = 2 + 0.459 (Re)p (Pr) ------------------ (3'5)

for l < (Re)p-< 70,000 and 0.6 < Pr < 400

Whether the heat transfer mechanisms of single particle can be
extended to multi-particle system is still unclear. Heat transfer
data on multi-particle systems obtained by Johnstone gt a1. (1941)
and Joukovski gt 31. (1940) were reported by Zenz and Othmer (1960).
They indicated that there is some departure from any asymptote at
(Nu)p = 2 for low particle Reynolds number, but 800 (1967) concluded
that the deviation is not significant. For dilute transport the void
which is defined as the volumetric ratio of particles to air stream is
greater than 99%. Due to the large void, each individual particle can
be assumed to be isolated from the others. It is reasonable,therefore,
to employ equation (3-5) for multi-particle system in dilute transport.

Relative Velocity Between Particle
and Fluid

 

As indicated in the previous section, heat transfer during pneumatic
conveying is influenced by the particle Reynolds number:

Vd
(Re)p = _I;£

V
a

which is based on the relative velocity,
V = V - V

r a p
Particle motion in fluid can be classified into three regimes: (l) Stokes

law regime, (2) intermediate regime, and (3) Newton's Law regime. In

any regime, the particle velocity depends mainly on the drag force.

Description of the drag coefficient is therefore the necessary condi-
tion to derive particle velocity.

Within the Stokes law regime, Stoke (1850) obtained the drag
force from the Navier-Stokes equations. The drag coefficient was

derived as:

24

cD = (Re)p

For the case of a free falling sphere, the terminal velocity given

by Stokes law is:

 

2 I
d -
v gimp pa)g
t 18 “a

 

------------------------------------- (3-6)

Allen (1968) indicated that equation (3-6) can apply only to the
particle Reynolds number less than 0.2.
Within the intermediate regime, Ossen((cited by 500, (1967))

obtained the drag coefficient as:

24
C a:
D R

( e)p

 

3
(1 +'IE (Re)p) ------------------------------ (3-7)

for (Re)p < 5.

Schillar and Nauman (1933) fitted the experimental values of CD and

(Re)p to obtain an empirical equation:

= 24 _ 0.687
CD (RE); (1 0.15(Re)p

> ------------------------ (3-8)

for 01< (Re)P-< 700.

Since a unique relationship exists between CD and (Re)p, it is
impossible to evaluate Vr from equation(3-7) or equation (3-8)

by knowing dp. Jones t l. (1966) gave the empirical formula of

relative velocity in this regime,

I 0.673 1.02
2.4 - d
v = (DP pa) ( ) (x)
r

0.7

 

----------------- (3-9)
0.372 0.347
pa Fla
Heywood (1962) combined Stokes and Newton's law with a graphical
solution and constructed tables. From Heywood's Tables, terminal
velocity can be obtained by knowing dp.

Within Newtons law regime, the drag coefficient has been confirmed

to be 0.44. Belden and Kassel (1949) suggested the formula for

relative velocity as:

 

I
vr 1.32 «fgdp (pp - pa)/ pa ---------------------------- <3-10)

Tchen (1947) studied the motion of a particle suspended in a turbulent
stream using the assumption that the resistance was described by Stokes
law and derived an analytical equation for the mean square relative
velocity. For estimating the equation, turbulent intensity has to be
determined. $00 (1967) determined that the Reynolds number of a 50
micron diameter carbon particle which was suspended in turbulent air flow
was 2.52 based on Tchen's result and 0.53 based on the free fall terminal
velocity from Stokes law. This comparison indicates that the free fall
terminal velocity cannot be used for evaluating relative velocity of

particles conveyed in a turbulent stream.

- 10 -

For the multi-particle system, Hinkle (1953) used a high speed
photographic technique to study the relative velocity in gas-solids

mixture systems and expressed an empirical relationship as:

_ 0.3 I 0.5
Vr 1.4 Vadp (pp / pa) ---------------------------- (3-11)

These results indicate that the loading ratio does not affect the
relative velocity. Hariu and Malstod (1949) and Wen and Simons (1959)
investigated the relative velocity during dense phase transport (high
loading ratio) and concluded that the air velocity is approximately
double the particle velocity:

Va 2: zvp ------------------------------------------------ (3-12)

The loading ratio used by Wen S£.§l' (1959) and Hariu _£ _1. (1949)
ranged from 80 to 750 and 2 to 18, respectively.

By comparison of the particle Reynolds numbers obtained from
single particle and multi-particle systems, it appears that the particle
Reynolds number of multi-particle system is much greater than that of
the single particle case. The deviation is obviously due to the
momentum transfer of particle collisions with wall and other particles
in the multi-particle system. However, Wen and Miller (1961) and
Yannopoulos gt 31° (1966) suggested that free fall terminal velocity
can be used to evaluate the relative velocity in gas-solids mixture
systems. According to Tchen's (1947) and Hinkle's (1953) investiga-

tions, the application of the free fall terminal velocity to gas-solids

mixture systems is questionable.

_ 11 -

There is an obvious lack of information about the relative
velocity within an acceleration region. Hariu £5 31. (1949) and
Hinkle (1953) concluded from their investigation of the pressure drop
that the distance of 14 feet is required for particles to attain a
constant velocity. Zenz and Othmer (1960) analyzed Russ and Culgan's
data (1950) and concluded that the acceleration rate increases as
loading ratio decreases. At high air velocities and low loading ratios,
the total acceleration distance is only about two feet. Allen (1968)
presented the following statement:

" It can be shown that under the conditions likely to be

found in sedimentation of elutriation, the assumption that

the terminal velocity is reached instantaneously introduces
a negligible error."

Heat Transfer From Wall
To Gas-Solids Mixtures

 

Heat transfer in gas-solids suspensions within a confining vessel
has been the subject of many studies because of the addition of suspended
particles having not only the advantage of higher heat capacity, but
also that of promoting heat transfer rate. The first investigators
in this field, Farbar and Morley (1957), illustrated that the reasons
for increased heat transfer rate due to adding particles to an air
stream are (l) the high volumetric specific heat of the solid
particles and (2) the reduction of the boundary layer thickness.

In addition, they found there exists a critical loading ratio of
unity. Above the loading ratio of one, heat transfer rate is consider-
able increased. The heat transfer rate increases only slightly,

however, when the loading ratio is less than unity. They correlated

- 12 -

the experimental results and give the mixture Nusselt number as:

(Nu)... = 0.14 (Re)2°6 (GP/ca)°°“5 ------------------------ (3-13)
for (Gp/Ga) > 2

Tien and Quan (1962) and Jepson and Smith (1963) conducted additional
investigations and indicated that, due to inertia, the introduction of
particles reduced the scale of turbulence of the gas as long as

the loading ratio was between zero and one. Above a loading of

unity, turbulence was increased. Results from Farbar and Morley
(1957) and Jepson and Smith (1963) are shown in Figure 3.1 and 3.2.
The investigations of previous researchers and several others which
will be mentioned later are limited to the fully developed region
(aerodynamically) as well as equal inlet temperatures of air and
particles. Danziger (1963) correlated the data of air-catalyst

mixtures in vertical transport and presented a correlation,

DG
= a 0.66
(Nu)m 0.0784 (13-)

(GP/Ga)0’4S -------------------- (3-14)
a

2
for (GP/Ga) >

Equation (3-14) is in good agreement with equation (3-13). Tien
(1961) made the first analytical attempt to predict the effect of
the addition of solid particles to an air stream on the heat trans-
fer coefficient. A solution was obtained for the constant wall
temperature, which was then extended to the case for arbitrary wall
temperatures. The limitations of Tien's analytical solutions are:

(1) particle and air are assumed to be at the same velocity,

Mixture Nusselt Number
(hmD/ka)

200-

m

Heat Transfer Coefficient - h

_ 13 -

Figure 3.1 Mixture Nusselt number vs. solids loading
for constant air flow rates ((Farber and
Morley (1957)).

 

00 - -
80 - (Re)a - 27 000 _"///,/////////zzzrr
60 -

‘ 13,500

I l l L
0.1 0.5 1.0 2 4 6

Solid Loading Ratio -- (GP/Ga)

 

J.“
O
A
w
(0
v
O)
I

11
= 60 ft/sec

  
    

Va = 40 ft/sec

 

2 l I 1 l l I l
0 2 4 6 8 10 12 14

Solids/Air Loading Ratio -- (GP/G )
a

Figure 3.2 Heat transfer coefficients at various

solids loading ratios((Jepson and Smith
(1963))

- 14 -

(2) particle and air are at the same inlet temperature, and (3)
the flow must be fully developed. Wen et al. (1961) correlated several

investigators' results and gave the equation,

m

C
h = E£(pm/pp)°°3 (Vt/gdp)0'21 ------------------------- (3'15)
P

As discussed in the previous section, since the free fall terminal
velocity of single particle is utilized in the multi-particle

system, the validy of the equation (3-15) may be questionable.

Farber gt 31. (1963) and Jepson _£.a1. (1963) investigated the effect
of particle size on the heat transfer coefficient. Both investigators
indicated the same relationship that the increase in particle diameter
reduces the heat transfer coefficient. The maximum diameter for
increased heat transfer coefficients were determined as 200 micron and
500 micron for glass beads and sands, respectively.

For the case of constant heat flux, Depew and Farbar (1963)
derived an analytical solution forthe mixture Nusselt number utilizing
techniques similar to Tien (1960). The limitations are the same as
Tien's solutions. In the work sponsored by the Atomic Energy Commission,
a group at the Babcock & Wilcox Co., Atomic Energy Division (1959)
studied flowing gas-solids mixtures as a heat transfer media, using
finely divided graphite with nitrogen, helium, and Freon 14. In
their final report, they presented a correlation of their data for
heat transfer from an electrically heated tube to gas-solids mixtures

vertical flow with the equation,

0.802 c c 0.45
(Nu)m = 0.0195 ((Rea) Pr)) (1 + 5953) ---------- (3—16)
88

- 15 -

Wilkinson and Norman (1967) presented heat transfer data obtained in
a vertical heated tube on suspensions of glass beads and graphite in

the air. A plot of (Nu)m/(Re)3'8 versus Tw/Tm revealed a negative

cube root dependence.

Fluid Mechanics of Gas-Solids Systems

Fluid mechanics of gas-solids systems is complex with respect
to the distribution of particles across the pipe and the velocity pro-
files of both phases. $00 and Regalbutp (1960) summarized their
investigation with the following statement:

"Due to the inertia of solid particles, transport of solid

particles across the velocity field of the turbulent stream

by turbulent diffusion results in deceleration of particles

in the region near the wall by the stream and acceleration

of particles transported toward the high velocity core.

In the core, the particles have lower mean velocity than the

stream while near the wall the mean particle velocity is

higher. This gives rise to a non-uniform distribution of

solid particles in fully developed turbulent pipe flow."
However, Soo'_t al. (1960) indicated that the constant distribution can
be assumed for the low loading ratio, high stream velocity, and small
particle size. Depew, gt al. (1963) indicated that due to a non-
uniform distribution of particles, the velocity profile will be
affected. The~change could be either toward a more plug-like or a
more streamline profile. 800 (1967) showed that within the range
of loading where the presence of solids contributes to lowering of

friction factor in fully developed turbulent pipe flow, the velocity

profile of the gaseous phase was unaltered.

- l6 -

CHAPTER IV

DEVELOPMENT OF THEORY

Assumption and Justification

 

Due to the complexity of the fluid mechanics of gas-solids
mixture systems, the analytical solutions for the temperatures of
air and particles are very difficult to obtain. Tien (1960) attempted
to solve the problem of utilizing the single phase techniques
which were employed by Sleicher _£ 31. (1957). Tien (1960), however,
found that insufficient mathematical theory has been developed for
its solution. For the case of adiabatic conditions, Sissom t l.

(1965), Bakker-Arkema 35 21. (1969) and Heldman £5 31. (1969) have
developed the mathematical expressions to evaluate the temperatures of
both phases. The former dealt with the dense transport; the latter
considered the dilute conveying. All investigators have assumed the

system to be plug flow in order to simplify the complexity. For the

arbitrary wall conditions, the following basic assumptions are utilized:

1. Fluid and Particle Properties Egg Independent pf Temperature.

 

 

 

Thermal properties of fluid or particles are usually a function
of temperature. However, thermal properties can be assumed constant if the
temperature range is small and the bulk mean temperature is used to
evaluate the properties. This assumption has been utilized previously

in general heat transfer calculations.

- 17 _

2. The Particles are Sufficiently Small £9 Allow the Assumption 9f
‘Np Temperature Gradient Within the Particles.

 

 

This assumption can be checked with the lumped-capacity proposed

by Holman (1968):

h V b d
lumped-capacity == RBAQ = 35-2
P P P

where V0 and Ap are individual particle volume and surface area
respectively. The distribution of particle sizes of non-fat dry milk

has been investigated by Hayashi _£ 31. (1967). The average diameter was
determined to be 60 micron. Particle heat transfer coefficients can

be evaluated with equation (3—5). At a particle Reynolds number of 10
which is an approximate value in this investiation, hp was calculated

to be 250 Btu/hr—ft2 - °F. From Chen's investigation (1969) the thermal
conductivity of the non-fat dry milk particle was measured as 0.26
Btu/hr-ft-°F at 3.5%.moisture content and 100° F. By the substitution

of the values of hp, dp, and kp, the lumped-capacity was calculated

to be 0.068 which is less than 0.1 and meets the assumption of no

temperature gradient.

3. Heat Transfer from the Pipe Wall £9 Ehg Pargiglg§.i§_gegligiplg.

 

Farber and'Morley (157) indicated that due to small contact
area between particle and wall as well as short contact time, the
possibility of any heat transfer from the pipe wall directly to the
particles can be eliminated.

4. Particles are Uniformly Dispersed and Reach Their Terminal Velocigy
25 Eh; Entrance Position_gf the Heat Exchanger.

 

 

 

 

The distribution of particles across the cross section of a

- 18 -

conveying tube and the acceleration of particles have been discussed
in previous sections of Chapter 3. This assumption appears to be
valid for the conditions of low loading ratio, high average stream

velocity, and small particle size.

5. Velocity Profilg lg Plug-Like.

 

The plug flow is usually considered to be a first approximation
in turbulent or non-Newtonian flow. It is readily seen that this
assumption is considerably different from the actual case. However,
in the case of entrance regions or turbulent pipe flow, at least a

major portion of the tube cross section approaches plug flow.

 

6. .N9 Mass Transfe; Occurs.

Spray-dried products are maintained at low moisture content.
Adsorption or desorption occurs only during the falling rate period
if the products are exposed to the air stream of low relative humidity.
Since mass transfer during the falling rate period is small, it can be
neglected if the retention time of particle in the air stream is
short. It can be realized that the latent heat of water vapor is
approximately 1000 Btu/1b. If mass transfer occurs during the process,
the temperature history of particles and air must be affected signifi-

cantly.

Mathematical Models
The following analysis will deal with three different wall condi-
tions; (1) uniform wall temperature, (2) linear wall temperature, and
(3) uniform heat flux. By derivation of equations which describe

both the enthalpy content of the air and the particles as the mixture

moves through the conveying tube, the change in temperatures of air and

 

particle
The proc
temperat
uniform
presente
(i) Ent
By

as a ref

The Chan
differen

Va;
As the a
tlansfe]

air, the

The Met

_ 19 -

particles can be predicted based on properties of air and particles.
The procedures for setting up the governing equations of linear wall
temperature or uniform heat flux case are similar to those of the
uniform wall temperature case. The detailed development will be
presented in the latter case only.

Case 1. Uniform wall Temperature

 

(i) Enthalpy balance of air
By using a control volume within the conveying tube (see Figure l)

as a reference, the enthalpy of air entering the control volume de is:

VapaCaSTadx ---------------------------------------------- (4-1)
The enthalpy of air leaving de is:
dTa
Vapacasaa + ‘33:) dx ------------------------------------- (4-2)
The change in enthalpy of air with respect to axial distance is the

difference between equation (4-2) and (4-1), or

dTa
Vapacasig ----------------------------------------------- (4-3)

As the air moves through the control volume, two types of heat
transfer may occur. The first is the convection between wall and
air, the other is the convection between air and solid particles.

The heat flux from pipe wall to the gas-solids mixture is defined by:

qw = hmpn ('rw - rm) dx ----------------------------------- (4-4)

20 -

.woumwmumo>cw Eoummw mLH H.¢ muswflm

lLsTl

u)—

NQG>

 

 

nl.l.lil

 

 

oEDHo> Houucoo

a a IAIIIII. HH<
q>,_

moHuHuumm

 

 

Th1 mixi

 

T
m

l
The ent!

particln

qp

Tne tote

then,

- 21 -

The mixture temperature Tm is defined as:

V p C T + V p C T
T = a a a a P P P P ________________________________ (4_5)
m V p C + V p C
see PPP

 

The enthalpy change resulting from the heat transfer between air and
particles can be written as:

qp = hpAS (Ta - Tp)dx ----------------------------------- (4-6)

The total change in enthalpy of air within the control volume is

then,
dT

a .. - - - _
vapacasdx dx hm"D(Tw Tm)dx hpAS(Ta Tp)dx (4 7)

Since 8 is cross-sectional area of the pipe, it can be represented

by: 2

=32. ................................................ ..
s 4 (4 8)

By substitution of equation (4-8) into (4-7), a new equation is

obtained:

dTa 4hm h A
71;- = m“. ' Tm) ‘ 4"“; p c (T. ' T.) """""""" (4'9)
a a a a a a

(ii) Enthalpy balance for solid particles
The enthalpy of solid particles entering the control volume de
is:

V C ST dx --------------------------------------------- 4-10
pppp P ( )

The enthalpy of solid particles leaving the control volume de is:
dT

v csr+——de ..................................... 4-11
pppp(p dx) ( )

The enthalpy change of solid particles with respect to axial direction

is the difference between equation (4-11) and (4-10), or

dT
v c s—P-dx -------------------------------------------- 4-12
app 9 dx ( )

 

 

The cha
de res-
neglect

(4-6).

d1“
EC
Equation

Boundary

 

 

 

With the

t0 Obtai‘

”vs”
”C
,_.-

- 22 -

The change in enthalpy of solid particles within the control volume
de results from the energy transmitted from air to particles by
neglecting the effect of the wall to particles is similar to equation

(4-6). Combining equation (4-6) and (4-12) gives:

dT h A
de -- —P——Vpppcp (Ta -Tp) -------------------------- “'13)

Equation (4-9) and (4-13) are the governing equations of the system.
Boundary conditions of the system under study are:
r'Tp = Tpi
x=o, 1T”. ------------------------------------ (4-14)
a a1
With the boundary conditions, the governing equations can be solved

to obtain: (see Appendix A)

 

 

T - T (St)
_£L___2L. = (.____e_ ) ( A )
Tpi - Tw w ¢(l+w) (sc)d 4(St)m
w ' (1+w)
T i - Tai 4(St)m x
{[(W)'('fp‘."-—i—)3EXPE’WBJ‘

p1 w

4w(St)m

0(l+w) (St)d x
[ o<1+w) (sc)d

. 51}

 

 

T . - T .
<H> 1 Exp [-

pi w

......................... (4-15)

 

 

 

 

In the 1

14-15)

- 23 -

- Tw (St)d

 

 

 

 

 

___a___=_____ D
Tpi - Tw ( w ) (0(1+w) (St)d 4(sr)d)
w ' (1+w)
T i - Tai 4(St)m x
{W 1 (1 + W) - (EEj-:—§—-) ] EXP 1 - 73;;3- '5 J -
pi w
4w(St) 4w(St) T . - T .
[l‘ouw “‘ ”101+. “‘ _(_,.___._,H
( ) (St)d ( ) (St)d Tp1 - Tai
P(l+w) (St)d x 1
Exp [ - - J r ------------------------- (4'16)

 

w J
In the case of the equal inlet temperatures of both phases, equation

(4-15) and (4-17) are simplified as follows:

 

 

T - TW (St)d 0
T . - T = ( w ) ( 0(1+w) (St) 4(St) )
p1 w d - m

 

w (1+w)

 

 

 

 

 

 

 

 

4(St)m x 4w(St)m
{ (1+w) EXP [ ' 7I¥§3_' B J ' [ D(1+W) (St)d J
0(l+w) (St)d x
Exp E - w ~15 1 } --------------------------- <4-17)
———-—Ta - T“ = <——(St)d) ( 0 )
Tpi - Tw w ¢(1+w) (St)d - 4(St)m
w (1+w)
4(St)m x 4W(St)m
1 - — - -
i“ MEX" [ (1+w) D J L 1 ¢(1+w) (sod 3
4w(St)m 0(l+w) (St)d x
~ _ 1
L 0<1+w) (socl 3 EXP L ‘ w n J J

............................ (4-18)

 

918:2-

 

Ac
I

wall te:

T

w
By fol}?
for air
(i) Enfl

The

 

 

(4'9).
0f equat

PGIature

- 24 -

Case 2. Linear Wall Temperature
According to Sleicher, _£._l. (1957), the definition of linear

wall temperature is:

Tw = Tai - (1x ------------------------------------- (4‘19)

By following the same procedures as for Case I, the enthalpy balance
for air and solid particles can be obtained.
(i) Enthalpy balance for air

The governing equation for air temperature is similar to equation
(4-9). The only difference is the wall temperature. By substitution
of equation (4-19) into equation (4-9), the new equation for air tem-
perature is obtained:

dTa 4hm h A
(T ”fax-T)-
ai m

dx =VpCD
a a 3
(ii) Enthalpy balance for solid particles
The heat transfer mechanism between air and particles is as same

for the case of uniform wall temperature. Equation (4-13), therefore,

is employed here as follows:

dT h A
de = —P———vpppcp (T. - 'rp) ---------------------------------- <4-21)

Equation (4-20) and (4-21) are the governing equations of the
linear wall temperature system. Boundary conditions of this system are:
{_T = T i
x = 0, P P -------------------------------------- (4-22)

Ta = Tai

 

with t

to obt

LH

- 25 -

With these boundary conditions, the governing equations can be solved

to obtain: (see Appendix B)

w (St)
(—————— + -——JP
4(St)m 4

 

E l
0(St)d 4(St) ]
0(St)d + - m

w (1+w)

 

 

w(T - T .) D(St)d 4(St)m

 

W

‘0(1+w) (St)d

 

) +

pi a1 ______ _ ______ _ _ _
{t at, ( (1+w) > (1 w)

W

““5”... one.) (51;)d

 

 

2 J Exp ( -
0(1+w) (St)d w

T - T . x W (St)
_£____2l =._ m
a1) D

+ (4(St)m + 4

 

 

E l
¢(St)d 4(St)mJ
”(3% " —w— ' (Tm—

{L‘Za'_.1(___-
D w (1+w)

4w(St) ¢(1+w) (soc1
Ill

T . D(St)d 4(St)m

 

)

_) } ------------- (4-23)
D

W

o<1+w) (St)d +

 

X

— ) 1 -------- (4-24)

 

 

J Exp ( - w

o<1+w)2<St)d

D

1f air

i.e. T

 

 

PH

 

 

 

 

 

 

- 26 -

If air and particle temperatures are equal at the entrance position,

i.e. Tai ' Tpi’ equation (4-23) and (4-24) can be simplified as:

 

 

 

 

 

 

 

 

T - Tai =.x + w + (St) - w
GD D 4(St)m 4 0(1+W) (St)d
[ 1 [ 4w(St)m
0(St) _ 4(3:) 3 1 2 3
9)(se)d +—__..w ‘1 ____(1+w)m M I“) (Sod
0(l+w)(St)d x
EXP ( - w D ) -------------------------- (4’25)
T - T . x w (St) w
._a___._.€_l. = — + ————.—— + ____I-II .. ..
01D D 4(St) 4 {Ml-m) (St)d
m
. 1
L 0(3t)d 4(St)m 3 [ ‘1’“) +
”(30.1 + —w— ' n+7)“—
4w(St)m 0(l+w)(St)d x
2 EXP ( - w 5') ----------- (4'26)
MIN) (St)d

Case 3. Uniform Heat Flux

The procedures for setting up the governing equations for the
case of uniform heat flux are similar to those of the uniform wall
temperature case. The detailed derivation is omitted.

(i) Enthalpy balance for air

The governing equation for air temperature is similar to equation

(4-9). The only difference is the first term on the right hand side

of eqoa

 

 

is tons

01

d4

 

(ii) El
It
as the

is empl

 

Es

Uniform

 

 

 

 

 

with t'

10 Obt

-27-

of equation (4-9). Since heat flux from pipe wall to gas-solids mixture

is constant, equation (4-9) can be written as:

dTa 4qw h A
T. '- {776—5 ‘ —E—v . c (T. " T.) """"""""""" “’27)
a a a a a 3

(ii) Enthalpy balance for solid particles
The heat transfer mechanism between air and particles is as same
as the case of uniform wall temperature. Equation (4-13), therefore,

is employed here as follows:

dT h A
de = —LV.P.C. (T. - T9) ------------------------------- (448)

Equations (4-27) and (4-28) are the governing equations of the

uniform heat flux system. Boundary conditions of this system are:

X
II
C
r-“n
II

T T i
TP TP .................................... (4-29)
a ai

With the boundary conditions, the governing equations can be solved

to obtain: (see Appendix C)

 

T - T 1 4x 4w (T .-T .)
_E__El = _ {_ _ __2_1__a_1_ +
qw/sflca 1+1» D 0(1+w) (St) d qw/GaCa

[ (T 1 'Tai) 4w

 

«aw/cat:a + commend]

 

P(l+w)(St)d x .}

Exp ( - w D )

-28-

 

 

 

T - T . 1 , 4x 4w2 (T .-T .)
_a_21=_{_- ___e_1__a.1_
qw/GaCa l+w D 0(1+w)(3t)d qw/GaCa
(T i—T .) 4w
w L __E___§l_ + ]
qw/Gaca 0(l+w)(St)d
0(l+w)(St)d x 1
Exp(- —) -------------------------- (4-31)

w D 1

If air and particle temperatures are equal at the entrance position,

 

 

 

i.e. TPi = Tai’ equation (4-30 and (4-31) can be simplified as:
T -T . 1 . 4x 4w
_P___£i = ___ { .__
qw/GaCa l+w D p(1+w)(sc)d
4w 0(1+w)(St)d x
< > Exp < - — )} -------- <4-32)
0(1+w)(St)d w D
2

Ta - Tpi 1 4x 4w
qw7caca = II; {'IT ' 0(l+w)(St)d

 

4w D(l+w)(St)d x
w ( 0(l+w)(St)d ) EXP ( ' w B

 

 

) } ----- (4-33)

-29-

CHAPTER V

EXPEREMENTAL

Experimentalggystem

 

A diagram of the experimental system is shown in Figure 5.1.
The system consisted of three separately controlled systems; particle
feed and recovery system, air metering and flow system, and heat
transfer system. The wall of the heat exchanger was held at the uniform

temperature for the experimental portion of this investigation.

1. Particle Feed and Recovegy System

A stainless-steel powder collector I was used as a source of
product in the experiments. This container has approximate dimensions
of a 5 ft. height and a diameter ranging from 0.5 ft. at bottom to 3 ft.
at the top. An agitator which turned around a vertical axis and a 2-in.
diameter auger were added to the container. The auger was turned by
a variable speed motor which allowed the solids feed rate to be varied
from 1.5 to 9 lbs/min. The particles were carried by air through the
tube to the separator and collected in the container II. The particle
feed rate was measured at the immediate outlet of the auger before
introducing the air to the tube. In order to check for influence of
air flow on particle feed rate, the particles collected in container II
were weighed after each experiment. The particle feed rate measured

by both methods were in agreement that indications of air flow did not

influence feed rate.

 

 

 

 

   

 

 

 

 

“\
“‘
HH nocmmucoo m “
\
X
X
3. d
masm N "w
non-m3 \“u u“
= “I“
. “:w
0 mop-5m “u"\
a. noumz \u u“
"5.. 3"“
u _ s;
uoamumATm \n u“
_ \
_ “ .u-Itu-uux \ u n“
_ _ \ _\
_ h.“
“I“
“I“
. \ .-\
I“
e
G
c.

\\\

\\\\\\\\\\\\\\\\\\\\\\\‘

   

       

 

 

 

 

 

    

Q.‘

E-\\\'?'

g n\\\\\\ \\\\\\\\\\\\\ s\\

\\\§§§\\\\‘\\\§\\\\\\\\\\\s\ .\

‘\\.

Houoaocoz

onsuua nouoom

“-\..~“-‘§§h“~\\~\.§‘~“§~.“A ‘\\. “‘§..§.~‘-‘.§..§‘.‘\\n

| "Moon” v

.s\\\\§\§§\§|§n~h~\\\\~\\\\§\‘\\\\\\\\\\\\.\\\\\\\\

 

 

JaSusqoxa mean

 

Husuco> o>~o>
u. .41 0...“. __
pow=< 1: 1| acuoz
mooam
ofinmmum>
" +|l
poumumw<
_
H Hocwmuaoo
Honcho

Hmucoamuomxm

 

H.m muawfim

_ 31 -

2. Air Meteripg and Flow System

 

Air was drawn from the room by the fan connected to the inlet of
conveying tube. The air flow rate was controlled by an adjustable valve
located at the outlet of the fan and was measured using a venturi tube
and U-tube manometer. A 1500-watt resistance heater was placed in the
air supply tube upstream from the location of product injection. The
aluminum tube with 2-in. diameter (o.d.) was used in this investigation.
The conveying tube length fromtfim fan to separator was approximately
30 ft. The whole conveying line was insulated by fiberglas with a
2-in. thickness. The properties of air were obtained from "Mechanical

Engineering Handbook" by Baumeister (1958).

3. Heat Transfer System

The heat transfer system consisted of a shell-and—tube heat
exchanger and a hot water source. The experimental heat exchanger
was 18-8 (chrome-nickel) stainless-steel. The hot water from the
source was circulated by a water pump through the annular space between
shell and tube and returned to the source. The flow rate of water
was determined by introducing the returned water from the heat exchanger
to a bucket which was then weighed. The heat exchanger was arranged
for co-current flow in this investigation and was insulated with 2-in.
fiberglas, also.

Before any solids were run through the heat exchanger, many
experiments were conducted to determine the characterics of the exper-
2hnental heat exchanger. According to Eckert (1963), the uniform wall

temperature is defined by conditions when the heat transfer coefficient

- 32 -

of one side approaches infinity compared to the Opposite side. Since
uniform wall temperature was assumed in this investigation, heat transfer
coefficients on the water side had to be much higher than on the mixture
side. In order to maintain a high heat transfer coefficient of water
side, the water flow rate was fixed at 3460 lbs/hr corresponding to a
Reynolds number of 1.23 x 104. Hsu (1963) suggested an equation to
evaluate the heat transfer coeffici nt for the forced convection in turbu-

lent flow inside annular space;

h 0 0
(Nu)w = I e = 0.02 (Re)2°8 (Pr)°°33 (5%) ------------- (5-1)

W

 

With Reynolds number at 1.23 x 104, heat transfer coefficient of water
side was calculated to be 312 Btu/hr-ft2-'F. The heat transfer
coefficients on the air side for Reynolds number of 4.7, 5.6, 6.4,
7.3, and 9.5 x 104 were experimentally determined to be 11.5, 13.6,
14.9, 16.7, and 20.0 Btu/hr-ft2-'F, respectively. The detailed
procedures for calculating air side heat transfer coefficients will be
presented in Chapter VI. Comparing the heat transfer coefficient

of water side with those of the air side indicates that the former
was much higher than the latter. This result confirms the assumption
of uniform wall temperature. Kreith (1966) suggested an equation

to evaluate the heat transfer coefficient of air flow passing through
a duct with uniform wall temperature.

T n

(sc)a (Pr) = 0.02 (Re);0°2 cfm) ------------------ (5-2)
W

UH

where: n 0.575 for gas heating

n 0.15 for gas cooling

- 33 _

In case of the entrance region, the heat transfer coefficient obtained
from equation (5-2) can be modified as:

hea = ha (1 + 6D/L) ------------------------------------- (5-3)

for D/L > 20
With the previous Reynolds numbers of the air side, heat transfer
coefficients calculated from equation (5-2) and (5-3) were compared
with the previous experimental results and are presented in Figure 5.2.
Since acceptable agreement was observed, these results support the
assumption of uniform wall temperature for the experimental set-up in this
investigation.

The characteristics of the experimental heat exchanger are listed

in Table 5.1.

Temperature Measurement

 

According to Juveland _£‘_l. (1964) and Zarbrodsky (1966), there
are diverse opinions regarding temperature measured by unshield
thermocouples in a gas-solids mixture system. Other researchers assumed
the unshield thermocouples measured some mean temperature between the
temperature of the air and solids. Ciborowski and Roszak (1966) indicated
that bare thermocouples measured the particles temperatures. It is
difficult to decide which of the three opinions are correct.

It is obvious that when a particle at a given temperature is
submerged in a fluid at another temperature, the temperature of both
phases must come to equilibrium after certain period of time or distance.

If the temperatures of both phases attain equilibrium, the bare thermo-

couples measure the unique temperature -- the mixture temperature.

 

Nusselt Number of Air - (haD/ua)

300

250

200

150

100

- 34 _

Calculated from Kreith Proposed Eq.
0 Experiments 1 Res ul ts

 

 

t._/\, l I l 1 1 L_J 4

l 3 4 5 6 7 8 9 10x10
Reynolds Number

Figure 5.2 Experimental Nusselt number of air compared
with Kreith proposed equation for constant
wall temperature.

35

:um\m.1NDM1H£ Nmooo.o

:um\moINuMuH£ omooo.o

houumuu£\=um ¢.m

mo- umnus\aum Nam

N
«0- x mm.-
H;\QH coon
um Nomno.o

Now mommmo.o
mum mnawao.o
um mmmm.o
mum ¢¢m0.~
mum mmom.~
um u

um noma.o
um ~moo.o

um mmoma.o

Howcmsuxm

um .oocmumamop Hams 03:9

3
m .mocmummmmu memo umumz

M .Haws menu mo muw>wuunucou HmEuosa
3: .omwm nouns mo uaomowwwmoo powmcowu Doom
memo noum3 mo amass: mvfiocmmm
mama Beam nouns
m
o .uouoamam ucoam>wnvm
momma nuances mo mono coHuoom mmouu
m .onsu Macaw mo mono cofluuom mmouo

“Na .Hfiozm mo Handsome umccH

oH< .onsu mo mono mummpsm House

HH< .onsu mo mono commune noccH
A .onsu mo :uwaog

can .wnsu mo umumemwo nouso

n .mmmcxuwnu AHmB

man .onnu mo Handsome umccH

ummm mo momumfiuouomumso H.m manna

- 36 -

This mixture temperature was employed to determine heat transfer
coefficients in this investigation.

The thermocouple locations are shown in Figure 5.3. Temper-
ature measurements were conducted at either entrance or exit positions
of each phase using copper-constantan thermocouples and a Honeywell
lZ-point recording potentiometer. Temperafures of each thermocouple
were automatically recorded every 60 seconds. For the case of equal
inlet temperatures, the temperatures which were measured at locations
1, 2, and 3 were equal and were assumed to be the inlet conditions. For
the other case, powder and air were held at different temperatures.
The exposed thermocouple at location 3 was assumed to measure the
inlet temperature of air. The particle inlet temperature was calculated
from the initial temperatures of air and powder which were measured at
locations 1 and 2, respectively based on enthalpy balance under the

adiabatic conditions. The equation for particle inlet temperature is:

 

Vapaca
= +——____ - ---------------------------- -
Tpi T2 v p 0 (T1 T3) (5 4)
P P P
The inlet temperature of mixture was obtained by:
npacaTfi pr pp T2
T , = ------------------------------- (5-5)
ml p C+ V PpC
Va 8 a pp p

Since the temperature profile within the inner tube of the heat exchanger
was not plug-like, the outlet temperatures were measured at three
different locations: 1/16, 4/16, and 15/16 in from the inner wall.

The temperature of wall was assumed to be that of water within the

annular space. From the investigation of Heldman t 1. (1969), the

- 37 -

equilibration distance for non-fat dry milk was approximately 4 ft.
from the particle feed position. To be sure that the equilibration
temperature was being measured, two thermocouples were located far down
stream -- 7 ft. from the exit of heat exchanger.

Due to the centrifugal force, particles tend to concentrate at
the outside curvature as it moves through an elbow. In order to
obtain the uniform distribution of particles within the heat exchanger,
3 particle redistributor was built at the entrance position of heat
exchanger as indicated in Figure 4. The particle redistributor was
a ring with l/8-in. thickness. Before the heat exchanger was connected
to the conveying tube, a plexi-glass tube was connected with the conveying
tube at the same location as the exchanger in order to observe the
effectiveness of the ring. Based on observations at the transparent
plexi-glass tube, it was found that the ring provided adequate distribu-

tion of the particles.

Experimental Procedures

 

There were two types of experiments conducted in this investiga-
tion. One type was with equal inlet temperature of air and particles.
The second type Was conducted with the particles and air at different
temperatures. For the latter case, the inlet air temperature was
varied. In the case of equal inlet temperatures, non-fat dry milk
was exposed to the room air for 24 hours in order to obtain the
uniform temperature. For the case of different inlet temperature, hot

air from the resistance heater was utilized.

-38-

 

 

 

 

 

\
J),

 

 

 

 

 

 

 

 

 

 

;, -
N
4 9
water ‘0 0 o O
outlet—5112: 567 T 10
room
temperature
0
8 - mixture to
q separator
powder
feeding particle redistributor
hot water
."—" inlet
. . .
O
2 3 I

 

 

313 inletr ’ I
—_‘T 6" |‘——

Figure 5.3 Location of thermocouples

 

 

- 39 -

Uniform wall temperature was obtained by introducing cold running
water into the hot water source which was heated by steam. The hot
water was then circulated through the annular space and returned to
the source until the water temperature attained about 180' F. The
water flow rate was fixed at 3460 lbs/hr. The temperature difference
between inlet and outlet water was no more than 2' F.

Solids introduced into the system required additional time for
establishing steady-state conditions. Gas-solids stream temperatures
were observed for steady values, which normally started 10 minutes after
feeding of solids began. Data which were taken after reaching steady-
state, were used to calculate heat transfer coefficients.

Powder which was conveyed from container I, through the heat
exchanger to the separator was held in container II for 24 hours. Before
running the next test, powder was manually moved back to container 1.
Because of the 24 hour exposure to the ambient air, the temperature of

powder was equilibrated to that of air.

Product Characteristics

 

Non-fat dry milk was used in all experiments conducted. Most of
product characteristics were taken from the information available. The
mean particle diameter was assumed to be 60 micron based on results
reported by Hayashi _£ a}, (1967). A particle density of 1.46 was
used based on data reported by Hall and Hedrick (1966) and verified
by Chen (1969). A specific heat of 0.36 Btu/lb-‘F was used in heat

transfer calculations and prediction of equations. This value was

- 40 -

predicted based on product composition and was verified by experimental
determination in a Differential Thermal Analyzer. However, the experimen-
tal results showed that the specific heat of non-fat dry milk increased
rapidly when the temperature was above 130' F. The maximum value of

0.6 Btu/lb-°F occurred at 170° F and higher temperatures resulted in
decreased specific heat. Below 130° F the specific heat was independent
of temperature and 0.36 Btu/lb-°F was used. The particle thermal
conductivity reported by Chen (1969) was 0.26 Btu/hr-ft-°F at 3.5%
moisture content and 100° F. The particle heat transfer area was

computed by a method proposed by Zenz and Othmer (1960) as:

where pp and ptp are weight density per unit volume of fluidized powder
in air and true particle density, respectively. The fluidized powder

density can be written as:

L.

Pp = G

------------------------------------------- (5-7)

-€°Joc’

._2 +
pa

For the case of dilute transport, Ga/pa is much greater than GP/pp. The
term of Gp/pp , therefore, can be neglected. The particle heat transfer

area can be simplified to:

6paG

A T—2 --------------------------------------------- (5-8)
p G
P P a

According to this expression, the particle heat transfer area can be

estimated by knowing particle size and loading ratio.

- 41 _

Dry milk is highly sensitive to moisture as indicated by Heldman
.g£._l. (1965). The experimental powder was determined to be 3.5%
moisture content by a standard "Toluene Moisture Test", as indicated
by Triebold _£.fll° (1963). In order to check whether mass transfer
occurred during the experiments conducted, tests were conducted as

a part of this investigation. The results indicated that there was

no detectable change of powder moisture content.

- 42 _

CHAPTER VI
RESULTS AND DISCUSSIONS

The experimental results of this investigation can be placed in
two categories; heat transfer results and temperature histories.

The experimental data are summarized in Appendix D.

Heat Transfer Results
According to Eckert (1963) and Kreith (1966), heat transfer

rate in a shell-and-tube heat exchanger can be defined by:

qw = UnDLATm -------------------------------------------- (6-1)

where ATm is the logarithmic mean temperature difference (LMTD) and

can be expressed as:

 

 

ATmi - ATmo
AT =
m AT
1n( mi)
ATmo
where AT.=T.'T
m1 m1 w
ATmo g Tmo - Tw

Heat transfer rate (qw) can also be determined by the enthalpy increase

of the gas-solids mixture,

qw = (Gaca+ cpcpxtrmo - Tmi) ---------------------------- <6-2)

_ 43 -

By combining equation (6-1) and (6-2) the following equation is obtained:

= (GaCa+GpCp)(Tmo-Tmi)
fiDLATm

 

u ------------------------------- (6-3)

0n the right hand side of equation (6-3), the parameters can be either

measured or obtained from the information available as indicated in

F,
I
Chapter V. The overall heat transfer coefficient U, therefore, can i
i
be evaluated. The overall heat transfer coefficient (U) is also defined i
i
by: i
i
11 1 1 i
—=—+—+— .................................... - 1
U h k h (6 4) i
w t m L

Equation (6-4) can be written in another form:

= + ....................................... ..
R0 RW Rt + Rm (6 4)

where R0: l/U, overall resistance
Rw: l/hw, water side resistance

Rt: b/kt’ tube wall resistance

Rm: 1/hm, mixture side resistance

For 18-8 (chrome-nickel) stainless steel, thermal conductivity (kt) is
equal to 9.4 Btu/hr-ft-°F as given by Eckert (1963). The wall thickness
(b) was 1/16 in. resulting in a tube wall resistance (Rt) of 0.00056
hr-ft2-°F/Btu. The heat transfer coefficient on the water side was
evaluated by equation (5-1) as 3l2 Btu/hr-ft2-°F. The water side
resistance (Rw) was then determined to be 0.00032 hr-ft2-°F/Btu. By
knowing Rt’ Rw’ and U from the experimental results, the mixture side

resistance was evaluated from equation (6-5) which can be rearranged as:

R = a - 0.00088 --------------------------------------- (6-6)
m 0

- 44 -

From equation (6-6), heat transfer coefficients of the mixture side
(hm) were computed and hence the mixture Nusselt number,

hmD
(Nu)m = .12—
a

The mixture Nusselt numbers calculated by these procedures are presented
in Appendix D and E and are plotted on log paper in Figures (6.1) to
(6.3). Figures (6.1) and (6.2) give the results of equal inlet temper-
atures (EIT) and the results in Figure (6.3) were obtained with different
inlet temperatures (DIT). In these figures, mixture Nusselt numbers are
plotted versus either mixture Reynolds number or the convective capacity
parameter (CCP). The slopes were obtained by a least square fitting
procedure on a Mathatron 4280 computer.

The slapes of mixture Nusselt number versus Reynolds number for constant
convective capacity parameters of 1.0, 1.54, and 1.84 were obtained to
be 0.8, 0.62, and 0.59, respectively, and are presented in Figure (6.1).
For the air stream alone( convective capacity parameter equal to unity)
the slope of 0.8 is the same as equation (5-2) as proposed by Kreith
(1966). For the case of uniform wall temperature, Farbar 3E 21. (1957)
and Danziger/(1963) indicated that the slopes of mixture Nusselt number
versus mixture Reynolds number in the fully developed region were 0.6 and
0.66 respectively. According to Depew _t'al. (1963), the gas solids
flow is fully developed if there is a constant pressure gradient along
the axial direction. At the entrance region studied in this investigation,
the average slope was determined to be 0.61 which is between the results
of Farbar t 1. (1957) and Danziger (1963). The average slope was

computed as the arithmetic mean of 0.62 and 0.59. By comparing the

Mixture Nusselt Number (hmD/ka)

- 45 -

400 r.

300-

250-

200L-

150?-

 

 

 

100‘ A 1 I L J 1 l L L J
1 3 4 5 6 7 8 9 10
Reynolds Number (DVa/ua) x 104

Figure 6.1 Mixture Nusselt number vs. Reynolds number for
various convective capacity parameters.

- 46 -

results of this investigation with the results of Farbar 35 El (1957)

and Danziger (1963), good agreement is indicated. From equations

(5-2) and (5-3), proposed by Kreith (1966), it is obvious that

heat transfer rate for the air stream at either the entrance of the fully
developed region depends on the 0.8 power of Reynolds number. The

same relationship was obtained in this investigation. Heat transfer

rates of the gas-solids system depends on the 0.6 to 0.66 for both
entrance and fully developed regions.

The other relationships between mixture Nusselt number and convective
capacity parameter were determined. For the case of equal inlet temper-
atures, the slopes of mixture Nusselt numbers versus convective capacity
parameter for constant Reynolds numbers of 4.7, 5.6, and 7.3 x 104 were
found to be 0.79, 0.73, and 0.72, respectively. Figure 6.2 illustrates
those results. For the second case, the inlet temperatures of particles
and air were held at the different temperatures. In order to investigate
whether there was a difference in heat transfer rate between equal and
different inlet temperatures, the mixture Nusselt number was plotted
versus the convective capacity parameter. The slopes of mixture Nusselt
numbers versus convective capacity parameter for constant Reynolds
numbers of 4.7 and 5.6 x 104 were determined as 0.72 and 0.73, respectively
as illustrated in Figure 6.3. Comparing the results of two cases
indicates that there was no significant difference in heat transfer
rate between equal and different inlet temperatures. The experimental
results of both cases, therefore, were combined in order to establish
the general correlation. If the combined data of mixture Nusselt number

tunnber are divided by Reynolds number to the 0.61 power [(Nu)m/(Re)£°61 ]

Mixture Nusselt Number (hmD/ua)

400 -

300

250

200

150

 

- 47 -

 

 

1.0 1.5 2.0 2.5 3.0 4.0 5.0
G C
Convective Capacity Parameter (l +-E£E£)
a a
Figure 6.2 Mixture Nusselt number vs. convective capacity

parameter for various Reynolds numbers with equal
inlet temperatures.

Mixture Nusselt Number (hmD/ua)

400

300

250

200

150

100

 

 

\x
\Q
(0+
IC).
’ v
F
‘10 Q
a \
® 4-
“3‘
__ i
X 0
1— O a
"a ,'X
‘L
9"“ 3
—9\,0 Q
l J l l J
1.0 1.5 2.0 2.5 3.0 4.0
G C
Convective Capacity Parameter (1 + CECE)
a a

Figure 6.3 Mixture Nusselt number vs. convective
capacity parameter for various Reynolds
numbers with different inlet temperatures.

- 49 -

and plotted versus the convective capacity parameter, it reveals a

0.73 power relationship as illustrated in Figure 6.4. The value of the

vertical intercept for the convective capacity parameter of unity

was found to be 0.17. The standard deviation was calculated to be 17%.
For the same uniform wall temperature system, a considerably different re-

sult was observed between the entrance region of this investigation

and the fully developed region in the research by Farbar _£ 31. (1957),

Tien _£__l. (1962) and Jepson £3 al. (1963). Farbar _£‘_l. (1957)

showed that there was no increase of heat transfer rate at a loading

ratio of less than one. Tien £5 al. (1962) and Jepson _t al. (1963)

indicated that due to the inertia, solid particles could conceivably

reduce the scale of turbulence in the air when the loading ratio was

less than one, also. The results of this investigation illustrated

that any addition of solid particles increases the heat transfer rate

from wall to the mixture. This phenomena may be due to the addition

of particles at the entrance region causing the change in air flow

characteristics. After the addition of particles, the air stream must

accelerate particles to the constant velocity as well as distri-

l>ute particles within air stream according to the momentum balance.

Tfliis change of the air flow characteristics may continue from the

particle entrance location to where the fully developed flow is attained

arid consequently increase the scale of turbulence in the entrance region.

vae heat transfer rate could be increased, also.

0.61
a

(Nu)m/ (Re)

0.50

0.40

0.35

0.30

0.25

0.20

 

-50-

   

0.73
0 experimental data

correlated line

 

0.15s-
0 61 G C 0.73
= . ' + °.
(Nu)m 0 l7 (Re)a (1 Gaca) (i7/)
0.
1 1 J 1 1 1
1.0 1.5 2.0 2.5 3.0 4.0

G C
Convective Capacity Parameter (1 + CECE)
a a
Figure 6.4 The combined Nusselt number versus convective
capacity parameter.

- 51 -

From the literature review, most investigators correlated mixture
Nusselt number in terms of Reynolds number (Re)a and loading ratio
(Gp/Ga)' The exception is the Babcock & Wilcox Co. (1959) work which
correlated the mixture Nusselt number in terms of Reynolds number and the
convective capacity parameter. Tien's (1960) theoretical analysis
showed that mixture Nusselt number was a function of Reynolds number
and convective capacity parameter instead of loading ratio. He also
indicated that the increase of heat transfer rate by addition of particles
to an air stream was due to the high volumetric specific heat of particles.
From Tien's (1960) investigation, it was illustrated that the different
heat transfer rates would be caused by different materials which possess
different specific heats. If the loading ratio is used to correlate
the experimental results, the correlation may be valid for a specific
material only. In order to generalize the correlation, heat transfer
results were correlated in terms of mixture Reynolds number and convective
capacity parameter in this investigation. The correlation is:

G C 0.73

G C ) (i7%) ---------------- (6-7)
aa

(Nu)m = 0.17(Re);)°61 (l +

Equation (6-7) is valid for convective capacity parameter greater one

land for the entrance region of a constant wall temperature system.

Temperature Histories

 

Due to the existing disagreement regarding temperature measure-
metit in gas-solids systems, the measured temperature profiles at exit
POEBitions of the heat exchanger cannot be assigned to any one phase.

Th6! only unique temperature which can be obtained is the mixture temper-

_ 52 -

ature of both phases at a downstream location. In order to avoid the
controversy, this mixture temperature was used to calculate heat transfer
coefficients. The measured temperatures are presented in Appendix D I
and E. With the aid of equation (6-7), the temperature histories
of gas-solids system can be predicted from the derived equations
of Chapter 4.
For the case of equal inlet temperatures, the temperatures of air
and particles can be predicted from equation (4-17) or (4-18) respectively.
The mixture temperature can be predicted by the computed temperatures
of air and particles according to equation (4-5),
= VapaCaTa + vpppCpr _____________________________ (4_5)

+
m Vapaca VpppCp

T

 

The computed temperatures are presented in Appendix F. Because mixture
temperature is the only property which can be measured accurately, the
mixture temperatures which were predicted at the exit Position of the
4-ft heat exchanger were compared with those from the experiments.

The comparison are presented in Figure 6.5 and 6.6 where good agreement
is indicated. As indicated previously, heat transfer rate is a function
of the convective capacity parameter rather that the loading ratio.
However, the loading ratio can be used in place of the convective
capacity parameter if a specific material is considered. In order to
illustrate how the degree of loading of particles influences the
temperatures of both phases, the loading ratio will give a more concise
explanation than the convective capacity parameter. In Figure 6.5

and 6.6, the temperature histories, therefore, are plotted versus

the loading ratio. For the case of different inlet temperatures,

- 53 -

the outlet temperatures of air and particles can be predicted from
equations (4-15) and (4-16) respectively. The mixture outlet tempera-
ture can be predicted by the computed temperatures of air and particles
according to equation (4-5). Equations (4-15) and (4-16) illustrate
that the dhmensionless temperature ratios of air and particles are

in terms of (Tpi - Tai)/(Tpi - Tw)' It is impossible to plot the results
of the different inlet temperature case since (Tpi - Tai)/(Tpi - Tw) was
not controlled as a constant value in the experiments. For this reason,
the predicted temperatures of mixture and both phases are presented

in Table 6.1 instead of plotting. By comparing the predicted and
measured temperatures of mixture in each individual experiment, good
agreement is obvious.

Since the temperatures of air and particles cannot be measured
directly, there is no direct method to verify the derived equations.
Fortunately, the mixture temperature of air and particles can be
obtained experimentally or predicted theoretically. The predicted
mixture temperature comes from the computed temperatures of air and
particles. If the measured and predicted mixture temperatures are in
agreement, the predicted air and particle temperatures can be assumed
to be the true temperatures of both phases. The derived equations, there-
fore, can be verified by this indirect method. From previous comparisons
of the measured and predicted temperatures, the derived equations from
(4-15) to (4-18) provided a good prediction of the air and particle

tempera tures .

54-

numaEmu

 

.«oH x
vmuoflkua msu sums

o.m u mAwmv you new one .musuxae .mmaoauuma mo mmusum
wusuxwa mo mopsumumaamu umauso omuswmms wsu wawumaaou m.c muswwm

m a
A 0\ 0V oHumm wcawmoa

 

 

 

 

 

o; as we so as m5 10.0 m5 «.0 no
1. _ a _ _ _ a _ _ _ so
an. 1 an. m
3 m u m m.o
B 1 B
3 .—
a - .a so
3 s n
H 1 B 0.0
3
a - an. no
3| 0
B I H
mm «HHQ co
3
1IAWI1, “W1 Em .musuxwaxlli
mm .mm ofiuuma
fl w.o
338363 33ow consumes o
mmusumumaamu vmumaaono
o.o

o.H

 

s
( 9 ‘me ‘d9) sarnnexadwal ssaIuorsuamra

55

. OH x n.¢ u mAwmv now Ham was .wusuxwa .mmaowuuma mo mmusumumaamu
vouowomwa wsu Lugs musuxfls mo mousumumaEmu umfiuso umusmmwe mau wswumaaoo 0.0 muswwm

A00\000 00000 wca0moa

 

 

 

 

 

0.a 0.0 0.0 h.0 0.0 m.0 0.0 0.0 ~.0 3.0
_ _ _ _ fi _ — fl a _
3a - as 1 00
0 1
3H - e
3a - He 1 50
3a - as
3H - as 00
mflllld1.u
a - e
111 10. O
0111111 E0 .9533":

 

am .maowuuma

salnnexadmal ssaIuorsuamrq

OHDU QR UQEUU wHSUKHB kan—wflwa

mmuaumumaamu omuownmua

m.o

m.o

 

- 56 -

 

 

 

Table 6.1 Comparison of the predicted and measured mixture temperatures
(Different Inlet Temperature)
Calculated Measured
Loading (°F) _ (°F)

Tw Tai Tpi Ratio C.C.P. Tmo Tmo Re
185.8 140.5 115.7 0.31 1.47 145.8 146.5 5.6 x 10
177.0 141.5 117.5 0.36 1.54 144.6 145.5 5.6
178.0 135.0 114.0 0.41 1.62 139.8 140.3 5.6
181.9 130.9 112.3 0.56 1.84 136.1 137.0 5.6
176.8 133.0 103.3 0.65 1.91 133.0 133.8 5.6
186.8 127.0 105.9 0.8 2.2 133.1 133.5 5.6
197.3 130.0 108.7 0.82 2.23 137.3 140.0 5.6
183.5 127.0 111.8 1.04 2.56 133.5 134.0 5.6
180.8 137.4 126.7 0.36 1.78 145.7 146.4 4.7
183.3 139.5 120.5 0.39 1.82 147.1 146.8 4.7
179.3 138.1 113.8 0.47 1.85 142.4 141.6 4.7
184.0 140.0 113.7 0.53 1.92 142.3 141.8 4.7
179.0 130.8 116.0 0.56 1.84 138.9 141.0 4.7
184.3 133.5 111.5 0.67 2.0 139.5 139.5 4.7
182.5 129.0 113.5 0.82 2.23 137.1 136.8 4.7
184.0 129.0 109.7 0.93 2.4 134.3 136.0 4.7

- 57 _

A technique for measuring the temperatures of air and particles
has not been developed. The temperature profiles were measured by four
bare thermocouples at the exit location of heat exchanger in an attempt
to obtain valid comparisons. According to Farbar gt 31. (1957),
the volumetric ratio of solid to air stream in gas-solids system is
so small that the bare thermocouples which are exposed to the mixture
stream measure the air temperature only. This assumption seems plausible
at low loading ratio. By assuming the measured temperature profile is
the air temperature profile, the outlet temperature of air can be
calculated from the measured profile by the method preposed by Henderson
(1955). This average outlet temperature was determined by dividing the
conduit into a number of small areas, and finding the average over

the total area. A system for doing this is shown in Figure 6.7. The

method of calculation can be expressed as:

= l. ........................... -
Tao S (T851+T382+T5S3+T484) (6 9)
__1_ 2 2 2 2 2 2 2

R
......................... (6-10)

The measured outlet temperatures of air from the outlet temperature
profiles are presented in Appendix F and compared with those predicted
from the derived equations (4-17) and (4-18). In Figure 6.8 and 6.9,
the predicted and measured temperatures are plotted versus the loading
ratio. The comparisons illustrate an interesting result. For a loading
ratio of less than 0.56, the measured outlet temperatures of air are
exactly as same as those predicted from the derived equation. At a

loading ratio of 0.82, the measured outlet temperatures of air are

- 5g -

R = 15/16 in.
R1 = 13/16 in.
R2 = 7/16 in.

 

 

 

 

 

 

 

 

Figure 6.7 Thermocouple locations at the exit position
with its represented areas.

m
. 0H x 0.0 u Awmv you Ham 0cm .wusuxfle .mmaowuuma mo mwusumuwmeuu uwfiuso wwuuwwwua

 

 

 

 

 

0 wsu sues enqu0E was mwaoauuma mo mousumuwmaou uwauso vwnsmmme msu weaummeoo 0.0 wusmam
a
Amo\ 00 oaumm 0000000
0.0 0.0 0.0 5.0 0.0 0.0 0.0 0.0 N.0 0.0 0
_ _ a _ _ a 0 _ _
a
3e - .0.
3 m
H u H
3 a
H 1 .H 1
3 E 1
H - H
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M... a - a. -
milldfi.1
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- 61 -

similar to the predicted or measured mixture temperatures. When the
loading ratio is increased to unity, the measured outlet temperatures
are close to the particle temperature as predicted from the derived
equation (4-17). According to the assumption in the theoretical
development, there is no direct heat transfer between particles and tube
wall in the case of arbitrary wall temperatures. Particles can only

be heated by obtaining heat energy from air. The temperature of the
particles, therefore, never reaches that of air as long as the temper-
ature gradient exists between mixture and tube wall. However, the
measured outlet temperatures of air at a loading ratio of one is lower than
the predicted and measured mixture temperatures and close to the
predicted particle temperature. The temperature reading of bare thermo-
couples at a loading ratio of one, therefore, did not measure the air
temperature, but the particle temperature. These comparisons indicate
that all three controversial opinions are reasonable depending on the
loading ratio considered. Since only four thermocouples were used to
establish the temperature profiles and the loading ratio was limited
within the range from zero to unity, further study is necessary in order
to completely explain the reason for the influence of loading ratio on

bare thermocouple measurement.

- 62 -

CHAPTER VII
ANALYSIS

The following analysis for the cooling systems of spray-dried
food products will be based on the previous experimental Nusselt
numbers which were obtained during a heating process. According to
equation (5-2), proposed by Kreith (1966), heating and cooling do not
follow the same correlation of Nusselt number. The deviation due to
using the heating correlation instead of the cooling one is, however,
sufficiently small. For example, when air and wall temperatures
are 120 and 50° F respectively, the deviation of Nusselt number is less
than 5% when comparing heating and cooling correlations of equation(5-2).
The types of cooling systems. used for cooling spray-dried food
products will vary considerably depending on facilities available.
In most cases, the cooling is accomplished by mixing the dry product
with room temperature or cool air. In the dry milk plant, powder
leaves the drier at approximate 120° F as indicated by Farrall _5 a1.
(1968) and 600 pounds powder per 15,000 cubic foot air per hour are
conveyed through a tube to the separators ((Hedrick (1969)). This
loading rate is equivalent to the loading ratio of 0.56. If the con-
veying tube is exposed to the room air and heat transfer of mixture to
the surrounding depends on natural circulation of the ambient air around
timeconveying tube, the heat transfer rate due to natural circulation is

t 31. (1969). After the initial

Very low as indicated by Heldman

Ifxaur feet of conveying tube, the mixture temperature was reduced by only

 

 

 

 

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ambie

COUW

In 1
air
C851
air
sin

is

Ce

- 63 -

4% of the initial temperature difference between the mixture and the
ambient air. This is the obvious reason that dry milk plants use
conveying tube lengths of more than 100 feet in order to cool the heated
powder. The natural circulation of air, therefore, can be approximated
by adiabatic conditions for short sections of conveying tube.

There are two types of heat transfer systems to be analyzed.
In the first case, heated powder mixes with room temperature or cool
air and the mixture is conveyed at the adiabatic conditions. The second
case describes heated powder mixing with room temperature air or cool
air and the mixture is conveyed through a shell-and-tube heat exchanger
similar to the one used in this investigation. A uniform wall temperature
is assumed. Both systems have the following characteristics:

loading ratio, Gp/Ga = 0.56

Reynolds number, (Re)a = 5.6 x 104

powder inlet temperature, Tpi = 120° F

room air temperature 80° F

cool air temperature 50° F

water temperature, Tw ' 50° F

Using this information, temperatures of air and powder can be predicted.
IFor the adiabatic conditions, air and particle temperatures can be

calculated by the equations proposed by Heldman gt a1. (1969),

:1:

-i%; { l - Exp [ -A(St)d(l+1/w)x ] } ------- (7-1)

- 64 _

7f“??? = T113 { 1 - Exp 1 -A(St)d(1+1/W)X J } .......
ai pi

(7-2)
For the case of uniform wall temperature, the temperatures of both phases
can be calculated from equations (4-15) and (4-16). The calculated
temperature histories are plotted versus axial distance and are pre-
sented in Figure 7.1.

The results (Figure 7.1) illustrate how the particle and air temper-
atures vary with the axial distance and the temperatures of each phase
at the exit position of a 10 ft. long conveying tube. For the
adiabatic conditions, heated powder at 120° F was mixed with air at
80 or 50° F. The equilibrium temperatures were calculated from equations
(7-1) and (7-2) and were found to be 98 and 82° F respectively. For
the case of uniform wall temperature, the particle and air temperatures
at the exit of a 10 ft. heat exchanger were predicted as 66 and 74° F
for particles and 64.5 and 72.4° F for air from equations (4-15) and
(4-16). For the adiabatic conditions, the heat transfer potential
is the temperature difference between particles and air. After the
temperature of both phases comes to the equilibrium,the heat transfer
potential no longer exists. The temperatures of both phases remain
at the equilibrium temperature. For the case of uniform wall temper—
ature, two heat transfer potentials exist. The first is the temper-
ature difference between particles and air. The second is the temper-
ature gradient between mixture and tube wall. Each temperature poten-
tial varies with the axial distance. Theoretically, the distance for the
temperatures of air and particles to come to equilibrium is infinity

for the adiabatic condition. In the actual situation, the temperature

65

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- 66 -

difference between two phases is not detectable after a given distance
of mixing. If the distance for the temperatures of both phases to
be within 1% of their equilibrium temperature is assumed to be the
equilibration distance, this distance was calculated from equations
(7-1) and (7-2) to be approximately 3 ft. A value of 4 ft. was measured
experimentally by Heldman §£_a1. (1969).

Due to temperature lag between particles and air as indicated in
the previous chapter, the temperature of particles never reaches that
of air as long as the temperature difference exists between mixture and
tube wall. For this reason, the characteristic temperature in the case
of uniform wall temperature is not the equilibrium temperature of the
adiabatic system. From examination of Figure 7.1, it is obvious
that a maximum air temperature occurs during the process when heated powder
mixes with room air or cool air and the mixture is then cooled by the
tube wall. The location of this maximum air temperature can be obtained
by differentiation of equation(4-l6) with respect to axial distance

and setting equal to zero. The result is:

X = Dlnz ............................ (7-3)

4(St)m
1¢(St)d(1+1/W) - Tm]

 

where: 2 2
¢ (St)d(1+1/w) [ (1+w)ffgi:Tw)+(1/w)(Tai'Tgi) J

¢(Sc)d 4(St2;“_# 4(852; 1
-—;_—- -'(I;W)- ] ['ITTGI— ] [(1+1/w)(Tpi-Tw).w(Tai-TW) J

 

 

For the example under consideration, the location of maximum air temper-
ature was found to be 1.2 ft. Equations (4-15) and (4-16) illustrate

that after the air temperature attains the maximum temperature, heat

-67-

transfer rate between air and particles is small in comparison to that
between the mixture and tube wall. This result implies that the first
exponential term in equations (4-15) and (4-16) is much larger than the
second exponential term at some distance downstream from the location
of maximum air temperature. The equations for predicting temperature
of both phases downstream from the location of maximum air temperature,
therefore, can be simplified by neglecting the second exponential

term in equations (4-15) and (4-16) and the new equations are obtained

 

 

 

 

as;
: : :w = ((S:)d ) (0 l+w s g 4 s )
pi w ( )( t)d ( 1:)“.l
w - (1+w)
Ta. 4(St)m X .
{1 (1m) - <—"-———--) J Exp 1 - 7‘10.) 5 J j ------- (M)
Tpi Tw
Ta - Tw _ (St)d g
T .- T - ( ”) (0(1+w)(St)d 4(St) )
pi w _ .____JE
w (1+w)
T 1 - Tai 4(St)m
{w[(1+v)-(—E—-—T—) _ T ]Exp[- (1%,)- "‘DJ} ----- (7- 5)

pi w
These two equations can be employed to predict the temperatures of air
and particles for the case of uniform wall temperature, since the
temperature of interest is not within the first one or two feet from
the location of particle feeding.

From the previous analysis, it was illustrated that there are two
characteristic temperatures for two different cooling systems; the
equilibrium temperature for adiabatic system and the maximum air temper-
ature for the uniform wall temperature case. Both characteristic

temperatures can be attained very rapidly after particle feeding. At

the exit position of a conveying tube with 10 ft. length, the shell-
and-tube heat exchanger provides the most effective cooling result while
the cool air injection is more effective than the room temperature
air injection under the assumed adiabatic conditions.

The convective capacity parameter is a characteristic parameter
of gas-solids system during heat transfer. As indicated in the previous
chapter, heat transfer rate is dependent upon the convective capacity
parameter instead of loading ratio. However, the loading ratio can be
utilized instead of the convective capacity parameter if only one
material is considered and its specific heat is constant with respect
to temperature. Table 7.1 indicates the comparison of the outlet
temperature of powder at the exit position of a conveying tube with 10
ft. length for various loading ratios. This comparison was made to
investigate the difference between the adiabatic and the uniform wall
temperature systems when powder at 120° F is mixed with 80 or 50° F
cool air under adiabatic conditions, or then conveyed through the tube
with 50° F wall temperature. For the adiabatic conditions, the particle
temperature can be cooled below 80° F at a loading ratio of 0.36 and
with cool air injection. In the case of uniform wall temperature,
the heated particles can be cooled below 80° F at any loading ratio
1>elow 1.04 and with cool air or room air conveying. This comparison
illustrates that rapid cooling can be accomplished even at high loading
Instio and with uniform wall temperature. In order to increase cooling
reate under adiabatic condition, a low loading ratio should be employed
0!? a long conveying tube is required for natural circulation of room

airu However, the system of cool air injection with cold wall temper-

69

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-70..

ature can offer rapid cooling with a short conveying distance.
The dry milk industry is concerned about the airborne contamination

of finished products by Salmonelleae. Since the conveying tube for the

 

system of cool air injection with cold wall temperature is much shorter
than that of natural circulation system, the power required to overcome
the pressure drop of a long conveying tube can be used to draw air
through a high efficiency air filter. The dry milk industry, therefore,
can obtain high air purity and the more effective cooling at comparable
operating costs.

Reynolds number is an additional parameter which influences the
temperatures of both phases. It is obvious that Reynolds number influ-
ences the retention time of particles within the conveying tube. The
longer the retention time of particles within the conveying tube, the
more heat energy per pound of powder transmitted. This explanation
can be verified by the experimental results in Figures 6.5 and 6.6. These
two figures illustrate that at the lower Reynolds number, particle
temperatures are closer to the wall temperature. In order to obtain
cool powder, a low Reynolds number is suggested. However, Zenz (1964)
indicated that there was a minimum air velocity at which particles
can be conveyed. The lowest Reynolds number, therefore, cannot be less
than the minimum Reynolds number for particle conveying.

The particle size affects the equilibration distance for the
adiabatic condition and the location of maximum air temperature. It
does not influence the equilibrium temperature and maximum air temper-
ature, since the particle size determines the heat transfer area between

Particles and air stream. The smaller particle size, the more heat

- 71 -

transfer area. The smaller particles, therefore, require shorter

distance to attain the equilibrium temperature or maximum air tempera-

ture in the case of adiabatic condition or uniform wall temperature,
respectively. The unique parameter of particle size should be repre-

sented by the combined parameter, (St)d¢, because both (St)d and fl

include particle diameter (dp). Due to small particle size and small
relative velocity, particle surface parameter, fl, and particle Stanton
number, (St)d, are the order of magnitude of 10 and 1, respectively.

The mixture Stanton number, (St)m, however, has the order of magnitude of
10-3. By comparing the order of magnitude, (St)d¢is much greater than (St)m.
Based on this observation, the following relationship exists in equation (7-4).

(St)d9)(1+w) 4(St)m
>> ”(1”)-

 

Equation (7-4) can then be simplified further to:

ELL-.30 1 Tei'Tai 4(St)“. x
Tpi-Tw = T476 i0”) ' (Tpi- 1;) i EXP [ ' T355." '13]

------------------------- (7-6)
This equation clearly illustrates that particle size does not contrib-
ute to the particle temperature history at locations downstream from
the location of maximum air temperature.

Based on the previous analysis, particle temperature history at
locations downstream from the location of maximum air temperature is
controlled primarily by three parameters; mixture Stanton number,
(St)m, initial temperature ratio, (Tpi-Tai)/CTpi-Tw), and convective
capacity ratio, w. As mentioned earlier in this chapter, the particle

temperature of interest is not necessarily that at locations upstream

- 72 -

from the location of maximum air temperature. The temperature history
at downstream locations can be illustrated by plotting equation (7-6)
as illustrated in Figures 7.2, 7.3, 7.4 and 7.5. Since the parameter
(St)d¢ does not influence the particle temperature history, the charts
can be applied to any practical range of (St)d¢. The charts of
Figures 7.2 to 7.5 show the dimensionless temperature,(Tp-Tw)/(Tpi-
Tw), as a function of the axial distance parameter, x/D, for various
values of w, (Tpi-Tai)/(Tpi-Tw), and (St)m. Each chart has a fixed
convective capacity ratio, w. In each chart, there are three separate
graphs for initial temperature ratios, (Tpi - Tai)/(Tpi - Tw) of

0, 0.5, and 1.0. Since the charts are not continuous for the values of
w and (Tpi - Tai)/(Tpi - Tw)’ interpolation can be used for the first
approximation.

In summary, the analysis indicates that the shell-and-tube heat
exhanger will provide better cooling effectiveness. It should be
emphasized that the shell-and tube heat exchanger can cause the vapor
condensation on the inner surface if the wall temperature is lower than
the dew point temperature of air stream. The powder, therefore, may
stick to the wall and hence reduce the heat transfer rate. In order
to prevent the vapor condensation, inlet air temperature should be
maintained several degrees lower than tube wall temperature. Since
air is cooled before entering into the heat exchanger, the condensation
Will occur on cooling coils before passing over the cool surface. In
addition, the inlet air, even with 100% relative humidity, never attains
the dew point temperature as long as the temperature of tube wall is

higher than that of the inlet air. For the actual cooling design, it is

- 73 -

necessary to check the relative humidity of inlet air and its dew point
temperature. The condensation problem, however, can be neglected if the

temperature of inlet air is maintained lower than that of tube wall.

- T )/(T 1 - Tw)

(T

G)

W = 0.45
91= 1.0

5 10

Figure 7.2

_ 74 -

 

15 20 25 30 35 40 45 50 55
x/D

Dimensionless temperature history of particles along
the axial distance during pneumatic transport of
spray-dried food products.

60

(Tp - Tw)/(TPi - Tw)

DJJ-‘UIO‘QO

.9 Ln a~ <0 23

w = 0.84
91= 1.0
5 10
Figure 7.3

 

- 75 -

15 2 25 30 35 40 45 50 55 60
x/D
Dimensionless temperature history of particles along

the axial distance during pneumatic transport of spray-
dried food products.

(Tp - Tw)/(TPi - Tw)

 

 

61= 0.5

- _. a- .—

I

. tit-1} fit
_ +_ WT-..
I
—.-'.-_.‘ ._.,-.. ____.

I

----

5 10 15 20 25 30 35 40 i 45 50 55 60
x/D

Figure 7.4 Dimensionless temperature history of particles along
the axial distance during pneumatic transport of spray-
dried food products.

('1:p - Tw)/(Tpi - Tw)

_ 77 _

 

W = 1.5
9.“ 1.0
i

5 10

Figure 7.5

 

15 20 25 30 35 40 45 50 55 60

x/D

Dimensionless temperature history of particles along the
axial distance during pneumatic transport of spray-dried
food products.

_ 7g -

CHAPTER VIII
CONCLUSIONS

In a gas-solids mixture systems, mixture Nusselt number depends
on 0.6 to 0.66 power of Reynolds number disregarding to entrance
or fully developed region.

There is no significant difference in the mixture heat transfer
coefficient between equal and different inlet temperatures of
air and particles in gas-solids system.

The mixture Nusselt number was correlated as:

G C 0.73

(Nu)m = 0.170103“ (1 + 59-53) (177.)
a a

This empirical equation is valid for the convective capacity
parameter greater than unity and within the entrance region in
the case of uniform wall temperature.
Heat transfer rate as expressed by the mixture Nusselt number increases
with loading ratio in the entrance region, as compared to the
decrease in rate between loading ratios of 0 and l for the fully
developed region.
Bare thermocouples measure the air temperature when the loading
ratio is below 0.56, the mixture temperature at the loading ratio
of 0.82, and the particle temperature at loading ratios greater

than unity.

Based on the verification of the indirect method, the temperatures
of air and particles are adequately predicted by equation (4-15)
to (4-18) for the case of uniform wall temperature.

After the air temperature attains the maximum temperature, heat
transfer rate between air and particles is small in comparison
to that between mixture and tube wall.

Particle size only determines the location of maximum air temper-
ature and does not influence the maximum air temperature itself.

Based on the analysis of different cooling systems, a shell-and-
tube heat exchanger should provide better cooling effectiveness

at a high solid loading ratio.

-80-

CHAPTER IX

RECOMMENDATIONS FOR FUTURE WORK

Determine the mixture Nusselt numbers for linear wall temperature
and constant heat flux systems and verify the derived equations.

Investigate the influence of loading ratios on the bare thermo-
couples reading more thoroughly.

Determine the mixture Nusselt numbers with different particle
sizes and find out the dependence of mixture on the particle
sizes.

Determine the mixture Nusselt number with different materials and
verify the dependence of mixture Nusselt number on the convective
capacity parameter.

Measure the temperature histories along the axial direction and
compare to the calculated results.

Measure the scale of turbulent in the entrance region within the
loading ratio of unity and check with that of fully developed

region.

-31-

BIBLIOGRAPHY

- 32 -

BIBLIOGRAPHY

Allen, T. 1968. Particle Size Measurement. Chapman and Hall, LTD.,
London.

Babcock and Wilcox Co., Atomic Energy Division. 1959. Cited by
Danziger, 1963.

Bakker-Arkema, F. W., J. R. Rosenau, and W. H. Clifford. 1969.
Measurement of Grain Surface Area and Its Effect on the Fixed
and Moving Beds of Biological Products. American Society of
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Baumeister, T. 1958. Mechanical Engineers' Handbook. McGraw-Hill
Book Company, Inc. Sixth Edition.

 

Belden, D. H. and L. S. Kassel. 1949. Pressure drops encountered
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lines. Ind. and Eng. Chem. 41-1174.

Breiman, L. 1952. Theory of heat and mass transfer from a slowly
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Chen, A. C. C. 1969. Mechanism of Heat Transfer Through Organic Powder
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Ciborowski and Roszak. 1954. Cited by Zabrodsky, 1966.

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Davidson, J. F. and D. Harrison. 1963. Fluidized Particles. Cam-
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Depew, C. A. and L. Farbar. 1963. Heat transfer to pneumatically
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Drake, R. M. 1961. Cited by 800, 1967.
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Translated by Joseph, F. G. McGraw-Hill Book Company, Inc., New
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Farbar, L. and M. J. Morley. 1957. Heat transfer to flowing gas—solids
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Farbar, L. and C. A. Depew. 1963. Heat transfer effects to gas-solids

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Farrall, A. W., A. C. Chen, P. Y. Wang, A. M. Dhanak, T. I. Hedrick
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Farrall, A. W., A. C. Chen, P. Y. Wang, and D. R. Heldman. 1968b. Field
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Hall, C. W. and T. I. Hedrick. 1966. Drying of Milk and Milk Products.
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Hanrahan, F. F., R. L. Selman and B. H. Webb. 1967. Experimental
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Hayashi, H., D. R. Heldman, and T. I. Hedrick. 1967. A comparison
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Hedrick, T.I. 1969. Personal Communication.

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Heywood, H. 1962. Symposium on the interaction between fluids and
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- g4 -

Holman, J. P. 1968. Heat Transfer. McGraw-Hill Book Company,
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Hsu, S. T. 1963. Engineering Heat Transfeg. Princeton, New
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Jepson, G. A. and W. Smith. 1963. Heat transfer from gas to wall
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Kreith, F. 1966. Principles of Heat Transfer. International Textbook
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Kubnek, G. R., P. C. Alien, and W. H. Gauvin. 1968. Heat transfer
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Sissom, L. E. and T. W. Jackson. 1966. Temperature distribution of
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Society of Mechanical Engineers Paper No. 65-HT-l7.

Sissom, L. E. and T. W. Jackson. 1967. Heat exchanger in fluid-dense

particle moving beds. Trans. of the ASME, J. of Heat Transfer.
89:1-60

 

- g5 -

Sleicher, C. A., Jr. and M. Tribus. 1957. Heat transfer in a pipe
with turbulent flow and arbitrary wall-temperature. Trans of the
ASME. 79:789-797.

800, S. L. and J. A. Regalbuto. 1960. Concentration distribution in
two-phase pipe flow. The Canadian Journal of Chem. Engr. 160-
166.

800. S. L. 1967. _F1uid Dynamics of Multi:phase Systems. Blaisdell
Publishing Company. Toronto, London.

 

Stoke, G. C., Cited by $00, 1967.
Tchen, C. M. 1947. Cited by $00, 1967.

Themelis, N. J. and W. H. Gauvin. 1963. Heat transfer to clouds of
particles. The Canadian Journal of Chem. Engr. 1-6.

Tien, C. L. 1960. Heat transfer by a turbulently flowing fluid-solids

mixture in a pipe. Trans. of the ASME, Journal of Heat Transfer.
83C:183-188.

Tien, C. L. and V. Quan. 1962. ASME Paper No. 62-HT-15. Cited by
800, 1967.

Triebold, H. 0. and L. W. Aurand. 1963. Food Compnsition and Analysis.
VanNostrand, Princeton, New Jersey.

 

Wen, C. Y. and H. P. Simons. 1959. Flow characteristics in horizontal
fluidized solids transport lines. Inc. & Eng. Chem. 53:51-53.

Wilkinson. G. T. and J. R. NOrman. 1967. Heat transfer to a suspension
of solids in a gas. Trans. Inst. Chem. Engrs. 45:T314-T318.

Yannopoulos, J. C., N. J. Themelis, and W. H. Gauvin. 1966. An
evaluation of the pneumatic transport reactor. The Canadian
Journal of Chem. Engr. 231-235.

Zabrodsky, S. S. 1966. Hydrodynamics and Heat Transfer in Fluidized
Beds. The M.I.T. Press, Cambridge, Massachusetts, and London,
England. (Translated by Zenz).

Zenz, F. A. and D. F. Othmer. 1960. Fluidination and Fluid Particle
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I and EC Fundamentals. 3(1):65-75.

-86-

APPENDICES

- g7 -

APPENDIX A
SOLUTIONS FOR THE CASE OF UNIFORM WALL TEMPERATURE
In equation (4-9) and (4-13), there exists three unknown such that

equation (4-5) must be used to obtain the solutions of Ta and Tp. Before

solving the governing equations, the following parameters should be

 

 

 

 

defined:
d (Re)d Pr m (Re) Pr
m
V p C
w =._E_E_E p = AD
V p C
a a a
4h 4 4q
= ____EL_. = _. w
B1 v p c D D(St)m Be v p c D
a a a a a
h A
B2 ' v p c = (St)dA
a a a
h A (St) A
3 V p C w
P P P
B = vapaca =___£__
4 V p C + V p C 1 + w
a a a p p p
V p C
B = P PHPI
5 V p C + V p C
a a a P P P

With the defined parameters, equation(4-5), (4-9), and (4-13) can be
rewritten as:

rm = 134 + BSTP ------------------------------------------ (A-l)

dT
a a:
7‘;- B1(Tw - Tm) - B2(Ta - TP) ------------------------- (A'Z)

-88-

dT
__E = - ....................................... -
dx B3(Ta Tp) (A 3)

Equation (A-3) can be rearranged as:

T =:
a 3

By substitution of equation(A-l) into (A-Z), the new equation is

dT
__2 + ........................................ -
dx Tp (A 4)

w'H

obtained as follows:

dT
a

———+ = - -------------- -

dx (B2 + B1B4)Ta B Tw (B1B + BZ)Tp (A 5)

l 5

Eliminating air temperature Ta from equation (A-5) with equation (A-4),

the differential equation for particle temperature is:

dZT dT
+ + + __2.+ = ------------- -
dxz (32 B3 3134) dx BIBBTP 1311331:w (A 6)

Equation (A-4) and (A-6) are the new governing equations. By using

the method of operator, where m =-§§;2, equation (A-6) is written as:

2 = ................ .-
[ m + (B2+B3+BIB4)m + 3133 JTP 311331:w (A 7)

The solutions of equation (A-7) should be composited by the particular
solution and complementary solution. The particular solution can be
obtained by the method of undetermined coefficients. Let the particular

solution and complementary solution by Y1 and Y2, respectively and

let Y1 = C3, where C3 is constant. By substitution of Y1 into equation
(A-7), the result is:
= ----------------------------------------- A-8
C3BIB3 B1B3Tw ( )

- 39 -

Therefore, the particular solution is:

Y1 = C3 = Tw

The complementary solution is obtained by the following procedures. Let

m2 + (B2 +B +B1B4)m + B B = 0 --------------------------- (A-9)

3 1 3

The solution to the equation (A-9) is:

 

-(B2+B3+B1B4) i: 2vrB2+B3+B14B - 43133
m = ------------ (A-10)

 

The particle surface area per unit mixture volume is defined by

equation (5-6). By substitution of equation (5-6) into parameter B

 

 

2
and B3, equation (A-10) can be simplified as:
--(B2 +B 32'1'B1B4) ifBz +83 -B 134 )2
m = ---------------------- (A-ll)
Therefore m = -(B +B )2 B B ------------------------------ (A-12)
’ 2 3 1 4
Since B1B2,B3, and B4 are real number, the complementary solution

are two distinct real roots and can be written as:

Y = ClExp [ -(B2+B3)x ] + c

2 Exp [ -B1B4x ] ------------- (A-13)

2
Then the complete solution for equation (A-6) is

TP = Y1 + Y2 -------------------------------------------- (IX-14)

or, Tp = ClExp [ -(B2+B3)x ] + c Exp [ -B1B4x ] + Tw ------- (A-15)

2

The boundary conditions of the system are:

0{T:= = Tp _____________________________________ (A-16)

T31

l and C2 are determined to be:

-(BlB4) (T -T w) - B 3(Tu-1:91)
c1 = B ”+3 _B B -------------------- (A-17)
2 3 1 4

With the boundary conditions, C

 

- 90 -

_ (B2+B3)(Tpi-Tw) + (Tai-Tpi)
- B +3 -3 B ----------------------- <A-18)
2 3 l 4

C2

Therefore, the final solution for particle temperature is:

- + -
_ _ [ B1B4qpi Tw) B3(Tai T ')

 

 

1
- _ ] Exp [ -(B +B )x ] +
p w B2+B3 B134 2 3
(B +B )(T .-T ) + B (T .-T .)
E 2 3 P; +; _B BB 81 9‘ ] Exp [-BlBax] ------- (A-l9)
2 3 1 4

By substitution of equation (A-l9) into equation (A-4), the solution

for air temperature is obtained:

 

 

B B B (T -T ) +'B (T .-T .)
2 l 4 - w 3 alAyfll
T -T = (——) [ -91 J Exp [ -(B +B )x J +
a w B3 B2+B3-B1B4 2 3
(BB-BlBa) [ (B2+B3)(TPi-Tw) + B3(Tai-TEi) 1
B3 B2+B3-B1B4
Exp [ -BlB4x J ------------------------------------- (A'ZO)

The parameters B are only used for the shorthand symbol and do not have

physical meaning. Now we substitute B parameters in terms of (St)m,

(St)d, and w as defined previously, equation (A-19) and (A-ZO) can be

written in the dimensionless forms,

 

 

 

 

 

T -T (st)
Tpi- w w 0(1+w)(St)d 4(St)m
w ’ (1+w>

' [(1+w) (Kai-T81) J ( 4(St)” x)

a. .— Exp ._ ._._._._._.._ ——
1 Tpi-Tw (1+W) D

4w(St)m T .-T . ¢(1+w)(3t)
_ pi a1 _ d x

[9(l+w)(St)d (Tpi-Tw ) J Exp ( w D) }

......................... (A-Zl)

- 91 -

-T (St)d

 

 

 

 

 

a w 9
---- = (fl d) ( )
TPi-T ¢(l+w)(St)d 4(St)d
w (1+w)
, T i-Tai 4(St)m X
i W [(1+W)-(fflf:E--)] EXP (- (1;;3"5
p1 w
4w(St)m 4w(St)m T .-T .
[1-1 1 ~6—3i——§1) J
0(1-iw)(St)d 0(1+w)(St)d T .-T .
p1 a1
0(1+W)(St)d x -.
Exp 1 - w B J j ------------------------ (A'ZZ)

In the case of the equal inlet temperature of both phases, equation

(A-21) and (A-22) are simplified as follows:

 

 

 

 

 

 

 

 

T -T (St)
Tpi- w ( )( t)d - t m
w (1+W)
4(St)!“ X 4W(St)m
{11+W) EXP [-'fj:;3"53 ‘ p(1+w)(St)d J
o<1+w)(5t)d
EXP [ - w B] j ------ ' ------------------- (A-23)
Eé_:_EE_ = ((S:)d *) ( p )
Tpi- Tw ”(1W)(St)d Egg-[B
w - (1+W)
4(St)m x 4W(St)m
{B(1+“) EXP [ (1+w) n 1 ' [1 ¢(1+w)(St>d J
4w(St)m ¢(1+W)(St)d x

 

[Wig] Em- . 53} --------- (“4)

- 92 -

APPENDIX B

SOLUTIONS FOR THE CASE OF LINEAR WALL TEMPERATURE

With the parameters defined in Appendix A, equation (4-20),

(4-5), and (4-13) can be written as:

dT
a _ _____________________ _
dx B1(Tai +'ax Tm) B2(Ta Tp) (B 1)
= + .................................... ..
Tm 34Ta BSTP (B 2)
dT
de =33“. - Tp) ------------------------------------ (B-3)
1 dT
or, T = —- '—-B + T --------------------------------------- (B-4)

a 3 dx p
By substitution of equation (B-2) into (B-l), the new equation is

obtained:

dTa
+ (B2+B1B4)Ta + (BB

3;- anp = 31(Tai‘1'ax) -------- (Is-5)

1 5
Eliminating air temperature Ta from equation (B-5) with equation (B-4),
the differential equation for particle temperature is:

d2T

—-£+(B23+B+BB)-E+BI3BT=BB(T.+X) ------- (B-6)
dx2 1 4 3p 1 3 31

Equation (B-4) and (B-6) are the new governing equations. Using the

method of operator gives:

[m2231+(B+B+BB41)m+B1Bp3]T=BBBa(Ti+ax) ------ (B-7)

- 93 -

By comparison of equation (B-7) with equation (A-7), it is obvious that
the complementary solution is exactly as same as that of case I which
is:

Y2 = ClExp [ -(BZ+B3)x J + C2Exp [ -B134x J ------------ (B-8)

The particular solution can be obtained by the method of undetermined
coefficients. Let the particular solution of equation (B-7) be:

Y1 = C3 + Cax ------------------------------------------- (B-9)

where C3 and C4 are arbitrary constant. By substitution of equation

(B-9) into (B-7), it gives:

++ +
[ (B2 B B B4)c4 BIB

3 1 ] + B B c x = B B T . +

3C3 1 3 4 1 3 a1

BlBjIX --------------------------------------------- (B-IO)

Comparing the coefficients of equation (B-lO) gives:

C =a ------------------------------------------------ (3'11)

4
(B 2+BB+B 18401
3 Tai ’ B1B3 """"""""""""""""" (3'12)

C

 

Equations(B-S),(B-ll), and (B-12) give the complete solution of the

particle temperature,

 

= - + + -
Tp ClExp [ (B2 B3)x ] CZExp [ Blth ] +
(B2+B3+B184)1
ax + T . - -------------------------- (3'13)
a1 BlB3

The boundary conditions of the system are:

Tp = Tpi
x = 0, {T = T
a ai

With the boundary conditions, C1 and C2 are determined to be:

 

- 94 -

_ 1
Cl - (32+B3-B1B4) [ a +'(B3'B1B4)(Tp1 ' Tai) +

 

(134
—§— (B2+B3+B1B4) J --------------------------------- (B-14)
3

l
c=< )[-a+B(T.-T.)+
2 32+B3 B1B4 2 p1 a1

 

(1

gig; (B2+BB)(BZ+BB+B1B4) J """""""""""""" (3'15)

Therefore, the final solutions for air and particles are obtained:

 

a 1
T -T . =(1x - —“- (3 +3 -B B ) + ( )
p 31 B1B3 2 3 l 4 B2+B3-B1B4

_Q__
{ E ‘0' + B2(Tpi - Tai) + B B (B2+B3(B2+B3+BIB4) J

 

1 3
Exp [ -B1B4x ] +
aB4
[ a I (B3’BlBa)(Tpi ' Tai) ‘ ‘3;— (Bz+B3+BlB4) J
Exp 1 -(B,+B,>x J } -------------------------------- (B-Ie)
a 81 1 3 2 3 1 4

{(53) [ +B (T -T ) +-—9H— (B +B )(B +B +B B ) ]

B3 *1 2 pi ai B133 2 3 2 3 1 4
Exp [-BlB4x J +

B -B B

3 1 4

_0_____. + - _ _
E ( 33 ) E a (B3 3134) (Tpi Tai)
a34
'33- (32+BB+BIBA) J Exp [ -(BZ+B3)X } --------------- (B-17)

-95-

Now we substitute B parameters in terms of (St)m, (St)d, and w as

defined in Appendix A, equation (B-l6) and (B-17) can be written in the

dimensionless forms,

T - T . (St)
a a1 _ x ( w m w

aD ” B'+ 4(St)m + 4 ' ¢(1+w)(sc)d) +

 

[ 1
0(St)d 4(St)mJ
9<St)d + w ' (I163-

 

{ w(TEi-Tai) .9EEE:§_ 4(St)m - w) -

aD ( w - (1+w) ) - (1

11<l+w><sc>d x }

4w(St)m
] Exp (-- w D ) ----------- (B-18)

 

 

¢(1+w)2(3t)d

T - T . (St)
__E_._ + .___JE - w ) +
4(St)m 4 0(1+w)(St)d

 

E 1 —J
M812)d 4(St)m
”(St)d + ’ (1+w)"

 

 

, (rpi - Tai) ( 0(St)é - 4(St)-lE )
1 dB w (1+W)

 

4w(St) 0(l+w)(St)
m 1 Exp ( - w d g ) 1 ----------- (B-19)

 

11(1+w>2<sn)d

 

 

 

 

- 96 -

If air and particle temperatures are equal at the entrance position,

i.e. Tai = Tpi’ equation (B-18) and (B-l9) can be simplified as:

 

 

 

 

 

 

 

 

 

 

T -T . (St)
_2___ag. = z +.___21__ ,. ____1. - w _
aD D 4(St)m 4 9)(1+w)(St)d
[ ¢(s:) 4(St) J E 4W(S:)m
0(St)d + d "YIIGSE 0(l+w) (St)d
p<1+w><3c>d X
EXP ( - w 5 ) ---------------------------- (B-ZO)
T -T . (St)
a ££=§+__L_+__fl - W _
aD D 4(St)m 4 (0(1+w)(St:)d
E 1 J [ <1-w) +
0(St)d 4(St)m
”(5% + “7;“ ' Tm)—
4w(St) 0(1+w)(St)
m ] Exp ( - w d g ) ------------ (B-21)

p<1+w)2<8t)d

 

- 97 -

APPENDIX C

SOLUTIONS FOR THE CASE OF UNIFORM HEAT FLUX

With the parameters defined in the Appendix A, equation (4-27)

and (4-28) can be written as:

dTa
75; = B6 - 32(Ta - Tp) --------------------------- (0’1)
dT
——de = B3 (Ta - TP) ------------------------------------ (M)
1 dT
= _ + ........................................ -
or, Ta B3 '75? Tp (C 3)

Eliminating air temperature Ta from equation (C-l) with equation (C-3)

gives:
d2T dT
—d—}{'§ + (B2+B3) dx = B3B6 ---------------------------- (0-4)

By using the method of operator, equation (C-4) can be written as:

2
1 m + (B2+B3)m J Tp = B236 ---------------------------- (c-s)

The complementary solution of equation (C-5) is readily seen as:
m = 0, -(B2+B3)

Therefore, Y1 = c1 + C2Exp [ -(B2+B3)x ] --------------------- (C-6)

The particular solution can be obtained with employing the undetermined

coefficient method. Let the particular solution by Y2 = C3x. Sub-

stituting Y into equation (C-4) gives:

2

 

B B
C = 3 6

--------------------------------------------- (c-7)
3 B2+B3

 

The particular solution is then:

The complete solution of equation (C-5) is obtained as:

 

 

B3B6x
= - + ................ ..
Tp c1 + 02 Exp [ (B2 B3)x ] + 32+B3 (c 9)
The boundary conditions of the system are:
. Tp = TPi
x = 0’ 1 T = .
a 31
With the boundary conditions, C1 and 02 are obtained to be:
_ BZTpi+B3Tai B3B6
c1 - B +3 - ——--§ --------------------------- (MD)
+
2 3 (32 B3)
B (T .-T .) B B
CZ 23—9—2... +0197 ........................... (c-11)
2 3 (B2+B3)

Therefore, the final solutions for air and particles are:

B3 ’ Be
. = (--- - (T .-T .) - (
p p1 32+B3 ‘1 p1 a1 B2+B3

 

) + B6X +

B

6 x
[ (TP.-Tai) + B2+BB ] Exp [ -(B2_B3)x J j --------- (c-12)

 

 

- 99 -

 

B3 Be B6
Ta-Tpi - (B2+B3) 1 '(Tpi'Tai) ' (BE-TBS) + B; (1+8?) +
B2 Be
(3;) [ (Tpi-Tai) +‘E;;§; J Exp [ -(B2+B3)x ] } ----- (C-13)

Substituting B parameters in terms of (St)m, (St)d, and w as defined

in Appendix A, equation (C-12)and (C-13) can be written in the

dimensionless forms,

 

 

 

 

 

 

{Lies - _1._ {BB _ -__.4B_._ _ ‘Tpi'Tai’ +
qw/GaCa 1 + w I) p(1+w)(sr.)d qw/GaCa
[ (Tpi-Tai) 4W J
qw/GaCa 0(1+w)(3t)d
¢(1+w)(5t)d
EXP ( - “-T— B ) ,1 ------------------------- (C-14)
5-22.1 - __1-_ {52- w2 , $213.3_
qw GaCa 1 + w D 9(1+w)(St)d qw/GaCa
w [ fzpi-TaiZ 4w 3
qw/caca ¢(1+w)(3t)d
0(1+W)(St)
Exp ( - w d-E )} -------------------------- (C-15)

If air and particle temperatures are equal at the entrance position,

i.e. T . = Tai’ equation (C-14)

p1 and (C-15) can be simplified as:

T -T
..P_.L1.. ...1..- ”£5 _ ".4111.--“
qw/caca 1 + w 1 D 0(l+w)(St)d

 

-100-

(0(1+w)(3t)d

 

 

 

4w X
(0(l-lw)(St)d) EXP (' w B ) 1 """""" (0'16)
. 2
._P.}. .-.l.. {333 - 4W .._
Q Ca 1 + w D 0(1+w)(St)d
4w mwxsod

 

w (mm-ESE) Exp ( - w %)}--" (C-17)

 

101

 

 

 

 

 

 

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102

 

 

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103 -

 

 

 

 

 

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