SIMULATION OF COUNTER-FLOW DRYING

Thesis for the Degree of M. S.
WCHIGAN STATE UNIVERSITY ,
TIMOTHY WENDELL EVANS
1970

   
     

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ABSTRACT

SIMULATION OF COUNTER-FLOW DRYING

by

Timothy Wéndell Evans

A theoretical model of counter—flow drying of
biological products was developed. The model was solved
by invariant imbedding, making certain estimations. A
new mathematical technique was developed (invariant
programming), which allowed direct solution of the model
without trial and error. The theoretical results com-

pared well with the experimental data of Ives (1967).

Mflo/z AZ 4/

firm Professor /7¢; , 7L1“
Approved

Department Chairman

SIMULATION OF COUNTER-FLOW DRYING

by

Timothy Wendell Evans

A THESIS

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

MASTER OF SCIENCE

Department of Agricultural Engineering

1970

6.1 ACKNOWLEDGEMENTS

The guidance and the financial assistance of Dr. F. W. Bakker-

Arkema is acknowledged with gratitude. The help and support of the

other committee members, Dr. J. V. Beck and Dr. J. B. Holtman, is

greatly appreciated.

A special thanks is extended to Pam Lombard for the splendid

preparation of the manuscript.

ii

TABLE OF CONTENTS

Page
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . 1
OBJECTIVES . . . . . . . . . . . . . . . . . . . . . . . . . . 3
MODEL OF SYSTEM. . . . . . . . . . . . . . . . . . . . . . . . 4

INVARIANT IMBEDDING. . . . . . . . . . . . . . . . . . . . . . 15
EQUILIBRIUM STATES . . . . . . . . . . . . . . . . . . . . . . 22
BOUNDARY CONDITIONS. . . . . . . . . . . . . . . . . . . . . . 26
NUMERICAL SOLUTION . . . . . . . . . . . . . . . . . . . . . . 33
INVARIANT PROGRAMMING. . . . . . . . . . . . . . . . . . . . . 37
RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . 43
SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . 61
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . 64
APPENDIX A . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Computer program which solves for the outlet properties
using invariant imbedding. . . . . . . . . . . . . . . . . 67

APPENDIX B O O O O O I O O O O O O O O O O O O 0 0 o 0 O 0 O O 78
Computer program which solves the system model . . . . . . 78
APPENDIX C o o o o o o o o o o o O o o o o o o o o o o o o o o o 81

Computer program which solves for the outlet prOperties
using invariant programming. . . . . . . . . . . . . . . . 81

iii

LIST OF TABLES

Table Page

1. Experimental Versus Theoretical Results for Three
Dryer Lengths. . . . . . . . . . . . . . . . . . . . . . 52

iv

Figure
l.
2.

3.

10.

ll.

12.

13.

14.

15.

16.

17.

LIST OF

An Elemental Dryer Volume.
Grain Kernel . . . . . . .
Counter-Flow Dryer . .
Equilibrium State 1. . . .
Equilibrium State 2. . . .
Equilibrium State 3. . .
Equilibrium State 4. . . .

Equilibrium State 5. . . .

Errors in the Invariant Imbedding Solution near
the T1 = 91 Boundary o o o o o o o o o o o o o 0

Errors in the Invariant Imbedding Solution near

the Hi = H

.eq Boundary. . .

Errors in the Invariant Imbedding Solution for

a Center Mesh Point. . . .

Schematic Diagram of the Solution of a Counter-Flow
Drying Problem, Using Invariant Programming.

Comparison of Theoretical and Experimental Product

Contents within the Dryer.

Comparison of Theoretical and Experimental Air

Temperatures within the Dryer.

The Outlet Product Moisture Content for Various
Inlet Air Temperatures Versus Dryer Length for

FIGURES

a Given Inlet Air Humidity Ratio . . . .

The Outlet Product Moisture Content for Various
Air Humidity Ratios Versus Dryer Length for a

Given Inlet Air Temperature.

The Outlet Product Moisture Content for Various
Inlet Air Humidity Ratios Versus Inlet Air Temper—

ature for a Given Dryer Length . . . . .

O

Page

10
10
23
23
23
23

23

45

46

47

49

53

54

56

58

59

Kl

M1, M2, M3

atm

Psat

XINF

NOMENCLATURE

cross-sectional area of the dryer, ft2
Chu's diffusion coefficient, ftz/hr

air flow rate, lbm dry air/hr ftz

product flow rate, lbm dry product/hr ft2
air humidity ratio, lbm water vapor/lbm dry air

average product moisture content, lbm water/lbm dry
product

local product moisture contents, lbm water/lbm dry
product

atmospheric pressure, psi

saturated vapor pressure, psi

feasible set of outlet product properties
feasible set of outlet air properties

air temperature, F

equilibrium moisture content, lbm water/lbm dry
product

width of the dryer, ft

length of the dryer, ft

specific surface area of the product, ftZ/ft3
specific heat of liquid water, Btu/lbm F
specific heat of dry air, Btu/lbm F

specific heat of dry product, Btu/lbm F
specific heat of water vapor, Btu/lbm F

2

convective heat transfer coefficient, Btu/hr ft F

vi

convective mass transfer coefficient, ft/hr
latent heat of evaporation, Btu/lbm water
relative humidity, decimal

coordinate distance from product inlet, ft
arbitrary position within the dryer, ft
coordinate from the surface of the kernel, ft

3 product/ft3 total

porosity, ft
density of particles, lbm dry product/ft3

product temperature, F

as subscripts:

i

O

inlet

outlet

as superscripts:

*

minimizes the dimensionless norm

desired value

vii

INTRODUCTION

Heated-air grain dryers are used on many farms and elevators
in the Midwest. These dryers have been typically of the deep bed type.
Due to the seasonal nature of grain drying most farmers and elevator
owners are not capable of handling the magnitude of grain to be dried
during the harvest season for economic reasons. Attempts have been
made to speed up the drying process by increasing the inlet air temper-
ature. This process usually lowers the quality of the dried grain. The
deep bed grain dryer requires time for dumping, cleaning and refilling.
Since continuous-flow dryers do not require a shut—down time, their use
will speed up the total drying process.

Another problem of the stationary deep bed grain dryer is that
the moisture content of the grain is not uniform throughout the bed
depth. The product is overdried at the air's inlet, but is underdried
at the air exit. This is also true of the continuous-flow cross-flow
grain dryer. However, the concurrent and the counter-flow dryers exit
the grain at a uniform moisture content.

It was observed by Thompson (1967) that counter flow dryers
remove more moisture per foot of dryer depth than the other continuous—
flow dryers. It should also be noted that the counter-flow dryer makes
less efficient use of the internal energy of the inlet air, since a
larger quantity of the air's energy is used to heat the grain and thus, less

energy is available for the evaporation of moisture.

In order to optimize a given drying system, a concise model of
the dryer is essential. Deep bed grain dryers have been successfully
modeled for a wide range of inlet and initial condition by Thompson
(1967) and Bakker-Arkema et a1. (1969). Thompson's model utilizes an
empirical drying rate equation. Bakker-Arkema considers the diffusion
of moisture within the kernel using a variable diffusion coefficient
for corn. Thompson has had moderate success in modeling of concurrent
and cross-flow grain dryers. The Bakker-Arkema model should work well
when extended to these two systems.

It is mofe difficult to model the counter-flow dryer than
either the cross-flow or the concurrent-flow grain dryers. In a counter-
flow dryer the product enters at one end of the dryer, while the air
enters the system at the other end. Therefore, both the air and the
product properties are not known at any one location. Thompson solved
this problem by assuming that the air temperature is equal to the pro-
duct temperature for all points within the system. This assumption
will yield accurate results only if the thermal capacity of the product
flow is approximately equal to the thermal capacity of the air flow.

Ives (1967)‘considered a special family of counter flow dryers,
in which the air flow rate is much greater than the product flow rate.
He applied thermodynamic relationships to determine the equilibrium
values for the outlet air temperature and humidity ratio. The usefulness
of his research is limited to dryers in which the outlet air and product

properties have come to equilibrium.

OBJECTIVES

The objectives of this research are the following:

1)

2)

3)

To study the theoretical mechanisms of counter—flow drying
of biological products.
To develop a mathematical model of this system.

To develop a concise and direct solution of the model.

MODEL OF SYSTEM

Mathematical modeling of counter-flow drying consists of for-
mulating balances of the conserved properties (mass and energy) of
the system. The equations resulting from these balances will repre-
sent the system if the correct assumptions are made. The assumptions
made in this model are the following:
a) Temperature and moisture ratios are the controlling poten—
tials.
b) The temperature gradients within the particles are negli-
gible.
c) The temperature and moisture gradients in the y and 2 dir-
ections are zero.
d) The conduction by particle to particle contact is negligi—
ble.
e) The system is in steady state.
In the following balance six dependent variables are considered:
product temperature (6), three local product moisture contents (M1,
M2 and M3), air humidity ratio (H) and air temperature (T). All of
these variables are functions of the position within the dryer (x) only.
In Figure 1, an elemental volume of the system, AAx,,is iso-
lated for consideration. Since it is assumed that the system is in steady
state only the input and output air and product properties are essential
for the evaluation of these balances.
Within the differential volume there are two modes of energy

transfer; changes in internal energy of the product and the air and

4

 

 

P-_—-—c—»-——_-— —————————————————————————

 

 

 

 

 

——-———--—--—---——--------—_--—--—-———fi

latent energy of evaporation. The incoming air at x' +13x has an internal

energy which can be expressed as:
(Ca Ca + cv H Ga) A Tx' + Ax (1)

When the air exist the elemental volume at x', its internal can be

represented by:

(Ca Ca + cv H Ga) A Tx' (2)
The entering product at x' has an internal energy of:

cG+cMG A9 3

(pp p) x. <)
The internal energy of the outlet product can be written as:

— (4)
(cp Gp + c M GP) A 9x, + Ax

The latent energy of the entering air can be expressed as:

A Ga hfg Hx' + Ax (5)

The air exiting the control volume has a latent energy equal to:

AGh H, (6)

Since it was assumed that the system is in steady state, the inputs
to the elemental volume must equal the outputs. Therefore, equation (1)
plus equation (3) plus equation (5) is equal to equation (2) plus

equation (4) plus equation (6). Combining similar terms yields:

+
(c G cv H Ga) (T

a a - Tx') = (cp G + c M G )

x' + Ax p p

H .) (7)

(Q " 9 g) " G h (HX' + Ax - x

x' + Ax x a fg

Dividing equation(7) by Ax and allowing Ax to approach zero, results

in equation (8):

 

 

dT cG+cMG d9 Gh dH
.__ = P D p __. _ a fggi ___ (8)
dx c G + c H G dx c G + c H G dx

a a v a a a v a

Consider an energy balance for a single particle of grain within
the elemental volume. There is an inflow of heat to the product due

to convection at the particle surface:

a' h (T ,

X + 3Ax - gx' +%Ax) Ax A (9)

The product's internal energy at x' and the product's internal energy

at x' + Ax have been previously expressed by equations (3) and (4),
respectively. The change in internal energy of the product in passing from
x' to x' + Ax is equal to the energy gained by convection. Therefore,

equation (4) minus equation (3) equals equation (9):

= a'h (T

(ca Ga + C M Gp) (9x' + Ax - Ox') x' + TAX -

(10)
gx’ + NAX) Ax

Dividing equation (10) by Ax and allowing Ax to approach zero, yields:
d9 a' h

K=cc+cHG (13-9) (11)
p P P

 

Due to the asumption of steady state, the mass lost by the
product in the elemental volume must equal the mass gained by the air.

The moisture held by the product at x' is equal to:

G A M , (12)
The mass of water contained by the product at x’ + AX is:
G AM, (13)
p x +13):
The total water in the form of vapor held by the air at x' +'Ax can

be expressed as:

14
Ga A Hx' + Ax ( )

The mass of water vapor held by the air exiting the elemental volume
can be represented by:
G AH, (15)
a x
Performing a mass balance the inlet mass equals the outlet mass. There-
fore, equation (12) plus equation (14) must equal equation (13) plus

equation (15). Combining similar terms yields:

fix.) a GP (M - M ,) (16)

G (H x' + Ax x

a x' + Ax-

Again, dividing equation (16) by Ax and allowing Ax to approach zero,

results in the following equation:

dH

EE'

§1_§h

_ G
- _J1
Ga (17)

The corn kernel was modeled as an infinite flat plate. In Figure
2 an elemental volume 4A2AxW is isolated for consideration. Three node
positions were chosen within the flat plate: the surface, the midway
point between the surface and the center of the flat plate, and the center.
The surface node represents a volume of kAzAxW. There are
inputs of moisture to this elemental volume: the moisture of the product
at x' and the diffusion of water from the interior portions of the kernel.

The moisture of the product x' can be represented by:
GP (AZ W/Ze) (M1)x' (18)

The diffusion of moisture from the interior portions of the kernel

can be expressed as:

- (19)
(0 ”Ax w) [(M2)x' + gAx (M1)x' + gAxJ / AZ

There are also outputs of moisture from the elemental volume: the moisture

of the product at x' +-Ax: and the convection at the surface of the

kernel. The moisture of the product at x' +'Ax can be represented by:

20
GF (Az W/2e) (M1)x, + Ax ( )

10

 

 

 

___ __ __“e.\___ ___---
be
\

 

 

14-le +1

Figure 2. Grain Kernel

 

”/7771 0771/0015 pow no”;

'"XfX’fAX'

:XEZI 1:. '

I , Hw +———— __ _-_......_.---.___ _ _ng if“ _'

 

 

agmi
1.

Figure 3. Counter-Flow Dryer

11

The convection of water at the surface of the kernel can be modeled as:

(p hD Ax W) [(M1) - XINF (21)

x' + gAx x' + gAx]

Performing a mass balance the inputs must equal the outputs. Therefore,

equation (18) plus equation (19) equals equation (20) plus equation (21):

GP(Az/2€) [(M1)x, + Ax - (M1)x,] a (pDAx) [(M2)x. + 3511" _

22
(M1)x. + JEAXJ/Az + (phDAx) [XINFX, + kAx - (M1)x, + kAx] ( )

Dividing equation (22) by Ax and allowing Ax to approach zero, yields:

dM1 zpDe zphDe
dx ' 2* (M2 ' ”1’ + Azc (XINF ' M1) (23)
(AZ) Gfi ' P

XINF, which was introduced in equation (21) is calculated from
the empirical moisture equilibrium isotherm equation developed by

Thompson (1968) from the data by Rodriquez - Arias (1963):

XINF = 0.01 i-ln (1-rh)/[3.82 x 10-5 (0 + SOUP5 (24)
Where rh is the relative humidity:

rh = HPa /[Psat (H + .622) J z (25)

tm

D represents Chu's (1968) diffusion coefficient for corn which is
a function of the product temperature and the average moisture content

of the corn kernel:

0 = 0.001629 EXP [ (0.00045 0 - 0.05485) 100.0'M

- 2513.0/¢ 3 (26)

12

where D is 8 expressed in degrees Kelvin. It should be noted here that
the Chu's diffusion coefficient was determined from data between the
product temperatures 120 F and 160 F. It has been shown by earlier
research (Bakker-Arkema et al., 1969) that this coefficient fits thin
layer drying data down to 100 F, but one cannot assert that Chu's
diffusion coefficient would also represent data above 160 F.

The node located at the midpoint between the surface and the
center in Figure 2 has a volume of AxAzW. There is an input of moisture
to this elemental volume at x'. This input is the inflow of moisture

carried by the inlet product, which can be represented as:
(cpAzwe) (M2)x' (27)

There is also an inflow of mass due to diffusion from the innermost node.

This diffusion of mass can be expressed as:

(DDAXW) [ (M3)xt + kAx " (M2)X' + kAx J / AZ (28)
The diffusion of moisture from the node being analyzed to the surface
node has been previously expressed by equation (19). The outlet product
at x' + Ax contains the quantity of moisture expressed by the following

equation:

(GPAzw/e) (M2)xc + Ax (29)
Formulating a mass balance, equation (27) plus equation (28) equals equa-

tion (19) plus equation (29). Combining similar terms yields:

(GpAz/3) [(M2)X' + Ax - (M2)}I'] a (pDAX) [(M3)X' + %AX

-2(M2)x' + kAx + (M3)x' + kAx] I AZ (30)

13

Again, dividing equation (30) by Ax and allowing Ax to approach zero, yields:
dM p D e

___ = ._____—— - 31
dx 2 (M1 2M2+M3) ( )
G (AZ)
P
The center node also has a volume of AxAzW. The only input of
moisture to this elemental volume is the inlet moisture at x'. This inlet

product moisture can be expressed as:

(GPAZW/e) (M3)x, (32)

The moisture held by the product at x' +-Ax can be written as:

(GPAZW/e) (M3)

X + AX (33)

There is diffusion of moisture to the neighboring nodes on both sides of
the center node. This transfer of moisture is equal to twice equation

(28). From a mass balance, equation (32) must equal two times equation

(28) plus equation (33):

(GPAZ/e) (013),“ + Ax - (M3>x.] = 20%: (max: + gAx
(34>
- (M3)x' + kAxJ / Az

Dividing equation (34) by Ax and allowing Ax to approach zero, yields:

dM 2 De
0

' M ) (35)
dx Gp(Az)2 2 3

Noting in Figure 2 that the node midway between the surface and
the center of the flat plate has twice the volume of the other two nodes,

the average moisture content can be represented by:

M = (M1 + 2M2 + M3) /4 (36)

14

Similarly, the total derivative of the average moisture content with

respect to x can be expressed as:

- dM dM dM
dM _ 1 2 3
3;. - ( dx + 2dx + dx ) /4 (37)

A typical counter—flow dryer is shown in Figure 3. The product
enters at the x equal zero end of the dryer with a known moisture con—

tent and temperature:
141(0) = 142(0) - M3(0) = E1 (38)

9 (O) = 6 (39)

i

The air enters at the x equal a end of the dryer with a known humidity

ratio and temperature:

H(a) H (40)

1
Ti (41)

T(a)

The nonlinear differential equations (8), (ll), (17), (23), (31),
and (35) plus the boundary conditions (38) through (41) represent a two
point boundary value problem. In its present form, this problem can
be solved only by trial and error. In the following sections these

equations will be cast into a form which allows direct solution.

15

INVARIANT IMBEDDING

It is a common practice to formulate a system equation as a
function of time or position. If sufficient initial and boundary con-
ditions are known this equation can be solved analytically or numeri—
cally for the state variable as a function of time or position.

Considering this system equation, invariant imbedding expresses
the state variable at a given position or time as a function of the
length of a continuous mechanism or the time of an event and the boundary
conditions of the system equation. The initial condition of an invariant
imbedding problem is the system's re8ponse when the time of an event
or the length of a continuous mechanism is zero. In most cases, there
is no response at the initial condition. The boundary conditions of
an invariant imbedding problem usually are known system responses to
specific boundary conditions of the original system equation.

The system equations for a counter-flow dryer are equations (8),
(ll), (17), (23), (31), and (35) and the independent variable is the
position within the bed (x). The boundary conditions to the system are
represented by equations (38) through (41). In the invariant imbedding
formulation it is sought to express the state variables (T, H, 9, M1,

M2, and M3) as functions of the dryer length a and the boundary conditions
of the system equations.

Consider a basic dryer as shown in Figure 3. The inlet product
properties are 81 and Mi at x = 0 and the inlet air properties at x = a

are Ti and Hi. It is desired to express the outlet prOperties of the

16

product (90 and Mg) and the outlet properties of the air (Ho and To)

as functions of the dryer length and the inlet (air or product) proper-
ties. Looking at the x equal a end of the dryer, the outlet product
temperature (80) can be represented as a function of the inlet air

temperature, the inlet air humidity ratio and the length of the dryer:

90 = 60(Ti’ Hi, 3) (42)

The total derivative of equation (42) with respect to the independent

variables Ti’ Hi and a can be written as:

39
O o O
dso aT clT1 + 5| dH1 + aa 1 da (43)
1 H1, 1 T1,a T1,H1

The partial derivatives in equation (43) can be thought of as sensitiv-
ity coefficients depicting the rate at which the state variable (90)
changes as a function of the inlet conditions (Ti and Hi) and the dryer

length a. Dividing equation (43) by the incremental distance dx yields:

dx 5T1 H133 dx 3H1 T1,a dx 5a T1,H1 dx

Since equation (42) was written for the location where x equals a, the
total derivative of a with respect to x must equal 1. Thus, equation

(44) can be rewritten as:

d o - 39° | dT1 + 39° | dI-Ii + 6 0|
dx 3T1 Hi’a dx 3H1 T1,a dx 3a Ti’Hi (45)

17

The total derivatives with respect to x can be calculated from equations
(8), (11) and (17) using the properties at the x equal a end of the dryer.
For a dryer of length zero the outlet product temperature must

equal the inlet product temperature:

9 (T

o i’ H

1’ 0) = 91 (46)

The boundary conditions for equation (45) will be discussed in a later
section.
The three local moisture contents (M1, M2 and M3) at x equal a

can also be expressed as functions of the inlet air temperature, the

inlet air humidity ratio, and the dryer length.
<Mj>o = [Mj<Ti. Hi. a>10 3 = 1, 2, 3 <47)

Following the same development as presented in equations (43) through

(45), results in:

d(M) MM) (11' 304) cm a<M>
"‘3§'3 = 'Sfl'gl a;£' +' "fil'g'l a;i'+"'"§'2|
i Him a i T ,a a T ,H.
i i 1
j I 1’ 2, 3 (48)

The total derivatives with respect to x can be calculated by equations
(8), (17), (23), (31), and (35) using the properties at the air's inlet.
For a dryer of length zero the outlet product moisture content

is equal to the inlet product moisture content:

[Mj (Ti, Hi, 0)]o = M1 j -- 1, 2, 3 (49)

18

The two boundary conditions will be discussed in a later section.

It is also desirable to know the outlet air temperature and
the outlet air humidity ratio as a function of the inlet air temper—
ature, the inlet air humidity ratio and the length of the dryer. The
relationship of the outlet air temperature as a function of these

independent variables can be written as:

T0 = To(Ti’ Hi’ a) (50)

The total derivative of the outlet air temperature with respect to

the independent variables can be expressed as:

3T0 d 6T0 d 3T0 d ( )
dT = ——— T ‘+-——- H +--—- a 51
° aTi Hi,a 1 5H1 T1,a 1 5a T1,H1

With a given value of Ti’ Hi and a at x equal a there exists a unique
value of the outlet air temperature. If the air humidity ratio and
the air temperature at x equal a are held equal to Hi and Ti’ respectively,
while the dryer length is increased to a + da, the outlet air temper-
ature will remain unchanged. Therefore, if the appropriate values of
dTiand dHiare chosen in equatifip (51), dTo can be assumfi? to be equal

1

zero. Selecting dTi equal tog;— da and dHi equal to Kda will satisfy

this criterion:

3 dT 3T :1}! 3T
'Fq'l ridH'n—q' 3.4.1,...32 “'0 (52)
5 1 Hi, x a 1 T1, x a Ti,H1

l9

Dividing equation (52) by da, yields the following equation:

5T0 dTi 3T0 dH1 5T0
aT' arm—fl "—dx +—,.' =0 ‘53)
i H1,a i T1,a Ti’Hi

The total derivatives with respect to x can be calculated by equations
(8) and (17). For a dryer of length equal zero the inlet air temperature

is equal to the outlet air temperature:

To (Ti, Hi, 0) = Ti (54)

The boundary conditions to equation (53) will be discussed in a later
section.

The outlet air humidity ratio can also be expressed as a function
of the inlet air temperature, the inlet air humidity ratio and the dryer
length. Following the procedure outlined by equations (50) through (53),

equation (55) results:

3H0 I dH1 3H0 dT1 3H0
3H1 T dx + 5T ' dx 3a ' = 0 (55)
1,a i Hi,a Ti’Hi

Again, the total derivatives with respect to x can be calculated by
equations (8) and (17). The initial condition to equation (55) is

expressed as:
H (T , H.. 0) = H (56)

Reversing the dryer such that the air enters at the x equal zero
end and the product inlet is at the x equal a end of the dryer, the
outlet air and product properties can be expressed as functions of the

inlet product temperature, the inlet product moisture content and the dryer

20

length. This relationship for the outlet air temperature can be written
as:

T = Tomi. iii. a) (57)

0

Following the procedure used to derive equation (45) yields:

dfi T
—dT° = —3T° Lei --3T° | —1 + L o l (58)
dx 39 - dx 5M dx 83 e _
i Mi,a 1 61,111 i. Mi

Similarly, the outlet air humidity ratio can be expressed as
a function of the inlet product temperature, the inlet product moisture
content and the dryer length. For this property the following rela-

tionship can be written:

dHo ___ 5110 I <191 5H0 dM1 3H0
3x— 553 amwuj' EH?" ,3 ‘59)
i’ 91" 91’ i

The total derivatives with respect to x can be calculated by equations
(11) and (36) using the properties at the product's inlet.

The outlet product properties can also be represented as functions
of the inlet product properties and the dryer length. Following the
procedure outlined by equations (50) through (53) yields equations

(60) and (61):

ago —-°-| 11+ 660 ' Si 690' _
5—9 +3M:' + 3;— — ‘ 0 (60)
i M1,a 91,a 91,M1
a M) d (M )
.(..1_°.|_ 21194.0] _.+3..L2‘ .0 (61)
391 Mi,a d" 5M1 91,3 d" 9131

j-1,2,3

21

Once the appropriate boundary conditions are chosen equations
(45), (48), (S3), and (55) can be solved numerically for the outlet
air and product properties as functions of the inlet air properties
and the dryer length. Similarly equations (58), (59), (60) and (61)
could be solved for the outlet properties as functions of the inlet

product properties and the dryer length.

EQUILIBRIUM STATES

A great deal of insight into the mechanisms of counter—flow
drying may be gained by an investigation of equilibrium states.
Equilibrium states refer to values of the outlet air prOperties (To’Ho)

and the outlet product properties (9 ‘36) such that these properties

0’
remain constant while the dryer is increased in length. It should be

noted again that the dryer was assumed to be in steady state.

Equilibrium State 1

In Figure 4 the inlet product temperature (91) is equal to the
outlet air temperature (To) and the outlet air humidity ratio (H0) is in
equilibrium with the inlet product moisture content Cfii). Since the
outlet air is in equilibrium with the inlet product with respect to
both heat and mass transfer, it is not possible for the product to increase

its outlet temperature or to decrease its outlet moisture content.

Equilibrium State 2

In Figure 5 the outlet product temperature (60) is equal to the
inlet air temperature (T1) and the outlet product moisture content (i?)
is in equilibrium with the inlet air humidity ratio (Hi). It is physi-
cally impossible for the outlet product to absorb any more thermal energy
from the air or transfer additional moisture to the air. Therefore,
the air cannot decrease its internal energy or increase its humidity ratio

by heat and mass transfer.

22

 

190

23

-EQHLZBEZMSTAZTEJ :f i

we

t-'-‘-'-‘1'

 

 

t---.
162:

Figure 4.
Equilibrium State 1

 

IE. ------- ilflfim
ZZFC7it- ----- '1 -ZZEZEEL
1a.
Figure 7.

Equilibrium State 4

 

 

 

Figure 5.
Equilibrium State 2

 

 

4e

+—--—-—--—-{I

 

 

- FHHwJ

Ga

Figure 6.

 

 

___};-i

Figure 8.

747:5

Equilibrium State 5

.ALfiuIEEF

Equilibrium State 3

 

24

Equilibrium State 3

In the case shown in Figure 6 the outlet air temperature (To)
is equal to the inlet product temperature (91) and the outlet product
moisture content (is) is the equilibrium value with respect to the inlet
air properties (Ti and Hi) and the outlet product temperature (90). It
is obvious that the air cannot transfer anymore thermal energy. Since it
requires thermal energy to evaporate moisture from the product, which is
already in moisture equilibrium at its outlet, no further moisture can
be exchanged between the air and the product. Similarly the product
cannot increase its internal energy for the air cannot transfer anymore

thermal energy.

Equilibrium State 4

In Figure 4 the outlet product temperature (90) is equal to the
inlet air temperature (Ti) and the outlet air humidity ratio (H0) is in
equilibrium with the inlet product moisture content (ii). The product
cannot increase its internal energy by heat transfer from the air. Since
the air is already in moisture equilibrium with the product, it is

not possible for the air to use its thermal energy for evaporation.

Before considering the fifth equilibrium state two assumptions must
be made. Again, it should be noted that the system's response is assumed
to be steady state. The additional assumptions made are the following:

a) The outlet air temperature must asymptotically approach
the inlet product temperature as the dryer length is increased. Similarly,

the outlet product temperature must asymptotically approach the inlet air

25

temperature as the dryer length is increased.
b) Energy transfer controls over mass transfer. In other
words, if there is no energy available for evaporation, there will be

no evaporation.

Equilibrium State 5 Ea-..
In Figure 8 the inlet product temperature (Si) is equal to

the inlet product temperature. From assumption (a) the outlet product ' -

 

temperature must equal the inlet product temperature as the dryer length i}
is increased. The outlet air temperature is equal to the inlet air

temperature for a dryer of length zero. Thus, the outlet air temperature

must equal the inlet air temperature regardless of the dryer length.

Thermal energy is required to evaporate moisture from the product,

therefore, there can be no mass transfer.

BOUNDARY CONDITIONS

If the appropriate boundary conditions are determined, the
invariant imbedding equations (45), (48), (53) and (55) along with
the initial conditions (46), (49), (54) and (56) can be solved numer-
ically for the outlet properties (air and product) as functions of the T]

:— JL

inlet air humidity ratio, the inlet air temperature and the dryer length.

‘ I. 4' .59

From equilibrium state 5, the outlet properties (90, E T

o , Ho) are

0!

 

equal to the inlet properties (91, E , Ti’ Hi) if the inlet air temper— LJ |
ature (Ti) is equal to the inlet product temperature. Therefore, the

following set of boundary conditions can be written:

90 (91,Hi,a) = 9i (62)
[Mj (9131,6010 = ii :1 = 1,2,3 (63)
T0 (91,Hi,a) = Ti (64)
Ho (9i,Hi,a) = Hi (65)

Given another Set of boundary conditions at a Specific inlet air humidity
ratio, the invariant imbedding equations can be solved numerically.
Consider the equilibrium value of the air humidity ratio Heq for

an air temperature 6 , a product temperature 91 and a product moisture

1
content fii. If the inlet air humidity ratio is equal to Heq’ it was
assumed that the outlet air and the outlet product properties are independent

of the inlet air humidity ratio. Therefore, when the inlet air humidity

26

27

ratio is equal to H , it was assumed that the second term on the right-
eq
hand side of equations (45) and (48) and the second term on the left-

hand side of equations (53) and (55) are zero. Rewriting these equations

 

yields:
de es dT ae
dx 3T1 H ,a dx aa H ,T (66)
eq eq 1
d(MI) (M.) dT 5(MI) 1'
d o a 3 T 0' d 1 0' j 1 2 3
x a X 33 = : a
i Heq,a Heq’Ti (67)
939. 9.2.9104 . 0
3T1 H a dx as H T (68)
eq’ eq’ 1
a dT 3
35:3 3:} 3:9 ‘0 <69)
1 Heq,a Heq’Ti

With an inlet air humidity ratio Heq’ equations (66) through (69)

along with the initial conditions (46), (49), (54) and (56) and the boun-
dary conditions (62) through (65) can be solved numerically for the outlet
properties as functions of the inlet air temperature (Ti) and the dryer
length (a). The solutions to these equations will generate the necessary
set of boundary conditions to solve equations (45), (48), (53) and (55).

The assumptions made in deriving equations (66) through (69)

should be considered in more detail. It was assumed that the second term
on the right-hand side of equations (45) and (48) and the second term on

the left-hand side of equations (53) and (55) were neglible when compared

28

with the other terms of these equations. For equation (45) this term
can be assumed small, while for equation (48) the assumption is reason-
able only for low outlet relative humidities. The assumption applied
to equations (53) and (55) is also valid if the outlet relative humidities
are low; however, for high relative humidities this assumption is very
poor.

As the dryer length a is increased a counter-flow dryer can be
said to be approaching one of the equilibrium states (1 through 4).
In other words, the system is approaching a mass equilibrium at one end
of the dryer and a thermal equilibrium at an outlet of the system. One
end of the dryer is limiting the degree of mass transfer, while the same
or the other end is controlling the heat transfer. For example, the
outlet product temperature may be very close to the inlet air temperature,
while the outlet air temperature may be much greater than the inlet product
temperature. Therefore, as a is increased the quantity of heat transfer
will be determined by the heat transfer at the air inlet. Thus in this
case, the rate at which the product absorbs energy from the air dictates
the rate at which energy is transferred in the system.

Equations (66) through (69) assume that both heat and mass
transfer are controlled by the x equal a end of the dryer. Therefore,
the system must be approaching equilibrium state 2. Since the outlet
air temperature (To) is usually lower than the inlet air temperature (T1),
the x equal zero end of the dryer limits mass transfer in most cases. This
means that equations (66) through (69) alone cannot predict the outlet

properties with the inlet humidity ratio equal He

29

Equations (58), (59), (60) and (61) express the outlet properties
(To’ 60, fig, Ho) as a function of the inlet product temperature, the
inlet product moisture content and the dryer length. The total deriva-
tives with respect to x in these equations are evaluated at the x equal
zero end. Therefore, these equations assume that the x equal zero end
of the dryer controls both heat and mass transfer. In other words, the E_..,
system is approaching equilibrium state 1. Again, it cannot be assumed

that equations (58) through (61) alone will accurately predict the out-

 

let properties at H1 equal Heq'

(61) and equations (66) through (69) can be used to calculate the outlet

A combination of equations (58) through @J

properties.

Both equations (69) and (59) can be evaluated for the outlet air
humidity ratio. Equation (69) assumes that the x equal a end of the
dryer controls the mass transfer, while equation (59) assumes that the
other end of the dryer limits the mass transfer. The maximum feasible
mass transfer is depicted by the minimum of the Ho's calculated by equa-
tions (59) and (69). This value is the correct HD for a dryer of length

a with the inlet properties 61, M1, Heq and T1. The appropriate values
of (Mlj)o j = 1,2,3 can be calculated by the same procedure using equa—
tions (61) and (67). Since the degree of mass transfer is almost always

controlled by the x equal zero end of the dryer the errors induced by

the assumption prior to equation (66) can be ignored.

Prior to the evaluation of 90 and To, a system energy balance
must be derived. Consider the system as a black box with inputs and
outputs of energy at both ends. There is an input of thermal energy by

the product at x equal zero of:

30

A (c G + Eric )9. (70)
p p 1 p 1

At the output the thermal energy of the product can be expressed as:

 

A c G + cM G 9 71
(pp 0p)0 <>
The inlet thermal energy of the air at x equal a can be written as: F31
A (caGa + CVHiGp)Ti (72)
The outlet air has a thermal energy equal to: I
L.
A (caGa + CvHoGp)To (73)

There is also energy transferred within the system in the form of latent

heat of evaporation. The latent energy of the air at its inlet is:

AhngaHi (74)

The latent energy of the air a x equal zero can be expressed as:

AhngaHo (75)

Performing an energy balance equation (70) plus equations (72) and (74)

euquals equation (71) plus equations (73) and (75):

(cc +cHG)(T-T)=-(cG +c'fic)(e-9,)

a a v i a o i p p o p o 1
+ G — - G - + G —

ahfg(Ho Hi) Cv a(Ho Hi) To c a(Ho Hi>ei (76)
8 can be calculated by equation (66) assuming that heat transfer

0

is controlled at the air inlet. Since H ,lfi , 9 , T and H (H ) are
0 o o . i eq

1

known, equation (76) can be solved for To. It seems possible to evaluate

31

equation (68) for To; however, it has previously been noted that this

equation cannot be trusted at high outlet relative humidities. Assuming

that heat transfer is controlled by the x equal zero end of the dryer,

To can be evaluated by equation (58). This value of To is compared with

the To calculated by equation (76) and the largest value is considered to

be the appropriate value. a
The correct value of 60 can be obtained by selecting the smallest ‘

60 generated by equations (66) and (60).

 

In order to evaluate the partial derivatives of the outlet proper— LJ

ties with respect to 91 and M1 in equations (58) and (61) it was necessary

to calculate the outlet properties for the inlet conditions (91, Mi)’

(91 + d61,fii) and (91, E: + dig). The partials in equation (60) can be

approximated by:

 

 

2:] = (6°)91H91’fii - (6°)91’fi1
59, Eva <19, (77)
390I (eo)91,fi1+dfii - (€991,121
3M1 61,8 dfi1 (78)

Similar expressions can be written to calculate the partials of the other
outlet properties with respect to the inlet product properties.

Using the procedure outlined in determining the outlet properties
(90, i%, To and Ho) at a specific inlet air humidity ratio (Heq) plus
the initial conditions [equations (46), (49), (54) and (56)] and the

boundary conditions at a specific inlet air temperature (91) [equations

32

(62) through (65)], the invariant imbedding equations (45), (48),
(53) and (55) can be solved numerically for the outlet properties as
functions of the inlet air properties and the dryer length. It should
be noted here that the technique used to calculate the outlet proper—
ties at a specific inlet air humidity ratio (Heq) is by no means unique.
Undoubtedly, there exist better methods.

If the partial derivatives of the outlet properties with respect

to the inlet air humidity ratio in equations (45), (48), (S3) and (55)

 

were known at a specific inlet air humidity ratio (Heq in this case),
the invariant imbedding equations could be solved directly for the

outlet properties at this boundary. Unfortunately, this is not the case.

NUMERICAL SOLUTION

Equations (45), (48), (53), and (55) were backward differ-
enced in the (a) direction and forward differenced in the (T1) and
(H1) directions. The finite-difference approximation of equation (45)

can be written as:

a+Aa _ a dTi a Aa a+Aa
+A _ (60 I + dx AT 60 +
T1 T1 1+AT1L T1+ATi 1 T1
H1+AH1 H1+OH1 H1+AH1 H1+AH1
dH
dxi a As o a+Aa Aa) / (1.0 +
H1+AH1 i H1+AH1
T1+AT1 T1+AT1
dT a Aa dHi a ég_)
__1; AT dx AH (79)
dx T1+AT1 1 T1+AT1 1
H1+AHi H1+AH1

It should be noted that the total derivatives with respect to x are
evaluated at (a). Similar expressions can be written for the remaining
equations (48), (53) and (55). These finite difference equations are
stable for large values of Aa(.l ft), AT1(10 F) andAui (.01 lbm water/

lbm dry air), if T1 and Hi are chosen such that:

——2; —2;
E;—' T1+~AT1 i and E;- T1+AT1 i are positive.
H 14-611 1 H 1MB 1

33

34

Since the total derivatives with respect to x at the x equal a end of
the dryer rarely change signs when (a) is increased, it is easy to
choose Ti and H1 approximately. This is not true for the x equal zero
end of the dryer. The total derivative of the inlet moisture content
[equations (58) through (61)] with respect to x changes sign when conden—

sation occurs. Therefore, it is difficult to maintain stability when I _1
these equations are used. Since these equations are employed in the

estimation of the outlet properties on the Hi equal Heq boundary, the g, .j

 

1

solution at this boundary is not stable when condensation occurs within :3
the dryer.

The finite difference approximations of equations (45), (48), (53)
and (55) with their initial conditions and the boundary conditions can
be solved numerically for the outlet properties. Using these outlet pro-
perties, it is possible to solve equations (8), (ll), (17), (23), (31)
and (35) for the air temperature, the product temperature, the air humid-
ity ratio and the local product moisture contents as functions of x.
Solutions to these can be obtained by any numerical integration routine.
Stability problems may occur if the appropriate initial conditions are
not chosen. Consider equations (8) and (11) written in state variable

form:

 

 

 

 

 

 

 

- a'h -a'h
d T G c + G Re C c +VG Hc W T -
'E- , - a a a v a a a v
x 9 a'h -a'h 6
Gc+cflc Gc+cfic
P P P P P P
fig
Gahf dx (80)
G c + G Be
a a a v 0

35

If the elements of the square matrix are asumed constant and the total
derivative of the air humidity ratio with respect to x is assumed zero,
the eigenvalues of this square matrix must be less than or equal to zero.
Therefore, if the system of equations is stable the following criterion
must be satisfied:

G c + G'fic

I’LL 1P s 1

G c + G He (81)

a a a v

 

Since the total derivative of the humidity ratio with respect to x is
negative, the criterion for stability is more conservative than equa-
tion (81). The stability criterion (81) is for the case when the initial
properties are chosen at the product inlet. If one chooses to solve
equations (8), (ll), (17), (23), (31) and (35) using the known prOper-
ties at x equal a, the following criterion must be satisfied:

G c + G Hc
a a a v 1
Gpcp + Gp‘Hc S (82)

 

In this case, the derivative of the air humidity ratio with respect
to x is positive. Thus, the actual criterion is less conservative than
equation (82). Therefore, a region exists where Gaca + Gchv is slightly

greater than C + G Mc in which one must guess which initial conditions

pcp p
to use in the solution of the system model.

If condensation occurs within the dryer, the system must be modeled
using the initial properties at the air inlet. Since condensation usually
occurs when the air flow is small with respect to the product flow, this

is not a severe limitation on the system model. It should be noted that

there are special cases [equation (81) is satisfied and condensation occurs

 

36

within the dryer] for which it is not possible to solve equations
(8), (ll), (17), (23), (31) and (35) for the air and product properties

as functions of x.

 

INVARIANT PROGRAMMING

The motivations for using the invariant imbedding concept for
the solution of the counter-flow dryer are the following:

1) This formulation yields the outlet air and product proper-

ties directly and in a known number of steps.

2) No estimates of the unknown air temperature, air humidity
ratio, product temperature and product moisture content
are needed.

3) In generating the desired results other pertinent information
for system design and system use is calculated; e.g. the
outlet properties as functions of the dryer length and the
inlet air or product properties.

Since the calculation of the outlet properties at the specific
inlet air humidity (Heq) can be considered to be no more than an esti-
mation, the procedure presented does not satisfy the second motivation.
In the following pages, an alternative formulation (invariant programming)
will be analyzed.

Consider a set S of discrete feasible outlet air humidity ratios,
Ho and outlet air temperatures, To. In deriving equations (53) and (55)
it was observed that for a given dryer length (a) there exists a unique
ordered pair of inlet air properties (T1,Hi) corresponding to each ordered
pair of outlet air properties (TO,HO). Similarly for a given value of a
the ordered pairs (Io’qa) map one to one into the set of ordered pairs

(acme).

37

 

38

It is desired to determine the
for a given dryer length and given inl
as the desired inlet air properties.
(Ti’Hi) corresponding to the set S to

which minimize the dimensionless norm.

 

outletproperties (90, Mb, To’ Ho)
et conditions. Define Ti and H;

One can search the ordered pairs

find the discrete values (T:,H:)

 

— . . q a
T. - T 2 H - H 2
1 i i i
(———.——> + (—-1-——) (83)
L Ti H1

*
The air properties (TO,H:) are the ordered pair which mapped into these

* —*
values of (T:,H:). Similarly the outlet properties (9°,M0) are the

* *
ordered pair corresponding to the outlet air properties (TO,HO).

Generally the dimensionless norm (83) is not equal to zero.

Therefore, to obtain more accurate values of the outlet properties, it is

essential to interpolate between the discrete values of (Ti,Hi).

For a given dryer length the inlet air properties are functions

of the outlet air properties:

Ti = Ti(TO,HO)

H. = H (T ,H )
1 i o o

(84)

(85)

Expanding equations (84) and (85) by Taylor's Series about the point

(T:,H:), yields:

- T* + BT1 T T* + BT1 H*
Ti _ i 3T * ( o - 0) 3H (Ho - o) (86)
o H o T*
o o
H H* 3H1 (T T ) 6H1 (H H*)
* * 87
1 1 3T0 Ho 0 3H0 To 0 o ( )

 

39

Equations (86) and (87) can be evaluated for the desired outlet air

' t i 1
properties (TO,HO) corresponding to the inlet air properties (Ti’Hi)' Thus:

3T1 , 5Hi
H "LEE; (H1'H1)‘3'f;
3:1 -211. 93;.)
BBC 6T0 6H0
* 5T1 ! * l *
o-[Sfi—O-(Ho-Ho)-(T1-Ti)]/

' *
(Ti - T1) 1 /

(88)

3T1

-——- (89)
BTO

For a given dryer length the outlet product properties (90,M6) can be

expressed as functions of the outlet air properties (TO,HO).

C)
II

:3
u

60(HO,TO)

MO(HO,TO)

(90)

(91)

Expanding equations (90) and (91) by the first two terms of Taylor

* *
Series about HO and T0,. yields:

t * 560 u
e. = e. +5; (To
_. _* 5E0 .
M " M + --'(T

o o 3T0 o

390

* ' *
- To) +— (H0 - no)

830

(92)

(93)

Where T; and H; are the values calculated in equations (88) and (89).

For a dryer of zero length, it is obvious that the outlet preper-

ties are equal to the inlet properties:

cacao)

a (ei’Mi)

(94)

 

40

(T1,H1) = (TO,HO) (95)

When the dryer length is increased by As the inlet air prOperties and

the outlet product properties can be calculated by the following formulas:

 

 

 

_ d8 _ dM n

(e ,M ) - ( 6 +-——— Aa,M +'——— As ) u
o 0 As 0 d 0 0 dx 0 (96) ; ..A
. dT1 dH 3 ,
(Ti’Hi) = ( T1 + 21?- Aa’Hi + Ex— Aa ) (97) N

Ag 0 0 .

 

The total derivatives with respect to x can be calculated by equation

(8), (ll), (17) and (37). Equations (96) and (97) can be written for

an arbitrary dryer length (a) as:

 

_ deo _ (IMO (98)
(eo’Mo) a + As a ( 9o + F? aA"Mo + Fix— 58' )
dTi dH1
(T1,H1) = ( Ti 4" 21-;- A8,H1 + a— As ) (99)
a + Aa a a

 

Using the initial conditions (94) and (95) and equations (98) and (99)
the values of 90,MS,T1 and H1 which yield the outlet air properties To
and HD for a dryer of length a + As can be calculated.

The calculation of the outlet properties by invariant programming
can be outlined as follows:

1) Determine a set S of feasible To and HO values for the desired

'
inlet air properties (T1 and Hi). A recommended set would

include discrete outlet air temperatures between the inlet

2)

3)

4)

5)

6)

7)

To

41

product temperature and the maximum desired inlet air
temperature. The range of outlet humidity ratios should
include discrete values between the minimum desired inlet
air humidity ratio and the humidity ratio at the wet bulb
temperature of the maximum desired inlet air temperature
and the maximum desired inlet air humidity ratio.

Initialize the outlet product properties and the inlet air
properties by equations (94) and (95).

Step the dryer length and calculate the appropriate set of
outlet product properties and inlet air properties by equa-
tions (98) and (99).

Repeat step three until computer output is desired.

For each pair of desired inlet air temperatures and inlet air
humidity ratios, search the set of calculated inlet air
properties to find the order pair (T:,H:) which minimizes the
dimensionless norm (83).

Calculate the desired outlet prOperties by equations (88), (89),
(92), and (93).

Return the step three and repeat the procedure until the maxi-

mum desired dryer length is reached.

improve the accuracy of the solution choose more discrete ordered

pairs (TO,HO) or use more terms in the Taylor Series for equations (86),

(87), (92) and (93).

It should be noted that this procedure will only model the system

if criterion (81) is satisfied and no condensation occurs within the dryer.

If criterion (82) is satisfied, one must consider a set R of feasible outlet

42

product properties (60,M%). Following a similar procedure will yield
the desired outlet properties.

Invariant programming requires no estimates and will give very
accurate results if the number of ordered pairs in S or R is large.

This is a definite advantage over invariant imbedding.

RESULTS AND DISCUSSION

Errors

Most numerical techniques yield approximations of the original
mathematical models, therefore the numerical solutions are not identical
to the exact solutions. Due to numerical approximations both the invariant
imbedding and the invariant programming formulations of the counter-
flow drying problem are subject to errors. The invariant imbedding
solution is subject to errors due to the assumptions and estimations
necessary to calculate the outlet properties along the boundaries. The
errors induced in the invariant imbedding formulation can be summarized
as the following:

1) Errors due to the assumptions (a and b, pages 24 and 25)
made along the T1 equal 91 boundary of the invariant
imbedding problem.

2) Errors due to the estimation procedure (pages 26 through 32)
used to calculate the outlet properties along the Hi
equal Heq boundary.

3) Truncation errors.

If the steptsizes in T1,H1 and a are chosen small enough the truncation
errors are small with respect to the other errors.

In order to study the effects of the first two types of errors,
the values of the outlet product moisture content calculated by the
invariant imbedding solution were compared with the values of this
property which satisfy the model. In Figures 9, 10 and 11, the outlet

moisture contents are compared near the Ti - 9i boundary, near the

43

44

Hi = Heq boundary and for a point in the center of the mesh. The inlet

conditions and parameters for the dryer considered are the following:

Ga 200 lbm dry air/hr

GP 35 lbm dry product/hr

61 80 F Hi

3%. .33 .nrns
In Figure 9, the outlet product moisture contents calculated by the I

invariant imbedding and the invariant programming formulations are

 

1‘4_44

plotted. The values obtained from the invariant programming formulation

I

were tested with the system model and proven to satisfy the model. It
should be noted that the inlet air temperature is very close to the
inlet product temperature. Also the inlet air humidity ratio is much
less than Heq' Therefore, it was assumed that the errors shown in
this figure are due largely to errors of kind number one. Though the
absolute magnitude of the errors are small, the relative errors near
this boundary are quite large.

In Figure 10 the inlet air temperature is much greater than the
inlet product temperature, while the inlet air humidity ratio is only
slightly greater than Heq' In this case, the errors were assumed to be
of the second type. The estimation procedure appears to have over-
estimated the magnitude of the mass transfer. The relative errors
near this boundary are small, but the absolute errors are quite large.

In Figure 11, an inlet air temperature and air humidity ratio in
the center of the mesh are chosen. Even though the errors along the
boundaries are large, the errors for inlet air properties a distance
from these boundaries are small. Therefore, if the invariant imbedding

formulation is used the outlet properties calculated near these boundaries

01/ 21.5.7- PRODUC' 7' MOISIMEE COA/Z'EZVT

45

 

.330
—- '-.-//vm1?/ANT [MEDIA/G;
--- .JNWRWMM/AG
322 P
.326 t
324 -
322 ~

 

 

.04 -08 «1‘2 .16 .20
DR YEP LENGTH

Figure 9. Errors in the Invariant Imbedding Solution near
the Ti 8 Bi Boundary

Ti - 90.0 F

Hi - .0117 lbm.water vapor/lbm dry air

Cu
k»

A“

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46

-— -JNVAP/ANT “IMBEDDING" " I
— :7/vm 91m m‘GRAMM/m .J

 

 

Figure 10.

.2 i .2 ,g, z. 2..
23191419101le

Errors in the Invariant Imbedding Solution near the
Hi - Heq Boundary

T1 3 180 F

Hi - .02003 lbm.water vapor/lbm dry air

47

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h;
w

— flVVflPf/UVT 'IMEEDD—[ZZ'GZ
-— — lAl/A’RIAMT' PEDEWlA/G

 

 

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Figure 11. Errors in the Invariant Imbedding Solution for
a Center Mesh Point

Ti 3 180 F

H 8 .0117 lbm.water vapor/lbm dry sir

i

48

should not be trusted.

There exist two basic types of errors in the invariant pro-
gramming solution. These are the following:
1) Errors due to the linear interpolation between the dis-
crete ordered pairs (TO,HO). [equations (88), (89), (92)
and (93)]. '
2) Truncation errors in the numerical solution of equations
(98) and (99).
If the step size in a is small the truncation errors can be considered
small with respect to the interpolation errors.
In Figure 12 a typical set S of feasible outlet air properties
is shown. The inlet conditions and parameters for the dryer considered

are the following:

Ga 261.3 lbm dry air/hr

Gp 18.7 lbm dry product/hr
91 80.5 F

Mi .481

The desired inlet air properties are:

T. 180.0 F
1
Hi .0053 lbm water vapor/lbm dry air
The set S includes outlet air temperatures from 150 F to 80.5 F and
outlet air humidity ratios from .03 to .0053 lbm water vapor/lbm dry

air. Each dot in Figure 12 represents a discrete value of the ordered

pair (TO,HO). It should be noted that the line in the lower right-hand

49

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Schematic Diagram of the Solution of a Counter-Flow
Drying Problem, Using Invariant Programming.

Figure 12.

50

corner is the saturation line. For any point below and to the right

of this line, the outlet relative humidity would be greater than 100%.
Therefore, these points cannot be included in the set of feasible outlet
air properties.

Starting at the upper left-hand corner of this figure, the circled
dot is the ordered pair (To’Ho) which minimizes the dimensionless norm
[equation (83)] for this dryer with a length of one inch. The corres—
ponding cross (+) is the interpolated value of this order pair. As
the dryer is increased in length, one inch at a time, the ordered pair
which minimizes the dimensionless norm moves down and to the right.

For each circle there is a corresponding interpolated value of the
ordered pair (TO,HO). When the dryer length is equal to 14 inches the
interpolated values of (TO,HO) are approximately on the saturation line.
Since the air flow rate is much greater than the product flow rate, the
outlet product temperature is equal to the inlet air temperature. Thus,
the dryer has approached equilibrium state 4. Increasing the dryer
length will not change the outlet properties (TO,HO,90,M6).

It should be noted that the order pair (TO,HO) which minimized
the dimensionless norm is not necessarily the nearest discrete ordered
pair to the interpolated value of (TO,HO). This can be explained by the
fact that the dimensionless norm is minimized for the other end of the
dryer.

It can be seen that the ordered pair which minimizes the dimension-
less norm may be quite distant from the interpolated value of the ordered

pair. More accurate results may be obtained by considering more discrete

51

values of (TO,HO), thus reducing this distance. This is especially true
near the saturation line, where the inlet air properties (Ti’Hi) and the
outlet product properties (90,Mg) are very sensitive to the outlet air
properties.

Contrary to the invariant imbedding formulation the invariant
programming can yield results with as much accuracy as desired if the
step size in a is small the number of discrete ordered pairs (To’Ho) is
large. A reasonable estimate of the interpolation error is one—half
the distance between two neighboring values of the discrete ordered
pairs (To,Ho). For the reaons presented above the invariant programming

formulation should be used whenever possible.

Experimental Versus Theoretical Results

The theoretical model was compared with the experimental results
of Ives (1967) (experiments III-56, III-58, and III-60). The inlet
conditions, the flow rates and the parameters for the experiment con-

sidered are the following:

Ga 261.3 lbm dry air/hr

Gp 18.7 lbm dry product/hr
61 80.5 F

1711 .481

hD 2.12 x 10'3 ft/hr

In Table l, the experimental and the theoretical outlet product moisture

contents are compared for three dryer lengths.

52

Table 1. Experimental Versus Theoretical Results for Three
Dryer Lengths

 

 

Dryer Length W
10 inches .244 .256
14 inches .224 .215
24 inches .216 .215

 

 

 

 

In Figure 13 the theoretical product moisture content versus x is
compared with the experimental product moisture content for a dryer of
length 14 inches. In Figure 14, the theoretical and the experimental
values of the air temperature are compared for the same dryer.
The agreement between the experimental and the theoretical results
is quite good. From Table 1, it will be noted that for a 10 inch long dryer the
theoretical value of the outlet product moisture content is slightly greater than
the corresponding experimental value. It appears that the theoretical
model under-estimates the quantity of mass transfer. It was noted
earlier that the Chu's diffusion coefficient for corn was calculated
from data between 120 F and 160 F. Therefore, one cannot be certain that
the diffusion coefficient calculated by equation (26) at 180 F is the
correct value. Also, little is known about convective mass transfer
within a bed of particles. Therefore, the value of the convective

mass transfer coefficient used is no more than a crude estimate.

Quasilinearization
For a dryer of a given length and with given inlet air and

product properties a quasilinearization or a non-linear estimation pro-

53

o [mas/MEN TAL
-, 774509571an

2):

 

P/PCZDUC 7’ MO/J‘fé/PE CMZJE/Vf

 

4” 8' 12” ’6”
’X
Figure 13. Comparison of Theoretical and Experimental
Product Moisture Contents within the Dryer

T1 - 180.0 F

Hi = .0053 lbm‘water vapor/lbm dry air

 

5

AR?

A59 ‘

LEO

 

Figure 14.

54

   

0 [XPE/P IME/V 7A2.
—- -_ THEQEEI 1C AL

Comparison of Theoretical and Experimental Air
Temperatures within the Dryer.

Ti = 180.0 F

Hi = .0053 lbm water vapor/lbm dry air

55

cedure could also be used to determine the outlet properties (90, M ,

0
To and Ho). The techniques presented in this paper have two major advan-
tages over quasilinearization. First contrary to the invariant imbedding
and the invariant programming formulations, the quasilinearization pro-
cedure requires a good initial estimate of the outlet preperties or the
system equations (8), (ll), (17), (23), (31) and (35) will become
unstable. Secondly, in generating the solutions the quasilinearization
procedure yields no other practical information. The invariant

imbedding and the invariant programming formulations generate the

outlet properties as functions of the dryer length and the inlet air

or product properties.

Parameter Study
In Figures 15, 16 and 17 the outlet product moisture content is

shown as a function of the dryer length and the inlet air prOperties

for a family of dryers with the following inlet properties and flow rates:

Ga 200 lbm dry air/hr

GP 35 lbm dry product/hr
. F

91 80 0

Mi .33

In Figure 15 the outlet product moisture content is plotted
for various inlet air temperatures as a function of dryer length for
a given inlet air humidity ratio. With the inlet air temperatures equal
to 100 F and 120 F, the dryer approaches a condition where the inlet
product temperature is equal to the outlet air temperature and the inlet

air temperature is equal to the outlet product temperature. These

dryers reach this state at .3 and .6 feet, respectively. With inlet

0071 E 7' PPOQUCT MO/J‘ TZ/PE'CD/VIZWT

 

56

 

 

 

 

.33
“H,- :2015"
J; '-' n. /00F
.3] L—
/20 F
oZQr
#10 F
/50 F
-27»
IEUF
.3 i4 .3 is J...
113143 V5? A£7V677¥
Figure 15. The Outlet Product Mbisture Content for Various

Inlet Air Temperatures Versus Dryer Length for a
Given Inlet Air Humidity Ratio

57

air temperatures equal to 140 F, 160 F and 180 F the dryer approaches
equilibrium state 4. For inlet air temperatures of 140 F and 160 F
the dryer reaches equilibrium state 4 for a dryer length of .8 and
.95 feet, respectively.

The basic shape of these curves is typical for counter-flow
grain dryers. The dryer decreases its outlet product moisture content
rapidly as the dryer length is increased until an equilibrium state is
reached. As the dryer length is increased the outlet air temperature
is lowered due to the energy required for evaporation of the moisture
and the energy used to increase the outlet product temperature. The
outlet air humidity ratio is increased as moisture is transferred from
the product. Therefore, for the dryer which approaches equilibrium
state 4, the system rapidly approaches a condition where the outlet air
is saturated.

In Figure 16 the outlet product moisture content for Various
inlet air humidities is plotted as a function of dryer length for an
inlet air temperature equal to 180 F. All of these dryers approach
equilibrium state 4. For the inlet air humidity ratios .04 and .03
lbm water vapor/lbm dry air the system reaches equilibrium state 4 when
the dryer length is increased to .8 feet and .9 feet, respectively.

In Figure 17, the outlet product moisture content for various inlet
air humidity ratios is shown as a function of the inlet air temperature
for a given dryer length. For the inlet air humidity ratios .02, .03 and
.04 lbm water vapor/lbm dry air all of the dryers have reached equili—
brium. For the inlet air humidity ratio .01 lbm‘water vapor /lbm dry

air the dryer has reached equilibrium if the inlet air temperature is

58

  

 

 

 

g 3..

S

E .‘--'7,-"=i“780*r' '

g: 3/ »

k

‘2

%

K) ‘\5\

a 3m;

§ 27+ 53m
K Minor
3 + A . lag-2:1 _
g .2 .4 .6 .8 ’-0

.Zm'llfl/GTH

Figure 16. The Outlet Product Moisture Content for Various
Inlet Air Humidity Ratios Versus Dryer Length
for a Given Inlet Air Temperature.

OWL/57' [DADODOCT [MO/(5702475- CO/fo/Vf

.33

.27 —

59

    

DRYER! LENGTH IFZ'

.3!~

 

J

lrxa AZCD /4Z> A6C> nap

”YE 7' AM’ TEMPZ'PA WAY

Figure 17. The Outlet Product Moisture Content for Various
Inlet Air Humidity Ratios Versus Inlet Air Temper-
ature for a Given Dryer Length.

60

less than or equal to 150 F. When the inlet air humidity ratio is equal
to .03 or .04, all of the dryers have reached equilibrium state 4.

If the inlet air humidity ratio is equal to .02 the system approaches
equilibrium state 4 if the inlet air temperature is greater than 90 F.
Similarly, if the inlet air humidity ratio is .01 the system approaches
equilibrium state 4 if the inlet air temperature is greater than 150 F.
If the inlet air humidity ratio is equal to .01 and the inlet air temper-
ature is less than or equal to 150 F or if the inlet air humidity ratio
is equal to .02 and the inlet air temperature is less than or equal

to 90 F, the system will approach a condition in which the inlet air
temperature is equal to the outlet product temperature and the inlet

product temperature is equal to the outlet air temperature.

SUMMARY AND CONCLUSIONS

Invariant Imbedding Versus Invariant Programming

Both invariant imbedding and invariant programming are ideally
suited to solve first.order two-point boundary value problems. In
this research the first order equations are equations (8), (ll), (17),
(23), (31) and (35) and the two-point boundary conditions are equations

(38) through (41). In general n of these boundary conditions will be

 

at x equal zero. The remaining“m boundary conditions will be at x equal
a. For the model presented in this paper n equals 4 and m equals 2.
Invariant imbedding equations can be written for the m unknown
properties at x equal zero and the n unknown properties at x equal a.
There exist two basic forms of these invariant imbedding equations.

1) For given values of the n known properties at x equal zero,
the n + m unknown properties can be expressed as functions
of the known properties at x equal a and the dryer length.
The outlet air and product properties are expressed as
functions of the two known properties at x equal a and
the dryer length in equations (45), (48), (53) and (55).

2) For given values of the m known properties at x equal a,
the n + m unknown pr0perties can be expressed as functions
of the known properties at x equal zero and the dryer length.
The outlet properties are expressed as functions of the four
known properties at x equal zero and the dryer length in

equations (58) and (61).

61

62

For both types of invariant imbedding equations initial conditions
are known values of the n + m unknown properties at a specific dryer
length. Boundary conditions for the first type of invariant imbedding
euqations are known values of the n + m unknown properties at specific
values of each of the m known properties at x equal 3. Therefore, to solve
for the n + m unknown properties using the first set of invariant imbedding
equations it is essential that n + m initial conditions and m(n+m) boundary
conditions be known.

Boundary conditions for the second set of invariant imbedding
equations consist of known values of the n + m unknown prOperties at
specific values of each of the n known properties at x equal zero. In this
case n + m initial conditions n(n+m) boundary conditions are necessary
to solve the invariant imbedding equations.

In some cases not all n + m initial conditions and m(n+m) boundary
conditions are known for the invariant imbedding equations of the first
type. Similarly there exist cases where the n + m initial conditions and
n(n+m) boundary conditions are not all known for the invariant imbedding
equations of the second type. In these cases it is difficult to solve
the invariant imbedding equations. Therefore, invariant programming
should be used. For the model studied in this paper, there were insuffi-

cient boundary conditions. Thus, it is essential to use invariant programming.

Model Limitations

It was stated on page 35 that there exist special cases [equation
(81) is satisfied and condensation occurs within the dryer) for which
it is not possible to solve the system equations (8), (ll), (17), (23),

(31) and (35). In the design of a counter-flow dryer one is interested

63

in maximizing the degree of drying. Therefore, a system in which conden-
sation occurs is of no practical interest.

Since invariant programming utilizes the model directly, this
method can be solved for the outlet properties if the model can be

evaluated.

Suggestions for Further Study

The model presented would be capable of representing the system
for a wider range of inlet conditions, if the diffusion coefficient for
corn was known for higher product temperatures. More research is needed
in the area of convective mass transfer within packed beds of particles.

With further study the model could be written in terms of more
representative driving potentials (free energies and chemical potentials).
With this modification the equilibrium states should be redefined and
experimentally verified.

On page4l it was noted that if criterion (82) was satisfied, one
must consider a set R of feasible outlet product properties. Theoretical

work is yet to be done for this special class of dryers.

BIBLIOGRAPHY

Bakker-Arkema, F. W., T. W. Evans and D. M. Farmer (1969). Simula-
tion of Multiple-Zone Grain Drying. Transactions ASAE (in
press).

 

Chu, S. T. and A. Hustrulid (1968). Numerical Solution of Diffusion
Equations. Transactions ASAE, ll, 5, 705.

 

Ives, N. C., W. V. Hukill and H. M. Black (1967). Corn Drying Time
at Counter-flow Steady State. Transactions ASAE, ll, 2, 240.

 

Lee, E. S. (1968) Quasilinearization and Invariant Imbedding.
Academic Press, New York and London.

Rodriquez-Arias, J. H., C. W. Hall and F. W. Bakker-Arkema (1963). Heat
of Evaporation for Shelled Corn. Cereal Chemistry, 40, 6, 676.

 

Thompson, T. L. (1967). Predicted Performances and Optimal Design
of Convectional Grain Dryers. Unpublished Ph.D. Thesis, Purdue
University, 1967.

64

a)

b)

C)

APPENDIX

Computer Program which solves for the outlet pro—

perties using invariant imbedding.

Computer Program which solves the system model.

Computer Program which solves for the outlet pro—

perties using invariant programming.

65

67

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