.V..._.

DRYER

" NT

I‘COGUR”

 

 

7?...
I1 .I

......x a.

2 Hi.

V...” (...;
«u...

A»...

.......:».V...V
.....I.
.... ...
«:3
.va:

....Zfiiz

fVVV:

 

 

.r.. . ..re .,
1...... 2.2..

{V :r 1..
... €31.95... In... t .

L. Inhv ‘5 l

I...

. . ...”..m... 3....

 

LIBRARY

Michigan Stan:
Unim ‘ISity

 

This is to certify that the

thesis entitled

SIMULATION AND OPEN-LOOP CONTROL
OF A COCURRENT DRYER

presented by

Wayne Howard Clifford

has been accepted towards fulfillment
of the requirements for

Ph.D. Engineering

degree in

MM

i; major professor

 

Date Feb. 10, 1972

 

0-7 639

ABSTRACT

SIMULATION AND OPEN—LOOP CONTROL
OF A COCURRENT DRYER

BY

Wayne Howard Clifford

The dynamic simulation of a cocurrent moving bed
dryer for shelled corn involves at least three independent
variables: time, distance along the bed, and whatever is
used to model the moisture gradient within an individual
corn kernel. The first problem is to simulate a thin
layer of drying corn—-the mathematical equivalent of
simulating a single kernel. In this case the diffusivity
of water in corn is available as a function of average
moisture content and average temperature of the kernel.
Also the Biot number is small enough so that temperature
gradients within the kernel may be neglected. The model
divides the kernel into three concentric regions with the
same dry mass in each region. Geometric parameters in the
model were fitted from experimental thin-layer drying
data, using a hill-climbing parameter optimization program.

The parameter optimization requires that the model

be solved many times, so a highly efficient integration

Wayne Howard Clifford

routine is important in order that the cost of the opti-
mization not be prohibitive. A method is presented where
a set of a few nonlinear differential equations is solved
by linearizing and solving the linearized equations by
Laplace transforms. For accuracy the linearization must
be done at the center of the interval, but in practice it
is done at the front and end and the average of the two is
used. Iterations are needed since the values at the end
of the interval are not known.

Once the geometric parameters have been found,
the convective heat and mass transfer coefficients are
needed. The steady—state equations correSponding to the
thin layer model were derived and solved. This solution
made use of the same technique as the thin‘layer solution.
Parameter optimization was again used, this time to find
the convective heat and mass transfer coefficients.

The dynamic corn dryer model consists of 6 non-
linear partial differential equations. The backward dif—
ference approximation was used for the time partials, and
then the equations were solved by the same technique used
above. The dynamic solution was used to evaluate the net
cost during the transition from one steady—state to
another, corresponding to changing from one batch of corn
to another at 5% higher moisture. The goal was to find
the best input air temperature function during the trans-

ition. Several one— and two-parameter functions were

Wayne Howard Clifford

tested and optimized in terms of their own parameters.
Since the best control found was not significantly better

than a single—step (or bang—bang) control, the latter

would be the best to use under current normal operational

conditions.

SIMULATION AND OPEN~LOOP CONTROL

OF A COCURRENT DRYER

Wayne Howard Clifford

A THESIS

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

DOCTOR OF PHILOSOPHY

Department Cf Chemical Engineering

1972

ACKNOWLEDGMENTS

The author wishes to express his thanks to
Dr. George A. Coulman for his guidance and help through—
out the course of this work.

The author is indebted to Dr. Fred W. Bakker-Arkema
for his assistance and interest.

Appreciation is also extended to the other members
of his committee: Dr. Myron H. Chetrick, Dr. Donald K.
Anderson and Dr. William G. Bickert.

Thanks are due to several of Dr. Bakker's graduate
students for help with various aspects of this work,
especially Daniel Elzinga for thin-layer experiments,
Ralph Gygax for steady—state data, Dave Farmer for intro-
ducing the author to Rosenbrocks hill—climbing program,
and Lloyd Lerew for getting programs into the computer.

The understanding and patience of the author's
family, especially his wife, Mary Kay, is sincerely

appreciated.

ii

TABLE OF CONTENTS

 

Page

ACKNOWLEDGMENTS. . . . . . . . . . . . . . ii
LIST OF TABLES . . . . . . . . . . . . . . iV
LIST OF FIGURES. . . . . . . . . . . . . . v
LIST OF SYMBOLS. . . . . . . . . . . . . . vi
INTRODUCTION. . . . . . . . . . . . . . . 1
LITERATURE REVIEW . . . . . . . . . . . . . 3
APPROACH TO THE PROBLEM . . . . . . . . . . . 6
NUMERICAL TREATMENT OF THE DIFFUSION EQUATION IN

APPROXIMATELY SPHERICAL PARTICLES . . . . . . . 13
A THIN-LAYER MODEL FOR SHELLED CORN . . . . . . . 22
STEADY- STATE SIMULATION AND CONVECTIVE HEAT AND MASS

TRANSFER COEFFICIENTS IN THE DEEP BED. . . . . 37
DYNAMIC SIMULATION OF A COCURRENT DRYER FOR

SHELLED CORN . . . . . . . . . . . . . 43
(DPEN-LOOP NEAR-OPTIMAL TEMPERATURE CONTROL. . . . . 61
CONCLUSIONS . . . . . . . . . . . . . . . 74
REFERENCES . . . . . . . . . . . . . . . 77
APPENDICES . . . . . . . . . . . . . . . 80

10.

ll.

12.

13.

LIST OF TABLES

Thin-layer data and simulation average
moisture contents as a function of time. . .

Thin-layer moisture contents as a function
of time for low temperature Troeger data . .

Thin—layer moisture contents as a function
of time for medium temperature Troeger
data 0 O O O O O O O O O O O 0 O

Thin-layer moisture contents as a function
of time for high temperature Troeger data . .

Differential equations for a steady—state deep
bed corn dryer . . . . . . . . . . .

Values of the proportionality constant between
convective coefficients for heat and mass
transfer . . . . . . . . . . . . .

Steady—state dryer data and best fit simulation
results . . . . . . . . . . . .

Dynamic equations for a cocurrent corn dryer. .

Substitutions used in the linearization of the
cocurrent dryer dynamic equations. . . . .

Linearized cocurrent dryer dynamic equations. .

Open-loop temperature control results,
Runs 1—22. . . . . . . . . . . . .

Open-loop temperature control results,
Runs 22—38 . . . . . . . . . . . .

Additional single—step temperature control
results . . . . . . . . . . . . .

iv

Page

30

33

34

35

38

41

42

48

57

58

65

68

72

.10.

LIST OF FIGURES

Page

Cocurrent corn dryer diagram . . . . . . . 7
Division of a spherical particle into pyramidal

segments . . . . . . . . . . . . . l4
Locating diagram for thin-layer model

parameters . . . . . . . . . . . . 20
Flow chart of the method used for single-step

integration for initial value problems . . . 25
Thin-layer moisture contents as a function

of time for Data Set #3 . . . . . . . . 31
Variation of moisture content with time for

selected sets of Troeger data . . . . . . 36
The linearization used for the previous time

step . . . . . . . . . . . . . . 50
Input temperature function for the best

single-step and overall . . . . . . . . 69
Exit moisture contents in response to the best

single—step and the best overall input

temperature functions. . . . . . . . . 70
Variation in penalty with time of single—step . 73

AorA

LIST OF SYMBOLS

3

area of corn kernel in ftZ/ft , 239.0 at e = 0.5.

area enclosing regions 1 or 2 in ft2/ft3
ratio of area to Ar, ft

heat capacity of corn, air, Btu/lb of dry
matter/°F

diffusivity of water inside the corn kernel,
ftz/hr

diffusivity of water in air, ftz/hr

equivalent particle diameter, ft

mass flow rate of corn, air, lb of dry matter/
hr/ft

absolute humidity of air, lb of water/lb of dry

air
heat transfer coefficient, Btu/hr/ft2/°F

mass transfer coefficient based on moisture
content differences, lb of water/hr/ft2

heat of vaporization of water, Btu/lb
thermal conductivity, Btu/hr/ft/°F

ratio of heat to mass transfer for coef—
ficients, 16°F/Etu

proportionality constant in Equation 44

dry basis moisture fraction, lb of water/lb of

dry corn
average moisture content over region 1, 2, 3
moisture content in equilibrium with air

relative humidity

vi

radial distance inside the corn kernel, ft
Laplace Transform variable

temperature of corn, air, °F

time, hours

overall mass transfer coefficient,
lb of water/hr/ft2

volume, ft3

length along the dryer, ft

ratio of area to Ar for region 1, 2

void fraction of the bed

dry basis particle density of corn, 72 lb/ft3
density of air, lb of dry air/ft3

viscosity, lb/sec/ft

vii

 

INTRODUCTION

As mankind moves toward the last quarter of the
twentieth century, the message of R. Buckminster Fuller,7
". . . the physical resources of earth can support all
of a multiplying humanity at higher standards of living

3

than anyone has ever experienced or dreamed,‘ presents
both an optimistic outlook for the future and the largest
challenge ever to our technology: ”So it is also part
of the great message to humanity . . . that the world's
problems cannot be solved by politics and can only be
solved by a physical invention-and-design revolution."7

Mr. Fuller even mentions part of the method of
approach. "We find that man is deveIOping an increas-
ing confidence in the way in which computers are re-
solving heretofore vexing and seemingly unsolvable
problems."7

This work represents the author's concept of the
next steps to be taken so that computers may be used to

bring about one part of that invention-and—design revolu—

tion.

*Footnote numbers refer to Bibliography entries.

 

A cocurrent corn dryer was chosen for this re—
search for several reasons. Corn is a very important raw
material for various foods; therefore any method of in—
creasing the supply, such as improving methods used in its
preservation, has humanitarian as well as economic value.
Most optimal control work has involved linear systems, so
a nonlinear system was sought. In addition, relatively
little control work has been done with distributed systems
such as tubular reactors or moving bed dryers. Also, most
optimal control uses the minimum time to reach steady
state, a minimum amount of control action, or the time
integral of the squared error from some desired value as
the objective function, and all of these seem rather
artificial. On the other hand, a straight economic objec—
tive function can easily and accurately be applied to a
corn dryer. Finally, the strong interest in corn drying
over the years has produced a wealth of basic technology.

The author feels that the following are important
contributions in this thesis. First, a new thin layer
model is presented based on a different way of looking at
the corn kernel. Next, this model is used with some
steady state data to find heat and mass transfer coeffi-
cients for the deep bed dryer. The corresponding dynamic
deep bed dryer equations are derived and solved to find a
practical optimum open loop control. In addition, a highly
efficient method of solving sets of a few differential

equations is presented and demonstrated.

LITERATURE REVIEW

The most significant factor in any simulation of
corn drying is the method used to treat the drying of a
single kernel of corn. Single kernel drying is also known
as thin‘layer drying since in a thin layer, each kernel
will be exposed to the same drying air. Many reviews are
available for such work, such as Bakker-Arkema and Lerew.l
Two approaches are used in single particle models: single—
lump models and moisture—gradient models.

In single—lump models the results of thin—layer
drying tests are summarized in empirical expressions.

l3 express the drying rate as a function

Troeger and Hukill
of equilibrium moisture content (which is a function of
air temperature and humidity), temperature, initial
moisture content, current moisture content, and air
velocity. Thompson et a1.ll use an empirical expression
to calculate the time required for a certain amount of
drying as an explicit function of all the same above
parameters except air velocity. In each case the empiri-
cal constants found in the model have very little physi-
cal meaning.

A moisture gradient model with results like a

Single-lump model was used by Chittenden and Hustrulid.5

They used a prefectly spherical model with no surface‘
resistance and a constant moisture diffusivity. Under
those assumptions, a closed form solution can be found
'for moisture content versus time, involving an infinite
series of exponentials.* Their main conclusion was that
the diffusivity is not constant.

The work of Chu and Hustrulid6 represents an
advancement in that the requirement of a constant dif—
fusivity was dropped; however, the perfect sphere and no
surface air resistance assumptions were kept. The dif-

fusivity was then estimated from short-time drying

results. This model is a pure internal moisture gradient
type. However, it also uses a "dynamic equilibrium
moisture content" and a " seudo initial moisture content."

While each of these is a plausible physical phenomenon
and each is useful for making the data fit their model,
they are both somewhat difficult to determine except in
terms of a specific model. The "dynamic equilibrium
moisture content" may arise from physical changes in the
corn at different temperatures and moisture contents,
perhaps related to the hysteresis effects found in re—
peated equilibrium moisture sorption isotherms. The

"pseudo initial moisture content" is said to arise from

 

*This solution is then used in the same manner as
the completely empirical models, but the constants have
physical meaning, e.g. the equivalent spherical radius or
diffusion coefficient.

 

the fact that the kernel is somewhat cooler than the dry-
ing air for the first part of the drying.

15 which does not

The work of Whitaker et_al.,
involve corn directly, uses a variable diffusion coeffi-
cient on a spherical body with finite surface resistance.
They divided the sphere into ten shells numerically and
used a linear function of moisture content for the diffu-
sion coefficient. This work was performed to prove the
method before attempting to apply it to biological systems.

The most important work found on control of a
cocurrent dryer for corn is that of Zachariah.l6 He
controlled the flow rate of corn in response to exit
moisture content using four different control systems,

and concluded that a proportional—integrated controller

with on-off override would be best.

 

 

APPROACH TO THE PROBLEM

The goal of this work is to find a control scheme
for a cocurrent corn dryer. In such a dryer, as is
depicted in Figure l, the wet corn and drying air enter
at the same end. Although any configuration is possible,
the flow generally is downward. An unloading mechanism
is provided at the bottom to control the corn flow rate.
Plug flow will be assumed for both corn and air, as well
as perfect insulation at the sides, so there will be no
gradients in planes orthogonal to the direction of flow
and the cross-sectional shape will not be important.

The independent variables in a mathematical
description of the dynamic operation of a deep bed corn
dryer will be time and distance along the bed as well as
whatever is used to model the moisture gradient inside
the corn kernels. It is clear, then, that a thin—layer
model must form the basis for any deep bed simulation.
Once it is established, the extensions to a steady—state
deep bed dryer and from there to the dynamic dryer are
mostly straightforward.

The thin—layer model used here has three internal
zones to determine the moisture gradient. Since each zone

adds a differential equation to the steady-state

 

 

     
   
     
   
 

____> Grain Flow

--——9 Air Flow
Wet Corn

Storage

H,

   

 
   

Air
Heater

     
    

 
  

Drying Air
Entrance

\\\\N\‘Qd

 

i \
)\ ’A// ’\/\
/ \
é ¢ ‘\Z.m/ Air Exits

J,

/~Z/,Corn Unloading
Mechanism

Figure l.—-Cocurrent corn dryer diagram.

W h

 

simulation (and a partial differential equation to the
dynamic simulation), the number of zones should be kept

to an absolute minimum. It was decided that the diffusiv—
ity of Chu and Hustrulid6 would be used, but that it

would be based on the average moisture content rather than
the average between each pair of zones. The assumption of
zero surface transport resistance was dropped, as was the
spherical shape requirement. The resulting model has
three shape parameters and the two outside tranSport
coefficients, all of which must be determined by fitting
experimental data.

The thermal conductivity in the kernel is high
enough so the Biot number (th/k, the ratio of internal
to external resistance to heat transfer) is small. For
this reason temperature gradients inside the kernel are
neglected.

Since most chemical engineering processes are
carried out in counter current fashion, a brief explana-
tion of the use of a cocurrent dryer is in order. One
benefit of cocurrent operation is the tempering effect of
the end portion of the dryer, making control easier and
the product more uniform. There are two drawbacks in
countercurrent operation. The first is the tendency of a
countercurrent dryer to heat the exit product. This can
damage the starch so it cannot be used for certain appli—

cations. It also wastes energy since the corn should be

 

stored cool and so must be cooled in a subsequent opera-
tion and since the heat could be used to evaporate water.
The other difficulty with countercurrent dryers is the
tendency of the exit air to become saturated so that
moisture is actually added to entering corn by condensa—
tion.

The method used here for solving sets of nonlinear
ordinary differential equations is analogous to the
improved Euler method, but the exact solution to the
linearized set is used where a simple Euler step would
be made. The set of nonlinear differential equations is
linearized over short intervals and the exact solution to
the linearized equations is used. The intervals, although
short, are an order of magnitude longer than would be used
for a corresponding numerical technique such as the pre-
dictor corrector. For good results the linearization must
be dOne at the center of the interval. In practice it is
done at both ends and the average value of each lineariza-
tion coefficient is used. This requires that an estimate
of the final values for the interval be available, so the
method iterates until the values just calculated agree
with their estimate. The value of this method is that it
is both fast and accurate. The drawback is that it is
complicated to program, requiring the "hand" calculation
of the solution to the linearized equations. In addition

to the thin—layer and dynamic dryer simulations described

 

10

here, a steady—state program was written and implemented.
It ran about 50 times faster than the corresponding pro-
gram using the Bulirsch-Stoer (4) method.

Control of a cocurrent distributed system is dif-
ficult due to the long time lags involved. For a corn
dryer the air-side dynamics are very fast—-generally 1000
or so times faster than the product dynamics—~50 that the
air is very nearly at steady—state with respect to the
instaneous corn temperature and moisture distribution.
Consider an attempt to read the exit corn moisture content
continuously and use this to control the input air temper—
ature. If the exit corn is too wet, the input air temper—
ature must be raised. However this causes the corn near
the entrance to dry faster, so the air near the exit
becomes both cooler and more humid, making the exit corn
wetter still. Lowering the input air temperature instead
would be worse yet, since corn near the entrance would be
dried less. When that corn reached the exitythe dryer
would probably be incapable of recovering to properly dry
it; certainly the control would be asking for still lower
temperatures.

For the above reason an Open loop control was
used, where the input corn moisture content is used to
control the input air temperature. It was assumed that
a certain production rate is required, so corn flow is

not manipulated. The input corn temperature has little

ll

effect, since sensible heats are small compared to heats
of vaporization. The air flow rate is determined by the
fan and motor used and these are difficult to change.

The only input variable left is the air temperature, which
is easily controlled. The specific control problem is as
follows: find the input air temperature as a function of
time during the period when a new steady-state is achieved.
It is thought that a given batch of corn would be fairly
uniform in moisture content so a single input air tempera-
ture would achieve steady state drying for an economic
optimum. The problem concerns the input air temperature
over the time of transition from one batch to another;
what is sought is the economic Optimum.

The problem is well—posed once the simulation
model is available. However, at this time optimal control
theory has advanced only to the point where systems of
order 10 or maybe 15 can be handled. The model would
result in a system of 300 order if the dryer is divided
into 50 length segments. In addition, this system is
very highly nonlinear, while most of the optimal control
work has been with linear systems. For this reason it is
Virtually impossible to apply current optimal control
techniques; and, therefore, the results should not be
termed an optimal control. The term "near—optimal" best

describes the practical optimum which is actually found.

 

 

12

One possible solution would be to set up the input
temperature profile as a parameter vector where each com-
ponent is the temperature used in the numerical method
at the time step in question. This would give a vector
of about 150 elements. The excellent parameter optimiza—
tion technique used in the thin-layer section takes about
50 function calls for each component, so 7500 calls would
be required. Since the resulting program takes 5 minutes
to carry out each approach to steady state when using the
CDC 6500 computer, over 26 days straight would be
required for this computation.

The more practical method used to find a near
optimal control was to express the input temperature

profile as a function of 2 parameters and find the

 

optimum of those parameters.

 

NUMERICAL TREATMENT OF THE DIFFUSION
EQUATION IN APPROXIMATELY

SPHERICAL PARTICLES

The purpose of this section is to give the thin-
layer method a stronger theoretical basis. Consider a
particle which starts with a uniform moisture content
and is treated in such a way that conditions at its sur—
face vary only in time but are identical over the whole
surface at any instant. Such a particle may be considered
pseudo-spherical if there is a single point within, called
the center, where the moisture gradients are zero in each
direction (VM=O), and if all net moisture transfer occurs
in the radial direction. While corn may not fit these
conditions exactly, one may assume that it comes close
enough so that the results will apply.

The particle is divided into n2 pyramid—shaped
segments, each segment having angular widths 2fi/n, as is
shown in Figure 2. Then each segment is divided into m
zones in such a way that each zone within a segment con—
tains the same amount of material. Note that n will tend
to be large, while m is usually between 3 and 10. In

what follows the first subscript refers to the segment,

13

 

l4

 

 

 

 

 

 

/
The iEE
\ Segment
\ —"/
Zone 3
+ 4
5 4 r1,3
1 1 r.
r 1 1,2
i,l
Figure 2.—-Division of a spherical particle

into pyramidal segments.

 

 

 

lil||V||ll ; A

15

the second to the zone, so rin will be the outside radius
of the j—th zone in the i—th segment, Min will be the
average moisture content, and Ai,j will be the outside
surface area of the same zone. As n is made larger the
moisture moving between segments becomes negligible com-
pared to that moving along a segment. Then each will
behave as if it were part of a sphere of radius ri,m'

If D, the diffusivity, is independent of r then

D

8M1 2 BMi
:3?— = 7 (r a?" ‘1’
r

HP’

The average moisture content of the j—th zone

 

 

will be:
r.
— l i 1’] 47Tr2
i . — r 1 ——7— dr (2)
’3 1,] i,j-l n

where V. . = 4“ (r? . - r3 . ), the volume of the j—th

ll] 31.12 1!] 113—]-
zone in the i—th segment. Of course, ri 0 = O for all i.

I

Next each side of equation 2 is differentiated

with respect to time.

 

lrj o
der3 = 1 iIEI (3M ) dr (3)
at V. . 2 5t
1’3 r R
1,3—1

But the aMi/at is given by equation 1, so on substitution

 

 

17

Consider the case where there is a bulk outside
moisture content, Me’ which is the equilibrium moisture
content of the product at the temperature and humidity
of the bulk air flowing past the product. In addition,
there is a convective mass transfer coefficient, hé. The
subscript m is used to denote a mass transfer coefficient
and the prime is used to differentiate between this co—
efficient which is based on moisture content differences
and hm, the one more commonly used, which is based on
humidity differences.

Then the rate of mass transfer per unit area is

given by:

 

where Dq:is the inside moisture conductivity and Mi 5
I

is the moisture content at the surface. After some

algebraic manipulation, Equation 9 yields

 

3M1 hr; —
(——-) = —— (M — M. ) (10)
Sr ri,m D0 + h’ Ar. e l’m
c m i,m
where Ari m is the distance between the surface and the
I

point where Mi = Mi m' Substituting this into Equation 5
I

results in

 

 

 

 

 

 

l6
_ ri J
dMi,j _ 1 41Tr2 D a 2' 3M1
T‘v Tics” 5T)dr} (4)
i,j n r
r. .
1,3-1
Carrying out the indicated integration
dMi,j 4WD {(r2 aMl) _ (r2 8M1
dt _ 2 8r r. . r r. . }
n Vi,j 1,3 1,3-1

(5)
Next ri’j is defined as the radial center of mass of
I
its zone:

Then ri’j is the best available estimate of the point
I

where M. = M. .,
1 1,3

ference form of the partial:

and is used to construct a divided dif-

3M. Mi '+l _ P.1 i
(——l) =—',3'—— :J,j=l,2...m—1 (7)
8r r. r. . - r .
1,] l,j+l 1,]
Noting that A. . = 4nr2 /n2, and substituting into

1,] i,j’

Equation 5 there results

 

 

dM.. M. . -1\—4. . M. .—H..
1,] = D {A. i,j+l 1,3 _ A. . 1,1 1,3-1}
d Vi,j 1,3 rin+l - ri,j 1,3-1 ri,j - ri,j-1

for j = l, 2 ... m—l (8)

 

17

Consider the case where there is a bulk outside
moisture content, Me, which is the equilibrium moisture
content of the product at the temperature and humidity
of the bulk air flowing past the product. In addition,
there is a convective mass transfer coefficient, hé. The
subscript m is used to denote a mass transfer coefficient
and the prime is used to differentiate between this co-
efficient which is based on moisture content differences

and h the one more commonly used, which is based on

ml
humidity differences.

Then the rate of mass transfer per unit area is

given by:

3M. Do

Do (—-—l .
c 8r ri m m e l,S Arim
I I

 

 

(9)
where Dq:is the inside moisture conductivity and Mi 5
I
is the moisture content at the surface. After some
algebraic manipulation, Equation 9 yields
8Mi hr; -
(gr—4r = -——— (Me — Mi,m) (10)

i,m DpC+ hé Ari m
I

where Ari m is the distance between the surface and the
I

point where Mi = Mi m' Substituting this into Equation 5
I

results in

 

 

 

 

18
—1 m l hmDAi m Mi m- Mi m l
d”, = { _,_z____ (M — M. )-DA. —J—;—' "}
dt Vi,j Dpc+ hm Ari,m e i,m i,m l ri m ri,m-l

The quantities of interest will be the m average

moisture contents throughout all n2 j—th zones. This is

denoted by dropping the first subscript.

M. = ____2_____._ (12)

n
volume, 2 V. . = V/m, then
i

 

v dM. n2 mi].
5 dt =21 (Vi,j ‘ dt) (14)
1—1
For j = 1,2 ... m-l, the result from Equation 8 is

 

 

1% = 7 {A31 (Mj+1 — Mj) — Aj_l (Mj - Mj_l)} (15)

 

where
n2 Ai .
A]. = ;_ f,—-—;—3f.——, j = 1,2, m-l. (16)
1-1 l,j+l 1,]
And for j = m,
dfim m DhglA _ _ _
t V .{Dp + h Ar (Me _ Mm) _ DAm—l (Mm Mm-l)}
c m m
(17)
where A is the total outside area and
DD
_ A _ ‘c
Ar — ‘2 h'—r (18)
n Ai m m
h‘ E ———-—-+—————
m i=1 DQ2+ hmAri,m

 

The result of the above is that the effects of

 

particle geometry on diffusion are captured in m parameters,
Ai, Ag, ... Aé_l, and Arm. Of course, it is not practical
to determine these parameters directly. Instead, they are
estimated by fitting the model to experimental thin—layer
drying data.

Another way of looking at this model of the corn
kernel is as in Figure 3. Here the kernel is divided into

three concentric layers with the same dry mass in each

 

% . X: \\‘ Ar3A

   

 

Figure 3.—-Locating diagram for thin—layer
model parameters.

 

 

 

21

layer, and with iso—moisture content surfaces between the
layers. The solid lines in the figure denote these sur-
faces. Then within each layer there is another iso—
moisture content surface where the moisture content is the
mass average for that layer, denoted by dotted lines in
the figure. The Ar values will be some kind of average
separation between the average moisture content surfaces
in the two layers. The same set of geometric parameters

will result.

 

 

 

A THIN—LAYER MODEL FOR SHELLED CORN

Using the techniques of the previous section, the
corn kernel is divided into three concentric Spherical
layers with the same dry mass in each layer. To aid in
visualizing the process and for ease in applying the re-
sults to deep bed work, a calculation basis of a cubic
foot of corn kernels with a void fraction of 0.5 is used.

Heat and mass balances inside the corn yield the

following equations:

 

 

 

dMl 3Ai
dM2 3A2 3AiD
= V D (M3 — M2) - V (M2 - M1) (20)
dM3 3A hé D 3A’
dt = VTDp + H7 Ar )(Me _ M3) - V D (M3 _ M2)
c In 3
(21)
dTC hfg A hé D
dt=CV(Dp+h7Ar)(M—M3) CV(TC-T)
pC c 3 c
(22)

22

 

 

 

23

In the above set of equations the value of A/V is
taken from Bakker—Arkema et a1.,2 namely 239 ft-l. Chu
and Hustrulid6 provide a formula for D as a function of

M and T .
c

 

4523.4 }

..3 _'
D — 1.629 x 10 EXP {(0.025 TC + 6.008) M - Tc + 459.7

(23)

where TC is in °F, and M is the average dry basis moisture
fraction. Three other quantities are assumed constant,

p = 72 lb of dry matter per cubic foot, Cp = 0.6 Btu per
c

°F per pound of dry matter, and hfg = 1040 Btu per pound
of water. It is understood of course that the heat
capacity is a function of moisture content. However, it

was decided that the extra accuracy in using the function
would not offset the difficulties encountered by the
added nonlinearity in the system. For this reason all
heat capacities here and in the deep bed sections are
”stream” values, based on average moisture contents and
pounds of dry matter in each stream. This value of Cp
corresponds to M = 0.2081. C
The unknowns are Ai/V, Aé/V, Ar3, hé, and h, the
heat transfer coefficient. Although estimates of hé and h
are available, the use of such estimates would mean that

any error in the estimate would color the determination

of the three geometric parameters.

_______________:J-IIIII-IIIIIIIIIIlI-l-I"Il

 

24

The five unknowns were determined by fitting
solutions of the differential equations to three sets Of
experimental data in a weighted least squares sense. The
hill climbing method Of parameter optimization Of

Rosenbrock9 was used tO minimize the function

E = z t [174' (t) — M (t)] (24)

CALC

The squared error at each point was multiplied (or
weighted) by the time to that point because the moisture
content changes fastest early in the drying and because
errors in measuring time are relatively less significant
at longer times.

Since each evaluation Of the error represents 36
hours Of system time (3 runs at 12 hours each) and a 5—
parameter Optimization normally requires over 200 trials,
it is important that the solution be carried out as
efficiently as possible. The equations are "almost"
linear, so a Laplace transform may be used over short
time intervals when the coefficients (diffusivity actually)

are the average Of the endpoints.

As may be seen in Figure 4, a front or starting
:point diffusivity is calculated from the known tempera-
‘ture and average moisture content at the start Of the time
interval. Next the moisture content at the end Of the

interval is estimated. This estimate need not be

 

 

 

 

 

1. Calculate coefficients Of the
linearized equations at the front
Of the interval from known initial
values.

 

 

 

 

 

2. Estimate values Of the dependent
variables at the end Of the interval.

 

 

 

 

 

 

3. Calculate the coefficients at the
end from the current estimate Of the
end values.

 

 

 

 

 

4. Use the average Of front and end
coefficients tO perform an exact
solution Of the linearized equations
to the end Of the interval. Laplace
transforms are used here.

 

 

 

    
 

    
 
 
 

5. Is the
step complete? Need
either prOper number Of trials
or convergence Of the end
value estimates.

   
 

 

 
       

YES

 

6. GO on tO the next step.

 

 

 

Figure 4. Flow chart of the method used for single-
step>integraton for initial value problems.

 

 

26

particularly good, in fact, the values at the front may

be used. The one used here was

M. = M. — 0.2 At (M.
1!

l, t+At l, t — Me)' (25)

t

The end diffusivity is calculated from this estimate and
the coefficients are calculated from the average Of the
front and end diffusivities. Next, a single step is made.
This provides much better values for the end moisture
content and temperature, from which a much better end
diffusivity is calculated. A new average diffusivity
provides new coefficients so a second step may be made
over the same interval. This loop is continued until it
converges. For these runs the second and third trials
always agreed within :rlO_5 moisture fraction.

The following substitutions were made to simplify

the Laplace transform solution.

 

 

3A’D 3A’D
_ l _ 2 _
a- V b— V OL—a‘i'b
D h’
c — 35 m B = c + b (26)

._fL- =.—h__A___
pc Ccpcv

 

 

 

 

 

 

27

So the equations are

H2

(27)

“I
fl
ll
m
3
N
I
m
3
H

N%

- b M3 - a M2 + a M1 (28)

Q)
fl.

CL
2
w

— CMe — 8M3 + bM2 (29)

Q.)
r...

Q19:
('i'I-Zl

= dMe - dM - 6T + dTa (30)

3

 

A bar over a variable will denote its Laplace transform, a

O

superscript will denote value at the front of the
interval, and S will be the transform variable. Then

the transformed equations are

 

SMl — M1 = aM2 — aMl (31)
_ _ o = _ _ _ _

8M2 M2 bM3 dMZ + aMl (32)
_ CMe _ _

SM3 - M3 = —§— — 8M3 + sz (33)
_ dMe + dTa _ _

ST - T° = ————§———— — dM3 - ST (34)

 

28

Solving these (Cramer's rule works rather well)

 

 

 

s [(S+d)(S+B) — b2] M: + s(s+s) aME + ab(SM§ + CM )
M = e
l s [(S+a)(S+d)(S+B) — b2 (S+a) - a2 (s+B)]
(35)
M = as (8+8) M: + s (s+a)(s+B) M3 + b(s+a)(SM§ + CMe)
2 s [(S+A) (S+d) (5+8) - b2 (S+a) - a2 (s+s)]
(36)
O O 2 O
M = abSMl + bS (S+a) M2 + [(S+a) (S+O(.) - a ] (5M3 + CMe)
3 s [(S+a) (S+d) (s+s) — b2 (S+a) — a2 (s+B)]
(37)
-d {abSM° + bS(S+a) Mo + [(S+a) (S+O() —a2] (SMo + CMe)}
T =

+ [(S+a)(S+d)(S+B)—a2(S+B)-b2(S+a)](ST° + dMe+STa)

2 (s+a) -a2 (s+B)]

 

s(s+6) [(S+a)(S+d)(S+B) -b

(38)

Writing the Heaviside expansion theorem in this notation,

If u (s) = gig} where Q(S) = (S-al)(S—a2) ...... (S-am)

and P(S) is a polynomial of degree m—l or less

In
Then U(t) = Z —d—§-— e
i

 

 

 

29

So the only difficulty in the solution was

finding the roots of the polynomial

(s + a) (s + a) (s + s) - b2 (s + a) — a2 (s + B).

It should be clear that these roots are all real
and negative, so a special subroutine was written to
solve a general cubic with all real roots.

The programs used to perform the integrations
and the optimization are shown in Appendix A. Experi-
mental thin layer drying data taken by D. G. Elzinga were
thus fitted to the model. The results are shown in
Table l and a representative set of results is plotted in
Figure 5. The drying was carried out with air at 98°F
and 50% relative humidity. The air flow rate was not

determined, but was quite high. The resulting values were:

Ai/V = 12.58 x 104 ft—2

Aé/V = 9.46 x 104 ft—2

3

Ar3 = 3.931 x 10‘ ft
h = 12.33 Btu/hr/ft2/°F
hé = 1.193 lb of water/hr/ftz.

There are two problems with the data Of Elzinga.
First, the whole set was taken with the same drying air
temperature. Second, the air flow rate was not recorded.

In order to give the model a broader foundation eleven

 

30

TABLE l.—-Thin—layer data* and simulation average moisture
contents as a function of time.

 

 

Time, Set 1 Set 2 Set 3

hours EXP SIM EXP SIM EXP SIM
0 .3605 .3605 .3377 .3377 .3813 .3813
0.5 .2863 .2916 .2712 .2805 .2895 .3007
1 .2619 .2582 .2493 .2510 .2649 .2639
2 .2243 .2195 .2143 .2155 .2263 .2226
3 .2003 .1965 .1929 .1938 .2017 .1984
4 .1810 .1806 .1747 .1787 .1817 .1821
5 .1712 .1690 na .1675 .1720 .1701
6 .1607 .1600 .1555 .1589 .1612 .1608
8 .1454 .1470 .1421 .1463 .1464 .1476
12 .1330 .1318 .1312 .1314 .1339 .1321

 

*

The data were taken by Mr. Daniel G. Elzinga Of
the Michigan State University Department Of Agricultural
Engineering on September 11, 1969.

Dry Basis

o
6

Moisture Content,

 

31

40

 

C) Experimental

—— Simulation

32—

28-

24-

20

16

 

 

 

A . 1 i

l I l I
0 2 4 6 8 10 12

12

 

Drying Time, Hrs.

Figure 5.—-Thin—1ayer moisture contents as a
function of time for data set #3.

 

————'

32

selected runs from the data of Troeger12 were also used.
Three of these were at about 90°F, five at about 125°F,

and three more at about 160°F. All were at an air velocity
of 160 ft/sec.

In checking the Troeger data with his thin layer
model, errors of up to 1.6% were found. Also, his raw
data could not be used since they were not taken at the
same times for each run. Therefore, a correction factor
was applied to the model assuming that the error would be
a linear function of time between the data points.

The first action taken with this data was to fit
it in the same manner as the Elzinga data, both grouped
by temperature and with all eleven runs together. The
results by temperature group varied a great deal from one
group to the other, not only in geometric parameters, but
also in the convective coefficients. For all eleven
together the result was a very poor fit. Finally, the
geometric parameters found from the Elzinga data were used
and only the convective transfer coffficients were
adjusted to fit the set of data at each temperature. This
gave a better fit of the data.

The results of fitting the Troeger data are shown
in Tables 2, 3, and 4, with selected runs plotted along

with their raw data in Figure 6.

33

TABLE 2.--Thin—layer moisture contents as a function of
time for low temperature Troeger data.*

 

Run #1404310 Run #1404410 Run #1504310

 

 

 

 

 

Time,
min. Sim Troeger Sim Troeger Sim Troeger
O .3458 .3548 4276 .4276 .3481 .3481
10 .3462 .3418 .4128 .4074 .3409 .3350
20 .3393 .3309 .4031 .3939 .3348 .3239
30 .3330 .3217 .3950 .3825 .3292 .3158
40 .3273 .3146 .3876 .3729 .3240 .3091
50 .3219 .3080 .3807 .3645 .3191 .3033
70 .3116 .2961 .3677 .3496 .3099 .2928
90 .3021 .2857 .3555 .3364 .3012 .2848
110 .2931 .2768 .3440 .3245 .2930 .2766
130 .2845 .2691 .3329 .3139 .2853 .2694
150 .2764 .2614 .3224 .3043 .2780 .2629
200 .2578 .2452 .2984 .2831 .2613 .2490
250 .2415 .2314 2772 .2653 .2467 .2375
300 .2272 .2200 .2586 .2503 .2339 .2274
350 .2145 .2104 .2422 .2375 .2228 .2187
400 .2034 .2014 .2278 .2253 .2130 .2114
500 .1850 .1870 .2039 .2065 .1970 .1987
600 .1706 .1757 .1854 .1909 .1846 .1889
700 .1591 .1667 .1709 .1790 .1750 .1811
800 .1500 .1594 .1594 .1701 .1675 .1748
900 .1427 .1535 .1502 .1626 .1615 .1696
*Using h = 2.054, hé = .1002.

E x o
mmaa. u .a m4 4 u : anamo4

 

 

 

 

 

 

nova. nova. mmva. mova. mvma. mnna. Nona. mmoa. mvma. vmva. oom
aoma. mnma. omma. mnma. bmom. mnma. vvma. Nona. omoa. mooa. omm
whoa. aana. mmoa. aaha. omam. mmma. Nvma. moma. whoa. hvha. omm
moma. anma. amha. anma. mvmm. mvam. vmom. nmma. vmma. mama. ovm
huma. omom. ooma. ooom. mmmm. hmmm. mmam. mmam. mmam. omam. oom
vmam. momm. mham. momm. vmnm. mmmm. oomm. mamm. aomm. momm. ooa
momm. mvvm. momm. ovvm. wmmm. mmom. nmvm. mmvm. aomm. ammm. ova
mmvm. aoom. mmvm. aoom. vmmm. mvmm. momm. mvmm. mmom. omom. oma
mmmm. mhhm. nmmm. momm. mvam. haom. whom. whom. oamm. onmm. ooa
whom. momm. ammm. momm. momm. momm. aamm. mmmm. maom. mnom. om
ommm. Nmam. aoom. mmam. onvm. vam. ommm. mmmm. mamm. momm. oo
vmom. mnmm. mmom. mnmm. ammm. vamm. amom. onom. mamm. oovm. mm

M omam. anmm. omam. Nnmm. Nmom. oaom. omom. nvam. movm. momm. vv
mmmm. mwvm. momm. mnvm. omnm. aanm. nmam. ommm. mamm. moom. om
mavm. whmm. vmvm. mnmm. vamm. oamm. mvmm. momm. mmom. ammm. mm
ammm. mmom. mmmm. mmom. mmmm. ommm. ammm. mmmm. momm. mmmm. om
mmmm. ovum. Nmom. ovum. mmmm. vmmm. movm. mmvm. mmmm. oomm. oa
Baum. momm. manm. momm. nvov. vvov. movm. mmvm. oamm. vomm. Na
momm. momm. momm. momm. aaav. ooav. ammm. mmmm. aoov. amov. m
vomm. mmmm. momm. vmmm. nnav. mnav. onmm. anmm. omov. moav. v
moov. moov. voov. voov. ovmv. ovmv. Naom. Naom. mmav. mmav. o

i Hmmmons Sam Hmmmoae 8am Hmmmone Eam Hmmmoab Sam Hmmmone Eam .QHE
_ .mEaB

ovvvammw qsm omvvammv ssm oavvammm com oamvamm# ssm oavvommv csm .

 

«.mamo HommOHB mndumnmméwp
EdanE MOM wfiau mo coapoc3m m mm mpcmpcoo musumaofi mommalcanell.m mamde

 

35

 

TABLE 4.——Thin-1ayer moisture contents as a function of
for high temperature Troeger data.*

time

 

Run #3204310

Run #3204410

 

 

Run #3404320

 

 

 

Time,
min. Sim Troeger Sim Troeger Sim Troeger
0 .3630 .3630 .4314 .4314 .3665 .3665
2.33 .3591 .3535 .4244 .4201 .3629 .3600
'4.67 .3541 .3443 .4166 .4092 .3581 .3538
7.0 .3483 .3354 .4084 .3986 .3527 .3478
9.33 .3422 .3272 .4002 .3882 .3468 .3421
11.67 .3359 .3197 .3921 .3782 .3408 .3367
16.33 .3235 .3053 .3764 .3606 .3288 .3266
21.0 .3115 .2917 .3615 .3441 .3172 .3156
25.67 .3002 .2798 .3474 .3286 .3063 .3045
30.33 .2895 .2703 .3341 .3150 .2959 .2941
35.0 .2794 .2614 .3215 .3033 .2862 .2844
46.67 .2565 .2420 .2931 .2773 .2641 .2626
58.33 .2367 .2259 .2684 .2563 .2451 .2439
70 .2195 .2115 .2471 .2386 .2286 .2293
81.67 .2045 .1995 .2285 .2234 .2143 .2169
93.33 .1913 .1890 .2124 .2097 .2018 .2044
116.67 .1695 .1696 .1859 .1867 .1812 .1856
140.0 .1523 .1565 .1652 .1698 .1651 .1699
163.33 .1384 .1448 .1488 .1555 .1522 .1575
186.67 .1269 .1355 .1356 .1439 .1417 .1471
210.0 .1174 .1275 .1246 .1338 .1331 .1382
*Using h = 9.623, hé = 0.4188.

2.. ....“-

 

_a~o-T-_‘:--—— ......

 

. .a- -..-r

36

.mamo Hmmmoue mo

mumm pmaomaom paw mEau Saaz ucmuzoo mHasmaoE mo coaumaam>an.o madman

oom

mmuscaz .mEaB mcamao

 

 

 

com com 004 com com
D r _ _ _
G
O
mcsm coaumazfiam
moooa .ommvovm a cam .mumo 36m
momma .oaqqomm # cam .mme 36m
moam .oavvova # cum .mumo 36m

 

 

 

'1uequ03 eansrow sIseg Ala

dues 18d

 

STEADY-STATE SIMULATION AND CONVECTIVE
HEAT AND MASS TRANSFER COEFFICIENTS

IN THE DEEP BED

Once a thin-layer model is established the dynamic
equations for a deep bed may be derived assuming plug flow
of both air and corn and no kernel to kernel heat or mass
transfer. This is done in the next section. The steady—
state equations are found by setting the time partial
derivatives to zero and making the distance partial deriva—
tives into total derivatives. This results in the set of
equations shown in Table 5. Generally input conditions are
known and the problem will be to determine temperatures,
moisture contents and the humidity along the bed. For a
cocurrent dryer the input conditions may be located at
z = 0, making the simulation an initial value problem of
six non-linear ordinary differential equations.

This set of equations is similar to the thin-layer
set, except that air temperature and humidity equations are
added. The same technique was used to solve them. Since
it is illustrated in both the thin—layer section and the
dynamic simulation section, it will not be repeated here.

The program to do this is found in SUBROUTINE MBCDS in

Appendix B.

37

 

38

 

 

 

 

 

 

 

TABLE 5.--Differentia1 equations for a steady-state deep
bed corn dryer.
d
Ml 2'8 Dalpc (M —M )
dz GC 1 2
0M2 _ 3 DdlpC (M _M ) - 3 DdzpC (M _M )
az _ G 1 2 G 2 3
c c
U _ D pc hm M 2 V 1n (l-RH)
_ D p + h7 Ar ’ e —O.382 (T + 50)
C m 3 a
dz G 2 3 G 3
c c
dT UA h
C h A _ _ ____§q _
dz 7 G c (T Tc) G C (M3 Me)
pc c pa
dlI _ UA _
d? _ G_ (M3 Me)
a
d
Ta _ _ hA (T _ T )
az — G C a c
a pa
_ —3 — 4523.4
D _ 1.629 x 10 EXP {(0.025Tc + 6.008)M — Tc + 459.7}
R = H ' PTOT , M'— M1 + M2 + M3
H (H + 0.622) - P “ 3

sat

 

39

The only things needed at this point are values for
the convective heat and mass transfer coefficients in the
deep bed. A great deal of work has been done on heat and
mass transfer in packed beds, as may be seen in the review
of Barker.3 The work which most closely parallels a corn
dryer is by Gamson et a1.8 and Wilke and Hougen,l4 since
they involve drying. There are two results from these
papers. First, a relationship has been found for the heat

transfer coefficient by dimensional analysis:

0 u . D G _ D G
h = 1.064 ch (—E—)‘2/3 (JUL) 0'41, _E_ > 350 (39)

cu_ DG_
h=1.95CG (—p—) 2/3 (—E-) 0'51
P k u

I

D G
4;; < 350 (40)

Using the following constants at 1200F

C = 0.24 Btu/lb°F
u = 0.0461 1b/hr ft
Dp = 0.0322 ft, from Bakker et_al.2
C U
312—: 0.735

and for G in 1b/hr ft2, these formulas yield

 

 

 

40

h = 0.363 G'59, G > 500 (41)
.49

h = 0.690 G , G < 500 (42)

This last formula will be of most use, since G will usually
be less than 500. Roughly speaking, the top formula corre—
sponds to turbulent flow and the second to the transition
region. If a continuous function is required, then a G of
615.78 should be used as the break point instead of 500.
For G less than 60, corresponding to laminar flow, the heat
transfer coefficient should be constant at its value for
G = 60.

Under the assumption that the log mean partial
pressure of air in the boundary layer, the mean molecular

weight of the air water mixture and the Schmidt number “D

a w
are all constant, the mass transfer coefficient should be

 

directly proportional to the heat transfer coefficient, so
h' = K - h (43)
m

Then K may be estimated from the thin—layer results. The

h
various values of Rh determined from the thin—layer data
are shown in Table 6. In the absence of more direct esti—
mates Equations 41, 42, and 43, along with a value of Kh
from Table 6, could be used to estimate these coefficients
for deep bed work.

To obtain better values of h and hfi for deep bed

work, however, it is advisable to use deep bed data directly.

 

 

41

Steady—state deep bed data were available from R. Gygax, so
a program was written, using Rosenbrock's9 hill—climbing
subroutine, to find the best values of Kh in Equation 43 and

KG in Equation 44:

h = K - G ' (44)

Here the objective function was based on the absolute value
of the moisture content error and the temperature error,
summed over all five runs,

. . _ 4 _ _
Objective — 2 x 10 Z I MEXP — MSIM'

and weighted so that a 0.1% error in moisture content counted

as much as a 200E error in temperature.

TABLE 6.-—Values of the proportionality constant between
convective coefficients for heat and mass transfer.

 

 

 

Data Source h hi Kh = h$/h
Elzinga Data 12.33 1.193 0.0967
Troeger-—Low Temp. 2.054 0.1002 0.0488
Troeger——Med. Temp. 4.430 0.1955 0.0441
Troeger—-High Temp. 9.623 0.4188 0.0435
Troeger——Average 0.0455

The results of this work are shown in Table 7, along

with the experimental data. The best values of Kh and KG

were found to be 0.0553 and 0.2803, respectively. The

theoretical value of KG in Equation 42 is 0.69. Since that

 

 

 

42

TABLE 7.——Steady—state dryer data and best fit simulation
results.*

 

Run Number 6 8 11 14 15

 

GC, Corn flo rate
lb/hr/ft dry basis 193 216.5 144 115.4 140.8

Ga’ Air flowzrate
1b/hr/ft dry basis 560 408 357 338 336

Entrance Conditions:

TC, Corn Temp CF 42.5 65 42 53 50
M, dry basis moisture

fraction .3141 .3141 .3403 .2697 .2697
Ta, Air Temp, OF 350 300 350 250 300
H, Abs. Humidity .0047 .0045 .0043 .0048 .0048

Experimental Exit Conditions (2 ft of bed length)
Ta, Air Temp, OF 134 107 112 102 106.5

M, Moisture Content .2213 .2557 .2498 .2136 .2194

Simulation Exit Conditions:

 

Ta, Air Temp, OF 141 113 117 113 122
M, Moisture Content .2301 .2601 .2498 .2120 .2132
* = I = . o
For KG .2803 and (h 0553

value refers to fixed beds of spheres, the results here may

be considered to be in agreement with the theoretical values.
The final optimization program executed in about

85 seconds and used 25 trials. Since each trial needed 5

SOJJitions of the steady—state equations, each solution for

2 feet of bed length took about 0.68 seconds of CDC 6500

running time.

 

 

DYNAMIC SIMULATION OF A COCURRENT

DRYER FOR SHELLED CORN

Several assumptions are used in deriving the equa—
tions for a cocurrent corn dryer. First are plug flow of
both corn and air, and perfect insulation at the dryer walls;
the result of these is that there will be no radial gradients
inside the bed. Also heat and mass transfer by conduction
along the bed (z—axis) are assumed negligible.

Each region of the corn is treated as a continuum
rather than as individual kernels. To do this the geometric
parameters are expressed per cubic foot of dryer. Here 0i
is substituted for the Ai/V of the thin—layer section. Since

the thin-layer calculations were based on a value of 8 of 0.5

they are
4 -2
col = 12.58 x 10 (2) (l-e) ft (46.)
4 —2 .
a2 = 9.46 X 10 (2) (l—s) ft (47)
—1
A = 239 (2) (1—8) ft (48)

A moisture balance on the innermost (Ml) region
from z to z + Az is as follows
(FLOW INPUT AT 2) - (FLOW OUTPUT AT Z + AZ) -

(OUTPUT BY DIFFUSION) = (ACCUMULATION RATE)

43

 

44

C)
O

_ M )

C —
- M — §—-M D alpcAz (M1 2

l I
Z+Az

1) (49)

——(

Note that each term is in units of pounds of water
per hour per square foot of bed cross-section. Each term

is divided by Az and the limit is taken as Az-+0:

G M - M (3 8M
_g lim 1 z+Az 1 z _ _ = _g _ 1
(3 ) A240 {" L‘T—‘Lz } D O‘1"c (M1 M2) 3‘1 E)at

(50)
which gives
G 8M 9 (1-e) 3M
c 1 _ _ _ _ c __1.

In a similar manner, a moisture balance is made on the

M2 segment
Gc Gc
M + D d p Az(M -M ) - —— M - D d p AZ(M -M )
3" 2]z 1c 1 2 3 2|Z+AZ 2c 2 3
O
— d—t (AZT (1 e) M2), (52)

which gives

 

 

 

 

45

 

GC 8M2 pc(1-s) 3M2
3— 8—2 = D O‘1‘3c (M1’M2) ’ D O‘2"c (M27M3) ' 3 at
(53)

The overall moisture transfer coefficient at the edge of the

kernel, U, is defined as

 

c h
_ D p + h Ar (54)

Then a moisture balance on the outer segment is

G G l
C C
2 z+Az
_ d 0c ,
— 3E (AZ?- (1-8) M3); (55)

where Me is the equilibrium moisture content of corn at

the bulk air conditions. This gives

0

c 8M3 pc(l—€) 3M
3_ 32 = D OL200 (M2_M3) _ UA (Ms—N2) 3 at

3.

 

(56)

Under the assumption that the temperature is
essentially uniform inside a kernel, a heat balance is
made on the corn. Here the units of each term are Btu's

per hour per square foot.

 

46

GCCpCTq/Z+ h A AZ(Ta-Tc) - GCCche/Z+Az- UAhngz(M3-Me)
= d—- (C p (l—€)Az T ) (57)
dt pc C c g
which gives
8T 8T0
CpCGC F = h A (Ta-TC) - U Ahfg (M3—Me) Cpcpc(l—g).§.E_

(58)

 

Note here that the heat capacity of the corn is a dry
basis "stream" value of 0.6 Btu/lb/OF. This corresponds to
about 21% moisture content dry basis. The only value of the
heat balance equation is to estimate the corn temperature
needed in the expression for diffusivity of moisture in the
kernel, so it was felt that the gain by having a linear equa—

tion would offset the loss in accuracy.

A moisture balance on the air is

_ d
GaHI + UA Az(M3"Me) - G HI — dt (paeAzH)
z z+Az
(59)
xvhich gives
8H _ _ _ 29.
Ga ——Z— — UA<M3 Me) pae at (60)

IX heat balance on the air is

 

47
GaC aTaI - GaC aTal - h A Az(Ta-TC)
p z p z+Az
= Q_ (p 8 AZ C T ) (61)
dt a pa a
which gives
BTa BTa
Gana 5;- = - h A (Ta—Tc) - pas Cpa 5E— (62)
In addition, the following relation from
Thompson et a1.12 is used for Me'
1n (1 - R )
M = —— —H (63)
e —0.382 (Ta + 50
vvhere R is the relative humidity,
R = H PTOT (64)
H P (H+ O.622)'
sat

“fliere Psat is the saturation pressure of water at Ta,

EUPOT is the atmospheric pressure, and H is the humidity

ir1 pounds of water per pound of dry air.

The complete set of equations for the dynamic
Inociel is shown in Table 8, where they are rearranged by
SOJrving for the z-partial. The strategy for solving the
eqllations was to substitute a simple backward—difference

fOrTnula for the time partial, for example

_ P
8M M1 M1 65
———7fi7——— ( )

1
8t

 

 

 

48

 

 

 

 

 

 

 

 

 

 

 

 

TABLE 8.--Dynamic equations for a cocurrent corn dryer.
3M1___3 DdlpC (M —M ) _ pc (l-e) 8M1
32 — G l '2 G at
c c
8M2 = 3 Ddlpc (M —M ) _ 3 Ddzp (M —M ) — pc (l-s) 8M2
82 G 1 2 G 2 3 G at
c c c
h’ / 1
U _ D pc m M = V n (1-RH)
— D p + h' Ar ’ e -0.382 (T + 50)
c m 3 a
8M3 = 3 D012pC (M —M ) _ 325 (M -M ) pc (1—8) 8M3
52 G 2 3 G 3 e G at
c c c
BTC = h A (T T ) _ EA hf (M -M ) - pc (l-e) BTC
82 G C c G C 3 e G at
c pc 0 a c
p s
8H _ Me _ - -.a._ a
82 _ G (M3 Me) G at
a a
333 = _ hA (T _ T ) _ 52: 333
32 G C a c G at
a pa a
_ —3 — _ 4523.4
D — 1.629 x 10 EXP {(0.025Tc + 6.008)M Tc + 459.7}
R _ H PTOT . fi_Ml+M2+M3
H 7 (H + 0.622) P ‘ 3

sat

 

 

 

49

where M1 is the function of z in the current time step

and M p is that of the previous time. The result of

l
substituting such a formula in each of the equations is a
set of 6 nonlinear ordinary differential equations at each
point in time. Those were solved by the same method used
in the thin layer work, namely, linearizing followed by an
exact (Laplace Transform) solution, where the linearization
is done at the center of the z-step.

The previous time-step function was linearized

into a slope and an intercept, so, for instance
M =M +zM (66)

where MlPZ is the intercept and MlPD is the slope. For
convenience, each z-step was treated as if it were from
0 to A2 rather than from 2 to z + AZ. This made it easier
to use Laplace transforms to find the exact solution to the
linearized equations.

The method of choosing the slope and intercept
of the previous time step is critical. As is shown in
Figure 7, the dotted line between the actual values stored
for the front and end of the interval overestimates the
function everywhere in between. The use of that line is
equivalent to adding an amount of moisture to the bed
equivalent to the area between it and the curve. To
eastimate the actual value of the function, the quadratic
vvas found which passes through the ends of the interval

23nd is a least squares fit of the next two closest points.

 

 

 

 

 

 

or T

M1, M2, M3, H, T

50

 

function in previous time step

values actually stored
point-to-point linearization

linearization finally used

 

 

 

z—Az z z+Az z+2Az

Length Along the Dryer

Figure 7.——The linearization used for the

previous time step.

 

 

 

51

Find the line y = A22 + Bz + C which coincides
with points yl and y2 in Figure 7, and where the z-axis
is shifted so that yl is at z = 0. Then C = y1 and

y2 = A22 + Bz + yl' Solving for B, one gets

B=——————--AZ =SI—AZ (67)

where

the slope. The least squares, LS will be

Min 2 2 _ 2 2
LS = A {[y3 -(Az3 + B23 + yl)] + [y4 (Az4 + B24 + yl)] }
(68)
Working with the above equation
_ Min _ 2 _ I _ 2
LS — A {[y3 A23 S 23 + A22z3 yl]
+ [Y ' A22 -' S'" + AZ Z - y ]2} (69)
4 4 “4 2 4 1
_Min _ _ . _ 2 2
LS — A {[(y3 y1 S 23) + A(2223 23)]
+ [( - - S'" ) + A( 2 - 22)]2} (7o)
y4 y1 ‘4 z2 4 4

The minimum should be found by setting the derivative

equal to zero

 

 

52
0 = 2[(y — — S'z ) + A(2 2 - 22)] (z z — 22)
3 y1 3 2 3 3 2 3 3

2 2
+2[(y4 — yl — S'z4) + A(2224 - 24)] (2224 - 24)

 

(71)
Which gives
( -y -S'z )(z 2 —22) + (y — -S'2 )(z 2 ~22)
y3 1 3 2 3 3 4 Y1 4 2 4 4 (72)
(z z -22)2 + (z 2 —22)2
2 3 3 2 4 4

The point is to find the line y = I + 28, where I is
the intercept and S is the lepe, which most closely fits
the curve y = A22 + Bz + C. One way to do this is to

minimize the integral of the squared difference

Z
. 2
M1“ J [(A22 + Bz + c) - (I + 23)]2 dz (73)

22 2 2 2 2 2
= (A2 +Bz+C) d2 + {(I+zS) -2(Az +Bz+C)(I+zs)}dz
(74)

Since the first integral involves neither I nor S, it

will not be involved in the minimization

2
Min 2 3

= {—2ASZ + (s2

- 2AI - 2BS)Z2

+ (213 - 2BI - 2CS)z + (12 - 21C)} dz ] (75)

 

 

 

53

(52-2AI-2BS)2:

2
1,8 [ 2 3 + (IS—BI-CS)2

2

 

 

+ (IZ-ZIC)22]

(76)

Setting the derivative with respect to S equal to zero

- A24
_2+
2

ZS — 2B 3 2 2 _
————3—-——z2 + I22 — C22 — 0 (77)

This is simplified by multiplying by 6/23

_ 2
61 + 4822 - 4B22 + 3A22 + 6C (78)
But 3(Az2 + B2 + C) = 3
2 2 y2
so 6I + 4322 = B22 + 3c + 3y2 (79)

Setting the derivative of equation 76 with respect to

I equal to zero

-2A23

2 2 2 _
3 + 822 — B22 + 2122 - 2C22 — 0 (8m

 

This is simplified by multiplying by 3/22

_ 2
6I + 3822 — 3B22 + 2A22 + 6C (81)

But 2(Azg + B2 + C) = 2y2

2

so 61 = 3522 = B22 + 4C + 2y2 (82)

 

 

 

 

 

54

Subtracting equation 82 from equation 79

822 = y2 - C but C = y1
Y ‘ Y
so S : iZ—__.]; : S'
2
Solving equation for I
61 = B22 + 4yl + 2y2 - 38 22 (83)
But B = S' - A22 from equation 65 so
6I = (S - A22)22 + 4yl + 2y2 - 38 22 (84)
6I = — A22 + 4 + 2y - 28'2 (85)
2 y1 2 2
_ _ 1 2
I — yl '6 A22 (86)

This means that the same slope is used but the
intercept is moved down slightly, as shown by the broken
line in Figure 7. In practice it was found that A23/6
tended to move the curve down too much, so a value of
about Azg/B was actually used.

The presence of Me in the H equation presents two
problems. First it occupies a position where H is expected.
.Anytime a dependent variable is missing from its own
equation there is great danger of numerical instability.

In the method used here it means that an individual step

might not converge. The second problem is that of finding

55

an expression for Me in the linearized equations. In what
follows M: and HI are initial values in the z-Step interval
and M: and HL are final values. One possible expression

might be

2 (87)

 

This is not particularly good because the dependence on
H is lost.

Another possible expression is

Me = CZH + C3 (88)
M: ' M: I I
where C = ——————— and C = M — C H . This expression
2 L I 3 2
H - H
works most of the time. However, there are situations

where C2 may be negative, since the air temperature may
drop considerably over the interval. When the C2 defined
above is negative it artificially introduces a positive
eigenvalue and makes the system very unstable. The

expression finally used was

Mi; Mi
Me = 02H where c2 = 0.5 {£3 + H5}. (89)

The strong dependence on H makes for good system stability

so the method used converges very quickly.

56

The substitutions used in the linearization are
shown in Table 9, while the set of linearized equations
is found in Table 10.

The Laplace transform is taken of each equation.
Here a bar over a quantity will signify its Laplace
transform, 8 is the transform independent variable, and
a superscript ° signifies the initial condition. Then

for instance the first equation from Table 4 is

SM —M°+—AM +BM +CMlPZ/S+clMlPD/S2

l l 1 l l 2 l
(90)
When all 6 equations are transformed in this manner
there results:
I 7 I - ‘ ‘
— 2 P2 PD
S+Al —81 0 O 0 0 M1 8 M: + SClMI + ClMl
_ _ — 20 P2 PD
Bl S+A2 B2 0 O 0 M2 5 M2 + SCle + ClMZ
0 -B S+A -B c 0 L M 52M° + sc MPZ + c MPD
2 3 32 3 3 1 3 13
= l_2 (91)
_ — s 2 n PD
0 0 B4 S+A4 0 D H s H° + SC4H + C4}!
-. — 2 P2 PD
0 0 B5 -BSC2 S+A5 v35 Tc S T: + SC Tc + Cch
— 2 PZ PD
0 O O 0 —C6 S-v’A6 Ta S T; + SC4Ta + C4'1‘a

 

 

 

 

 

 

The solution of this set of 6 linear equations is

straightforward, and is found in Appendix C. The Laplace

 

57

TABLE 9___Substitutions used in the linearization of the
cocurrent dryer dynamic equations.

 

3 dlp D pC(l—e)
B1 = G C1 = G At A1 = B1 + C1
C C
3 a p D
_ 2 C _
B2 — Gc A2 _ Bl + 32 + cl
M: M:
c = 0 5 {—— + ——}
2 HL HI
3AU
B3—G— A3—B3+B2+Cl
C
p E
UA __ a _
B4 G" C4 ‘ G AI A4 ‘ B4C2 + C4
a a
h AU
= f3 Ah =
B5 G c C5 G c A5 C5 + C1
c pc c pc
hA _
C6 G c A6 ’ C4 + C6

 

 

 

TABLE

 

58

lO.—-Linearized cocurrent dryer dynamic equations.

 

.5

93
N

I 8

Q:
N

D.)
b.)

M

9|
N

0..
CE

PZ P
+ B1M2 + ClMl + z ClMl

D

PZ PD
A2M2 + B2M3 + ClMl + 2 C1M2

— A3M3 + B3C2H + ClMEZ + 2 clMgD
A4H + C4HPZ 4 z C4HPD
ASTC + BSCZH - B5M3 + clT:Z + z cngD
A6Ta + C4T:Z + z C4T:D

 

 

 

 
 

1....-.mfl—1La-e-ggfnrfwr:“'t‘ .. - .- ~ -- 1

59

transform of each dependent variable is expressed as the
ratio of two polynomials. In this case a double zero
appears in the denominator, so a modified Heaviside

expansion theorem is used:

P(S)

If U =
520(8)

, where Q(S) = (S-al)-(S-a2)--°(S-am) (92)

and P(S) is a polynomial of degree m + l or less then

 

V I _ u m P(a.) a.
0(2) = 657%“ + Q(O)P (0) M3) Q (o) + z 2 .‘L e i
(Q(O)) i=1 aiQ'(ai)
(93)
where P'(S) = g% and Q'(s) = 3%

The first four equations of the linearized set
are clearly independent of the last two. This is because
the only temperature involved in the mass transfer equa-
tions is TC, the corn temperature, which is only involved
in the diffusivity, D. This direct dependence is lost in
the linearization.

Appendix D contains the resulting program.

It was checked for accuracy in two ways. First, the
corresponding steady-state equations were solved using
the Bulirisch-Stoer integration method,4 which has
step-by-step error control and can be considered as the
current standard. A program to do this was obtained from

Mrs. Nancy Clark of Argonne National Laboratories and

Z

 

. ..." --.... ___._-._- .
j—fiL‘iv- _ -- - 7’

60

rewritten for use on the IBM 1800. The program came to
steady state with better than 5—place agreement with the
standard. Second, the whole program was run with twice

as large a time-step. In that case 4 to 5-place agreement

with the original was found.

 

OPEN-LOOP NEAR-OPTIMAL

TEMPERATURE CONTROL

The control method one expects to use is a funda—
mental part of the design of a corn dryer. The concept used
here is that corn flow rate is specified, perhaps by the
number of bushels per day (or week) one expects to have to
dry. For best use of electricity in the blower a single
motor will be chosen and run at its most efficient operating
point, so the air flow rate is specified indirectly. Both
of these will then be constant at their specified values.
Since input air humidity or input corn moisture content or
temperature cannot be controlled, the only remaining variable
available for manipulation is the input air temperature.

In this section a single control situation--step
increase in input moisture content——was chosen. Then several
control functions were tried to find the "best" one in a
strictly economic sense. This "optimization" took 38 trials,
and each trial took 5 minutes of running time on the CDC 6500.
So it should be clear than an exhaustive search of various
coontrol situations——steps of various sizes from various
starting points and both up and down-—would be extremely
(Expensive to run. This does, however, illustrate the method
to use.

61

 

it... 42...... - .
62

The problem chosen for the determination of a
near-Optimal control is the best input temperature as a
function of time for a cocurrent dryer while changing
steady-states from drying corn of 33% dry basis moisture
content to 38%. The dryer is 1 foot long, the corn flow
rate is 15 pounds of dry matter per hour per square foot,
and the air flow rate is 200 pounds of dry air per hour
per square foot. The input air has 0.014 pounds of water
per pound of dry air, the input corn temperature is 72°F,
and the desired exit moisture content is 18.3% dry basis.
Under these conditions the initial input air temperature
is 155.71°F and the final input air temperature is
176.02°F.

The objective or cost function has two components:
First, corn of higher than the Specified moisture content
is less valuable. Second, drying past the specified
moisture content means using too much heat. The penalty
for insufficiently dried corn is given by Scott10
as; one cent per bushel for each half per cent wet basis
(aver 15.5%. Since there are 56 pounds of dry matter per

tnishel of corn, there would be 10.2722 pounds of water at
115.5%. At 16% wet basis a bushel would have 10.6667 pounds
CHE water, so 0.3945 pounds of excess moisture costs 1 cent.

5%) the cost of underdrying is 2.535 cents per pound of

 

 

63

excess water. The cost of overdrying was calculated

assuming 90 cents per million But cost of heat (natural gas
is the probable energy source) and 1040 Btu per pound of
water evaporated. Then the differential cost of drying
(meaning change in cost for small changes in amount of
drying) is 0.0936 cents per pound of water dried. So the

net cost of underdried corn is 2.441 cents per pound of
eXcess water. Then the objective or penalty function will be

d(penalty) :{2.441 (M-EO)I_M>MO (94)
dt 0.0936(MO-M). MEMO

 

where MO is the target moisture content. Note that since the
air flow rate is constant the cost of blowing air through the
bed will not depend on the exit moisture content.

The following function was used as a normalized

input temperature:

_ T - 155.71
Y " 176.02 — 155.71 (95)

 

The first type of control tried was of the ”bang—bang" or
single—step type. Here y is zero until the switching time,
ts’ is reached, after which y = 1. There are three reasons
for choosing this type of control: (1) It is simple to
hnplement. (2) Many control situations have such a control

as the Optimum. (3) It is easy to evaluate the temperature

integral

for this control. The temperature integral will represent
the total amount of heat energy applied to the system, so

it is expected that I will be approximately the same at the

T
optimum for various control functions.

The second type of control function used was an

"S-curve" of two parameters:

B (B+l) _ B (97)

y=_
eat+B

Note that this function is zero at t = 0 and approaches

1.0 as t increases. The value of IT for this function is

B + 1 ‘
a (98)

 

IT:

 

When the value of IT is held constant the S-curve becomes
a single—parameter function.

The results of the 5 bang—bang control trials are
shown in Table 11, Runs 1-5. These were fitted to a quad—
ratic least squares of the same type as in the linearization
of the previous time—step on page 49. Here the two smallest
values were fitted exactly and the others were weighted by
one over the square of their X—distance from the average of

the two lowest points. The result was that the best estimate

of the minimum was at IT = 0.0361.

 

65

TABLE llr-Open-loOp temperature control results, Runs 1-22.

 

Run

 

# Parameters IT Penalty*
1 tS = 0.025 .0125 7.522501E-4
2 tS = 0.05 .0375 7.473014E—4
3 tS = 0.075 .0625 7.449360E-4
4 tS = 0.1 . .0875 7.829919E-4
5 t8 = 0.2 .1875 1.230722E-3
6 a = 40 B = 0.8370035 .0361 7.390341E-4
7 a = 30 B = 5.69979 .0361 7.366338E-4
8 a = 28.38209 B = 20 .0361 7.361246E-4
9 a = 27.83887 B = 100 .0361 7.359437E-4
10 a = 27.76997 B = 200 .0361 7.359203E-4
11 a = 27.71468 B = 1000 .0361 7.359016E-4
12 a = 27.70269 B = 7000 .0361 7.358974E-4
13 a = 28.57334 .035 7.365738E-4
14 a = 27.02884 .037 7.353200E-4
15 a = 25 .04 7.332408E-4
16 a = 20 .05 7.277053E-4
17 a = 15 .0667 7.194253E-4
18 a = 10 .1 7.130775E-4
19 a = 5 .2 1.299459E-3
20 a = 9 .101 7.184637E-4
21 a = 11 .0909 7.131513E-4
22 a = 10.23 .09775 7.127858E-4

 

Runs 1-5 are single-step;
Runs 6-12 are S-curve;
Runs 13—22 are exponential.

*The penalty is in cent—hours per pound of dry corn.

 

 

 

 

. '-.zir'_'7'g';:c:_.—_;v; i. . - .__

66

Next IT was held constant at 0.0361 and various
pairs of a and B were run for the S-curve. This series
appears as Runs 6—12 in Table 11. The penalty decreased

as larger and larger values of B were used. It is clear

that for very large values of B Equation 97 could become
unstable, so this search was stOpped at B = 7000. Then

Equation 97 was rearranged to

 

 

 

y=B {_:§_Ei_l__ 1} = fin _ {at}, (99)
e + B e + B
For very large B, this reduces to the following:
y = 1 - e’at, I = 1/a (100)

T

so this exponential was tried. Runs 13-22 used this
function; the Optimum was estimated with the quadratic
fit as above to be at a = 10.23.

The final set of functions used started out to
be simple time lag, followed by the exponential of

Equation 100. If te is the time lag this is

y -{0' t i te (101)
T e a

 

67

However, this equation actually went the wrong way, so
instead an equation was tried which keeps y above some

minimum value, ym, and approaches one exponentially:

-at

y = ym + (l-ym) (l-e ) (103)

_ _ .1 104
IT _ (l ym) a ( )

This function was used in the final optimization. First
IT was held constant at 0.09775 as shown in Runs 22-28 in
Table 12. Here the Optimum was estimated at a = 6.45.
Then a was held constant at 6.45 and IT was varied for
Runs 29-34. The Optimum for that was estimated to be at
IT = 0.076. Finally, Runs 35 and 36 show that the best
value Of a was still near 6.45.

The last trial, Run 38, is very near the Optimum
for these functions. The input temperature for this
Optimum is shown in Figure 8. The exit moisture content

_as a function Of time is shown in Figure 9 for this run

and for the best single-step.

68

TABLEJJL-—Open-lOOp temperature control results, Runs 22-38.

 

 

Rgn Parameters IT Penalty*
22 A = 10.23 ym = 0 0.09775 7.127858E-4
23 A = 9.5 ym = 0.071375 0.09775 7.045864E-4
24 A = 8.0 ym = 0.2180 0.09775 7.065060E-4
25 A = 5.0 ym = 0.5125 0.09775 7.065060E-4
26 A = 7.38 ym = 0.278605 0.09775 6.839579E-4
27 A = 5.7 ym = 0.442825 0.09775 6.879560E-4
28 A = 6.45 ym = 0.3695125 0.09775 6.814373E-4
29 A = 6.45 ym = 0.3550 0.1 6.840006E-4
30 A = 6.45 ym = 0.38725 0.095 6.787003E-4
31 A = 6.45 ym = 0.4840 0.08 6.716968E-4
32 A = 6 45 ym = 0.6775 0.05 6.834780E-4
33 A = 6.45 ym = 0.9355 0.01 7.355592E—4
34 A = 6.45 ym = 0.525925 0.0735 6.715480E-4
35 A = 7.0 ym = 0.4855 0.0735 6.724805E-4
36 A = 6 0 ym = 0.5590 0.0735 6.725858E-4
37 A = 4 0 ym = 0.7060 0.0735 7.274961E-4
38 A = 6.45 ym = 0.5098 0.06 6.710516E-4

These control functions are from Equations 100 and
1014

*The penalty is in cent—hours per pound of dry corn.

 

f" ‘ "‘"m‘m “"

y Of Equation 95

 

 

 

 

 

69
1.0 _____ 1
///
//
//
/
0.8 a //
/
//’ Nr\‘Best Overall
/
/
/
/
0.6 — /
/
/
/
0.4 v
0.2 A kQ//// Best SingleaStep
0 I I I I I

 

 

0 0.1 0.2 0.3 0.4 0.5 0.6

Time, Hrs.

Figure 8.—-Input temperature function for the best
Single-step and overall. '

 

_fifi-fia~

70

 

18.35

18.3 _

18.2 a

    

Best /

\\Vp//Overall
\

     

18.1 —

EXIt Average Moisture Content, % Dry Basis

18.0 '—

 

' 17. l 1
95 T I

O 1.0 2.0 3.0 3.

 

“I"

 

Time, Hrs.

. Figure 9.—-Exit moisture contents in response tO the best
SlIKJle-Step and the best overall input temperature functions.

 

 

 

 

71

The cost penalty shown is in cent—hours per
pound Of dry corn. At a flow rate of 15 lb of dry corn
per hour the net penalty at the optimum (6.710516 x 10-4
calculated penalty) is 0.010066 cents while for the best
single step (calculated penalty 7.449360 x 1074) it is
0.011174 cents. Since this was for 3.75 hours Operation,
56.25 pounds of dry corn were processed or 1.0045 bushels.
Then the Optimum here shows a net cost of 0.01002 cents per
bushel while the best single step net cost is 0.0111 cents
per bushel. In View of this, the best practical control
probably is to change the input temperature set point as
soon as the new moisture content corn reaches the bed.

TO implement this control policy the Operator
(either human or automatic) must estimate the time when a
new batch Of corn will reach the drying zone. Additional
single-step trials were run to test the sensitivity Of the
penalty to errors in this estimate. The results of these
trials are Shown in Table 13, and plotted along with Runs
1—5 in Figure 10. If a 50% increase in the penalty is

3), there is a 25 minute

allowed (to approximately 1.1x10_
band during which the step may be made, from about 14
minutes before to about 11 minutes after the new corn

reaches the drying zone.

 

72

TABLE 13.-—Additional single—step temperature control

results.

 

 

Time Of Step Penalty
tS=-0.3 1.226E-3
t3=-0.2 1.044E-3
tS=-0.l 0.885E-3
tS=—O.05 0.819E-3
ts: 0.0 0.769E—3
t = 0.3 3.400E-3

 

 

 

73

 

Of Dry Corn

cent-hours/lb.

1000 x Penalty,

 

 

 

 

-.3 -.2 -.1 0 .1 .2 .3

Time Of Single—Step, Hrs.

Figure 10.--Variation in penalty with time of
single-step.

 

ww— -..---" 1_ii__

CONCLUSIONS

The quality of a corn dryer simulation is limited
by its weakest link—-the thin—layer model. Although a
great deal Of work has been done, this problem may be con-
sidered unsolved in that no model accurately predicts
known thin—layer drying behavior. The author feels
strongly that a good model must carry the internal moisture
gradient, especially so for modeling nonstandard dryers
such as cocurrent—countercurrent or intermittent air flow
where the external history Of the kernel is vastly differ-
ent than in thin—layer experiments. On the other hand, if
too many internal segments are used the model is expensive
.tO run. The best number is probably in the range of three
to five; at three some Of the gradient is probably lost,
and at five it would be very difficult to apply the integra-
tion method used here tO the dynamic deep bed model.

Future thin—layer work along the lines of this
model should be pointed toward finding a better estimate
of the diffusivity, D. The one used here was determined
with a model where the convective mass transfer coefficient,
hfi, was assumed to be infinite. From the thin—layer
results here it appears that the D is tOO small at high

moisture contents and too large at lower moisture contents.

74

75

The values Of the convective heat and mass transfer coef-
ficients were higher for the high temperature Troeger data
than the low temperature data, so D is probably too small
at higher temperatures also.

Convective heat and mass transfer coefficients are
difficult to measure in the deep bed. The method used
here-—finding the set which best fits experimental data—-
is easy to apply if there is enough computer time available
or the model runs efficiently so that many trials may be
made. The result will, of course, depend on the thin-layer
model upon which the steady—state model is based. Future
work in this area should concentrate on Obtaining more
accurate and complete steady-state data, with several
replicates or redundant measuring systems for each variable.

The control work done here demonstrates one Of the
good properties Of the cocurrent dryer--it is relatively
stable to changes in inputs. The control results make it
clear that little would be gained in the use of a compli—
cated Open-lOOp control. The best choice might be to
measure the incoming moisture content, look up the needed
input air temperature, and reset the air temperature
controller to that value. This could be either manual or
automatic——even computerized. Future work along this line
could make use Of a more sophisticated objective function
where sensible heats are accounted for as well as the heat

of vaporization. Such work might also involve other Sizes

 

—_—-i .21..

76

of changes in input moisture content, a decrease instead
Of an increase, or multiple changes.

Other types Of controls, probably corn flow rate
manipulation, could also be simulated with this program.
That control has the interesting property that the corn

flow rate may always take its maximum value for the input

air temperature being used.

 

 

REFE RENCE S

77

10.

REFERENCES

Bakker-Arkema, F. W., and L. E. Lerew (1970).
Single Particle Cooling and Thin Layer Drying
Simulation. Proceedings of the Institute Of
Simulation of Cooling and Drying Beds of
Agricultural Products. Michigan State
University, November 2-4, 1970.

Bakker-Arkema, F. W., J. R. Rosenau and W. H. Clifford
(1969). Measurement of Grain Surface Area and
its Effect on Heat and Mass Transfer Rates.
ASAE Paper NO. 69-356.

Barker, J. J. Heat Transfer in Packed Beds,
Industrial and Engineering Chemistry 57(4),43,
April, 1965.

Bulirsch, R., and J. Stoer. Numerical Treatment Of
Ordinary Differential Equations by Extrapolation
Methods. Proceedings Of the 1965 IFIP Congress,
Vol. 2, Spartan Books, Washington, D.C.

Chittenden, D. H., and A. Hustrulid (1966).
Determining Drying Constants for Shelled Corn.
Trans. ASAE 9(2), 52-55.

Chu, S. T., and A. Hustrulid (1968). Numerical
Solution of Diffusion Equations. Trans. ASAE
11(5), 705.

Fuller, R. Buckminster. Utopia or Oblivion: The
PrOSpects for Humanity. Bantam Books (1969),
New York. I

 

Gamson, B. W., G. Thodcs and O. A. Hougen. Heat,
Mass and Momentum Transfer in the Flow of Gases
Through Granular Solids, Trans. AIChE, Vol. 39
(1943), P. l.

Rosenbrock, H. H., and C. Storey. Computational
Techniques for Chemical Engineers. Pergamon
Press (1966), pp. 64-68.

 

Scott, John T., Jr. "Economics of Corn Conditioning
and Storage Alternatives for Farmers." Ag. Exp.
Station Bulletin, University of Illinois,
January, 1969.

78

 

 

 

ll.

12.

l3.

14.

15.

16.

79

Thompson, T. L., R. M. Peart and G. H. Foster (1968).
Mathematical Simulation Of Corn Drying-~A New
Model. Trans ASAE 11(4), 582.

Troeger, J. M. (1967). DevelOpment of a Mathematical
Model for Predicting the Drying Rate Of Single
Layers Of Shelled Corn. Ph.D. Dissertation,
University Microfilms, Number 67-13,006.

Troeger, J. M., and Hukill, W. V. (1970). Mathe—
matical Description Of the Drying Rate Of Fully
Exposed Corn. ASAE Paper NO. 70-324.

Wilke, C. R., and O. A. Hougen. Mass Transfer in the
Flow Of Gasses through Granular Solids Extended
to Low Modified Reynolds Numbers. Trans AIChE,
Vol. 41 (1945), P. 441.

Whitaker, T., H. J. Barre, and M. Y. Hamdy (1969).
Theoretical and Experimental Studies of Diffusion
in Spherical Bodies with a Variable Diffusion
Coefficient. Trans ASAE 12 (5) 668.

Zachariah, G. L. and G. W. Isaacs (1966). Simulating
a moisture—control system for a continous—flow
drier. Trans. ASAE 9 (3) 297.

APPENDICES

80

APPENDIX A

THIN LAYER PROGRAM

81

APPENDIX A

THIN LAYER PROGRAM

This program is used tO find A’, A’ Ar h’,

2’ 3' m

and h from thin layer data by parameter Optimization.

The calling program, TL4F, is used to initialize the hill
climbing search. It calls HCLMB, the subroutine which
uses Rosenbrock's9 method. HCLMB calls DEL4Q, the sub-
routine tO calculate the total error from the experi—
mental data for a given set Of the 5 parameters. The
other subroutines are DIFF, which calculates the dif-
fusivity Of water in corn from Equation 23, and CUBIC,
which finds the roots Of a cubic equation.

Some of the variables in DEL4Q and their cor-

.respondence in Equation 26 are as follows:

AA = a DB = d
ALF = d QUE = d ° M + 0 ° T
e a

BB = b DEL = 0

SEA = C DIFUS = D

BET = 8 VOL = V

AREA = A RHO = p

HFG = h CPP = C

fg pc

82

83

In addition, the denominator polymonial
2 2
(s+a)(s+0I)(s-I-B)-6(s+a)-a (8+8)
is transtrmed into

s3 + 52 - (BZ2) + s - (BZW) + (BZZ)

where

BZ2 = a + d + B
BZW = a (d + B - a) + d - B - b2
BZZ = a (B (d — a) - b2)

84

mNomzaur
Nmomxaur
aNOQZJU:
ONomzauI
maomxaur
maomxaur
Naomzauz
OaOQZJUI
maomzauz
vaomzauz
Maomzauz
Naomtauz
aaOQtJUI
anmzaur
ooomzaux
moomzaur
hoomzauz
ooongur
moomxhur
voomZAQI
moomzauz
Noomzaux
aoomxgox

.zqeeoxe oz_33<o ex» >0 ewo~>oxe
we ems: ruezz .ex :emzma eo meopom> xeoxn ex.z.m.o.em
.mxoeom> m< omeoem mmumeeqx ex >e ex >3343eo<
me< >wxp .xeeooxe cz_33<o mi» 22 omzo_mzmzxo ez< >m
ome~>oee mm ems: xo_:: .~**ex :eezmx eo meoeow> xeoz: >.e<
zo_e<z_xemp
zzeoe «me o m3<oom .zo~e<z_xewp mew eoe a quoamu me>ez
mzo~e<23<>m zoaeozoe eo xmmzoz x:z.x<xu wepmz
zo_e<z.:eme
eOe mxmemm_zeme zo_euz:e exp eo e33<> emmwe<an mew
zoeeezczeme ace zoaeozae exp 10 mzoee<oaq>m
e>emmmooom zmmzemm zo_e<ue<> moqezmueme emm¢e<xu zzeue
mm3m<_x<> e2mozmemaz_ eo emezozn ex
memewzqeqe eo zoaeeeeummo

mmam<am<>
HZmOZwmmoza ax mo zoahuzam < mo zazazaz th Dzmu Oh
wmoaxaa

mzaux wzaknommam

*************%*******%#%*%**%%%***

021
hexw 33<o

.avxwo
.m:x.m3x.mx.e.>.z.m.o.em.e<.me>ez.meemz.mem.zzeue.ex.mzno: 334g
\me.oo~.o.oo~u.o.o.m\me>ez. meemz.mew.zzeoe.ex «bee
\mvemm.- .m¢m~o~.~ .mummeo~¢o.m.m.emm¢e .e.ooem~1\e eeeo
\mmm.m*m.o.o*m\e:x.m3x «pee
.m.e ..memox ..m.m3x ..mcem zonmzmzeo
.mmcmx ..m~.e< ..m.z ..m.m ..m.e ..m~.> zoemZmzao
ovamo aqzemexm
evxe zeeooee

UUUUUUUUUUUU UUUUUUUUUUQ

. _. ..J-fi." "'7"

 

85

.Gmomzaux

mmOQZJUI
hmomzauz
omomtauz
mmomzaux
vmomzauz
mmomxaux
NmOQZJQI
amomzaoz
omoszoI
ovomzauI
mvomzauz
hvomzaur
ovcmzauz
mvomZJQI
¢¢0m2401
mvomzaoz
Nvomzauz
avom2401
ovomxaur
omomzauz
mmomzaux
FMOQZJUI
omomzauz
mMOQZJUI
vmomzaur
mmomxauz
NmomZJUI
amomzaux
omomzauz
omomzaux
mmomzauz
hmomzauz
omomZAUI
mNomZJUI
vmomzaur

 

hmed<ewa>hzemahw2emawezzhuaeax.mZAU: 44<u
oxuamax

x<zxuamom2x

zmzxuaavmax

meaua 00a 00

auwa>hz

ommumahmz

Deccalumaw

ooouzzhua

mung

\Nooeooaeco0\oxex<zxezazx <k<o
.mvxeamomax ..momax e.m.hm..m.mx zoamZmzuo
.mm.a<..m.ze.m.meAmvoeamNo> zoamzwzao
xmmom 4<zmwhxw

z~<z z<moomm

00a

x<mooma wzaaa<u whaz<m

oxOu Oh maa<u dxfim >am>w

hDom< ezoah<hom m~x< z< Oh momma hmzo Zth mhzaxa >420

mzaux muzum exam toxm x 924 m hauhnw Oh «wow aoow 4 ma hm
oQXHOm hDOm< ma mmhmz do w34<> a<uah0dma coca

< owed: mm< xom Oh m44<u mahmz Aubz: whaazou 44a: x<mooma
th Om .anwa>bz oz< 104 >¢w> mam hwm Oh hmwm ma ha >aa<mmzww

ot<moomd zu<z th za hzmzwh<hm 4<zawhxw Z< mowwz xDu
oXOm za owoz<IU mm #02 hsz m0h0w>lx th
..x.m. ma hmaa awsz<m<a men ozoahoznm m>~humomo

th mh<34<>w Oh mzahnommzm a<zawhxm Z< mo wz<z thl

mw4m<~m<>

hzmonawoza th mo mozzom «mad: mo mohuw> haazal

mwam<aa<>

hzmQdewoza th mo mozaom mmzoa mo «Ohuw> hDazal

mm4m<a¢<>

thQmewoza wIk mo mm24<> 4<ahaza mo mohuw> hDazal

mxm<xm¢

x0e
max
max

x

UUUUUUUQUUUUUUUU’JUUUUQUUUUUUUUUUUUDU

 

 

86

moomZJUI
voomxaur
mocmzauz
Noomzauz

.hwt zomawhaxu m>~h<4wa OI¢N.P<Z¢OM
.mawhm >242 OOhIomaaOhm orvwoh<t¢ou
.\\\.h4180m
..woOam.e\.N

aocmxauzmwam<ad<> awozu «moo OINN.\.womaweu m an m34<> hzwmwma OINN.h<z¢O¢

ooomzaoz
omomzaux
mmomzaoz
hwomzauz
omomzauz
mmomxaox
vmomzauz
mmomzaux
Nmomzauz
amomzaux
omomzaux
oromzauz
mNOQIJUI
whomzauz
chomzaux
mhomzaur
vhomzaux
muomzaux
whomzaur
abomzaUI
Choctaw:
doomZJUI
moomzaux
NOOQZJUI
ooomzaux
moomzauz
voomxaux
MOOQZJUI
Noomzauz
abomtaur
ooomZJUI

.mzouh<hom Ianoaemmmhm uuzmz: Inaeouemawhm uuzm Iaaeo—.\.h<zxom
.mzoahdhom m~x< oz .mawhm uuzmzz Dz .mowhm 003m 02 oxamoh<z¢0u
.ooOamem mo m34<> bmz~OI0a.h<z¢ou

..moOam..\eh2~Oa wzahmdhmOImaoh<z¢Du

\m\x <h<o

aavzeaaomnxeaaomax zoamzwzao
.aohm..a.mx..aCoeaaomeaa.>..a.xeaaoa< zoamezuo

AXOm.me.OAXa

.mxex.>.z.weoehmea<ewq>bzemahmz .mam.22huaedxomzaux mzahnommam

mmm
omm

mNm

own
man
0am
mom
can

**********************************

.oooa. evmalmha on em oqo>
.4421306 mmhnmzou ..zoahuznm < mo m34<> hm<w4 mo hmwh<mmo
th wzaozam mow OOIth uahdzohpd zqe exuoxmzwmom 61.1

embivo om .OOOa .mwmxa zox<wmma .mmmwszZm a<uazwru
mod meOazzuwk Adzoah<h3azou. e>mehm .u 024 xoommzwmom oIoI
meZmemwa

02w
zmzhwx
Ahomawoexmohdxxcm ooN
x at .oom hzama
owNazazaz mm Oh Zoahuzzm mIPum
“mox zoamzmzao
Axemoxmmou wzahsoxmzm

02w
.xmuOuemaxemaxemx.xe>ezewe0a

UUUUUUUUUUUUUUUUUUUUUU

87

aMaQZqUI
OMamIJUI
omamzaux
mNamZAUI
hNamtaUI
QNaQZJUI
mNamzaUI
vmamZJUI
mmamzaur
NNamZAUI
aNamxaux
omamzauz
OaaQZJUI
waamzauz
haamZAUI
caamzauz
maamzauz
vaamZJUI
MaamZJUI
Naamzauz
aaamZAUI
Oaamzauz
00am1401
meamZAUI
hoaGZJUI
beamIAUI
mOamZJUI
veamZJUI
mOamzaux
NOaszUI
acaciaur
00amzauI
ooomzaux
mOOQZJUI
hoomxaox
boomzhuz

ova.oma.o>a .mlxxoz. ea
a+ Dan:—
.x.w *m.OI «xx.m
mva.mma.mma .oele. em
.oxeu. xou aaqu
mozahzou
ovaeova.mva ..Homxlxaooox. ea
mMaemmaemva ..aom4x1._.ox. e—
ex .au_ ova oo
.o.>*.x.m + .~.xu.~.ox
a+ ex*.atx.uv
ex .au_ oma oo
aux
vu._.z
o.ou.~.o
.H.x*a.ou.~.m
ex .au_ maa oo
omhmxo<xm mm zoz_sz < Baez: maxq Io<m ozon< ewhm

ouoa

ouma

Guam

onu—

AOamez. mead:
wazmhzou

ooan.fi.>

mOa.OOa.mOa .xlao um
ooouafio>

~+ dxfiaatxvufi

ax eaua mOa Do

ax .auz mOa Do

> 2a xamh<z >h~h2wou baa

Ouuhmxmu

Om Amomez. mead:

.mx eana eaavx. .oomez. whamz
Axeou. XOu 4440

.hmz zoammhamu mkDanm< oxvmoh<zzou

mva

ova
mma

OMa
mNa

oma
maa

Oaa

mOa
OOa

ovm

 

 

88

weamzauz
acaciaur
meamzauz
veamzauz
mcamxauz
Ncamzauz
abamzauz
ooamzauz
mmamzaur
wmamxaUI
hmamzauz
omaQIAUI
mmamxauz
vmamzauz
mmamzaur

Nmamtauz.

amamzaux
omamzaux
ovamzaux
mvamZJUI
hvamzauI
ovamzaux
mvamzauz
vvamxaux
mvamxaur
Nvamzauz
avamzauz
ovamZJQI
omamxaux
anamzauz
hMamZJUI
OMamzaux
mmamzaur
¢MamzauI
MMamxaux
NMamzauz

a+axfiaalaxoufi
ax eaua CNN 00
Omuhmzmm

>UZm~uamum awhmlmwa w>omaz~ Oh Axumh<xl>o mwx< mh<hom

zmohwe

.ovm.z. me—x:

mamemamemam .memlou. ea
zaabwm

.mmm.:. meme:

ma~.mo~.mo~ .thoenpoe. e—
0mxee~o*o.00a "hoe
oulhmaee "memo

oow.0a~ .oom .alme>hz. e—
zmahwx

.omm.z. meme:

oma.oma.moa .mepmzuo_+m~.e~
ommaoamx ma zoabvzaxmmh ea xuwru
. .mwm.z. meme:
.ex .aum ..Hox. .Oe .omm.z. meme:
om .om ewe .mam.z. meme:
menomme ezwmmee hoeeoo

mma oe ow

a+xux

oma.oma.oma .exlx. u—
moa.moa.m~a .eonH. ea
Nu.x.z

m6a.moa.o~a .mt .xoz. ea
euoe

.xow no.m u.x.w

xxom + .x.o u.x.o

a+ m_um_

.Homxuxaox

ex .aua ooa oo

oea oh ow

a+ o_uo_

ouxxoz

maN
NaN
Cam

mom

CON
mqa

00a

owa
mha
aha
mOa

oca
mma

oma

 

 

 

89

OONQIJUI
00am£auz
mOamIAUI
baamIJUI
00amzaux
mOamzauI
¢Oam2401
MOamzauz
NoamZAUI
aaamxaux
ooamtaux
omamzauz
mmamIJUI
hmaQZJUI
owamIAUI
mwamZAQI
vmamzauz
mmamZJUI
NmamXJUI
amamZAUI
Omamzaoz
meamZJUI
whamZJQI
whamzauz
chamzauz
mhamzauz
vhamzAUI
mhamzaox
Nhamzaur
ahamZJUI
owamzaux
opaQZJUI
moamzauz

cam
caa Oh ow
h<wemm 024 xo<m ow
a+oauua
zozwo\._.hmu.aa.>
_+ex*mexua_
exeaua com on
.e>.hm0mux02mo
._.»m*.~.po+e>ue>
ex.aua mmm oo
o.oue>
.m_.>*z I .aohou.~.pm
~+ex81alaouma
exeauu mvm oo
.NH.>*.aH.e<+zu3
~+ex*.aIJ.n~_
m+exwmexuam
ex.au_ ovm oo
coon:
«exeann mvm co
mmm.om~.mmm .aex.e~
.v.e<n._.eo
~+ex*mexno
ex.auu omm oo
anxuxex
ex.aux com on
.a_.>*xooo+.maoe< uxauoe<
exIm—uaa

H+ex8ouma
ex.aum mmm oo
ovla+exus
exewnow mum oo
.o.>*.exoou.o.e<

DON

mmN

omN
mvN

O¢N

mMN

OMN

mNN

ONN

 

 

 

90

Avcmawoe «mmIvexm.MaeNm .hDAhDO Cambuzoaexomvh<x¢9m
.Nomawm. n a h< «mew u 10aeoa.mau.Xa.h<I¢Dm

.Caomau. mam oz< aim» u 10a a

eoomauexvew.aaumexvemoNau. n w>< .mxI INa.¢.mu.xm.h<z¢Cm
wh<¢omhzu Db >o<wm wm< w: om .zoah<~_4<ahaza mwhwaazou mth
Cooumxah

C.®0H4dh

x<02wuaa.42w

meaua CO CC

Aoo.azazwnm<mxw

meauoo CNN Do

Coouxozmw
AwoluAmCN
Coouava

.ao>#dau*oxxv\<wm<*am.auawo

.aauaoom.\om1nuoumo

uHmDC {Dam mmDa<> FDAHDO 0a IChazm <h<o

mzoah<mmha CZ~¢DO mmDa<> PDQFDO o IChazm <h<C

awhm Iu<w h< hDQhDO m IChazm <h<o

\mamm..hhmmoemobm.\mzaxm (P40

\oowoehoaa.o\xa<1hexa<iw <h<C

\C.0MN.C.oeovaCa.OoNhem.o\<wx<.aauewmxeozxeho> <h<o

<h<o D<h2mzadwaxw wIp no mmohuw> wm< 12m oz< .mzw .wzw .oxw

\hOomN .Naohm .moomN\Ixm <h<C

\om6ma..o*m.¢o.¢aeoo.Na.oa.Nowaehaowa.haoomemoemmemvoomxuzm <P<C
\Na.Ma..o*m.aN.¢a..Cemmomaeooehv.ha.0N.®aemvoaN.mo.¢N\wzw <h<o
\MoMae.O%mevm.va..CehooOa.Na.ha.Ca.ma.mo.CN.m¢.NN.Oa.oN\ozw <h<o
\aeo*m.a.Cea*o\oz <h<o

.Aavmzweameavuzw. w02w4<>_30m

..a.wzw..N.aouzw. wCZw4<>~DOm

.AaCQIweaaeavuxw. wCZw4<>a30w

.mvrzm zoamezao

.mvho..vemoxoeamowe.m.xw zoametao
.Na.uxm..Na.wzweaNavozme.m.Na.CZm zoamzwxaa
am.azuzw..m.azw..m.tw.ANaooze.m.a ZOHmexao

Adexoxxw.0¢awo mzahaommzm

0am
mCm

com

C

000

 

 

91

<wm+wmuhmo

mo+<<uun<

<<*.a.eu<<

..m.e*.¢.e+o:m*m:e~o.\xv.e*<mx<*<<u<mm

<<*.N.eumo

no>\m:e_o*o.mu<<

zomh<30w eeuo n<zao_¢o za weZmHOMeemoo mh<43u4<u zoz
ehexwexaxmamemomhamz

omaemma.mma..oozmeo_.e~
x.qzxmeehoee_o+onoo.Hm.onm:u~o

each xa ecu pzaoe zmahmm

..moxm + Amoxm + .aozw.*mmmmmmmmmm.ou<zxw

aqu~

AmH<Zmeaonxmouhnmo*m.o I .aoozmu.~.zm

m.au~ cma on

aehue»

4<>eweza wrp eo o2w wrh #4 mmon<> mp<z~emm zoz

momxm .nep .nzm .mqozm.x .oomemowhaez

vnwm.o + mzahux
.maa.maa "mammwouoea
.mvmzm.4eh.eeaouoaoo
x<:o.auv com on
.x<zv.e<one\o.aupnmo

x<zv4mux<zo

moH.moa.oonmtme.e_

x<zv4mux<zv

moa.mOa.mm.>Iarx.e_

mux<zo

Na.aumrx cam oo

eemoo mz<xp e<mr wommeoo .xreru.m.e
memoo mz<xh mqu mommhoo.xzoxu.v.e
meanwou.m.e

«NxInwo.\~<ux~.e

.axInwooxaqu.a.e

mzoanoM m4 wx< mmwemz<m<e wrh
.o.mae¢.m_.xa .mw:n<> 3<Hmproa.xov.»<z¢0e

Ca

v-i

mMa
CMa

mNa

CNa

maa

Caa

mCa
CCa

mmo

mam

UUUUUU

 

 

92

m.au~ moa oo

.voeoueh

mozapzou

eZmo\.~mxm*<N*wm+aN*o~cu.~.~.zo
ezwoxxuxuea.zo
Nmszmo*<<+..~.4:m*<<*o~+.aonzw*.wm*moIo~*o~..«NNux
eZmo\xu.~.m.xo
.Anwo+-.*e2mo.\._.hou._.po
-o+-*.3~m+z~m+NN*.NNm+ezwo+ezmoo.ueZwo
NNo+NN+-ue2mo

ooa ch ow

AmOe*NN.\.~.pou._.ho

cha.o>a.m>a.mI_.e_
.w=o+neh*-.*mou+.aoeouxa.»o
-o+NNHAzNo+-*.~No+NN..umoe
moa.m>a.m>a.ml_.e_

mequxucho

Nmzm*.<<*<<Im~*<~C+o~8wmux
A.~.42m*<~+xa.me*<<.*-uo~
«Maxweqmw+nmenzmaaauwmxm

ewm + NNuON

en< + NNuoN

<< + NN u<~

1-4h4mooexwuxaoxm

.aeNuNN

m.an~ oma oo

N.xem.~_.oam.mowh_ez
oma.oma.ooa.>Imo.mu.a.N+eaw.e~
oma.ova.ova.-I.oa.zmpo_*ooaoe_
xN_.mmm.~ao.zNo.m~m..m.~..m.~..~.N. camoo novo
.mm8mo I A<<Ien<oupwo .*<<u-o
mm*wmlhwm*un<+.<<Ipmm+u4<.*<<uz~o
pmm+en<+<<u-o

mpzm_ofieumoo n<~zoz>4oe mon<>zmo~w mIp emu zoz
xm<r»*nmo+m~<zm*wouwoo

<wm8eoowoumo

Cma

mha

Cha

moa

Cba
Cma
Cva

 

oa<wm hoz whoom 444 esz mm — Ma oNN ehooa waooaz

th do zo~h<h3azou th za mzoahdxwh— mo mwmxnz th Ch 4<Dow mm a
oa<m¢ wm< whooa

th 444 2m13 on N0 + Nflxu + N**~*NC + m**~ zoah<DCm th mw>aom
.a .mxm .NC .30 eNC .mNeNNeaNCC—QCC wzahCmezm

92w

zxopmx

aexommm.momem.whax3

«exam .heh .sz .¢<mzw.x AOCmemomhamz
vam.o + wzmpux
mameommeomwxamozmhoa.ua

mnzahzou

mz~e*eem4eew + eoeewueoxew
.oo.exx.ozmqumzmuo.oo~nxxm
mo~.oam.mom .oanveeIx.eH

momeoaN.OaN .Aarxoaxoua

mozeezoo

wzme4xew8xew + eoeewuxoeem
.OsozzmIe4mxm*o.ooauxem

oom.>o~.~oa . mImo.~I .m.oImxH».mm< . eH
ammo + mzahumzah

<zzmue<mzm

ahuaeh

Aaozwuxaonzw

m.au_ mod on

omaemmaemNanaoua

a+x~ux~

3<>emezu LIHP ere eo ozm mzh.h< weqzeemm zmz w><z 2oz
.Amvzm+.N.zw+.a.zm.*mmmmmmmmmm.ou<zzm
.m.xw«.m.ho+eeuep
.xoxmxxx.aozo+.a.zmu._.zm

‘ ¢.Nux moa oo

.Hoxm*.H.ho+eeueh

xaoxmfixa.aozou.~.zm

93

CNN
maN
CaN

mCN
MCN

CCN

baa

mOa
00a

mma

UUUQ

 

 

94

 

 

:o + x*.~o + e~o1.ueo
~o+x+x+xue~oz

ANoezueNuexvquunm

zxux moa
Cufi CCa
QDIz<waC «Cm zomh<xwhu ZOHINZ Cw>oxdx~ th «thw 302 C

Coa Ch CC
x0 + amxuzx

zxuaN Cma
Coa Ch CC
XCImeuzx

.xoohmamuxo mva
mva.mva.oaa .xo.u_
3o I N~*xo I moxammxuxo
xoum.OIu¢ox
NN + Nouxo
zxuNN mma
ooa o» oo
zx*xo4>.OI¢mxuzx oma
caa.o~a.oma no.aIAzxomm«oe
Nu—
.eoIe.\xeo+e.nzx
.No .30 .mo .xo I moxoerMueo
.NC .30 emu .xo + amXCXMOuu u
.xooeeOmuxo maa
zmohmm
_Iua oaa
maa.maa.oaa .xooeu
zo*mmmmmmmmmm.o I aoxuaoxuxo
~o*mmmmmmmmmm.qu¢ox
au—
~Iu0u—
o+x*.o+x*.<+x.. u.u.m.<.x.xu0e
.wOZmomw>zoo mwpm<e mom woo >m owFOmmmoo zoah<ooeamhz~ o
ore lea: .z_z oz< x<z wzp zwmzhmm ozae<4oexmez_ >o owhqzahmm ma ha o
.m>_e<>~emo or» 1o moxmw m1» mezpmm owh<40m~ ma boom whooaz th o

 

 

95

02w

szhwa

.N0 + MNHNNHaNomm< + ANC+MN+NN+aNCWm< + xmwuxmw
Azu I mN#NN + AmN+NN.*aN.mm<u¢mw

szhmx momxw whDazou 302

zxumN
mm2.emx.mmx xeeuee_

e+eu~

a + eexnee_

A¢+Cuao*ooauo

mea.mpx.mex .mxuseex

oex.eex.ee2 ..zxeme44euem.o I .x.me<.e_
x I zxuzx

x+eus

.ee\e~e:*euee.\enx

mma

Cma
mha
Cha

 

 

APPENDIX B

STEADY-STATE PROGRAM

96

APPENDIX B

STEADY-STATE PROGRAM

The programs presented here are used to find h and
hfi for a steady—state corn dryer by fitting sets Of eXperi-
mental data. Three programs are shown here: WHCZS is a
calling program for HCLMB, the Optimization routine used.
SSCAL is used to store the data, call MBCDS, calculate the
error function, and return to HCLMB. MBCDS solves the 6
ordinary differential equations for the steady—state moving
bed corn dryer. Three other routines are not shown: HCLMB
and CUBIC are found in Appendix A. EMAIR is found in
Appendix D. The variables correspond as much as possible
tothose of DCMCD in Appendix D. C2 and c3 in DCMCD

correSpond to Equation 88.

97

 

 

98

Coouox

mu4
Cweoheoo a4vma
CoNnx<xN

AmoCauhexoaoh<thm
AmooauhexmemHChdszu
ANamaCP<szm
.\.XCNCh<szm

.oozm.xm.<o:~.xm.xuhzm.x~.mupzm.xe.xI4Izzm.xe.mIBszmm

.xo. CI» ImexoeIlma<Im.XCGXI—I1Imexoe<hIN.XNewCDCI¢.xC.hqzmou
.Nooamea.CaMMGmoaamNeaoaamoooaauemoaau.aoCaueoaaoh<zmou
.Comamve u xzv .XCa .momam. u mxmxcexoaoh<zmom

\ C \ 4 <k<o

ANaCImPCa zozzou

Nzw .41 “CC .46 euh .4» 202200

ACaohuh eAOaC><I eAanhha ..CaCm4ah ..oavm41m zoamZNzao
ACaohCo .ACaohvo ..Cavthm eaoavh<h .AmCx ZDHmZmzao

Axew. 4<CWm mzahnommzm

02w
haxw 44<C
A 4<CWm .me em4x .mx .x .> .2 .w ea
.hC ea< ewd>pz emdhmz .maw .ZZHCA aux C @2401 44<u
x .mahmz AOONeNvoqwm
Nuax
CMmd>hz
CoCummm
mlwoeaHZZHCa
Amooam>exmemaoh<zmom
\ weo .C.Ca .Cooa \ max <H<o
\ Noo .Coo .C.C \ m4x <h<o
.mox ..mvmsx ..mem4x eamCPm ..mvmx ZOHWZNZMQ
Aood< .Amvz ..m.w .Amoo eaov> zoamzwzao
4<Cmm 4<zmwhxw
mNCIX z<moomd

Ch

CCN
CmN
CvN
CMN

CNN

CaN
CCN

CON

 

 

99

 

02m
zmahwm
x .w ACCNeMthHaz
. wnzahzou Cma
..~.m4tw I NzNCmm<HCoCCN + mum
..aom4ahl<h.mm<#aooo + mum CMa
Aavhuo ..~CP<C ..avm4ahe<heaa.m4:w N
.sz .Ch e<I eaa.thw..—C><he.avhhaaoaNemakaxz CNa
CNa.CMa.CMa ..aavzmboaomu
A h<xtI .awaI .ECCII ex<zN C moumt 44<u
aaothmnsz
.mohuonuo
Aavh<®n<w
Aavh<Iu<I
Aavhuhuuh
Aavh<Hu<h
zsxzeana Cma Do
CoCum Caa
ACNNemvmhamz
“omNemvwhamz CCa
CCaeoaaeoaa .Aaaozmhoavua
thzaxd mm Oh mh43mmm NIH mmwsqu aa IChazm <H<o C
AmoxuameI
«vinh<xzz
AaoxquCII om
mzzahzou om
aaohuw exavh<w .Aavm4dh .Cx .Aavm4zw .N
0x .Aaowu» .AaC><I .Aaothm .Aa.h<h eaaowha ACaNem.mham3
.Hohuh ..a.h<I exavm4ah ACCNeNoC<wm
AHCNCC .Aa.><o ..a.h<h ..aom4tm ..avhwzm .Aaohha aomNeNvo<wm
Zszeana Cm Do
ACNNem.mham3
AomNemvmkamz
3meoa .2312 aovNeNvo<mm

 

100

.mcmawvexomoh<zm0u

Abomamoeum IvevoNawemmwxveouexwa Ivoh<za0m

aAvooamexvvmehoCaueaxmeooomvveXNemaeu mhhfi «Om 4<~¢hxoaexvoh<zm0m

Avooamexvevooamexvehooame.xmeeoomovexoemommexmCh<xxou
Animhlma<zwexCeQthlszCICexm.>haoazzzxmexo..mvzrv

excaeanzrvexoaeaa.zlvexoaex<mzw1meXNaezoaham0aImexm.\\Ch<zm0m

a ooomm< mh4=mmx wh<hml>0<whm thIameone\\.NIa.h<zm0m

AAfioom¢+ho\voMNm¢I.moooo+h*mNC.C.*twCdxm*mlm0NCoauAhezw.muao
\ 0N eCa .m .N .a \ IkZ4 <h<0

\ oveoaevaeoaeoa \ x4: <k<0

\ooNh evNoC .CoC \ 0012 e<a0 .CQC <h<0

\ CoOMN eC.C¢Oa \ m<d< eon: <h<0

\ mlmvoomevmovooevmmmoNa.mvoo \ mmoeN<m<ea<u<emaw <h<0

.NaCImhoa 202200

Nzwe<Ieuwe<0.Uh.<h 202200

.mox42 .Am.Ih24 zoamZmzao

ANCNOh ..Nuzob .ANCNCH ._Nom0k .Avaoh ZOHmZNZHQ

AvoNdh eavozah .A¢.Na_ ..vvm .2004: eACCND ZOHmezao

mezwaumummou 20Hh00 h0dh00 Na

Haqao 20mm mh40mmm .mwn4<>200Hm PDQPDD Ca
mamhm CZHKDQ mmD4<> 4<axh HDQHDO o
Admhm Iu<w m0 02w. mws4<> 44< PDQHDO w
mw>0 hm<Hm h

a004 woamza 20 m20~h<mwha 00< le

huwumw ICHHZm «#40

4<Hzoz>40d mmoao omIm < 00 mh00¢ th mwhnazou CHmDU
I Qz< H #4 ma< 0h maCu hzthOC mmbhmaoz ma4HDOw Arehvmaqzm
. o . owmazowm mwzaHDOmmDm

AH<NZI .dwaI .mOCII exqzwo moumz wzahaommzm

mvv
mm¢
Cm¢
va

an
0N¢

UUUUUUUUUUUQUU

AAdHWHVIHZ4CF<04ufiaooouN4wo
veaudhma mbN 00
40em<m2weN Avaem.wh~¢3
.anemvwhamz
ACNvemvwhamx
ovaehvaehva .amvxmhoavma
hDQhDD m04<> wuz<hma0 oamN th ma walk
CoCuN
d004 02H>02lN th «whzw 302
.avN0uaav40
Ceaua mva 00
hu4<lu2¢vx
CC + m0u>¢4<
.4d0*<C.\II*<wm<uCC
ACQCHCCC\II*<wm<um0
COIN\M¢0*Qquu00m
IIHH<¢ZIuazI
dXNIIflH<0H¢0011uII
QmNammo mm .meI <0 m0 mZDahCZDm m4 II 024 dz: 2“ had
.0d0*00.\<wm<*0¢1um0mm
<0\<mm<umovm
waflmmqfioomumumm
00\0011*N<M4<*oomum0Nm
00\0011*a<m4<*06mum0am
amdeN<m<uN<m4<
amamfia<u<na<u4<
dmdm*m<m<u<wm<
.mdmloeao806Nudmdw

lOl

N2wum<m2w
<HHACCN0
Chuamows
<IHA¢CN0

NimHAMCND

NZNHANCN:

NszAaCND

Nva

e1:

mva

0

102

Ne4< + ue3e<uvmae<
vu4<umu4<uvme<
~u4< + am uee3u<
¢e4< + am uue3u<
~o4mmu~mm
Nu*vouve4<

mm + Noume4<

mm + aou~e4<

.aNC + A¢C40\..¢.40e20040.¢H<ZmCHmcouNu
Coonmu

om*eomoumm

om4eovouvm

34*uomoumo

Aeoom + uaooxe4o*ezruom

u4o4eomonmo

u4o4uo4muam

AA.mo4oeeqmzwoeuao+ruo0*m.oneao

omm.omN.ONa .eeHooua

a + ee_1umehv

A N + .voxmeoa + AmCZmeoH + 2N42meoHqumeHo

ovm oz< mOa mezhwm e004 zo~e<¢whm mre «mezm
.4<>mwhza mzp do ezoze wre m4 wz<m

mrh mm oe ma mp42~pmm pmeae mare um omommz my zoaeov oz
.ommamwo e4 .mxmr mmo4<> oZm mre wh<zapmm ><z mzo
xxmoaaem<m2mveuaoureo

x¢.~:\.xvoao..o.aoom4<2wu_mo

ezoee mre #4 mavzm oz< >ea>4woee4o m1» hz<3 Om4< w:

NDZHPZDC
N I x<zNuN4mo
Cbaeocaemma .Avlmooa+N4mo+NC I x<iNvua

x<24eaudhmo mCN 00
AQHWHCX<ZHX<£4

Cha

mOa

0000

UL)

00a
mma

103

.aCNapuANoNah

ADNmu + Amownfiumzm<ofiNm + 00x*.N.N0 + udmu<%amflaa.NDuaN.3dh

Nm*.m.N: + vmau<*.mo~o + aoHAaoNouxmomee
N: «Ce mezmaoaeemoo doe mi» whoezoo

.CNmC + ZDmH¢m4<vflmNam<u.a.~dh
..moN08Nm + ANoNaumamm<CHam + <CxwaaCNOuaa.zdh
amHANCND + ¢MNm<fiaaoN0uAaCNah

a: 10m mhzwauammwou d0h NIH whDdzou

32m .mzo .mxo .mvvemoweaxz
nma.oma.moa .Amaozmhoaoua
xve4< + vu4< + vE4< + mmCHoNae<nzzm
ox + 1am + am.*ueme< + ovme< + mmaevumzo
vu4< + mmaevumzo
meZMHoaeumoo :oeeom mrh whoezoo zoz
. mormmummm
omaev + mmau< + mmae< + He3u<*mmuoox
eezu<8¢u4< + omaevuoox
mvmeq + ueme<*aoum0x
mvmev + ox + ueme<44mn<ox
.vu44 + moonmmuox
AmoNo + ANCN: + .aowouzom
xxvoaoamo + mo48monommo
ve4<4Nmumvmu<
Nm4amuomau<
Nm8Nmnm~o
Ne4< + uemuvuvmmuv
ve4< + me4<ueeme<
me4< + ue3e<uvmae<
vu4<*am uv4u<
me4< + hezuqummauv

mma

hwa

UQU

UUU

 

 

 

ommm*320 + ANCNOPHCunaaoNOH
ommmflsz + mCXH320 + .NCZOPHCC + “NCNOhuaaCZOF

ommmemzm + maxHNIm + 2m4~=*3zo

.NCNOPHCQ + ANCIOPHaaCNCP

....
Cmmm + 00x*m20 + .moNDHsz + ANCm0h*CC + .NCNOhuaaCmCH
+

00x Am

AAmCNdh
22m.zdh
..moNdh
AamCND

.N0*m20 + .NCmOhuaaovah

dmhm 02000m th 00 302

xvoaep4~044mmu.moaoe
.vozeh*~oo*mouxmozoh
xvomep4N044mou.momoe
I xv4~34~044monxmomoe

mawpm mmth 24 .0044am 302 ma< <9 024 Ch 00 Q0» th

C0*m0*mmuommm
muflmm + moxuwox
AOINDHmC + amCNDHCCHMCx

.0mm0 wm< m20~e0e~emmDm 02a30440m NIH

104

N20\120 I N400u02N0
32 HPM4<HN20

320 + N20H>m4<uzzm
N20 + m20HNM4<nN20
mtm + >¢4<um20

<k 024 CH 000 mhzmauaumwoo 20H500 NIP mh0a200

.22m0+mu+muol20m.%v0 +

A¢0N0%mmo%0Nam<uavoNdh

0OXHA¢VNO + 22N04<+Nu4<ofim0 I AMCNCHHQIM< + ANVNDHN0CH¢0HA¢CZQH

AvoNDHMNamq +

.m0 I AmoN0.*vmuA¢VNdp

I 000 mhzm~0H00000 d0h NIP whadzou

.aCNdhnamVNQH

3mm04ee3e< + ooxaxmoao + xxmvazauezuv + .ao~:*aooammu.m43ee
ammo + .moanuvmaeq + mm».m.~:n.momee
m2 mod mezwaoauemoo eop 01» whoezoo

UUU

105

 

ommmomm.omm vaone_

.xsz + sz + m*ax.*m.\xmuzxm
zzo+zzm+zzo + m». 40x + m*.mzo+ox+ox..uox
40x + voxuvox

sz + mzmuqox

mzm + mzm+m+m+muox

x~4mo8moexmuxw

m8mnmm

14x4eum

v.4u4x mmm oo

4: oz0444e mp14ezoo zoz
zo4e<N~444p424 ore mmem4ezoo mare

N20\..NCIOH + OZN0*~N.NOPVHAOO4D
N20\2.aC30h + OZNOHAaCNOh0uAmo40
320\4¢.Ndhu.¢o40

Oxuamo40

OXMANC43

0xn2a04:

320\AaCNmHqu

00Na4<apaza 00 Hm32 mmD4<> 020 th 302

ONNehONewON aawo3mh0aoum

0.200.004 Ammvemomeax3

mON.OON.OON Ammaoma

OON.OON.mma AA“aOxym0<HOImOoaIAmmmomm<qu
OONeooaeooa .Aanzmh0HomH

Ammaeaxm.310.N20em20 ..moa .ANom .AaCNCOH000 4440
¢Ia mh000 .m004<>zwoaw mIe H00 302

ANVNOPHOO + Omm0432mnaNoNOe

.mozoeuou + OMmmamzo + qu*32mn.Nozoe

2N4N0h800 + ommmrmtm + Moxamzm + .04N0432mu.momae
.Nomoe460 + ommm + momezm + 263N34m2mu.momoe

eox + .ooNoemzmuxmovoe

.<e do dob th m44au dwhm em<4 wIe

NON

mON
OON
00a
00a

U

106

 

020

zmzhwx

ACC4DH<h

20.40u0h

A¢C40u<I

m4020u~20

002~h200

OhNemONemCN 2x<2N I ¢I006a + N000
40ea<0zweox Avaemvwhaxz

mlwmoo + Nuax

OCNeNCNeNCN ..0.30H0~.0~

N400 + NuN

20.40naaoN0

Ceana mmN 00

002~h200

dwhm athOZ< 000 xu<0 00 ZOIh
00h04Q100 ma m0hmIN 02h hzaoa mth h<
moa 0h 00

40pm<020emhao .Omvemvwham3
mvNemanmCa ..OC3mh0HCOa
a.m.4z+amu4:+amv40ofimmmmmmmmMMoCum<mz0

.040<4a<>< ma 0k<2~h00 302 01h FZHDQ mth #4

zx0*ox + xeao43u.e4.4o

m4..4.30h + Nm*.xaom0p.+ NmHAeHCNDOO + oxuox
xaoaap + NmHAAHONOH + Nm4244voe4uox

v+4ne~

Neaua mmm oo

xm40x + .Ho4ouxao4:

m8..~.3ep + Nm*.HCNOO+ oxuox

Amoaee + mmaxaoweeuox

v.4u4 mmm 0o

OX\anxm

320 + m*.sz + Nzo + m*.mzm + ox + oxoouox
mzm + m + muox

CNN
mCN
NON

OCN

mmN

OmN

UL)

mvN

mmN

OMN
mNN

o-

 

APPENDIX C

SINGLE-STEP SOLUTION FOR THE

DYNAMIC SIMULATION

107

 

APPENDIX C
SINGLE-STEP SOLUTION FOR THE

DYNAMIC SIMULATION

The problem here is to solve Equation (91);

Cramer's Rule was used.

 

 

[ S”‘1 "BI 0 0 0 0 .A-7M171 _szMi + sclMI:z + clMED-fi
'31 S+A2 ‘32 0 0 0 M2 52M; + sclngz + C114:D
O 782 8”3‘3 'B3C2 O 0 M3 1 52M3 + SCIMEZ + CIMED (9].)
0 0 -B4 s+A4 0 0 H 52 s H° + SC4HPZ + C4HPD
0 0 BS 'Bscz S+A5 “C5 Tc SZT° + SClT:z + clT:D
0 0 0 0 -c6 s+A6 Ta LS2T° + SC4T:Z C4T:D
— .4- .4 —(

 

 

 

 

 

The top four rows Of this equation are independent,

so they can be solved by themselves for Mi, Mé, M3' and H.
75+A1 “Bl 0 0 7 FFQ_7 FSZMi + sceri’Z + clM‘ED-1
’Bl S+A2 -82 0 M2 — 1 S2M§ + SClMgZ + ClMgD
O ’32 S+A3 ‘B3C2 M3 82 82M; + SClMgZ + ClMgD
0 0 -B4 S+A4 H ' 52H° + SC4HPZ + C4HPD
i _ e 1 e. _

 

 

 

 

 

108

109

Each will be expressed as the ratio of two polynomials in

S.

theproduct of 82

(S+Al) {(S+A2

+ B {- B

1

(S+A )(S+A )(

1 2

._ E32

2(S+Al

((S+Al)(S+A2)

2
- B2(S+Al

 

The denominator polynomial for each will be the same,

and the determinate of the left matrix.

 

)(S+A4)

 

 

 

 

E S+A2 ~13j 0 -B1 0 0 i
O 1 I .
| = 82 (S+Al) -Ii S+A3 -B3C2 j+ His2 -82 S+A3 -B3C2
0 -B4 S+A4 0 —B4 S+A4
I
2
)(S+A3)(S+A4) - B4B3C2(S+A2) — B2(S+A4)}
l(S+A3)(o+A4) + BlB4B3C2}
S+A3)(S+A4) - B4B3C2(S+Al)(S+A2)
)(S+A ) - B2(S+A )(S+A ) + BZB B C
4 1 3 4 1 4 3 2
- B2}{(S+A )(S+A ) - B B c }
l 3 4 4 3 2

 

110

_ 2 4 3
— S [S + S (Al+A2+A3+A4)

+ A A -B2 + A A - B B C - B

4) 1 2 1 3 4 4 3 2 }

2 2
+8 {(Al+A2)(A3+A 2

2 2
+5 {(AlAZ-Bl)(A3+A4)+(A3A4-B4B3C2)-BZ(A1+A4)}
+ {(A A -B2)(A A -B B C ) — 52 A A I]
1 2 1 3 4 4 3 2 2 1 4

Then the denominator polynomial is

 

S (S + S ° BM3 + 82 ° BM2 + S ° BMW + BMZ) = Denominator

where BM3 = A + A + A + A

1 2 3 4
BM2 _ A +A — 52 + A +A - B B C - 32 + A A + A A
‘ 1 2 1 3 4 4 3 2 2 1 2 3 4
BMW — (A A -B2) (A +A ) + (A A - B B c ) (A +A ) - B2(A +A )
’ 1 2 1 3 4 3 4 4 3 2 1 2 2 1 4
BMZ — (A A -B2) (A A — B B C ) -‘B2 A A
‘ 1 2 1 3 4 4 3 2 2 1 4

As an example the numerator polynomial for M1 is

as follows:

Numerator

Of M1 = 82M° + SC MPZ + C MPD —B S+A -B3C

 

 

S H0 + SC H + C H 0 -B S+A

111

 

 

 

 

 

S+A2 -32 0
= (82M: + SClM]:Z + CIMID) -B2 S+A3 —B3C2
0 -B4 S+A4
82M° + SClMPZ + ClMPD "B2 '
+ 131 S2M° + SClMPZ + ClMPD S+A3 -B3C2
52H° + SC4HPZ + C4HPD -B4 S+A4
= (s2M°+sclM§Z+clmiD){(S+A2)(S+A3)(S+A4)—B§(S+A4)—B4B3G§S+A2)I
+ (SZME + SClMgZ + ClMgD)°Bl°{(S+A3)(S+A4) — B4B3C2}
+ (SZME + SClM§Z + ClMgD)-Bl° B2 (S+A4)
+ (82H° + SC4HPZ + C4HPD) - Bl . B2B3C2

This will be put in the form of a power series in S.

Here the coefficient subscript refers to the dependent

variable.
5 o 4 3 . 2 m
= S ~Ml + S -TP4(1) + S -TP3(1) + S °TP2(1)+-S-TPW(1)+-1PZ(1)
TP4(1) - C MPZ + (A + A + A ) - M° + B - M°
— l 1 2 3 4 l l 2

112

 

_ PD . PZ
TP3(1) — ClMI + (A2+A3+A4) ClMl
+ M° (A A - B B C - B2 + A (A +A ))
l 3 4 4 3 2 2 2 3 4
+ B (C MPZ + (A +A ) - M° + B - M°)
1 1 2 3 4 2 2 3
TP2(1) = (A +A +A )°C MPD + (A A -B B C -B2+A (A +A ))-C MPZ
2 3 4 l l 3 4 4 3 2 2 2 3 4 1 l
- _ _ 2. _ o . PD . PZ
+(A2(A3 A4 B4B3C2) 82 A4) Ml-I-Bl (ClM2 + (A3+A4) C1M2
+ (A °A -B B c )- M° + B (c MPZ + A M° + B C °H°))
3 4 4 3 2 2 2 l 3 4 3 3 2
TPW(1) = (A A - B B c - 32 + A (A +A )) . c MPD
3 4 4 3 2 2 2 3 4 1 1
2 PZ
+ (A2(A3A4 B4B3C2) ‘ B2 A4) C1M1
PD ’ PZ
+ Bl ((A3+A4) C1M2 + (A3A4 — B4B3C2) ClM2
PD PZ PZ
+ B2 -(ClM3 + A4 C1M3 + B3 C2C4H ))
—— ' _ ._ 2 0 PD
TPZ(l) — (A2(A3A4 B4B3C2) B2A4) ClMl
PD
+ Bl ((A3A4 B4B3C2) ClM2
PD . PD
+ B2 (A4 C1M3 + B312 C4H ))
The other three numerator polynomials for M2, M3
and H are found in the same manner. Then the last two

equations are solved in terms Of M3 and H. Again Cramer's

Rule is used. The solution below is for T5.

113

I— "'I ”r1 7‘ 2 o P Z PD 2 —_ — '1
S+A5 C5 1;) S T + SClTC + ClTC + S (B5C2H B5M3)
. _1
“‘2' 2 o PZ PD
- +
J C6 S+A6 ITa S S Ta SC4Ta + C4Ta
I. -J >- —I) >-- --I

 

 

 

 

 

 

 

 

2,,o . .Pz , PD . 2 0 P2 ,PD 2 — _ —
T = (S+A6)(S 1C + SCllC + Cllc I + CSIS Ta + SC4Ta + C41a } + S BS(C2H M3)
C 2 2
S (S + (A5+A6) - S + A5A6 - C5C6)
The term involving H and M3 is as follows
SS(C H° — M°) + S4(C :P4(4) — TP4(3)) + --~ + (C -TPZ(4) - TPZ(3))
.2 — — - 2 3 2 2
S (C I—M ) -
2 3 4 3 2
S - S - BM3 + S - 8M2 + s - BMW + BMZ

Then in order to make this substitution the top and bottom
Of the Té equation are multiplied by the denominator

above. Then the denominator of Té and Ta is

Denominator 2 2
_ _ = S -(S + (A.+A ) . S + A A - C C )
of Tc' Ta 5 6 5 6 5 6

°(S4 + BM3 ° S3 + BMZ ° 82 + BMW ° 8 + BMZ)

And the numerator of TC may be found

 

 

114

 

— _ 2 o PZ PD
Numerator of Tc — ((S+A6)(S TC + SClTC + Cch )
2 o PZ PD
+ c5 (S Ta + SC4Ta + C4Ta ))
4 3 2
- (s + BM3 - s + BMZ - S + BMW - S + BMZ)
+ B 85(C H° - M°) + s4(c - TP4(4) — TP4(3))
5 2 3 2

+ ... + (C2 - TPZ(4) — TPZ(3))

The numerator is then put in a polynomal form

Numerator of TC = S2 ”1‘: +56 - TQ6(l) +55 - TW5(l} + s4 - TQ4(l)

+ S3 °TW3(l)-+82 °TW2(l)+-S' TQW(l))4—TQZ(1)

 

where j
TQ6(l) = TCW + BM3 . T2
TQ5(l) = TCT + TCW ' BM3 + BM2 - T°
TQ4(l) = TC3 + TCT - BM3 + TCW ° BMZ + BMW - T2
TQ3(l) = TC3 ' BM3 + TCT ‘ BMZ + TCW ' BMW + BMZ ' T2
TQ2(l) = TC3 - BMZ + TCT - BMW + TCW - BMZ
TQW(l) = TC3 - BMW + TCT - BMZ

TQZ(l) = TC3 - BMZ

 

 

 

115

and
TCW = A ' T° + C . T° + C TPZ
6 C 5 a l c
TCT = A6 ' ClTiz + CszD + C5C4T:Z
TC3 = A6 - ClTED + c5 . C4T:D

The inverse Laplace transformation is as given in
Equations 90 and 91. The value of M1 at the end of a z-step

is as follows:

 

 

 

Ml(AZ) + Eggéil ~AZ + BMZ ' TPW<l) - T§Z(l) - BMW
(BMZ)
5 o 4 3 . 2
4 (rich‘Fri'TP4 (l)+ri-'1'P3 (l)+ri'TP2 (l)+ri.TPW(l)+TPZ (1))
+ 2
i=1 r? (4r? + 3r? -BM3 + 2r. °BM2 + BMW)
1 l l l

EXP(r.AZ)
l

where the ri are the roots of the denominator polynomial.

 

 

APPENDIX D

COCURRENT DRYER DYNAMIC PROGRAM

116

APPENDIX D

COCURRENT DRYER DYNAMIC PROGRAM

The dynamic program consists of a large main program,
DCMCD, and 5 subroutines. DCMCD does the main computation.
It is controlled by the IDTSW vector and the first data
card. It calls 4 subroutines: F4PT fits values in the
previous time step 4 points at a time as shown in section
4. EMAIR calculates the equivalent moisture content of the
air from its temperature and humidity. The relative
humidity is limited to 0.9998 in this subroutine because
higher values make the equivalent moisture content very
large. QUARF gets the roots of a 4-th order polynomial;
it uses QCUBE, a faster version of CUBIC in Appendix A,
to estimate the roots of the derivative polynomial. ENTR
supplies the values of the dependent variables at the
entrance of the dryer. Since ENTR receives the exit
conditions from the dryer, it could be used to implement
a feedback controller or, as in this case, calculate a
cost function.

The following are corresponding variables between

the dynamic simulation section or Appendix C and DCMCD.

ll7

 

t
. ...... rd».

         

«....

Y
\.

 

ll8

EPS = e RHOA = pa
1
AFAl — Al/V RHOC — pc
AFA2 = Aé/V ALFl = A1
DR3 = Ar3 ALF2 = A2
HFG ALF3 = A3
f9
ARA3 3 ALF4 = A4
CPC ALFS = A5
pC
CPA ALF6 = A6

 

The values below are the same in Appendix C and in DCMCD.

Here I = l is for M1’ 2 for M2, 3 for M3, and 4 for H.
Also J = l is for TC and J = 2 is for Ta'
BM3 TP4(I) TQ6(J)
BMZ TP3(I) TQS(J)
BMW TP2(1) TQ4(J)
BMZ TPW(I) TQ3(J)
TPZ(I) TQ2(J)
TQW(J)
TQZ(J)

The U-matrix in DCMCD is large enough to hold a

complete set of the 6 dependent variables along the dryer

for one instant in time.

At the start of a time-step it

holds the values for the previous time-step, and it is

filled.with the present time-step values as the integration

proceeds.

 

119

The step—size in the z-dimension is controlled by

 

the DLZ-vector. Since the variables change much faster

near the entrance to the dryer, Az is made smaller there.

 

120

 

.m~.1mhou zoumezmo

.0¢.Ngo ..o.a3 ..ovx ..o.om.3 ..N.o.o zommzwzmo

.N.~Oh ..N.30h .Amvmo» ..N.m0h ..N.¢Oh ..N.m0h ..Nvooh zommzwtuo
.¢.~ah ..¢.zah ..¢.Nab ..¢.mah aa¢v¢ah .AOVJD ..0-D 20—m2wzmo

.x<¢ooma th mwhqzmzmwh cod oho. 0<4m~ .moxqu roam dea wa<
mzouh~ozou 4<uh~z~ th 0<4mn m>mh40wz «Om ..hxwz th «Du m34<>
44~h~2~ th mu mzo «Om m34<> Adzum th mmeI mZDm whamhnat

«Om OmmD. mm m4 hum; wad >th o<4mu no mw34<> w>~h~moa «Om

comm th wzoa< hz<hmzou wad mzomhfiozou 44—h—ZH th ouoqamm «Om
.mzouhmozou 4<~h~2~ thmmumHo m «on a: hwm mm z<x¢0ma mnxh

mzmh IF“: mw34<> 02w 024 dprwu HDQPDO mg
awhm wzuh mo 02w th #4 xHah<le hDthO Nd
mhzwnummmwou zohhom FDQPDO ad

hmqao 20am mhnamwm .mw34<>2mo~w HDQhDD OH
mdwhm ozmmzo mm34<> Ad—«H HDQPDO

Aawkm 104m mo 02w. mw34<> 444 bamboo

om >m wnmnmm>~o tho mmwaza Nalm wmozou
om >m mhmumm>~o tho wwwAZD anm mmozoa
.N >m wmdmmuzfi .mwmzaz ZO~P<¢mp~ moz<Iu
.H >m mm4w102~ .mmmzsz zomh<mmh~ m02<Iu
wwz<Iu mwszZ zo~h<mwhm no zomm mwmw>wm
mmumou mmmmzth mm<z oz< pde 2H o<wa
hummuw , >mhzw xmhou

NM~TW~ON®O

ohawzah mo zombUZDu < m< xOhUm>lD ozmmmhzw AD.h.mhzw
.I 024 P #4 mH< Oh «moo thhzou wthmmoz mHJHDGw AI.H.m~<Iw
4<~ZOZ>AOQ awoxo Ibl¢ < mo whoom mm<DO
ommqao 2H owm: .>Joa «memo owlm < do thOm wmnuo
on<~zoz>40a zomhduOJJDU 4 Oh mhzuoa mDOu mhfim hd¢m

. . . me—DOma mmzmhsommbm

.hDdzmana<p .hDthOumwa<h .hDQhaoahDdzvaUIUQ z<mooxd

UUUUUUUUUUUUUL)UUQUUQUUQUUUQUQ

U

 

 

121

 

Imbom Amo¢amvmh~m3
. 3mh0~ amocamvo<wd
omDOI xma hDOu wa<DOm «ma
amhp<z >mo do mGZDOQ Z~ mm< mwh<d 304m Iwom

wh<m 304m ¢~< th mm «a

wh<m 304m 2100 th mm um

IkamA mw>mo 4<HOh th mm IhzwAN

mmDoI 2H awhm mzmh wa um Fame

mz—h AduthH th mm mznh

mawhm wx~h do mwmzaz th m— Xhfi

m>om< m4 mm o<4um

<0.uo.I»24N.h4mo.HZHH.XPo.w<4m~ Aoo¢.m.whmmz
40.00.1h24Nah4wo._zmh.x#o.o<4m~ «00¢.N.o<ma OOH

<h<o zomhqmomhzfi th Zn o<mm

AND.0.000ml.mk2m 4440

A~¢.ofil.x¢.m.>.ofiu..xm.o.ol.¢.xmm.h<zxoi om¢

.~.mflwo.xomcp<zx0u m¢¢

.m.oi.mz_p B4 Im.\.»<zxol o¢¢

A~.mfiwo.nm I¢.¢.mfim.amwx¢.o_.¢m_ I¢lp<zmoi mm¢

.A¢.ofii.x¢.m.w.ofii.Axm.o.om.¢.xw.mm.n apps moi nafixpzofl.xm_qum0u om¢

.¢.ofil.x¢.¢.o~i.x¢.>.odm..xm.o.oi.¢.xo.m.ml.xmlquxoa mm¢
.azwknm~<xm.xo.azmpuzaoozo.xm.>Haomzaxxm.xo..m.z:¢ a

.xofi..mczx¢.xo~..fl.z:¢.on.mqmzmxm.x~H.onBHmoaIm.xm.\\chqzmol H~¢

A...m¢< mhnnmmx mm» mane: Imm.m.o¢. u mz_p quofi.xom.\\.oxalh<zmol o~¢

Am.m~mo.xw.b<zmou o~¢

.mfimficpqzxoi mo¢

.m.mfilm.odsm.h<zm0u oo¢

AN+>+xvfimmmmmmmmmmoou.~.>.x.¢mw
AAhoom¢+h.\¢.mNm¢l.moooo+h*mmooo.*zm.axm*mlm0N0oHuAkazm.mumo

\ mmo.o*om .Ho.o*m .muwo.¢um\Nno <p<o
\mko.o.o.~>.¢~.o.o.ox<o:¢ .ooxm .4ao .oao <p<o

\ o.om~ .o.o¢o~ \ m<¢< .ou: <p<o

\mano.m .¢wo¢.o .¢mmm.-.m.o\mmo .mqiq .H<l<.maw «hqo

UUUUUUUUU

 

122

 

Omaummo mu

QII.II

AOH¢.m.wh~m3

atlazI .04¢.N404wm

0m.

04.
oom.mmm.mmm

o¢.o*
omm.m>m.m
BN4.om4.om

.wwa 40 mo m204huzam m4 II 024 QII 2“ had

.uau*uo.

uo
uw\uorm*
uo\uoxa*
a

a

.mam

mmd o» oo
Ixuwcoo.ouazz
o**4o*mom.ouxx
mom Oh oo
o**4o*ooo.ouxx
Am>.m4ol<o.m~
mom as oo
*o.oo*ooo.ouzz
hm Ao.ool<o.u_
4 A.~.3mhomcu_

\4m¢4*om1nuumm
<o\4mm<nuu¢m
\4wm4*o.muuumm
m<u4<*o.mnmumm
44¢4<*o.mumu4m
mamWN4m4nN4u44
mawfl44u4u44u44
amam%m<m4n4mm4
Ip24~*m.onIN
no.4.*o.mnamam
.4.N3u11.~.:
c.4uw omH Do
om.~u4 0N4 on
N: Ao4¢.m.o4mm
mNH o» oo
.4.om.:u.~.~:
0.4"“ ~44 on

.ouauo.46.4434 AOH¢.N.o<wm

mN4.m44

N04.m04.m04

om.HuH odd Do
.m04 .044uH4mm
thm 4440
.ooalo44m—4mu

mm~
NNA

mom
owm

mmm
omm

mhm
0N4

mNH
0N4

mad
N44
044
mO~

mod
NOH

o

123

oow44~¢wm m~ xmmh4zl3 th awhm Io4w P4 .aoo4 wimp th m— warp
Xhfi.~u:hfi CNN 00

wow Oh oo

Huzmm

.4N¢.m.m»~mz

Hzmp .o~¢.m.mh_mz

om~.mm4.mm4 ..m43mhoucm4
wm4 Oh om

4m~¢.m.mh_xz

.o.4u4..~.0m.:4 .m4m:w.Ih24~ 4m~¢.m.wp~mx
..m.0m.: ..N.Om.: .44.om.:4ammua4mzm
.o.~u~..4.om.:..m4mzm.zw .mm¢.m.mp~x3
..m.0m.3 ..~.om.: .44.om.:.mmmux4mzm
424* .o¢¢.m.mhumx

mm4.mm4.mm4 ..m442mpomvm4

mm244> 44Hh_zH psakao

oxu.o.m

om4.om4.om~ Axoxcmm<*mlmm.o I Azmxvmm4414
xxx I oxuox

4H4xxz4xlh4x4\z4xu3mx

camm4 + ox + oxnhqx

memm4 + ox*Aoamm4 + ox4u34x

x0 + mmxuox

xo 3 mmxnxmvm

Auom¢4 I mmx*xmx.pmomuxo
camm4Am.oanxmx

¢u*mo + 0144*4ouoomm4

@144 + mm44uoamm4

¢u + ouuou4<

Ho + muumu44

Ak4wo*4w.\mam*4ormu¢u
4h4wo*oo.\.mamlo.4.*ooxau40
.4auu4w.\11*4wa<uou
.uao*0w4\zx*4mm4umu
uoxmxmxo*azxumu:m

mm~

om~
mmH

mmH

omH

0N4

 

 

124

 

CNN Oh 00
. .Numvo .44.“.o .Am4N4o+.N4N4o+N4wo

.ANVN40+N4mo .N4wo .AH.¢.D .._.m42 ..~.N03 .44.H.D 0ha¢m 4440
mo>.00h.00h .HIN14m—

o.~u~ 004 Do

xHah4zlo th 24 44mm hmxmm

4N30~4QHN4mo

04.4nNo mom Do

4:.m4mzw.N .mm¢.m4mh_x3
.4N¢.m.wk4mz

wimp Aom¢.m.wpum3
o¢4.>¢4.>¢4 44m43mho44m~
mmo.>¢4.>¢4 410104.;—

HDQHDO m344> w024pmHo ome th m4 mmxp
cocuN

aOO4 ozH>ozIN wIp mmhzw zoz
440N3u4404:

B.4um m¢_ on
A4m4~:.ANVND..4.N:4mmmnm4mzw
4N3.wz~h4mhzm 4440

.mw>mo th 024>4w4 44Hawh4z
th ZO owm4m zomh023m m>Hh0wfimO Z4 mh4430440 Oh 0mm: mm
>4: ahzm >43 mmIH Z~ camhm mime mDDH>wxd m1? «Om mw344>

02w th wz~4h200 mo~0w>IND mIH «#2w 0» 4440 20 H4Ih whoz

h4wO#42ho4p404m + Hzmkuwzmh
onx010—

m~o.o~o.mao .m*.m\2hfi.lzho.m_
o~o.o~o.omo ..hvxmhom.¢_
Hux010~

Omoumooaomo 4N*.~\zhfi4lxhfi.u~
ooo.moo.moo Aacvzmho~0m~
oux0I0~

a

cow

>44

044
mmo

m¢4

0N0
map
040

moo
coo

UUUUUU

 

.|UIIV1]

125

4.¢.40\.4¢04D.4044:0¢~4zw + HN00*moouN0

Dm§u0m0umm
Dw§u0¢mu¢0
Dwfim0mmnm0
Am0Dm + m~00\u~0*a21u0w
m—0*u0NmHN0
m~0*u040u40
A4Am44zam4mzw4mm40+1m04*m.onu~0
omN.omN.o>4 Athw.u—
H + xbmontho
A.403mh04+4m.zmh0H+Am03mh0H0*2414m03mh0m+.m03mh0~4+mluthfi
0¢N 024 mOH zwmzhwm aDD4 zomh4mwhm th mthm
o44>mwhzm wIH m0 hzomm th m4 wt4m
mIp mm 0H m_ wh4zmhmw hmxmm mmrh mm 0m0wmz m4 20~h04 Dz
o0wa~mm0 mu .meI mw344> 02m wIH wh4zmhmw >42 wzo

..m4ND.x402w4mum0nIm0
.¢.ND\.A¢4ND.20.ND.KH4£mu~N0
hzcam wa M4 mm42m 024 >PH>HmDmm~0 wIH H243 Om44 w:

mszapzoo

¢o*.m.ulouim.4.o

¢o*.4.4.on.4._.o

064 DP ow

4o*.~.4.on1~.4.o

4o*.4._.ou44.mco

om4.mm4.om4 44:.mn04mm<_.l_

oxulm.~.o
. ox .44.“.o .14>¢.~4a+4m¢.N4o.u ..m¢.~non m
.N4mo ..~._lo .4406: .._.om.: .4~.o¢.: cha¢i 44<o
own ow oo
. .N.H.o .14.“.o ..41~fi.~4on ..4+~6.N40+N4wa N
.N4wo ..Hca: ..H.~+~1.: ..H.H+~sl: .44.Ns.3 0pa¢a 44<o

Simfifio: 40¢..N3“:

Chu

mod

004

mm4

oma
CNN

mfih

04h
mop

UQUUQ

 

126

4: mod mPZwuumumwou gob m1» mpaazou
Nu.~:a.zxm.mzm.mzm .m¢¢.m.mh_x3
>w4.om4.ww4 4.44.:m»o~.u_
omo.wm4.mm4 4x01000m4
.4m + 4m4<.*40*¢¢44*~m + ax*mm4m<n~zm
4¢m4<+¢m4<+¢¢44+4o+mm0*4m44*~m + ox*pazm4 + 44m+414<.*40*uamm4uxzm
m¢mm4 + m~414 + 4¢m4< + mm + 400*Nm + .4144 + 4&44.*uamu<n~zm
4&44 + mN4m4umzm
mkzw~o~uumou ickpom mIb wpzazoo :oz
440 + 4&444*40*~m + mm4m<*.40 + mm0upox
mm4m4 + .4m44 + 4u4<0uwm + .40 + mm.*ha3u<uzox
mm4u<*¢m4<nh0x
hazm4*¢m44 + mm4u4u30x
m¢mu<*4¢44uhmx
m¢mm4 + mamm4WHu44nzmx
0x*mm + 4m4<*m¢mm<uh4x
4¢m44 + 40 + mm4*mm + uamm4*4m4< + m¢mm<u3<x
¢u44*40 + ¢0*mmnox
¢0¢mm + ¢¢4<*440 + Nm4nm¢mu4
4H0 + Nm.%flm + N144W~0nmmau4
Nm*~mum~m
mm44 + mamm4u¢m~m<
¢m44 + MM44uuamm4
mm44 + ma3m4n¢m~u4
mm4< + ha3u<nmmfim<
Nm44 + ua3m4u¢mmm4
4&44*mu44n¢mu4
¢m44§~m44u¢~u4
Nm44 + 4&44nha3m4
¢m44 + 4&44nma3m4
Noummnmmm
.m204h3p4hmm2m szm mx4z ham .m4m4444>< zoz wx4 mhzm40411w00 wxh 444
¢u + No*¢mu¢u4<
40 + mm + Nmumm4<
mm + 4m4<u~u4<
40 + 4mu4u44

Fwd
omo

 

. --_--1 (ill: a

44¢.ND*0N4m4 + .H.¢.0*pa3u4 h
+ .N.¢.04*Nm0 + Am4N3%h0x + 24.m.0*30x + Am.m.0*¢mam4 N
+4.N.ND*¢4¢4+.H.N.0*ma3m4+.N.N40+A.H4ND#¢u44+.H.4404*H00*N0n.m.mah

42¢.N3*ka3¢4 + 4~.¢000*Nm0 + AMVND*30X + m
.4.M.0*¢N4u4 + AN.m00+A4N4N3*ma3u4+24.m40+.40ND*H0.*Nmn.m0mah

A¢4ND*Nm0 + .m.~:*¢N4m4 + .Hsm40 + .N.ND*NmuAm.¢ah
mt «Om mh2w40fimuwo0 QOP th mh3a200
A.N.¢40%Hu44*mmm+AN.m00*¢Hm44*N0+4N.N00*hmx+AN.440*0¢mm4*40u2NONah

A.AH.¢v0*4m44+AN.¢000*Nm0+44.m40*¢4m4+2N.m.0*ua3m44 H
* N0+AH.N40%h0x+.N.N00*30x+.44.440*m¢mm4+.m.440%mamm40*40uAN.3ah

.2240ND*4¢44 + 24.4004 N
*Nmm+4m4ND*¢Hm4+.H.m00*mazm4+gm.m.0vfimm+4N4Nsfikmx+24.NO0N
# 30x+Amam40fi¢m~m4+A.44NDWm¢mm<+.4.H.0%mamm4+4N.4004*Hmu2N4Nah

127

42¢0NDWNMO + AmVNDA¢QKu4 + ”asm000%m0+ m
.N4Nnfizmx + AH.N40*¢m4m4+AN.N.0+AAaszflmamu4+AH.4000%40u4N0mah

Am0ND*Nm + .N4ND*¢mHu4 + AH.N40 + .40ND*Hmn.N4¢ah
N: 10m mPZm~0~umm00 mow th wH0a200
AAAN.¢00*Nmm+AN.m00fi¢u440*mm+AN.N40*m¢mm40%40+am.4.0up4xn44VNah

AA24.440*Nm0+24.m40*¢m44+am.m.0vfimm 4
+ AH.N40*04mm4+fim.m40*mamm44*40+.4.4.0*H4X+.N.H.0*34xn44VIQP

A.A¢4N3*Nmm+2m.ND*¢¢44+44.m00.%mm+ANOND*m¢md4+44.N00 N
fiddMu4+4N.N000%40 + .44N3*H4x+.4.400#34x + AN.400*¢mNm4n.4.Nah

. .AmvNDWNm + ANVND m
*mamm4 + 44.N.0 .*40 + 240N3*34x + 24.400*¢mmm4 + AN.440n444mah

ANVND*40 + .4.ND%¢mN¢4 + 44.440u2444ah

004

0

 

 

A.m4NDI440ND*N04*mmn.N.mOh

mmmkm mwah z~ .0m44Hu 302 wx4 4b 024 0b do now th
.Nam.0*o0 + AN.000*mm44nm4h

AH.m40*o0 + Amaov0 + 44.000*mu44np4h

.m0Nnfio0 + 44.040 + ~00~D*mm44n34h

.N.o.0*m0 + .N.m.0*0u44nm0h

.4ao4ofim0 + .mam00 + .4.m40*om44uh0h

.00N0*m0 + Auamvo + .mvafiou44n30h

‘ o0wm0 wd4 wZOmHDkHHmmDm 02430440m m1»
sz\3zm I N4w0nOZNO

sz\320 I N4w0uozN0

sz*0omu4nsz

220*0omu4 + sz%oamm4n320

sz*0omm4 + 320*oamm4 + sznmzm

mzm*0omm4 + szfioamm4 + 320nmzm

00mm4 + mzm*oamm4 + Nimn¢zm

Qdmm4 + mxmnmzm

4b 024 0h mom meZw~0~mqu0 zohhom th wHDaZO0

128

.m.¢.0*h0x+4AN.m.0*0NHu4+.AN.N00*Hm44+AN.H00*400*N00%¢0HA¢VNQH

.4.¢40*h0x + .N.¢00*30x + AAH.MVO*0NHW4 + H
AN.m.0*ka3¢4 + 4.4.N40*Hm44 + AN.N00 + AH.440*404*N00*¢mHA¢03ah

A¢4N:*h0x + A4.¢.0*z0x+.~.¢.0%mmflm4+4.m.~3*mm~m4m
+ .H.m00*ha3m4+.mam00+44N4N0*HM44+44.N40+244N0*404*N00*¢mnA¢.Nah.

.¢.~D*30x + .4.¢40*mmfim4 m
+ Ama¢00 + .AmvND*ha3m4 + 44.m00 + 4NOND*N04*¢mnA¢0mah

A40N0*mm4m4 + AH.¢.0 + .m0N0*¢0u.¢4¢ah
I mow mhzw40~mmm00 BOP th wh3a200
AN.¢.0%0NHm4*Nm0+.N.m00*h0x+2.N.N00*¢4m4+2N.H40*4m44*~m4fimmu4m4Nah

AAH.¢.0*0N4u4+AN.¢V0*ha3m40*mmm+.4.m.0%h0x+.mam004
*30x+ A.H.N40fi¢4¢4+.N.N40*ua3m4+424.440*4m44+AN.H004*400*N0n.m03ah

UL)

 

 

 

129

4D 02~44Hu whw4a200 302

ZO~h4N444~hmzm mIH mwhw4a200 m—Ih

w024h200

N20\..~.30h + 02N0*AH4NOP4HA¢+H.4D

mHN.m4N.OHN .mlmvmm

sz\A.—43ah + 02N0§-vwah0u2_44:

4.4u4 mam 00

0w-44~h~za mm Ems: mm344> 02m th :02

m.mxm.mm4 Amm¢.m.wh_az

mON.OON.OON 4mm40m—

oom.oomam04 4.444m4m04fimlw0.wlmmw.¢~

CON.OOH.OOH .20443mh040mn

mmo.oo~.ooa 42010~4¢m
4mmHuxxw.sz.3zmamzm.mzm.A¢vmaamvx.4N0m..4024u1430 4440
414 whoox .mw:44>zw0_w th hwo 202

m4h*sz + ANVNOP*Q0HAN4NOh

m4h*3zm + p4hflsz + ~N430h*o0namvzah

m4h*N20 + h4h*zzm + :4H%sz + 4N4N0b*o0n.N.NOh

m4p*mzm + h4hfimzm + 34h*zzm + 400N:*~zm + .mvmahfio0n4N4m0h
m4h + h4h*m20 + 34H*sz + 404N:*3zm + AN4¢OPWQ0HAN4¢Oh

P4P + 34h*mzm + ~00ND*N20 + 4N4m0h*00n.mvmah

24h .o4~3*m2mnAN.cOh

o4h mo dob th W444; awhm hm44 th

. m0hk~£0 + 4N4N0hfiou44n~44NOh
m0h*320 + h0hflNxm + AN430h*om44 + .NvNoHn44430h
m0h*mzm+p0hkzzm+30h*sz+2NON0h*om44+amonhugaVNOP
m0h*mzm+h0h*mzm+30hfizzm+Am0ND*N20+.N0mOh*om44+AN.NOPHAHVMOP
m0h + p0h*mzm + 30P*sz + Am0ND*3£m + 2N.¢Ow*om44 + 4N0m0Hu44440h
h0h + 30h*mzm + 4m0ND*Ntm + 4N0m0h*om44 + ANO¢Ohn440m0h
30k + .m0ND*m£m + AmvaPMAHVOOh

amhm 0200mm th 00 302

A.mvNahl.¢.Nah*N00*mmu4NON0h

A.m43ahl2443ah*w00*mmnAN.IOP

AAm0NahlA¢0Nah*N0.*mmnAN.NOP

A.m0mahl440mapfimu0*mmuAN4m0h

AAm0¢QhI444¢aH*N00*mmu.N.40h

m4N
OHN

mom

CON
mwm
OOH
mmo

0U

 

130

 

4444DHAH.ND
440NDHAH.N303
A_.NoVDNAH.dD
.mvaau.m._v0
o.~u~ mmm 00

dwhm meh024 «Om x040 00 zth

ND DHZH 40 pad 024 .xmmh4zID th 24 w044a wk“ 24 N0 PDQ
00hm4azo0 m4 awbmlw th h2~0d mHI» p4

m04 0h 00

4D.m4mzm.mhmo Aom¢.mvmh~mx

m¢mam04.m04 Aonzmh0~0u~

0¢©.m04.m04 22010—4;—

A.m04D.AN440.2444:0ammum4mzm

.04044H4>4 mm wk424hmw 3m2 mIH PZ~OQ mmrh #4

zxm*ox + 4a0043u4a004:

.mvNoh + m*444030h + m*4.4.moh + m*4.44moh + 0x444uox
m*4444¢op + m*..44mo» + m*.xmcooh + m*.aH4ND...uox
¢+4na4

m.4n~ mmm oo
Xm*ox + 40.4:n4004:
.~.~ah + mfixx4vzap + m*ox.nox

.~.~ah + m*.4_4mah + m*444_¢ah + m».HcN:..uox

¢.4u4 mmm oo

.mWAsz+sz+ox44\anxw

m*.3zm+3zm+zzm + m*xmzm*o.¢ + m*.mzm+0x+ox44.nox
mzm+mzm+m+m+muox

omm.omm.o~m 4¢1014a0

oxxxmu2xw

mAAsz+sz + m*.zzw+zzm+zzm + m*ox04nox

mzm*o.¢ + m*.mzm*o.m + m*4¢zm*o.o + m*.mzm+0x+ox...uax
mzm + ox + oxuox

mzm + m + mnox

4~4wo*m.axmuxm

Zimum

6.4"44 mmm oo

mmN

0mm

m¢N
04o

mmm

0mm
mNN

ONN

UUUU

131

 

4¢xlmx.*¢xu00

¢xfiwao4m|4>u¢>u4<

Amxlmxcnmxuu

mxumao4w14>am>u4

~x\.4>lm>.umao4m
4ma04m.szx.¢x.mx.mx.¢>.m>.N>.4>4ha¢u m20H20mm3m

02w

OCH CH 00

w024h200

0m4.C>N.C>N Amw~.m_
.C.HHC.AC.H.00.a4me 2Cm¢am0whmmz
24m.40: ..N.H.0 .24.~Cavmmwum4mzw
Cmuaum CON 00

wink ACN¢um0whmm3

Cummu

hoNaohNaofiN .ANHCImh0~.m~

CmC.ChN.ChN .2010mvmm

4mN¢.m4wh—mx

Aoafium..~.Cm.Dv .2402m.1h24~ .mN¢.m4wh_mz
44m.Cm.C ..NaomCD .44.Cm42420wud4mzm
.0.4H~..~.Cm030.m402m.IN AmN¢am0wh~m3
..m.Cm.C ..N.Cm00 .44.Cm030m0wum4mzw
mink AC¢¢.m.m»~mz

m¢o.o¢o.o¢o ..mHCImHOHCmn

.0mkw4a200 zwwm m4: awhm wink 4 hzmoa mHIh #4
44.4Cufimaom03

Caaum CON C0

wCZHPZCC

4D.m402w.0x AmN¢.mvwh_a3

mlwm.C + Nnox

CON.mCN.mCN .vaxmh040mm

m¢o.moN.moN AXCICHCmm

~4w0 + NnN

CNN
OCN
CCN
NCN

CmC
o¢o

04C

CON

mCN

CON
m¢C

132

ou~
430*30 I mo*~0.\xo*~0nzx

4m0> 20~m2m2~0
2H.mam.N0.30.N0.m0.¢m.mx.Nma4m.mm4DC mthDCaCDm

024

4*14.xm44.+140»<z¢0u

zaahma

.quxmchmOmuam<zm
AAo.Om+p.*~mm.osV\.Imto.4.0044u¢~<zm

x<zImuIa

.oom.m0mh_mz

m44.om4.om4 .x4zranzm.¢_

zmapmx

o.oum442m

AooN.m.mp_mz

044.o04.oo Azmcmu

..ch.o + 10*p4ma4\hoha*1uxm
mmm40.4\.zmum4axwuh<ma

apx4m.~¢4m a mnm¢om¢.m*ab + m\.ah.oo4<*m.w I Hmoom.mmuzm
wam.mmm + p§mmmmmmmmmm.ouap

\moommmmom.m\m 4p<o

\ mooo.o .owo.o \ x4zzm.h0pa «~40
.N**:*.om+h0*Nmm.010aano.4ura
411*p4malpopa.\.p.zw01a*2».h<ma*mmo.ouz

.a44 >ao m4\CNI m4 24 .I.>h404221 024

i owe 20 .p.azmh 20mm .zo_ho<xm 20 2w .hzwpzoo
wmapmHO: tamam44_:ow mp<4304<0 0h mthzommDm zofipozau
.1.p4mh4zm zomhozsu

02w

szhwa
Nx*mx*0*mmm4.014>uh24x
Aoouoo+u*u4\400*<<+u*<Clnu

CCN

CNH

m-
C44

OCH
Co

00000

 

 

133

mm4.m¢4.m¢4 .mNI4044

om4.om4.oo4 4.2x0mm4*0Imo.m I 4x4mm4044

4 + was

x I 2xuzx

.uoxoz*¢ I mocxuux

N0 + x44m0 + or + QICuoI

30 + x«4~0 + m0 + x*.m0 + 40 + 40.0u40

x + 40u0I

m0 + x + xuuo

N0 + x*.30 + x*4N0 + x*.m0 + x400";

zxnx

onfi

zmnhmm

4N0 I N4*344m04 + .30 + N4*2x +.N4*344mm4 + mmwuxam
4~0 I N4 + N4*zx + 34 .mm4 + 4m0 + N4 + 2x0mm4uxmw
44*mmuN4

Nafl4mu34

4m + mmuN4

Na + Hauzx

.m.>u¢m

4~4>umm

440>umm

zxu4x4>

a + 4H4

044 Ch 00

4x0>u2x

m.4ux mm4 oo

4 + 4u4

zmzhwm

. om4.o44.o44 44.44
44.4mm.N4.34.m4 .4m0> ..m4> .44.>.mm200 4440
30 + 4mfiz4u~4

N0 + 41*N4u34

4m + mumm4

zxu4m

044 Oh so

004

m¢4
C44

mm4
Cm4

0N4
C44

004

 

ZXMNN

mo4.mo4.mN4 442x4mm4fimImm.o I 4x4mm4044
x I zxuzx

440\4N0I*1I104\4ux

30 + x*.N0 + uNCICnmo
No+x+x+xn4~oz
4N0.:0.N0.x.xm0unu
zxnx
40*xouhoolmmxuzx

044.0N4.CN4 40.4I4404mm4044
Nn4

.10I44444o+1.u10

.N0 .30 6N0 .x0 I mmXCXmCuumo
.NC .30 .NC .x0 + mmx.xu04n m
Axovhmownx0
zxzhmx

ml"

m44.m44.o44 4x0414
30WMMmmmmmmmm.o I xmx*mmxux0
N0*mmmmmmmmmm.olummx

4u4

o+x*.m+x*44+x44 u.o.m.4.x.xmol
42x.x04 002044>4000
.m0200xw>200 xmhm4m mom woo >0 omHCmmmCC 204h440amwhz4

134

th I>43 .242 024 x4: wIH wazhwm Cz4h<4oamwh24 >0 0wh4z4hmm m4 #4

ow>4>4>44m0 th do mommN m1» wazhmm 0wh440m4 m4 boom M4004: th
.44wm hCZ thCm 444 .sz w4 4 44 .NN .hoom m4004x

th mo 204>4k3a200 wIH Z4 mzo4h4xwh4 10 xmmzbz NIP Ch 44DCm m4 4
.44wm mm4 mbCCm

wxh 444 szz Cn N0 + N*20 + N**N*N0 + m**N 204H4DCm th mm>40m
44 .xxw .NC “30 .N0 .MN.NN.4N4w000C 024400103w

020
Cm4.CC4.CC4 44044
CC4nfi

 

mN4

mC4
CN4

m44

C44

C04
mw4

UUQUUUU

 

135

 

4hfi4I4axw I Co4u>

.pz4od m4Ih #4 24

had m4 204H0234 mm0h4mmaxm» h0a24 th
440>u4443

m.4n4 CN4 00

Ammaor Io .moou .mZHP h4I>.xm.>.C4w.n >h442deC.XCN4h4z¢Om
aah.>ha AmCNam4wh4mz

huah

ahuaah

Cx*4m*4aahlab0 I >hau>ha

C44 Ch 00

Oxnrmfi4aaklah0 + >han>Ha

CC4.C44.mC4 4CXC¢4

sz I 4.m40+4N0:+44004*mmmmmmmmmMoCqu
Zmthm

o0w0mmz m4 ZC4h4hDQZCO 444h4z4 wsz
2m13 0mm: m4 #4 .4440 >xzn0 4 m4 m4Ih
CC.CC.CC 4CoCCm¢+hCm4

\NCNC.C>4 .m444homm4 \4b.4h 4h40
\mCONC4.C\NZm 4h40

\Cmoooo .44¢.N .CoCfiM\4m.Im.>pa.ah.aah 4h40
\CoNh .44C.C .Cm.0*m\> 4h40

\ MN.C4 \ 4 4h40

4m4> .4000 204m20240

.0.h.dhzm mz4h30m00m

02w

Zmnhmm

x0 I mmxumN

x0 + mmx N4N
4x00hmomnx0

m¢4.m¢4.C44 4x0vm4

30 I NN%x0 I mmx*mmxux0
x0*m.CIu¢0x

NN + N0nx0

CN4

mCN

C44

mC4

CC4

CC
CC

m¢4

IIIIIIIIIHIII . ,1

136

 

moooooo4
040N.C44
mwa.C44
NmCm.444
¢CNN.N44
oCCC.m44
4mmw.M44
hom>o¢44
CMNhom44
C4Cwoo44
howooh44
mo4m.o44
Cmo>.CN4
4w>¢oNN4
CmN4o¢N4
owqoooN4
moN>.>N4
4440.0N4
>>mC.Cm4
>¢om.4m4
owhw.Nm4
C¢¢mo4m4
Nm¢¢.om4
4mmo.CM4
4000.0m4
NmNo.C¢4
CCC>.N¢4
mmoNoo¢4
N44>.mm4

C.CCN

>¢C4aoo4
Cmoo.qo4
oomN.C44
Nmom.044
400m.444
>NON.N44
Coco.m44
CNCO.M44
>¢4mo¢44
C4Cm.m44
M4®m.o44
NC>0.044
mowmoo44
4000.0N4
N4¢m.NN4
Comm.¢N4
wmmmomN4
4NomeoN4
040N.>N4
omom.mN4
NNm¢ooN4
Cm¢C.CM4
04m .4M4
om¢m.mm4
mmoc.mM4
o4mNomm4
CCM4.Cm4
Cw4Noh44
CCCC.N>

0.m4

CN40NNC.C
>40>NNC.C
N44CNNC.C
NoN¢NNC.C
©¢MNNNC.C
thCNNCoC
CCCC4NC.C
Chmm4NCoC
NNON4NC.C
NNC04NCoC
C¢CCCNC.C
mommoNooo
C¢moo4ooo
CCC¢C4C.C
CChom4C.C
>¢Cmm4ooo
N40Cm4oco
mmo>>4ooo
®m>¢>4ooo
m4N4>4CoC
4Nm>o4CoC
4>0No4ooo
>¢Cmm4ooo
wthm4C.C
oqu¢4ooo
Cm4>¢4CoC
mo¢¢¢4C.C
ww>4¢4ooo
CCCC¢4C.C

0.4

m4m>>4oC
mmmm>4oC
4hoo>4oC
o>¢4m4oC
CN4mw4oC
¢No¢w4oo
o4qom4oC
>N4Cw4oo
NCC4o4oo
oom¢o4oo
omm>04oC
m>N4CNoC
m4omCNoC
mme4NoC
¢CN>4NoC
oo¢mNNoC
4o¢oNNoC
C¢C¢MNoC
44momNoC
wm¢m¢NoC
mthmNoC
4N040Noo
m¢mm>Noo
¢>NCCNoC
000m0N0C
m0C¢CMoC
oo¢m4MoC
NCCMNMoC
CCCCmmoC

mNC.C

mw4oo
m¢oo
mN¢.C
4.0
mNMoC
mmoo
mNm.C
Moo
thoC
mNoC
mNNoC
Noo
m>4oo
m4oC
mN4oC
C4oC
oooo
wCoC
No.0
Coco
mCoC
4C.C
mooo
NCoC
C4C.C
N4C.C
woooo
40CoC
Coo

fledma

hmmw4m.o mow¢mmoo
©4oomm.o 4mo>m~.o
m¢4mmm.o mooo¢m.o
mmwmmm.o oN>m4N.o
oonmmm.o 4¢o>¢moo
¢m~4mm.o oomomm.o
moommm.o mmm¢m~.o
4oommm.o 4ommmm.o
mmmm¢m.o owomoN.o
4o>o¢m.o m¢m>om.o
mom4mm.o mmmmwm.o
o4momm.o o4w>>~.o
mwmmom.o mommwm.o
ooomom.o oomoo~.o
0N00pm.o m4¢pomoo
m¢m¢mm.o 404momoo
wommmm.o ommmomoo
¢¢mNoN.o N~o44m.o
mwooom.o m4o¢4m.o
omm4om.o 4omw4m.o
No¢oom.o ¢om4~moo
omm44m.o o>m¢mmoo
o>m>4m.o mnoomm.o
4m¢mmm.o mmommm.o
>4ommm.o >m¢ommoo
Nomnmm.o m»>omm.o
omoomm.o 4mmomm.o
omwomm.o oooowm.o
oooomm.o oooomm.o
o o o o o 4 4 4 o o o o o

0.0 4m4 o4I

020

zazhmx

>AA4HI444 + 4pu404:

 

 

137

omm4.NC4
mmomoNC4
ooqmoNC4
momw.NC4
N4oo.mo4
4mmm.mC4
tho.mC4
moCo.mo4
NmCN.¢C4
NM4m.¢C4
ommwe¢o4
o4>4.mo4
ommm.mo4
04ow.m04
>>>Nooo4
ono.oC4
Cwo4oho4
ommmowo4
N>No.mC4
4mNm.mC4
o4mC.OC4

mm¢oo4o4
4mo4omo4
CCCMoNC4
NmNoeNC4
moomoNC4
4044omo4
¢Nmm.mC4
Comoam04
C4qoqmo4
CPMN=¢C4
mh¢ms¢o4
mohma¢o4
meNomC4
>4om.mC4
44moomo4
CC4M.CC4
mmN>.oC4
th4.>C4
NNCC.>C4
Cm>C.004
omhmowo4

mooo¢Nooo
meC¢NC.C
o¢¢m¢Nooo
®C>>¢NC.C
4C4>¢NC.C
Nomo¢NC.C
Nmom¢NC.C
oww¢¢Nooo
>>C¢¢NCoC
omNm¢NC.C
¢CMN¢NC.C
4m¢4¢NCoC
CO¢C¢NC.C
NC¢CMNC.C
Cm¢mmNoco
4mMNMNCoC
>04CMNC.C
4004mmooo
mCNMMNC60
omMNmNC.C
CmoCmNooo

 

0Nmmo4co
NCCmC4oC
CmC¢C4oC
Cm¢¢o4oo
000404.C
MCmm04oC
mo>m04oo
mmNCo4oC
wohoo4oo
M4m>o4oC
omw>04oo
4meo4oC
o¢4oo4oC
hmwmo4oC
OCmC>4oC
m¢M4>4oC
o>4N>4oC
noCm>4uo
04C¢>4oC
>mom>4oo
Nm4o>4oo

om>>04oo
Chcwm4.C
>m0004.C
4¢oCo4oo
mmo4o4oo
mohmo4oo
omwm04oo
memo4oo
mNNCo4.C
N¢m>o4.o
Nowwo4co
C¢NCCN.C
0>C4CN.C
4m4mCN.C
mm>¢CNuo
NC¢CCN.C
4m4mCN.C
>¢COCN.C
omm44Nco
CCCM4N.C
4Com4N.C

 

>44
Cmo>o4oo
moooo4oo
CCmCCNoC
>¢m4CN.C
wmwNCNoo
o>4¢CN.C
mommomoo
NCCPCNoo
oo¢mCN.C
>¢CC4N.C
Coo44NoC
mmmm4NoC
Cwom4NoC
NCCC4N.C
womw4moo
N0>CNNoC
howNNN.C
omomNN.C
©4m>NNuo
CONCNNoC
w4NNmN.C

m>0.C
mCoC
mNooo

m>woo
mmoo
mNmoC

mhhoo
mhoo
mNhoC

m>©.C
mooo
mNooC

mhmoo
mmoo
mNm.C
moo

 

 

   

“'IIIIIIIIIIIIIIIIII

7