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This is to certify that the
thesis entitled

OPTIMIZATION TECHNIQUES FOR GRAIN DHYLR
DEJIGN AND ANALYSIS
presented by

David Michael Farmer

has been accepted towards fulfillment
of the requirements for

 

PhD degreeinAgLinultnnal Engineering

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ABSTRACT

OPTIMIZATION TECHNIQUES FOR GRAIN DRYER DESIGN AND ANALYSIS

By

David Michael Farmer

This study uses techniques of mathematical simulation and Optimr
ization in development of user-oriented algorithms for analysis and
optimal design of selected grain drying systems; digital computer
programs and the necessary documentation for their use are included.

In Part I, a prototype program is developed for use by equipment
manufacturers, extension personnel, and individual fanm operators
(via remote time—share computer terminals) in design and economic
analysis of batch-in-bin type drying systems. A specialized optimiza-
tion technique for minimizing operating cost, subject to constraints
on available equipment and product quality, is presented; an empirical
model, capable of rapid evaluation, for batch drying of corn is adapted
for computational efficiency in the search techniques. Potential use
of the package is illustrated by studies of the sensitivity of operating
costs to economic and ambient conditions, to design parameters, and to
variations in operating and marketing practice.

Selected results for the central-Michigan area are:
1. Fuel cost is a much larger component of total Operating cost than

telectrical cost. Thus, dryer design changes intended to improve thermal

David Michael Farmer

efficiency can substantially reduce operating cost. Alternate fuels
should be carefully considered by Operators, based on heating value
and price.

2. Replacing heat from.fossil fuels by resistance electric heat
is not economical for batch dryers at the current energy price levels.

3. Under adverse ambient drying conditions a greater percentage
rise in airflow costs than heating costs occurs. 7

A. If drying conditions are such that Optimal airflows are low,»
operating costs fall with increasing depth until excessive condensation
occurs: the Operating cost reduction per unit depth, hoWever, decreases
with increasing depth and must be weighed against increased construction
costs.

5. For an equivalent daily grain volume, drying a single batch
on a 20-hour schedule is appreciably cheaper than drying twp batches
on 10-hour schedules.

Part II of the thesis study is concerned with development and
digital computer implementation of two algorithms for Optimal design
of concurrent-flow dryer, counterflow cooler systems, with and withOut
recycle of cooler exhaust air. The Optimization technique of dynamic
programming is employed to insure overall Optimal choice of construction
and Operating parameters for each system; computer programming techniques
designed to minimize time and memory requirements are introduced.
The user is permitted maximum freedom in choice of constraints, compon-
ent models, and performance criteria in adapting the algorithm to his
needs. Examples of the use of both algorithms in the design of corn
drying systems are given. Under a single set of ambient and economic

conditions, Operating expenses for the dryer—cooler system without

David Michael Farmer

air recycle were found to be slightly lower than those for the reCycled'

air system.

mag/g. are

Major Professor 61/ym'72”

Approved

 

Dep mnt Chairman

‘OPTIMIZATTON TECHNIQUES FOR GRAIN DRYER DESIGN AND ANALYSIS

by

David Michael Farmer

A THESIS

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

DOCTOR OF PHILOSOPHY
Department of Agricultural Engineering

(1972)

This work is dedicated, in equal measure, to the two who have made it
possible: to my wife Ginny for unfailing affection and helpfulness; and
to Dr. Fred W. Bakker—Arkema, for inspiration, patience and friendship

freely given.

ii

ACKNOWLEDGMENTS

The author gratefully acknowledges the guidance and encouragement of
Dr. Fred W. Bakker-Arkema who, by his example, has shown the path of
excellence in personal as well as technical pursuits. The friendship
and unselfish assistance of students--both past and present-~in the
Processing Group have been most appreciated; in the preparation of this
work, special thanks are due to Lloyd E. Lerew, the "enigmatuer", and '
to Stephen F. DeBoer, Gonzalo Roe, and Mitchell G. Roth.

Appreciation is expressed to Dr. William G. Bickert, Dr. George
Coulman, Professor Robert L. Maddex, and Dr. Walter M. Urbain, not only
for serving on the guidance committee but also for their help during the
course Of study.

The partial financial support Of The Andersons, Incorporated, Maumee,
Ohio was much appreciated. A note Of thanks is also due to those who
assisted in keypunching, Ken Jones and G. R. Shidle, and to the typist,

Ms. Georgianna Farmer.

iii

II.

TABLE OF CONTENTS

AN ECONOMIC STUDY OF BATCH-IN-BIN DRYER OPERATING COST. . .
A. Introduction. . . . . . . . . . . . . . . . . . . . .
B. Batch-in-Bin Dryer Simulation Models. . . . . . . . . .
C. DevelOpment of the Cost (Objective) Function. . . . . . .
D. Optimization Technique. .
E. Constraints and Parameter Values Needed for Input . .
F. Program.Description and Use of the Algorithm. . . . . . .
G. Sensitivity Studies .

G.l. Standard conditions and results.

G 2. Effect of the heating price. airflow price ratio. . .
G 3. Effect of variation in ambient conditions. . . . . .
G.h. Effect of variation in grain depth . . . . . . . . .
G 5. Effect of variation in required drying time. . . . .
G 6. Effect of variation in marketing Options . . . . . .
H. Observations on the Performance of the Algorithm.

I. Proposal for Dryer Testing.

J. Suggestions for Further Study . . . . . . . . . . . . . .

OPTIMIZATION OF CONCURRENT DRYER-COUNTERCURRENT COOLER

COMBINATIONS USING DYNAMIC PROGRAMMING. . . . . . . . . . . .

A. Introduction.
B. Dynamic Programming . . . . . . . . . . . . .,. . . . .

C. Optimization for Concurrent Dryer-Counterflow Cooler
Configuration Without Air Recycle . . . . . . . .

C. l. Dryer- -cooler description . . . .
C. 2. Development of models for cooler and dryer . . . . .
C. 3. Dynamic programming formulation and computer

programming . . . . . . . . . . . . . . . . . .

iv

Page

15

22
27
27
BI
33
36
no
A2
A3

A6

A9
A9
53

6A

68
72

Page
C.h. Procedure for use of algorithm and an example. . . . 77

D. Optimization for Configuration with Air Recycle . .' . . . 87

D. l. Dryer-cooler description . . . . . . . 87
D. 2. Dynamic programming formulation and cOmputer
programming . . . . . . . . . . . 87

D. 3. Example and discussion Of results. . . . . . . . . . 104
D. A. Comparison of three example programs . . . . . . . .. 108

E. Summary and Conclusions . . . . . . . . . . . . . . . . . 110
F. Suggestions for Further Study . . . . . . . . . . . . . . 111
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
APPENDIX A--Computer Program for Part I. . . . . . . . . . . . . . 120
APPENDIX B--Computer Programs for Part II. . . . ... . . . . . . . 123

APPENDIX C—-Subprograms. . . . . . . . . . . . . . . . . . . . . . 144.

11t[([(‘

Table

LIST OF TABLES

Comparison of experimental data with model
simulations.

Comparison of QUICK and FXBD models with
experimental data. . . . . . . . .

Results of parameter studies.

Comparison of annual dryer costs for two 9000 bu/yr
systems. . . . . . . . . . . . . . . . . . . .

Summary of Optimal conditions for three drying
systems. . . . . . . . . . . . . . . . . .

vi

Page

12

30

37

109

LIST OF FIGURES

Figure Page
1. Variation in Moisture Content with Depth and Time . . . . . <5
2. Possible Isostere-Isocost Configurations . . . . . . . . . . l8
3. Flowchart of Necessary Data Cards . . . . . . . . . . . . . 23
A. 3 Drying Costs and Corresponding Isostere-Isocost SlOpeS

for Standard October Tests . . . . . . . . .'. . . . . 28
5. 3-Dimensional Schematic View Of the Economic Behavior Of

the Model with Increasing Bed Depth . . . . . . . . . . 3h

6. 3—Dimensional Schematic View of the Economic Behavior of
the model with Increased Elapsed Drying Time . . . . .. 39

7. Common Marketing Options for Shelled Corn . . . . . . . . . Al

8. Schematic Representation Of Crossflow Dryer and Dynamic
Programming MOdel . . . . . . . . . . . . . . . . . . . 57

9. Schematic Representation of Combination Dryer—Cooler
Without Air Recycle . . . . . . . . . . . . . . . . . . 65

IO. Schematic Representation of Combination Dryer-COOler
‘With Air Recycle . . . . . . . . . . . . . . . . . . . .88

vii

NOMENCLATURE

a = specific surface area of grain, ft2/ft3

a1,a2,a3,ah,a5 = empirically determined constants, Equation(I.B.lO)
bl’b2 = empirically determined functions

cl,c2- = empirical constants

c8 = specific heat of dry air, Btu/lb-deg F

cp = specific heat of product, Btu/lb~deg F

cw = specific heat of water, Btu/lb-deg F

C8 -= airflow cost, cents per square foot of bed area

Ch = heating cost, cents per square foot of bed area

DF = dimensionless grain flow rate

DM .= dimensionless moisture content

DX = dimensionless depth of bed

e a: base of Napierian logrithms

E = average pounds Of moisture removed per pound of dry air during a

specified elapsed time, Equation(I.B.l)

F = total value Of the Objective function, cents per square foot of
bed area

g. = vector functional of systems equations

Ga = mass flow rate of air per unit area, lb/hr/ft2

h = convective heat transfer coefficient, Btu/hr/ft2

hfg -= unit heat of vaporization of moisture in grain, Btu/lb

H := fuel heat value, Btu/gallon

HP = horsepower

i a: partial process cost incurred during stage n

viii

Wu.

{X

’3 3| 3‘»?

z

:3

optimal total process cost
Optimal process subtotal oost
exponent, dimensionless
exponent, dimensionless
time constant, hr-1
: exponents, dimensionless
moisture content, decimal dry basis
average moisture content, decimal dry basis
equilibrium.moisture content, decimal dry basis
initial moisture content, decimal dry basis
stage of the system
total number of process stages
time, dimensionless
depth, dimensionless
pressure drop per unit area of bed, inches of water
= atmospheric pressure, psia
saturation vapor pressure, psia
price of electricity, cents per kilowatt—hour
price of fuel, cents per gallon

airflow rate, cfm of dry air per square foot of horizontal drying
surface

relative humidity of air, decimal
elapsed drying time, hours
elapsed drying time, minutes

time in hours required to reduce the dimensionless moisture ratio
to 0.5

= ambient dry bulb temperature, deg F

 

T = dry bulb temperature of air at an infinite dryer length after a

G constant wet bulb process to an equilibrium relative humidity for
grain at the initial moisture content, deg F
Tinlet inlet air temperature, deg F
1LT = temperature rise of incoming air due to its passage over the motor
and fan, deg F ’
E = vector of control or decision variables
U = set of all feasible controls
V = air velocity, feet/minute

x. = bed depth or distance in bed from.air inlet, feet
3-: = vector of state variables

= set of all feasible states
dimensionless constant

dimensionless constant

-;‘US >‘
u

fan efficiency, decimal

<S'
u

electric motor efficiency, decimal

.zs
!
u

an. = thermal efficiency, decimal

Q
n

product temperature, deg F
/a; = density or dry air,1b/rt3
P, = dry grain bulk density, lb/ft3

water vapor density, lb/ft3

1°
ll

 

I. AN ECONOMIC STUDY OF BATCH—IN-BIN DRYER OPERATING COST

‘A. Introduction

On-farm.drying is a fact of life for most corn farmers today. The
advent of combine harvest of corn has increased the harvesting, drying
and handling rates to a point where many country elevators no longer
have the capacity to process the high moisture corn being delivered,
thus forcing the grower to dry his own. In addition, on-farm drying and
storage have given the farmer flexibility in selling his crop, enabling
him to wait until the initial glut on the corn market has passed. Also,
he may now avoid dockage charges for high moisture corn by drying his
own. In assuming the responsibility for drying, however, the grower
himself is liable for dryer operating costs and possible product
degradation.

Farmers and elevator Operators have usually taken care to avoid
.readily apparent deterioration due to rodents, insects or microorganisms
<iuring storage. Deterioration of processing quality due to overheating
<iuring drying has not been reflected in market prices as long as the
arverage moisture content fell within an acceptable range. Lately,
fulwever, increasing complaints from.cereal and snack food processors
afS'well as from the wet milling (corn starch) industry indicate that a
Prendmmimay soon be paid for prOperly dried grain.

Bussell(l969) attributes the wide diversity in corn quality to
several sources in the grain harvesting, conditioning and marketing sys—

tenh Improper dryer operation may occur on the farm.or at the elevator

2

because of ignorance of correct Operating procedures or due to overloading
a dryer too small to meet the flow rates in the harvest system. Dryer
manufacturers, who have not evolved a uniform.means of rating dryers,

are held accountable for the latter charge. Farmers are blamed for not
taking time to learn correct dryer operation or to teach it to their
employees.

These criticisms from a manufacturer of drying equipment point out the
need for a means of comparison of dryer capacities and for clearly defined
dryer Operating procedures. The recent deveIOpment of good dryer computer
models (Bakker-Arkema et al.,l97l) help to define Operating limits;
nonetheless, an appreciation and study of dryer economics is needed by
both the manufacturer for intelligent design and by the dryer Operator
for intelligent use.

To date, the studies of Mprey et al.(l969) and Bloome et al.(l970) on
the economic feasibility of layer drying cover the recently published
work specifically concerned with the economics of bin dryer Operation.
These, however, are necessarily based on long-range weather conditions for
a particular area and hence have limited applicability elsewhere. In '
this study, the intent is to develOp an economic Optimization model for
a batch-in-bin dryer which will be of use to both dryer operators and
Inanufacturers and which will be adaptable to the user's need, regardless
(if location. If such a program were inserted in a telephone-linked
ccnnputer system (such as in Michigan State University's Future Plan
Programs; Harsh et al.,l97l) it would be within reach of any potential
laser at minimal cost. A series Of similar programs, one for each dryer
type, would allow rapid comparison of dryer performance and provide a
rational means of rating dryer capacities.

The batch-in-bin dryer typically consists of a cylindrical metal bin

3

up to 36 feet in diameter or larger, which is filled during drying to a
depth Of from one to three feet. A high volume fan equipped with a burner
(usually LP gas) introduces heated air into a plenum chamber beneath . I.
,the grain from.whence it flows through a perforated metal floor and up
through the bed of grain. The rate of drying is adjusted to the harvest-
ing rate so that one or two batches are dried daily. Since the grain
first begins to dry in the lower portion Of the bin where the heated air
enters, the upper portion of the grain is last to dry. In practice,
drying is continued until an acceptable average moisture content is
reached; the grain is then cooled and placed in storage or is shipped.
Auger unloading of the dryer serves to mix the wet and dry grain after
which equilibration takes place in storage.

It should be realized that choice of type and size of dryer is largely
determined by the remaining components in the harvesting-handling-drying
storage system. Thus, an economic Optimization model of dryer Operation
should have the flexibility to be incorporated into large-scale systems
models (Farmer,l97l). Furthermore, Operating cost, which this algorithm.
is designed to evaluate, is only one component of total dryer cost;
.Bloome(l970) also includes cost of storage structure, heater, fan, mater-
ial handling components and their Operating cost, depreciation, interest,
:repairs, taxes and insurance. The present model is set up on a per square
:foot basis so that it may be easily adjusted for evaluation of dryers
liaving different diameters.

Some problems in dryer choice or Operation, for which the model is
applicable, are suggested:

1. Purchase of a new harvesting machine will increase the daily

input to an existing batch—in-bin dryer.

A

a. What are the implications of this change in terms of dryer
efficiency?

b. If the purchase of a new dryer is contemplated, what will be
the expected operating cost?

Is there a cost per bushel improvement in drying two or more

batches daily instead of a single batch?

What effect does the time of harvest have on drying costs, i.e.,

how do costs associated with higher harvest moisture contents

early in the season balance against increased field losses and
lower ambient temperatures expected later?

What effect will a change in the price of fuel or electricity
have upon Operating cost?

What are the Operating cost implications of a change in heat
source (e.g. LP gas vs. natural gas vs. electrical heat)?
How'much can be saved in dryer operating costs by marketing at

moisture contents above that needed for long-term storage?

The algorithm.can also be used to provide insight into the following

questions, which might occur in dryer design or sales and service:

1.

Are automatic controls which adjust dryer operating conditions

in accordance with changes in weather conditions worthwhile in
terms of expected savings?

Is it feasible to tailor dryer design to location, based on
expected weather conditions?

For a given location, what Operating recommendations should be
given to customers for most economical Operation?

Is the investment in high quality motor and fan components worth-

while in terms Of increased dryer efficiency?

 

 

5

5. What are the expected cost advantages of direct fired over
indirect fired dryers?
6. How can conStruction costs (depth vs. diameter) be balanced '
against Operating costs in order to size a line of dryers?
In the succeeding sections, the simulation models conSidered for
batch-in-bin drying are first discussed. Secondly, the cost (Objective)
function is developed, followed by an exposition of the Optimization
technique employed. Next, constraints and sources of information to
facilitate use of the algorithm are delineated along with a program
listing and instructions for use. Finally, several Of the questions
posed above are investigated for conditions prevailing in the central

Michigan area.

B. Batch-in-Bin Dryer Simulation MOdels

Use of the Optimization Option in the economic model requires that
the simulation be performed a number of times in order to approach
those Operating conditions for which cost is minimized. In order that
computing time and cost do not become prohibitive, a rapid process Slmr
'ulation model is needed to locate the vicinity of the optimum.

Two models from.the literature were chosen as candidates for rapid
simulation of the process. The first (Nelson,l960) utilizes a dimen-
:sional analysis approach to predict average moisture contents during
(irying of deep beds of various grains. The author reported a fit of
liithin A-5% on all experimental data. Corn drying trials were not in-
eluded.in the data; however, it was presumed that for a single type of
grain the model could considerably improve its prediction accuracy over

that fbr all grains. The form of the model is:

n c x/Qt
_ 2 m
E “ °1[(Mo'Me)(Tin1et‘Tc)/ Tinlet] [1"9 3 (1°34)
The final average moisture content is computed by:
M = M0 - EIOa/(thm/ag) (1.3.2)

In order to evaluate the effectiveness Of the model, a comparison
with deep bed corn drying experiments was needed. A thOroughsearch
disclosed little usable published deep bed corn data. In order to pro-
vide a balanced set Of information against which the model could be mea-
sured, the experimental data of Kirk(1958), Hamdy et al.(l969), and
Farmer(l969) were chosen, encompassing the inlet air temperature range
Of lOO.—165. deg F at a variety of airflow rates, elapsed drying times,
and ambient relative humidities. The predicted average moisture contents,
using Nelson's constants, were calculated. In addition, a non-linear
parameter estimation routine, GAUSHAUS (Meeter,l965) was used, retaining
the form Of the model, to estimate least-squares best-fit constants
(cl,c2, and n) using the available data. A comparison of the results
is shown in Table 1.

Between the inlet air temperatures of lOO.—165. deg F, the predictions
(of the model using the "GAUSHAUS" constants fell within 2.% of each data
Exaint; however, there were no additional data available with which to make
an independent comparison.

A second model, first proposed by Hukill(l953), was tried in an
attenmm.to improve the accuracy of the predictions. In essence, the
InOdelinits original form used the following relations between dimen-

sionless quantities:

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M-M Nx
N = e = 2 (I B 3)
m - M - Nx. Nt . .
0 e 2 ‘+ 2 - l

where:

N = - *ahra<“o'“e> (I B n

x 6000°Q/°acatH(Tinlet- TG)
and

Nt 5 t/tH (I.B.5)

The general shape of the moisture profile which develops during a
deep bed drying process, given constant inlet conditions and a uniform
initial moisture content (Figure I), can be well approximated (aside
from cases where large amounts of condensation occur) by'a function of

the form:

Nm = pj/(pj +yk - 1.) (1.13.6)

where:

fl,f> O

o iij<gao -;j-—a-cx> - j = o
’ x-voo ’ XFO
. _—_ . oo
0 S k<COO "1MF{) O , kt;::’

For fixei positive values Offl and X , the value of j determines
tdie lepe of the curve; the value of k regulates the shift of the curve
along the distance axis.

In Hukill's model,/3 = 2, IL: 2 and j = Nx’ k = Nt' Because a

c<Dmparison Of data with the prediction of the model showed considerable

 

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"a ouswwm

lNBlNOO BHDISION

lO

discrepancy, a later version of the model (Holtman et al.,l967; Barre et
al.,l970) was tried in which/8 = X: e; for this version j and k were

redefined by:

xpghfglflM -M )

 

J 0 ‘3 (1.3.7)
Q Pa°a(Tin1et‘TG)
k = Kt . (1.3.3)
where K is a time constant
m1 In2
and m1, m2 are each positive exponents less than one.

A good fit to a single drying test was obtained by Barre et al.(l970);
however, they failed to report an explicit functional relationship for
values of K. Since the available experimental data did not agree well
using several trial functional forms for K, a strictly empirical modifi-
cation was developed using the available data. The general functional
form adopted for K was:

£13 at.

hdiere a1, ..., ah are constants to be determined. Due-to the appearance
cut the parameter K in the expressions for both j and k, a multiplicative
Ccnistant as for the expression for k was used to further differentiate

tflleltwo dimensionless quantities; thus, in this study:

k = aSKt (I.B.ll)

Values of the constants a1, a2, ..., a5 are given in Table 1.
In searching for constants which would provide the best fit of the
model to the data, GAUSHAUS failed to work prOperly. An available pattern

Search optimization program, HCLMB (Rosenbrock et al.,l960), was used

11

to determine the constants, again using a least-squares best-fit criter-
ion. A comparison of the available data with the values generated by
this model is shown in Table 1. Since this model showed improvement
over the first considered and was essentially as feet for computer
evaluation, it was chosen to represent the process for the subsequent
sensitivity studies. This model will henceforth be referred to by its
FORTRAN IV subroutine name, QUICK.

Because all available experimental data had been used to evaluate
the empirical constants used in QUICK, none was on hand for an independent
review of the model's authenticity. As a precaution against distortion
of the model by the choice of data used in its construction, a secondary
check was required; this check was also needed to prevent extrapolation
of QUICK beyond the limits of its validity.

Tb provide this check on the optima predicted by QUICK, the Michigan
State University deep bed drying siimuation 3st (Bakker-Arkema et al.,
1971) was employed. This model, which is based on fundamental mass and
energy balances, accounts for phenomena (e.g. condensation) which are_
beyond the scope of QUICK. In addition, it can be considered valid over
the entire range of input parameters normally used in deep bed drying.
JFXBD, which utilizes an iterative method of solution, requires consider-
iatfle time for a single simulation; thus it could not be readily used
in the Optimization process.

Using the experimental data of Table 1 covering inlet air tempera-
tfllres from.lOO - 165 deg F and initial moisture contents from.O.296 -
C).432 dry basis, simulations were made using the QUICK and FXBD models.
From the summary of Table 2, the predicted dry basis final moisture

(Huntents of FXBD fall roughly in the range of 3% higher to %% lower

12
Table 2: Comparison Of QUICK and FXBD Models with Experimental Data*

Final Moisture Contents, decimal, dry basis

Data # ~ Source Data "QUICK" "FXBD"

(Farmer,l969)
#1 0.2280 0.2065 0.2172
#2 0.2625 0.2670 0.2187
#3 0.1150 0.1180 0.1300
#A 0.1472 0.1t39 0.1410

(Hamdy,l969)
#1 0.3335 0.3337 0.31116
#2 0 . 2935 O . 29111. 0 . 308A
#3 0.2530 0.2586 0.27h9
#1 0.2120 0.2265 0.2176
#5 0.1910 0.1983 0.2257
#6 0.1700 0.1739 0.2074
#7 O . 1533 O . 1531 0 . 1916
#8 0.1122 0.1359 0.1779
Sum of squares fit = 0.0855 = 1.1738

*‘Conditions for these tests are shown in Table l.

13

than the data values, while the predictions of QUICK are within this
range.

Optimal drying conditions predicted by QUICK in the subsequent,'
sensitivity studies were used as inputs for FXBD. Using the stated A
tolerances as a standard, the model QUICK was considered valid only for
those trials in which the difference of predicted final average moisture
contents fell within this range; for trials in which the difference was

greater, QUICK was presumed to be an inadequate model and the results

were discarded.

C. Development of the Cost (Objective) Function

In line with the objective or a unitized economic model from which
more complex models might be formed, the cost function was constructed
on a per square foot basis.

In conventional deep-bed drying, two sources of energy are required:
one (typically natural or LP gas) for heating the air, the second (typ-
ically electricity) for powering a fan to force the drying air through
the bed. A formulation of the heating component of the cost function

.for fossil fuels is (Bloome,l970):

60-PFQ(,OaCa +Pw°w)(Tin1et ’ Tam—Amt (1.0.1)

Ch = Hath

Vfllere [5T is the temperature rise of the incoming air due to its passage
OVer the motor and friction with the fan blades.

To derive an expression for (LT, the factors influencing its magni-
tdhie must be known. Bloome(l970) states that this temperature rise is a

f'unction of fan efficiency and Of the static pressure against which the

1A

fan operates. This reference cites the following field data: for
fans providing an airflow rate Of 12.8 cfm/ft2 through a 16-foot depth
of shelled corn, a temperature rise A T of approximately 2 deg F occurs.
A 30% fan-motor efficiency factor is given.

The corresponding pressure drop, 1.585 inches of water, was calcu-
lated from the empirical relation of Thompson (1967):

1.528 .

p = x[-5%.-l (I.C.2)
Assuming that temperature rise of the air is directly proportional to
pressure drOp and inversely proportional to fan motor efficiency, an

expression for’AxT was developed:

_ _2_ 9.3.9. 3
AT — 2.0 ( 1585 ”MT: ) (1.0.3)

If electrical resistance heating is substituted for the conventional
fossil fuel sources, the heating component of the Objective function
becomes:

60'Q(/Oaca + wcw)(Tinlet - Tamb -ATWEt

Ch "" 31.13. (1.0.4)

 

The cost component for airflow arises from the electrical energy
lised by the fan motor. The trend toward higher horsepower electrical
Inotors on farms, as well as the disadvantage of having an idle tractor,
Ilas almost obyiated the use of the power take-off for this application.
TPheoretical horsepower required on a per square foot basis for the

IRotor-fan combination is (Hall,l957):

HP = 33%. - (1.0.5)

,15

Using Equation (I.C.2) to calculate pressure drop, the cost of elece

tricity for running the fan and motor is:

(1.0.6)

 

0.746 (HP)P t
C _ E
a- 7‘f7hn
Total Operating cost per square foot is then given by the sum of

the heating and airflow costs (Equations (1.0.1) or (I.C.A) and (I.C.6)).

The objective fUnction thus has the following form:

F = Ch'+ Ca (1.0.7)
D. Optimization Technique

The following independent variables abstracted from the drying model
and objective function are known to influence the fixed bed dryer eco-
nomic performance: initial and final moisture contents; bed depth;
elapsed drying time; ambient air conditions; fuel and electrical prices;
thermal, fan and electrical-mechanical conversion efficiencies; inlet air
temperature; and airflow rate. Assuming that a dryer has been purchased
and harvesting policy decided upon, only inlet air temperature and air-
.flow rate remain amenable to control. Viewed in this perspective, the
11roblem.reduces to a two-independent-parameter minimization of the cost,
Ifiinction, subject to constraints on inlet air (or grain) temperature,
Eiirflow rate, final moisture content, and time.

(The first Optimization technique used on the problem was the pattern
Search technique HCLMB. This method, for several reasons, was unsatis-

factory for the present problem. For two reasons, convergence to an

<Diptimumw'as very slow: (1) because the optimum was necessarily located

C31 the final moisture content isostere, and (ii) because the line of

 

l6

constant final average moisture (isostere) and the iso-cost lines were

‘ often essentially parallel near the Optimum. In addition, it was dif-
ficult to ascertain the relationship between the isosteres and isocost
lines in the vicinity Of the optimum. Also, the technique did nOt indi-
cate whether convergence was to a local or global Optimum, leavingopen
the possibility of a line of alternate Optimal solutions or a better
solution than that initially found. Because of these difficulties, it
was advantageous to deveIOp a specialized Optimization technique which
exploits the characteristics of the given problem. This algorithm.will
be explained after some preliminary remarks.

One special feature of this problem is the use of the final average
moisture content isostere as a constraint. Unlike the constraints Spec- .
ified on airflow and inlet air temperature, the shape and location of
this isostere are not known beforehand but must be evaluated during the
course of solution.

Consideration of the drying process indicates that the isosteres
when graphed on inlet air temperature-airflow rate coordinates, must
exhibit a shape which is concave upward. That is, at high inlet air
temperatures and low airflow rates additional heat has little effect
on average moisture content; at high airflow rates and low temperatures,
additional air does not appreciably lower the average moisture content.
For the iso-cost lines, analysis of the objective function, Equation(I.C.7),
shows that for high temperatures and low airflow rates, the near-linear
fuel cost term predominates, while at low temperatures with high airflow
rates, the non-linear electrical cost term takes precedence; therefore,
the shape of the iso-cost lines is also concave upward.

Given these strictures, four different relationships between the

17

final average moisture content isostere and isocost lines are possible
(See Figure 2).

For Case 1, the slope of the isocost line remains flatter than that
of the isostere throughout the region. The unique least cost Optimum
occurs on either the maximum temperature or minimum airflow rate bound.
Case II, the inverse situation occurs on either the minimum.temperature
or maximum airflow rate bound.

I In Case III, the isocost lines cross the isostere at two locations
within the region of interest but are flatter than the isostere at the
left or top bound and steeper at the right or bottom.bound.~ Since the
isocost lines are essentially parallel, it is evident that the Optimum
will occur on the interior of the region and, depending on the sippes
of the functions, may exhibit a line, rather than a point, solution.

Case IV is simdlar to Case III except that the isocost lines are
flatter than the isostere on the bottom.or right-hand side and steeper
on the top or left hand side. In this situation_ two local Optima occur.
The minimum of the two (global Optimum) depends on the particular con-
figuration Of the problem.

With the foregoing background the mechanics of the algorithm can
be explained. In the first step the final moisture contents at each
of the four corners of the region are determined. Since all isosteres
exhibit the same shape, within any feasible region two and only two
bounds will be crossed by the final average moisture content isostere
(disregarding the unlikely degenerate case where it intersects a corner
Of the region). Examination of the four corners determines along which
bounds to search for the crossing points. For this purpose the method

I of Dekker(l967) was chosen, due to its property of supralinear convergence

 

 

 

18

   

ISOTERE ’ CASE'I

 

 

    

ISOTERE

CASE II

 

 

 

 

 

 

Figure 2: Possible Isotere-IBOCOBt Configurations

 

19

to the zero of an unknown function. Thirdly, at both crossing points

the slopes of both the isostere and isocost lines are sampled. By

comparison of lepes, CaSes I and II can be immediately recognized and

the Optimum identified. Case IV requires, in addition, a comparison

of the two local optima for identification of the global Optimum. In

the remaining situation, Case III, the Optimum occurs in the interior

of the region; hence, further exploration is necessary. In this phase

Of the algorithm, the equation of the diagonal from upper right to lower

left corner of the region is first determined. Again using the one-

dimensional search, the crossing point of the desired isostere is located

and the SIOpes of both the isostere and isocost line sampled. If the cost

SIOpe at that point is steeper than that of the isostere, the Optimum

must occur to the left and above the point sampled: hence a new'and smaller

rectangular region is defined containing the Optimum. In like manner,

if the cost slope at the point is flatter than that of the isostere, the

Optimum must occur to the right and below the point sampled: thus, a

large portion of the region may be discarded in this step of the search.

vThe preceding method is repeated on successively smaller regions until

the optimum.is approached arbitrarily closely.. Since convergence takes

place from.both sides of the optimum. a line Of solutions may be obtained.
For design purposes. it may be desired to investigate cost per bushel

as a function of depth or time. By imbedding the foregoing method into

a minimization search with respect to either of these, its applicability

can be extended. Alternatively, the method can be used selectively in

order to survey a number of possible design combinations without a

formal search.

20

E. Constraints and Parameter Values Needed for Input

In order to properly use the Optimization procedure, constraints
must be provided on the two independent variables, inlet air temperature
and air flow rate. Because the model is primarily derived for inlet air
temperatures less than 140 deg F, this is suggested as a maximum.upper
bound. Hall(l957) recommends air temperatures of less than 1A0 deg F
for grain to be milled. The lower limit on air temperature is arbitrary;
a lower bound of 70 deg F was chosen. An airflow range of 10-50 cfm/
bushel is considered normal for this type of dryer; the airflow con-
straints may be chosen accordingly. In addition to deterioration in
quality due to overheating Of the bottom layers of grain, molding may
occur in storage if the moisture content of the top layers of the drying
bed is too great, even though the average moisture content after mixing
is satisfactory. Maddex(l97l) suggests a maximum.moisture content of
16% wet basis in the upper layer of the drying bed if shelled corn is to
be stored on a long-term basis; this boundary condition Option can be
inserted into the solution if desired.

In order to specify the cost function, several parameters related
to the dryer components and Operating conditions must be known. 0f the
three efficiencies inherent in the objective function, only fan effici-
encies are normally specified in.manufacturers literature; electric
motor efficiency curves, however, can be developed by simple tests
independent of the dryer configuration. The thermal efficiency deter—
minations for a given dryer are a function of both the burner chosen and
the air distribution system.design. Thus this parameter must be measured
by tests on the dryer itself; for simple calculations these tests can

best be made without drying (simple heat transfer).

21

For general use of the algorithm, the following guidelines are
suggested: 3

Thermal efficiency of heater:

direct fired, fall operation 70%
direct fired, winter operation 50%
(Hall,1957)
indirect fired, fall Operation 50%
(indirect fired, winter Operation 30%
Electrical-mechanical conversion efficiency of motor:
65-85% (Anon.,l958)
Fan efficiency:
A0-70% (Perry et al.,l963)
Heating value: I
Natural gas (methane) 22,190 Btu/lb
Propane 19,9Ah Btu/lb
(Hall,l957)

For the simulation models, as well as for the cost function, it is
necessary to know the expected ambient air temperature and relative hump
idity (or equivalent prOperty). The ASAE Yearbook(l97l) features monthly
weather maps for the continental United States, giving mean wet and dry
bulb temperatures and standard deviations from.the mean. The algorithm
is written such that this information can be substituted directly if
desired. Mean barometeric pressure for a particular location may be

estimated from:

 

 

Altitude, feet Mean barometricpressure,_' . Hg
0 ' ’ 29.921
500 29.38
1000 28.86

5000 24.89

22

The remaining parameters necessary in the economic dryer model
include: electrical price, fuel price, initial and final moisture
contents, bed depth, and drying time. These inputs vary according to

the prevailing economic climate and the purpose of the user.

F. Program.Description and Use Of the Algorithm

The computer implementation of the algorithm may be applied in
several ways, subject to the discretion of the user. Available Options,
which are specified on the data input cards, include: 1) a single simu-
lation by the QUICK model, with or without a cost evaluation, 2) cost Op-
timization with either inlet temperature or airflow rate variable, and
3) cost optimization with both temperature and airflow rate variable.

To further facilitate use of the program, input data values can be
specified in any common units (e.g. either wet or dry basis moisture
contents), with conversion, if necessary, being done by the program. The
flowchart, Figure 3, is written to allow the user to easily supply the
necessary data cards.

The computer program, with accompanying comment cards, is listed in
Appendix.A. The psychrometric subroutines have been excerpted from.Lerew
(1971); the equilibrium moisture content subroutine is from.Bakker-Arkema
et al.(l971).

Several diagnostic messages have been included in the program, in
the event of user difficulty. A brief explanation Of eaCh of these
follows:

(1) Messages:
MOISTURE CONTENT IS NOT IN THE NORMAL RANGE

AIRFIDW RATE INPUT IS NOT IN THE NORMAL RANGE

‘

 

 

 

Card 1:

 

' IS
NLY A SING

DEL SIMULATION

DESIRED -

Yes
IS

THE COST EVALUATIO

WANTED?
No

 

 

Col. l-lO
NOGOST

 

 

Figure 3:

 

 

23

NOTES: 1. All alphanumeric characters (shown in
capital letters) are to be right justi-
fied in their fields; leave no blanks
,between words. ,
2. All numerical inputs are to be in
decimal form.and are to follow the units

 

N0 Col . l-lO

 

 

 

 

 

 

 

 

‘ OPTIMIZE
Yes
Airflow
only
varies
Col. '1-10 Col. 11-20
COST AEFIOW

 

 

 

 

 

 

 

specified.

 

 

only

varies

 

Yes

 

 

 

Col . 11-20

 

 

 

INLETI‘EMP

Col . 11-20
mm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

COI . l-lO
DRYBASIS

 

 

 

 

 

 

001. 21430:

001. 11-20: Initial moisture content
Final moisture content desired, if specified
001. 31-40: Upper bound on.m.c. of upper bed layer, if specified

 

 

 

 

Card 3:

 

 

 

  

 

, Col. 1-10

 

PSI

inches of mercugy

 

 

001. 1-10
INCHES OF HG

 

 

 

 

 

Flowchart of Necessary Data Cards

g,

 

24

7

Col. 11-20
ambient pressure

 

 

 

 

 

 

 

  

Col. l-lO

2
Cflm per ft ""‘ CFMPERSQFT

 

 

   

‘ SPECIFIED
IN

 

  

cfm/bushel

 

Col. 1-10
CFMPERBU ‘
l ,
Col. 11-20: Airflow rate to be used initially for
simulation or Optimization

 

 

 

 

 

 

 

 

 
  

 

001. 21-30: Max. airflow rate bound

PTIMIZATION es__
3' 001. 31-10: Min. airflow rate bound

TO BE DONE

    

 

 

 

 

 

 

 

Card 5 «~

 

 

Col. 1-10: bed depth, feet

Col. 11-20: elapsed drying time, hours

001. 21-30: ambient dry bulb temperature, deg F

001. 31-40: inlet air temperature, deg F, to be
used initially for simulation or
Optimization

 

 

 

 

Col. 41-50: max. air temperature bound

yes"1001. 51-60: min. air temperature bound

 

 

 

 

 

 

 

 

 

 

Figure 3 (cont'd)

 

25

 

 

 

 

 

 

 

  

 

 

  

 

 

 

 

 

 

 

Absolute Wetbulb Temperature__ Col. 1-10
Humidity WETBULB .
Col. 1-10 Relative Humddity
ABSHUM

Col. 1-10

RELHUM

i .
=1 Col. 11—20: humidity input, decimal]

 

 

 

 

l

 

 

Card 7:

 

 

 

 

 

 

 

 

Col. l-lO: fan efficiency

001. 11-20: motor efficiency

001. 21-30: thermal efficiency

001. 31-40: electrical price, cents per KWH

Col. 41-50: fuel price, cents per gallon or cents per
pound

Col. 51-60: fuel heat value, Btu/gallon or
Btu/pound, consistent with fuel
price units

 

 

 

 

 

END OF DATA

Figure 3(cont'd)

 

 

£0

‘WETBULB TEMPERATURE INPUT IS NOT IN NORMAL RANGE

ABSOLUTE HUMIDITY IS NOT IN THE NORMAL RANGE

RELATIVE HUMJDITY INPUT IS INCORRECT

. Explanation:

The reliable working range of the model has been exceeded by input
data. Change the parameter specified.
(2) Messages:

AIRFLOW RATE BOUNDS ARE INCORRECT

INLET AIR TEMP BOUNDS ARE INCORRECT

Explanation:

The limits permissible for optimization, inlet air temperatures
between 70 and 140 deg F and airflows between 10 and 50 cfm/ftz, have
been exceeded.

(3) Message:

THE M0 OF THE TOP LAYER CANNOT REACH THE MAX PERMISSIBLE MC WITHIN THE

STATED BOUNDS

Either the upper limit on airflow rate is too low to permit sufficient
drying of the top layer or the bound on moisture content for the tOp
layer is too stringent.

(A) Message:

IT IS IMPOSSIBLE TO REACH THE DESIRED MC WITHIN THE STATED BOUNDS

Explanation:

Even at the highest values of airflow rate and temperature, the
specified average moisture cannot be attained.

(5) Message:
THE MODEL CANNOT HANDLE THIS CASE
Explanation:

The search for the correct vapor pressure at the temperature Tb was

27

unsuccessful; therefore, the model QUICK cannot be used for the given
input data set.
(6) Message:
THE FINAL MC ISOSTERE CROSSES OTHER THAN 2 BOUNDS - THE METHOD FAIIS
Explanation:
This may be caused by an airflow rate bound which is too low.

Reset the bound and rerun the trial.

G. Sensitivity Studies

Tb illustrate an application of the algorithm, a set of economic
and climatic conditions typical of the Central Michigan area was chosen
and the sensitivity of the Optimal solution to expected variations in
these parameters was explored. The following sections summarize and
interpret the results obtained.
0.1. Standard conditions and results

For the month of October the Central Michigan area experiences,
on the average, a dry bulb temperature of 49.5 deg F and wet bulb temp-
erature of 45 deg F; mean barometric pressure is approximately 1A.3A
psi (ASAE Yearbook,l97l). The prevailing 1971 LP gas and electrical
prices are respectively, l6.5¢/gal and 2¢/KWH, when purchased in amounts
ordinarily consumed by individual farms. Mid-range efficiencies were
chosen for the fan (55%), electrical-mechanical conversion of the motor
(75%), and heating system (70%). An Operating policy of 10 hours (2
'batches/day) at two foot depth was chosen arbitrarily, with a uniform
harvest moisture content of 28.0% wlb. to be reduced to an average of
15.5% w.b. for marketing or storage.

Figure A Shows the set of inlet air temperature-airflow rate

28

.0009 nonouoo pumpcwum Mow nonpaw umOuOnHuopounonH mewoaoanouuoo can mueoo w:«%un "q ousmfim

....o .mm-fikmeZwF 13 Fun;—

 

        

o¢_, on. ON. 0: 00. 00
q _ . _ _ O.
N
8
.5
. low M
mxwhmofl ...sz200 ”$5.55: .32....— M
.0033; .0 ... 8 0...... 8
000.010.. . .003 .
2.0.0- w ram/.7 2.. w W.
000.0: A .0 . . I on 3
N¢n.OI u .m._ an}. on. v c
0000. u .0 ._ + «0.
000.0- A .0._ 0033. e W
000.01... 0 v
8.0.0.. u 0.. .00. l
a
.003 00....

  

 

 

29

conditions for which the desired average final moisture content is
achieved. The corresponding lepe of some isocost lines and their
values at selected intervals are also noted. With this choice of para-
meters, the optimal Operating cost will occur on either the minimum
temperature bound or the maximum.airflow rate bound (Case II), since
the slope of the isocost lines is steeper than that of the isostere
anywhere within the feasible region. The Optimal total cost/bu of
A.08¢ indicates a saving over the highest non-optimal cost within the
feasible region, A.18¢. However, relaxation of the maximum.airflow
rate bound would allow further improvement.

If the additional constraint of a maximum moisture content of
16.0% w.b. is applied, it is impossible for the top layer of the bed to
reach this moisture and still satisfy the remaining conditions on the
problem; within the bounds of the region shown (Figure A), the top
layer reaches, at best, 20.0% w;b. Either more time must be allotted to
the process, the airflow rate bound increased, the depth decreased, or
both in order to satisfy this top layer constraint.
0,2. Effect of the heating pricezairflow price ratio

The value of an isocost line at a given point of intersection with
an isostere will Of course be altered by a change in the price of energy
for heat or airflow, a change in thermal, electric motor or fan effici-
encies, or any combination of these. The slope Of the isocost lines
at the intersection also varies with a change in the heating price:
airflow price ratio (a change in efficiency is effectively a change in
price). If the change in SIOpe is sufficient to change the Optimization
problem from.one case to another (Figure 2), the location of the minimmm,

as well as its value, may be drastically altered. In Table 3, the

 

 

“M
o a

 

 

 

m3. Huetmm(MMuw.wu~mm-MM.MD

 

 

FIT-u mtmfia Mafia: Burma-a mun—M a... ‘3? M iii.
0.» him-om 01-, £2.82. arr. Moe Moe mun hmmme.I~e—e 2 h “than!”
me. n. In elm e/cal In 1. v.0. fl! peg! 2 cl“ W “I 2 Iver,
A f .
W 0.8 2. 10. 0.55 0.75 0.” 2. 16.5 65. 0.155 16.36 ‘D-w k” 6.” 6.“ ”.62 Q. 0.1”
hetwet.) . ‘
I . ' .
:1!» Mt. 0.60 0.65 - . 6.67 6.87 ”.62 ”. 0.190
0 .
:3. menu: 0.70 0.0: 6.65 6.03 95.62 50. 0.190
. 0
Indirect. 0.95 5.60 93.62 50. 0.190
fired 9.50
I. -
Min. Iaectrlc 1.5 6.61 6.01 93.62 90 0.190
Price
7.
Kin. mo; 16. 5.6L 3.50 93.62 50 0.190
Price
0
lax. Net 17. 6.71 6.19 93.62 50 0.190
Price
R
Inn. total 1.5 16. 5.69 3.63 93.62 50 0.190
Price
1
Electric 2.0 - 15.27 0.92 93.62 50 .0.190
Meeting .
J
°°°d°ch 53- 4.6. 5.99 3.60 92.92 so 0.190
Weather .
I
Poor Oct. 69.5 69. 6.05 6.20160. 26.37 0190
Weather
L ’ ' .
Inn "on 36 33. 7.71 6.02160. 25.23 0.120
Heather .
I
Gee bevel
Preeeure 1670 6.69 6.18 93.73 50. 0.190
I
Variation 1. 5.03 6.29 03.99 50 0.175
in depth
0
Variation 1.5 5.77 6.01 03.76 50. 0.107
menu: '
P
Variation 2.5 7.31 3.66 90.71 50. 0.209
indent:
Q
In Coat.
Added 1. 5.72 7.15 1.60. 21.69 J 177
0
Ian Coat 2. 7.05 6.61 160. 25 01 0.199
Added
3
Variation 6. 6.39 6.1” 160. 60.79 0203
in n-
7
Variation 0. 6.69 0061.06.79 50 0.200
in tile
U
Variation 12. 6.69 6.05 05.75 $0 0.170
in till
7
Variation 16. 6.17 3.06 76.0 50 0.195
in 11..
ll
Variation 2). 10-50 5.” 3.69 70. 66.99 0196
1:! till
1
Alternative
mun; 0.20 0.20 3.6. 2.25 73.12 50 0.237
Option
1
Alternetlve
maul“ 0.25 0.20 10-50 2.29 1.63 70 37.73 0220
Option .
z
Caee II! D. 0.60 0.60 0.0) 2.5 LO. 6.12 2.57 92.06 20.55 0190

31

sensitivity of the optimum to changes in several parameters is shown.
The first set of results stems from changes in the heating price:
airflow price ratio (Trials A through I):

1. For a Case I type isocost-isostere configuration the goal of‘
Ieconomy of Operation (associated with high inlet temperatures and low
airflow rates) is Opposed to the goal of high grain quality (provided by
low inlet temperatures and high airflow rates); an overall economic
model must simultaneously consider these contradictory objectives. ‘In
a Case II situation, the two goals are in harmony. No general statement
can be made about configurations III and IV.

2. Fuel cost is by far the largest contributor to total Operating '
cost in these trials; electrical cost plays only a minor role (a maximum
of 7%). In terms of dryer design, this implies that little is to be
gained by improvement of fan—motor efficiency. Emphasis should instead
be placed on improving thermal efficiency. Similarly, any possible
changes in the electrical price will not substantially affect the total
operating cost of gas-fired dryers; however, consideration of the price
structures and heat values for competing fuels may lead to a sizeable
Operating cost reduction.

3. Replacement of fossil fuel burning heaters by electrical resist-
ance heaters is economically unsound under the prevailing electrical:
fuel price ratio.

6. Indirect fired dryers Operate at a substantially higher total
<XDst per bushel than do direct fired dryers, due to the large contribution
017.fuel cost to total cost and the inefficiency of the heat exchanger.

.ngLL_Effect of variation in ambient conditions

Using values of wet and dry bulb temperature one standard deviation

32

from their October mean values and holding all other parameters at their
standard values, the variation in the Optimal solution due solely to
expected weather fluctuations was investigated (Table 3 - Trials J and K).
The average ambient conditions for November were likewise used, with
the remaining parameters held constant (Trial L). Finally the effect
of atmospheric pressure on the Optimal solution was explored (Trial M).
The effect of changes in ambient conditions is to shift the locatiOn
and orientation of the average final moisture content isostere relative
to the isocost lines. From consideration of Equations (1.0.1) and (1.0.6),
increased humidity and decreased ambient air temperature lower the
positions of the isocost lines on the inlet air temperature - airflow
rate graph; this shift is uniform, due to the nearly constant specific
volume of air at any ambient dry bulb temperature encountered during the
drying season.
A comparison of the results of Trial A in Table 3, in which an
ambient absolute humidity of 0.005 lb water vapor/lb dry air is used,
with that of Trial K, using 0.0075 lb water vapor/lb dry air, shows an
increase of 0.2¢/bushel at the higher humidity, using the same 49.5 deg F
ambient dry bulb temperature. The slope of the latter isostere is steeper
than that of the former at any comparable point. Clearly the comparative-
ly adverse drying conditions of Trial K also shift the isostere upward.
The analogous situation, in which absolute humidity is held constant
and ambient dry bulb temperature varied, is seen in Trials A and J.
Maintaining an absolute humidity of 0.005 lb water/lb air and raising
the dry bulb temperature from 69.5 deg F to 53.0 deg F lowers the Optimal
drying costs by 0.3¢/bushel, decreases the slope of the isostere, and

shifts it downward.

33

Since the slepe of the isostere becomes steeper due to adverse
weather conditions, while the lepe of the isocost lines remains the
same, the percentage contribution of the airflow component to total
cost increases. A season-long study of drying cOst fluctuation, however,
would account for the decline in the initial moisture content due to
field drying, which would partially offset the increase in drying costs
due to poor weather in the latter months.

An increase in atmospheric pressure to 16.70 psi, as shown by Trial
M, results in increased drying costs over the Central Michigan average
of 16.34 psi (Trial A). However, the magnitude of the increase, 0.l¢/bu,
is relatively small compared to that due to expected monthly variation
in wet and dry bulb temperatures.

0.4.‘ Effect of variation in grain depth

The depth of grain in a deep bed dryer is a major consideration
in design since increased batch volume can be achieved only by increased
structural costs, either horizontal or vertical. Furthermore, harvesting
machinery capacity and dryer Operating schedule will influence the depth
of grain to be dried in a given system. In order to analyze the effect
of increasing depth on dryer operating performance, Trials A, N, 0,
and P (Table 3) utilized depth as a parameter, with all other conditions
standard.

To prOperly perceive the problem, it is useful to visualize it in
three dimensions (Figure 5). The three—dimensional representation of
isocost and final average moisture isostere surfaces depicts the mode
of intersection inferred from the simulations. Any cross-section of
the inlet air temperature-airflow rate plane will exhibit one of the

four cases shown in Figure 2.

 

I-lT'DI'NO

34

AVERAGE FINAL M.C.
ISOSTERE SURFACE

ISOCOST

mum “‘1‘“

 

 

 

 

 

 

 

 

 

 

 

 

, «mm ..n‘
4 3931916 .
llé=iiahuh__~ V I

 

 

 

 

 

 

Figure 5: 3-Dimensional Schematic View of the Economic Behavior of

the Model with Increasing Bed Depth

35

The simulation results show a pattern of decreasing cost per bushel
with increased depth, but with a decreasing rate of change, 1.8. the
incremental gain in dryer efficiency becomes smaller as depth increases.
In these trials, the model was found to be inadequate at depths greater
than 2% feet, presumably because excessive condensation in the upper
bed occurred at the low airflow rates specified by the Optimal solution.
At the 2% foot depth, the rate of improvement in per bushel cost was very
small; since condensation is more pronounced at greater depths, per bushel
Operating cost can be expected to begin increasing as depth is increased
much beyond 21 foot level.

In order to use the algorithm for Optimal design it is necessary
to balance construction costs against performance for a dryer of speci-
fied volume. The following example, illustrating two hypothetical dryers
of equal volume but different geometries, demonstrates the relative I
magnitude of capital and operating costs for each as well as the net
cost differential between the two.

Assume a corn harvesting system with a 9000 bushel per year through-
put and a specified 10 hour per day drying schedule. The depth of Dryer
I is arbitrarily chosen to be one foot and its radius 15 feet. Dryer II
is filled to'a depth of two feet and has a corresponding radius of 10.6
feet, (Trials Q and R, respectively, in Table 3). Assuming that burner
capital costs will be approximately equal for the two solutions, the
total cost components which vary between the two dryers are: fixed cost
of fan and motor, Operating cost, horizontal structural costs, and
vertical structural costs.

Using the fixed cost coefficients for layer dryers given by Morey
et al.(1969), and the operating cost evaluation of the QUICK model, a

comparison of the two dryers was made. Due to the similarity in

36

construction of layer and batch-in-bin dryers, adoption of the layer dry-
er fixed cost coefficients for this example is a reasonable approximation.

Because specification of dryer dimensions automatically fixes the
construction cost components, these were not imbedded in the Operating
cost function. Fan and motor cost, however, is a function of Operating ’
conditions and hence was incorporated in the operating cost optimization.
- For this example, the standard October Operating conditions were taken as
average for the entire harvesting season. A comparison of component and
total costs is shown in Table A.

It is evident in the example that horizontal construction costs and
Operating costs are the major contributors to total annual dryer cost.

As shown previously, the per bushel drying costs are decreased by
increasing the bed depth (at least within a range of depths). The de—
creased horizontal surface area gives an economic advantage to the
deeper dryer. However, as depth increases, the moisture content of the
top layer becomes too high for safe long—term storage.

In an actual fixed-volume design problem, depth could be expressed
as a parameter and the algorithm restructured to search for minimum
total annual cost, with time-variant constraints and climatic and eco-
nomic conditions specified by the designer. In this manner, dryer
design could be tailored to specifications of the locale for which it
‘was intended.

GLS. Effect of variation in required drying time

One means of increasing system capacity is to dry two batches daily,
rather than one. It is also conceivable that a three batch schedule
could be employed, although the final moisture gradient for a bed of
the same depth would not favor long-term storage; materials handling

considerations also favor a fewer number of batches per day.

37

Table 6: Comparison of Annual Dryer Costs for Two 9000 bu/yr Systems

Dryer I: Depth = 1 foot; radius = 15 feet; standard October drying
conditions.

Construction costs, horizontal = 66¢/ft2/yr x 707 ft2 = $325.
Construction costs, vertical = lO¢/ft2/yr x 90 ft2 = $ 9.
Fan-motor cost = $lO/HP/yr x 1.311? = $ 13.
Optimal Operating cost = 7.1683¢/bu x 9000 bu/yr = $643.

(Trial 0) Total = $990.

Dryer II: Depth = 2 feet; radius = 10.6 feet; standard October drying
conditions.

Construction costs, horizontal = A6¢/ft2/yr x.352 ft2 = $162.
Construction costs, vertical = lO¢/ft2/yr x.130 ft2 = $ 13.
Fan-motor cost = $lo/HP/yr x 2 HP = $ 20.
Optimal Operating cost = 6.6077¢/bu x 9000 bu/yr = ‘§321.

(Trial R) Total = $592..

38‘

To examine the effect on operating cost of the drying schedule
alone, Trials P and w of Table 3 were compared. Using standard October
conditions. drying a two foot bed for twenty hours incurred a cost of
5.90¢/ft2. An equivalent daily volume can be dried in two one-foot
deepbatches each using a 10 hour drying time; the operating cost for
this process is 2(5.03) = lO.O6¢/ft2. Thus a saving of 1..l6¢/ft2
or 2.60¢/bu is realized by the twenty hour schedule. If this figure
were the average over the entire drying season for a 10,000 bu/yr
operation, the yearly savings would be $260. A 5-year cost amortization
for the dryer would then allow $l300 breakeven cost to be spent for the
additional height needed to dry by the twenty hour single batch schedule.

For a single drying depth it is also of interest to examine the
change in Operating cost with increased required drying times. The
Trials A (10 hours), and S (6 hours) of Table 3 correspond to doubling
and tripling the system capacity, respectively, over the capacity for
the 20 hour process Trial w. Three intermediate required drying times
(Trials T, U, V) were also simulated for additional cost information.

Again, a 3-dimensional schematic view (Figure 6) is useful in inter-
preting the simulation results. Because operating cost increases linearly
with time (for fixed inlet air temperature, airflow rate, depth, ambient
conditions, and price structure), the isocost surfaces change with time
as shown. According to the model, drying costs vary from a local mini-
mum of 6.39¢/ft2 in a 6 hour test, to a maximum of 6.53¢/ft2 in a 10
hour test, and again to a local minimum.of 5.9O¢/ft2 for a 20 hour test.
For this example, the alternation implies that at least two lines of
intersection exist between some of the isocost surfaces and the constant

final average moisture content surface. It is also possible, with a

39

FINAL AVERAGE M. C.

 

 

 

 

 

Isosnzaa SURFACE
ISOCOST
a - SURFACE A“
\
no“
“A ‘,tcgai:1031NK3
“W: k cos'r ,
T .
' V‘\
\
M \\ _
\
E \\
\x
\
\'\
r60" \ \al
V \
3‘" \ll
1‘
\ K

 

 

Figure 6: 3-Dimensional Schematic View of the Economic Behavior of the
Model with Increased Elapsed Drying Time

AO

flatter slope of the isocost surfaces relative to that Of the isostere'
surface for the minimum total cost to occur at the minimum.time bound;
for a steeper slope, the minimum total cost will pass to the upper time
bound.

Due to the small change in Operating cost with time, the duration
of drying for a single batch is probably of minor importance. Conse-
quently the dryer Operating schedule can be principally determined by
the considerations of increased system throughput with a multiple-batch
schedule or increased grain quality possible with a single batch per
day.

0.6. Effect of variation in marketing options

A suggested use of the algorithm is to assist in investigation of
the profitability of various marketing alternatives Open to the grower.
Several of the common choices are shown in Figure 7 (Scott,l969). To
incorporate this model into a overall economic picture, it would be
necessary to also include aeration, storage, and handling costs, a
penalty (dockage) function for failure to reach the specified storage
moisture, plus seasonal variation in rate of harvest, harvest moisture,
and market prices. Nonetheless, insight into the relative profitability
of marketing options can be gained by experimentation with the present
algorithm.

Trials A, X and I (Table 3) illustrate three of the options suggested
by Scott, using the standard set of parameters. For comparison, drying
to 15.5 and 20.5% w.b. (Scott, 1969), a net loss in Operating cost of
7.l7¢/bu is sustained on each bushel of corn sold at 20.0% w.b. The
third option, Trial Y, allows the corn to field dry to 25.0% wub. before

harvest, after which it is artificially dried to 20.0% w.b. for marketing.

1 Sold at harvest directly from.the combine

2 Sold at harvest
Dried to 20.0% w.b.

‘_\\\3 Stored, high

aeration, sold
March 15

Harvest with combine
beginning at 28.0% or
25.0% w.b. moisture

6 Sold at harvest
Dried to 15.5% w.b.

‘5 Stored low aeration,
sold July 15

Figure 7: Common Marketing Options for Shelled Corn

A2

Although this alternative cuts the Operating cost to 1.A3¢/bu the large
dockage penalty remains and the net difference per bushel is 6.35¢ over
the cost of the first Option.

The foregoing example demonstrates a simple marketing analysis'
which can be easily made using the algorithm. Because the optimum can
be located by the algorithm.very quickly, the model can also be combined
with weather and crop data in large—scale models to explore seasonal

harvesting and marketing strategies.

H. Observations Of the Performance of the Algorithm

Using the criterion for acceptable model performance (Section 1.8.),
admissible results were obtained with all values of input parameters
shown in Table/3? However, in Trials w and Y it was necessary to raise
the lower airflow bound to 10 cfm/bu. This procedure did not affect
the final result but was necessary in order for the algorithm.to begin
the solution properly; such a failure is indicated by a program diag-
nostic. In separate tests, a model deficiency was also detected at low
inlet air temperatures and airflow rates of 70 cfm/bu. The weakness of
the model in this region will ordinarily b/aavoided by use of the sug-
gested upper airflow rate bound of 50 cfm/bu.

The Optimization procedure, using the data presented, encountered
Case I, II, and IV type solutions (Figure 2) during the Trials A-Y. As
mentioned previously, the location of the optimum depends on the relative
SIOpes of the final moisture content isostere and the isocost lines.
If their point Of tangency lies outside the feasible region, as in
these trials, the optimal solution will occur on a bound; with this

condition, a small change in the constraints, final moisture content

A3

isostere, or isocost lines can shift the optimum.from.one boundary of
the feasible region to another. Case III, that of an interior Optimum,
is shown by Trial Z, using a similar data set. It is therefore necessary
to retain the generality of the algorithm, since any one of the four
cases may occur in a particular solution.

An indication of the speed of the algorithm is given by its per-
formance in the search of Trial Z, Table 3. For this search, requiring
10 reductions of the initial feasible region, execution time was 3.7

seconds using a CDC 6500 computer.

I. Proposal for Dryer Testing

As shown in prior sections, the model and optimization teChniques
can be used to investigate the adaptability of the dryer to individual
Operations or to study the effects of one or more variables on dryer
performance. These applications, however, require access to a computer
and manipulation of input parameters; at the present stage of farming
SOphistication, this may be too difficult or inconvenient for many grain
growers. Despite these drawbacks, this package and similar programs for
other dryers can still be useful tools for extension workers by enabling
them to quickly generate data which can be transformed into useful
publications.

In this light, there are two unfulfilled needs to which this type
of model may be applied: (i) a mapping of the expected relative per-
formance of each dryer type with respect to climatic variation, and
(ii) development of meaningful uniform rating procedures for performance
comparison of competing dryers.

For both applications, at least two indices of performance are

66

important for comparison:, capacity and efficiency. Because of differ-
ences in weather conditions, energy prices, and Operating parameters,
it has heretofore been difficult to compare the results of tests of
similar dryers; in the following discussion a standardization of test
conditions is proposed which, through computer simulation and limited
testing, will permit comparative studies to be made. I

A means of localizing dryer tests with respect to climate is sug-
gested by the air conditioning design procedure of the American SOciety
of Heating, Refrigeration, and Air Conditioning Engineers. .For this
system, long-term nationwide weather records for the season of interest
have been analyzed to determine the wet and dry bulb temperature below'
which 95% of the total hours of the season may be expected to fall;
these temperatures and their corresponding locatiOns are listed in the
ASHRAE Guide and Data Book. Adaptation of this procedure for dryer
analysis would require calculation of the wet and dry bulb temperatures
abgyg which a percentage of the total hours of the season can be expected
to fall. Mbreover, the length and occurrence of the harvesting season
could be varied geographically. Analysis of weather bureau data by com—
puter would reduce the amount of labor required to produce these tables.
Atmospheric pressure, which also affects the drying process, is mainly
a function of altitude. Thus, location—specific values could be included
in the table with the expected temperature data.

In order to fairly compare dryer performances, a wide initial to
final moisture content span should be used. A standard initial moisture
content of 28% wet basis (near the maximum.harvest moisture) and a mean
final moisture content of 16.5% wet basis (required for long-term.storage)

are suggested. In addition, for product quality consideration the

65

permissible final moisture content range could be specified as well
as an upper temperature bound.

For comparison of manufactured dryers, fan, motor and thermal
efficiencies can be determined by simple tests; these could be performed
_by an independent organization, similar to the Nebraska Tractor Tests.
Average efficiency figures can be assumed for testing the variation in
dryer efficiency with location.

'With these conventions, comparison of batch-in-bin dryer performance
(assuming model adequacy and uniformity of dryer air distribution)
could be done as follows:

. 1. For each location (two or three per state would probably be
sufficient) relatively severe weather conditions would be determined
by the procedure described above.

2. The standard moisture content range and published efficiencies
would be used in the simulation.

3. Bed depth, airflow rate, and inlet air temperature would be
specified by the manufacturer.

6. Simulation would be performed on a per square foot basis for
determination of the time required to dry to the specified final average
moisture content.

5. Output information, for comparison with similar dryers, would
include: bed depth, time required, and air and heat energy requirements.
Using these data and prevailing energy prices, total dryer capacities and
Operating costs could be easily calculated.

For a study of the variation in efficiency of batch-in-bin dryers
with location, the following modifications could be made:

1. Average efficiency values could be used.

66

2. A four-dimensional search (with appropriate bounds) in bed
depth, time, airflow rate, and inlet air temperature could be perfbrmed,
with the objective to be the minimization of the energy inputzdryer
capacity ratio.‘ For this search, a cost ratio for high grade (electrical)
to low grade (heat) energy must be assumed. Again, a oheésquare-foot

'bed cross—section would be assumed.

J. Suggestions for Further Study

Suggestions for further investigation in the field of deep fixed ~
bed drying economics center principally on improvement of the algorithm
and applications:

1. Because of the small data sample from which the model was
constructed, its accuracy and range of applicability are not firmly
established. Thus, there is a need for additional well instrumented
drying data over the entire range of model parameters. In addition to
serving as a check on the present form of the model QUICK, these data
can be used to improve the values of constants used in the model.

Other models which are developed should be considered as replace-
Inents for QUICK, provided that their computational times and accuracies
are better than or comparable to that of QUICK. For example, an extension
of the dimensionless parameter model of Bakker-Arkema et al.(l971)
accounts for condensation in its prediction of final average moisture
content and requires less time for evaluation than QUICK.

2. The user should be able to improve the speed of the Optimiza-
‘tion.routine, OPTWHIZ, without significant loss of accuracy by relaxing
the termination requirements for the one—dimensional searches and by

reducing the interval to be searched.

67

3. If optimization with respect to depth or time is desired for
purposes of design, an extension of the OPTWHIZ technique to three
dimensions might be made if the geometry of the isocost and iSOstere
surfaces were known for all possible cases. A

6. For purposes of control, it would be valuable to know precisely
under what conditions each of the cases of Figures 2, 6 and 7 occurs.
For example, if the optimal conditions are consistently at the upper
temperature bound for constant values of depth and time, only the air-
flow rate need be adjusted to maintain optimality.

5. Equation I.C.3, which gives the contribution of the fan and
motor to inlet air temperature rise, is an approximation based on avail-
able information. Since at low inlet air temperatures and high airflow
rates this factor becomes important, a study should be made to improve
or confirm.this expression.

6. Similar operating cost models should be develooed for other
common grain dryers. (If the QUICK model is shown to be valid for temr
oeratures higher than 160 deg F, the algorithm is directly applicable to
the column batch drying system.) Incorporation of these into their respec-
tive systems models would facilitate comparison of competing systems for
a particular enterprise, or enable the grower to determine the effect
of a proposed change in a given system.

A complementary study of capital costs for each type of dryer is
needed to extend the capabilities of the operating cost models. These
costs, however, unlike operating costs, vary somewhat from.manufacturer
to manufacturer and tend to rise faster with economic inflation than do
the utility prices. It would therefore be necessary to update the fixed

cost data frequently to maintain current capital cost models.

68

7. The model might be useful in the deveIOpment of a set of
Operating guidelines (e.g. nomographs) for a particular locale to be
used by operators to minimize drying cost while maintaining grain' .
(quality. '

8. This and simdlar models could aid in development of a rational
dryer performance rating scheme, as suggested in Section I. A testing
program generally accepted by manufacturers, elevators, growers, and
universities could be set up to determine equipment efficiency factors,
a concept similar to that of the Nebraska Tractor Tests. With the
availability of telephone-linked computer systems, evaluation of coma
peting dryers for a particular locale could thus easily and impartially

be made.

II. OPTIMIZATION OF CONCURRENT DRYER-COUNTERCURRENT
COOLER COMBINATIONS USING DYNAMIC PROGRAMMING

A. Introduction

Due to the large grain volumes handled daily by commercial elevators,
a high degree of automation is possible; this permits reduction in labor
costs but requires generally higher capital investments. Thus, in a-
highly competitive market the costs required to process the incoming
. grain for storage or shipment can spell the difference between success
and failure; small differences in Operating cost per bushel or dryer
capacity rather than capital cost can be the deciding factor in choosing
between two competitive designs.

The continuous flow convection-type dryers used in elevator Oper-
ations can be classified into three categories based on the relative
direction of air and product flow: (i) cross—flow, in which the air moves
perpendicular to the flow of the grain; (ii) concurrent flow, where the
air and product travel in the same direction; and (iii) counterflow, in
which the air flow is opposite to that of the grain. In addition to
these basic classes, commercial dryers are built which use combinations
of these in sequential sections.

Thompson(1967) has commented on the performance characteristics of
the three basic dryer types. Cross—flow type dryers, when built in a
single stage, overdry the grain on the side where the inlet air enters
and underdry on the air exhaust side. Counterflow dryers, in which the

hottest air meets the driest grain, have the highest moisture removal

69

 

all. III Vull'lllllllll'

 

41.:
r .3 >
' b

L.

50

rate per foot of depth of the three types but in doing so subject the
grain to considerable stress. Concurrent flow dryers typically remove
the majority of the moisture within a relatively short depth and then '
provide a tempering period through the rest of the bed. A

Improper drying of the product can cause two major types of damage:
reduced milling quality—~caused primarily by overexposure to high
temperatures--and stress cracking, with consequent kernel breakage,
caused by rapid moisture removal. Thus, choice of dryer design can .
greatly influence product quality as well as system capacity, and the
two objectives are rarely compatible. Some combination of existing
dryer types in a single machine will ultimately be the compromise required
to meet the rising raw product standards of the cereal products manu-
facturers.

A design currently of interest employs a concurrent flow dryer
coupled with a counterflow cooler for the grain (Anderson,l97l; Kline
et al.,l97l). The concurrent dryer, which exposes the grain to the least
stress, is used to remove the majority of the moisture. The counterflow
design, used as a cooler, allows the grain to gradually cool, thereby
slowly reducing the temperature and moisture gradients within the kernel;
this cooler also accomplishes a slight amount Of drying. Some sacrifice
in capacity, for a given sized machine, may be expected from.the use of
the concurrent dryer instead of the more efficient counterflow design.
It is claimed, however, that higher dryer temperatures with consequent
higher moisture removal efficiencies can be achieved with a machine of
this design combination. In addition, there exists the possibility of
recycling to the dryer all or a portion of the air exhausted from the

cooler, utilizing the heat energy which has been added to the air during

DJ.

the cooling process. Despite the promise of high product quality with
this machine configuration, Thompson(1967) found that "counterflow cooling
immediately following (concurrent) drying was not an adequate substitute

‘for delayed cooling in reducing the susceptibility of the corn to break

or form stress cracks." A I

Clearly, the Optimal design problem is not a straightforward one.
Variation in economic and ambient air conditions, product quality con-

' siderations, temperature, airflow, and depth parameters must all be
considered in making design recommendations.' Due to the extreme vari-
ability Of weather and product moisture over the short harvest season,
as well as the high labor costs and large number of variables involved
in the construction of prototype machines, another design technique was
required--the use of simulation and optimiZation.

Earlier researchers in the field of optimal design of grain drying
equipment have been hampered by the lack of adequate models to describe
the drying process. Although Hukill(l956) pointed out the inaccuracy of
his model of the batch drying process, Ahn et al.(l966) and Schroeder
et al.(l965) adapted it for modeling the crossflow dryer. Fortunately,
there now exist several more reliable models from which the researcher
can choose; these are summarized in Bakker—Arkema et al.(l97l).

‘Within the published drying literature, only two attempts at optimr
ization of multistage dryers have been reported. Both Ahn et al.(l966)
and Schroeder et al.(l965) used the dynamic programming technique to
determine Optimal airflow allocation in multistage crossflow dryers.
Because different performance criteria were used, the two studies were
not comparable.

In related, single-stage Optimization work on grain dryer design,

52

Thompson(1967) employed gradient and one—dimensional search techniques
for determination of depth, grain flow and airflow rates in all three
basic continuous dryer types. Thygeson et al.(l970) used the differential
algorithm technique of Wilde and Beightler(l967) to predict optimal
bed depth and gas flow rate in a dryer during the constant rate drying
period. The deficiencies of both techniques for the multistage problem
are discussed in the following section.

In designing an Optimization procedure, one of the initial steps
must be the formulation of a perfOrmance criterion or objective function
for the system.to be investigated. The criteria used in previous studies
Of grain drying optimization fit three separate classifications: min-
imization of cost (capital and/or Operating), maximization of throughput,
and maximization of quality. Usually these are stated in some combination;
the most comprehensive objective function (Thompson,l967) had the following

form:

drying_sp§ed )6

E = cl ( 6%/hr + 02.f(drylng temperature)dt

+ c3(heat energy supplied) + ch(fan energy supplied)

+ c5(size of dryer needed) I (II.A.1)

In this thesis study the goal is to develop user-oriented techniques
for optimization of two concurrent flow dryer—counterflow cooler combin-
ations and to demonstrate their use. Choice of simulation models, constraints,
and objective function will be left, insofar as possible, to the user.
Adequate instructions for use of the associated computer programs will

be given and possible modifications discussed.

53

B. Dynamic Programming

Dynamic programming (d.p.) is an optimization technique developed,
especially for problems characterized by multistage decision processes.
These problems arise, e.g., in a sequential manufacturing process when
decisions made during each production step affect the prOperties and
accumulated cost of the product at each subsequent step in the process.
Other problems of this type originate in such diverse fields as economics,
physics, and transportation scheduling.

Unlike linear programming, no standard a1gorithm.exists which can
be applied to the problem.at hand; instead, dynamic programming is es-
sentially a philosophy which allows the user to structure problems of
considerable complexity into a form amenable to solutiOn. That is, d.p.
is used for determination of all process inputs, outputs, and controls
necessary to achieve the objective in the most desirable way (least
cost, maximum profit, etc.) for a given process.

To be structured into a form for optimization by means of d.p.,

a problem.must have the following properties (Hadley,l966):-

(i) it must be possible to visualize the problem as a sequential
decision problem, where each step in the process requires one
or more decisions.

(ii) the system.must have a measure of performance which is
expressible in the same units at each stage of the process.

(iii) the parameters measuring the performance of the system and
its response to controls which are applied must remain the
same at each stage in the process.

Some alternative approaches to the optimization of multistage deci-

sion processes are discussed by Rosenbrock and Storey(l966), including

56

hill-climbing on parameters, the gradient method in function space,

and Pontryagin's method. 0f the available methods only Pontryagin's
method and d.p. will always yield the "best" overall solution; the other
techniques may converge to a "good", but suboptimal solution. Further
difficulties with the. other algorithms arise in the awkward manner in
which constraints are handled and in finding an initial feasible solution
from which to begin computations.

Pontryagin's method has not found wide acceptance in process control
problems due to difficulties in formulation of the solution and excessive
numerical computations necessary for solution. Although d.p. also suffers
from these two disadvantages, it yields a solution which is more useful
in the present context. A discussion of the "pros and cons" of d.p.,
will follow a discussion of the d.p. formulation.

In order to explain the dynamic programming principle, some defini-
tions and an example are useful; the following notation and description
is due to Larson(l968):

Let I denote the state vector, the components of which describe the
state of the system at each stage in the process.

Let 5 denote the control or decision vector, whose components are
those variables which may be manipulated to influence the values of the
state variables.

The stage of the system will be denoted by the scalar index n, which
varies monotonically through the N process stages.

The system equations describe how the state variables at stage n

 

are transformed into a new vector of state variables at stage n+1 by

application of a control vector 5 at stage n. The general relation is:

x(n+1) = §(;(n),u(n),n) _ (11.3.1)

55

The performance criterion (objective function) evaluates the per-
formance of the system when a given control sequence u(n), ned.,..,N is
applied. If the criterion is a cost function, the sequence of controls
which minimizes it are sought; a reward function is associated with a
maximizing sequence of controls. For the problems considered herein, a
cost function criterion is used, hence all further reference will be to
minimization of cost. The corresponding objective function takes on the

general form:

N
I = Zn i<n>,fi<n).n 1 I ' (11.3.2)

UFO
where I is the total cost for the process, which is to be minimized, and
i is the partial cost incurred at stage n.

The purpose of constraints on the problem is to limit the values
of the state and control variables at a given stage. TheSe controls
may be expressed as: xeX(n) and ECUU-gn).

The optimization problem can then be summarized as follows:
Given: 1) a system described by a set of system equations

§(n+l) = E(§(n).5(n).n)
2) the constraints
x€X(n)
u€U()-c,n)
3) an initial state 32(0)

Find: the control sequence u(n), n;0,...,N which minimizes:

N . .

I = E i[ 3E(n),u'(n),n 1 (11.3.3)
n=0

An illustration of a problem having this structure is provided by

Db

an exhaustive review of a crossflow grain dryer design problem.solved
by Ahn et al.(l966) using d.p. In this type of dryer, as shown in
Figure 8, a long column of grain, which moves steadily by gravity flow
through the dryer, is exposed to an airflow stream passing perpendicular
to the grain. The hot entering air, which picks up moisture during its
exposure to the grain, is then exhausted. Wet grain entering the tOp of
the column dries to an acceptable average moisture content before its
release at the bottom of the column. Prior to this study, the usual
practice in crossflow dryer design had been to use an approximately evenly.
spaced airflow pattern along the length of the column. These authors
wished to investigate the possibility of improving dryer performance by
partitioning the airflow into three equally spaced sections along the
column with each section receiving a different portion of the total air -
flow. Inlet and outlet moisture contents, entering air temperature and
humidity, grain mass flow rate, and column width were taken to be fixed.
In terms of dynamic programming, the crossflow drying process can
be drawn schematically as shown in Figure 8, where the three equally
spaced sections of the dryer are taken to be the three stages of the
process. In this case the state vector has a single component, the
moisture content of the grain, which completely describes the variable
of interest in the system. A constraint on the grain moisture content
(m.c.) is given by the final m.c.; no outlet m.c. above this point is
acceptable. Likewise, only the initial m.c. specified is used as an
input to the process. The type of control available to the process is
the same at each stage, namely the amount of air allotted to the grain
passing through the section. The control variables are subject to the

constraint that the total airflow be less than or equal to a value fixed

 

 

 

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by the available fans.

The system equations are provided by a crossflow dryer adaptation
of a batch dryer model (Hukill,l953; also Section 1.8. of this study)
which gives the average m.c. as a function of dryer inputs and grain
properties. The same system.equations thus may be used for all three
stages. Finally, the objective function to be minimized is given by the
ratio of the mass flow rate of drying air to the mass moisture removal
rate, or simply the sum of the airflow rates Of the individual sections,
since moisture removal rate is fixed by the specified inlet and outlet
m.c. and grain flow rate.

Tb implement the solution to this problem, it is necessary to intro-
duce the Principle of Optimality (Bellman,1965) and the basic iterative
relation associated with this problem. The Principle of Optimality
can be stated as follows: "an Optimal policy (i.e. sequence of decisions)
has the property that whats r the initial state and initial decisionsre,‘
the remaining decisions must constitute an Optimal policy with respect
to the state resulting from the first decision":i)From.this observation-
which can be rigorously proved mathematically- a technique for solution
of multistage decision problems was developed. Smdth et al.(l970) have
interpreted the Principle of Optimality by stating "every component in a
serial structure influences every downstream component; only the last
component is independent. The last component can be suboptimdzed inde-
pendently for each possible state of the input it receives. Once this
is done, the last two components are grouped together and subOptimized
independently for each possible state of input. This process continues
until the entire structure is included in the Optimization."

If the solution process begins at the final stage of the process and

59

proceeds iteratively backward to the initial stage, the method is called
"backward d.p."; using the earlier notation, the recurrence relation‘

‘can then be stated mathematically as:

I'(x,n) = ucU {i(x,u,n) + I'[ g(x,u,n),n+l ]} (II.B.6)

where I' denotes an optimal sum of the partial costs incurred in stages

up to and including the current stage n; i identifies the cost of a trial
solution during the current stage n. Thus the Optimal solution including
the present stage is found by minimizing the sum of the costs incurred
during the present stage and the Optimal cost associated with the resulting
state in the stage which computationally preceded it.

Implementation of this procedure on a digital computer requires that
the Optimal costs as well as the associated Optimal controls be stored at
discrete node points in the state space. These results are required for.
interpolation among nodes adjacent to the state point in stage n+1 resulting
from.discrete applied controls in stage n. I

Returning to the example of the crossflow dryer, the solution begins
at the third stage, which brings the grain to the desired final moisture.

The state space at the interface between the second and third stages is,

__._—---.—..rw mwd‘w‘“

partitioned into discrete, evenly spaced nodes representing moisture con-
tents in the range from which the final moisture content can feasibly be
reached during the third stage. The possible values of airflow to be
alloted to the third stage are also discretized; this range is bounded
above by the constraint on total airflow and below by zero, in which case
no drying would occur in the third stage.

Starting at each of the nodes designated at the interface of the

second and third stages, all possible discrete airflows are tried in

OU

turn. Of those which result in the desired final moisture content at
the dryer outlet, the Optimum.control is that which achieves this gOal
at the minimum cost (i.e. uses the least air to accomplish it). The
minimum cost and associated optimal control are then stored in the come
puter memory at a location designated by the node at the second-third
stage interface from which the solution proceeded. For those nodes fromI
which it is impossible to reach the desired final moisture content using
available airflows, an absurd value indicating their infeasibility is
stored in memory. .

The solution now proceeds to the second stage of the dryer. A new
grid of possible moisture contents is set up at the interface of the
first and second stages. As_before, each of the possible airflow controls
is applied at each of the m.c. nodal points of the grid and the resulting
m.c. (Obtained from.the model) at the interface of the second and third
stages is noted. In general, the resulting m-c. will not fall on a node
and a value of the cost and associated airflow will have to be obtained .
by interpolation between the two adjoining nodes at the interface of the
second and third stages. By summing the cost at the interpolated point
and the cost of the applied control of the current stage, a total cost
for the process through the current stage is obtained. Repeating this
procedure for each Of the possible controls and comparing all sums, the
optimal total cost for each m,c. node at the first-second stage inter-
face can be found. These values are stored in the computer memory along
with their associated node point. In order that the maximum.airflow
constraint is not violated, it is also necessary to store the total
accumulated airflow from.previously computed stages.

At the first or beginning stage of the dryer, the initial m.c. is

61

specified. Thus there is no need to set up a grid of moisture contents
from which to work. Starting at the initial moisture content, all feasi-
ble controls are again applied. For each control which results hn a feas-
ible state, the cost of the control is assessed and added to an interpolated
value from.the first-second stage interface. When all resulting sums are
compared, the minimum sum is the Optimal total process cost and the assoc-
iated airflow control is the Optimal control for the firststage. To 3
determine the Optimal controls for the remaining stages, the first control
is again applied to the model and interpolation is performed among the
nodes at the second-third stage interface to recover the optimal airflow
control for the second stage. Optimal airflow for the third stage is

then simply obtained by subtraction of the two Optimal controls from.the
accumulated total. The first, second, and third dryer stages were found
to have Optimal airflow rates of 6300, 5700, and 5600 lb/hr, respectively.

In addition to serving as an introduction to grain dryer design by

dynamic programming, (c.f. Schroeder et al.,l965) the preceding example
illustrates several of the advantages and disadvantages of the method:

(i) the algorithm for each process must be separately designed.
Although the formulation in the dryer example was straight-
forward, many more complex.problems require a combination of
"experience, intuition, and luck" (Smith et al.,l97l). Fame-
iliarity with the system to be optimized allows the designer
to choose relatively narrow ranges for the state variables--
grids which are fine enough to preserve accuracy but coarse
enough to speed computation--and to use all known constraints
to limit the number of computations necessary.

(ii) a large number of evaluations of the systems equations is

necessary even for a well designed problem. Therefore, it

(iii)

(iV)

(V)

62

is necessary that the models are capable of rapid solution by
the computer; in particular, this rules out long iterative-
type solutions. A

it is necessary to store a large number of results generated

at each stage in order to be able to interpolate in the state
space of previous stages and to recover the Optimal policy at
the end of the problem. This has been the primary limitation
in the uSe of the d.p. technique, since present computers are
limited in the amount of fast access memory available. As the
number of state variables increases, the storage location
requirement increases as the product of the number of levels

of all state variables. Most authors state a limit of three

as the absolute limit on the number of state variables that can
be handled by present computers.

in the solution of a dynamic programming problem, not just one,
but a family of solutions is generated. Thus at the end of the
example problem, Optimal solutions are available for a range

of moisture contents in either a one or two stage dryer as well
as the single solution for the three stage dryer. Mbreover,
because the grids of optimal costs and controls for the second
and third stages are unaffected by the inlet conditions spec- ,
ified in the first stage, they can be used to evaluate Optimal
cost and policy for any initial moisture content within a wide‘
range and to serve as a base for the evaluation of dryers having.
four or more stages.

once the problem.has been solved with a relatively coarse grid,

additional refinement is possible by narrowing the ranges of

63

states and controls to be considered and by employing a finer
grid mesh. '

Several variations on the basic d.p. technique have been explored
in the literature. Ameng these is "forward d.p." in which it is desired
to find the optimal policy for a problem.which starts at a known initial
state but for which a variety of desirable final states is possible
(Larson,l968). Optimization of stochastic processes has receiVed atten-
tion (Bellman,1965). Manufacturing processes having loops and branches
have been treated (Mitten et al.,l963) as well as countercurrent processes
(Dranoff et al.,l96l).

The develOpment of dynamic programming is still in a state of flux.
Several recent developments in diverse fields have shOwn promise in exp
tending the size and variety of problems which can be handled by this
technique. For example, special methods which minimize size requirements
for problems having continuous rather than discrete stages have been
developed (Larson,1968;'WOng,l967). A new type of magnetic memory using
orthoferrite crystals promises an inexpensive, fast-access, high-capacity“
memory for the current computers. Finally, a new generation of computers
(Illiac IV of the University of Illinois and Burroughs Corporation) is
being deveIOped expressly to handle the type of computations found in
dynamic programming. Although dynamic programming may become more useful
as a design tool, its use will continue to be limited by the skill in

formulation of the user.

66

0. Optimization of Concurrent Dryer—Counterflow Cooler

Configuration Without Air Recycle
C.l. Dryer-cooler description'

The concurrent flow dryer-counterflow cooler configuration to be
analyzed in this section is shown schematically in Figure 9. In this
design, inlet air to the dryer is first drawn over the motor Or engine
used to power the fan, thus serving to slightly increase the air temr
perature-as well as to cool the power source. The air then passes into
a combustion chamber where it is heated by combustion of a fossil fuel,

typically LP or natural gas. Next the hot air is forced through the bed
in the same direction as the grain, which moves downward by gravity flow.
As the air traverses the bed it cools by evaporation and accumulates
moisture until it is saturated or is exhausted from.the system. The hot
grain passes from.the dryer into the counterflow cooler where it is pro-
gressively cooled by an opposing airstream.being drawn by suction through
the moving bed. By means of this gradual cooling process, steep internal
temperature gradients in the grain are avoided and stress cracking is
minimized. The air used to cool the grain may perform.a certain amOunt
of drying, increasing in humidity during the process. At the exit of the
cooler, the cooling air temperature approaches that of the incoming
grain; thus the cooling air could be a potential warm air source for the
dryer. In this design configuration, however, it is exhausted from the
system by the suction fan.

To Optimize the design of this dryer by use of mathematical tech-
niques, it is first necessary to define the objective of the optimiza-
tion and its mode of measurement and secondly to identify the state and

control variables, their constraints, the stages of the system, and

0)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

GRAIN l “R lee—_—
....T ...... a £13,)
HOT .
, A”, AIR
INLET
’ . .
CONCURRENT FLOW
DRYER‘
DRY, WARM *EXHAUST
GRAIN AIR ‘
’EXHAUST
i {.0 P AIR.
- COUNTERFLOW
COOLER
Y .
DRY, COOL COOL AIR
GRAIN INLET

Figure 9: Schematic Representation of Combination Dryer-Cooler Without

Air Recycle

66

the system.equations necessary to model the components.

For the concurrent-countercurrent dryer system, it was decided to
minimize the Operating cost for a given grain throughput rather than to
maximize throughput under cost constraints. Thus the optimum.conditions
for airflow, depth, et cetera were sought. Capital costs were not included
in the analysis since, on a large volume Operation, small differences I I
in construction cost are overshadowed by Operating cost. Also, unlike
Equation (II.A.l), no penalty terms were included in the objective function,
although a constraint on maximum dryer air temperature was imposed.
Anderson(l97l) stated that at present no premium is paid by the cereal
industry for quality dried grain. Hence, the objective function used
reflects the actual processing cost; additional quality terms can be.
leasily added to the objective function should the marketing situation :.
Change. 6 .

Operating costs arise from.three major sources in the system, In.
the dryer, energy is required to heat the incoming air. The efficiency
of this process depends on burner and dryer design, the fuel used, and
on prevailing ambient conditions. The second energy requirement of the'
dryer is for driving the fan; its magnitude depends principally upon
the depth of the bed and the airflow rate required, as well as On the
efficiency Of the fan and driver designs. As noted before, a portion
of the "wasted" energy input-dissipated as frictional heat-is recovered
by using it to preheat the drying air. The operating cost of the cooler
fan, which exhausts the air by suction, is a function of the same vari-
ables. However, its inefficiency does not permit a savings on the heating
component of dryer cOst unless the inlet air to the dryer is routed

over the cooler fan and motor. In the configuration Chosen in the"

 

I‘ll-[lull "J

67

formulation, utilization of the ”wasted" energy of the dryer fan, but

not that of the cooler fan, is assumed. For the continuous flow Oper-
ation a time base of one hour was Chosen. By specifying the fixed grain-
flow'in bushels per hour, and measuring Operating costs in cents per

hour the common performance ratio of cents per bushel could be calculated.
The cost functions, which are the same as in Part I of the Thesis, are
given in Part I, Section C. A second common measure of performance,A

Btu per pound of water evaporated, can be calculated after the Optimization
is complete.

Since the product (grain) is the only flow component which passes
through both the dryer and cooler, the process divides naturally into
two major stages. The additional components, fans and heater, can be.
lumped with whichever major stage each is associated because they do
not operate directly on the product.

In many respects this problem is an analog of the cross-flow dryer
design example presented. Like that case, the air is not reused from
stage to stage; hence its prOperties need not be designated as state var—
iables. Again, because the primary function of the process is drying,
product moisture content is the natural choice as one state variable.
Unlike the example, however, the temperature of the product in this
problem is also required for assessment of the performance of the two
stages. This requires the addition of product temperature as a second '
state variable.

Controls on the dryer include its depth, the temperature of the
entering air, and the airflow rate. Cooler controls are provided by depth
and airflow rate.

It is possible to impose many constraints, both artificial and

natural, on the problem to limit the number of possible solutions under

68
consideration. In this context, experience with the system components
is ilealuable in choosing maximum airflow and temperature constraints,
placing upper and lower bounds on bed depth, and establishing feasible
ranges and increment sizes for the two state variables at both stages._
The major preliminary task remaining before the final d.p. formulation
is the choice or development of suitable systems equations, or models,
which adequately describe the performance of the two stages of the
process under all potential conditions of Operation. In the following
section, the development of these models is considered.
C.2. DevelOoment of models for cooler and dryer

For the two process stages of the d.p. formulation two separate
models are required, one for the dryer, the other for the cooler. From
each model, the resultant values of the state variables (product moisture
and temperature) are needed as a function of all possible controls over
the anticipated range of the model. In addition, the formulation also
required prediction of the air temperature and humidity in order to
satisfy physical constraints on saturation of the air.

The presently available models for these system.components are of
an iterative nature (Bakker-Arkema et al.,l97l) requiring time-consuming
computations for a single simulation. These fundamental models, however,
were used in generating data from which strictly empirical models could
be constructed. Two of these fundamental models, CONCUR and COUNT, were
chosen to provide information for the empirical dryer and cooler models,
respectively. Fortran IV coding and documentation for these models is
found in Bakker-Arkema et al.(l972). Theoretical considerations are
discussed in Bakker-Arkeme et al.(l97l).

Since product flow rate was not intended as a parameter in this

69

study, most of the models were constructed based on a single flow rate,
9 bu/hr/ftz. Since several different techniques were used in the
construction of the models, these will be discussed under separate
headings.

Concurrent dryer outlet moisture content model.-- DeBoer(l97l) con—
structed an empirical model for the moisture content of cOrn at the

dryer exit as a function of inlet air and grain parameters, air flow

 

 

 

rate, and dryer length. A plot of the dimensionless depth DX =;ga:

M - M a a
versus the dimensionless moisture ratio DM 5 Me“ Me for various
values of dimensionless grain flow rate DF 5 28¢ indicated that
all resulting curves can be approximated by thepfunctional form

A DX = bl(DM - l)b2, where'bl and b2 are each functions of DF. GAUSHAUS

(Meeter,l965). a non-linear least squares curve fitting subroutine, was
used to approximate the constants. The equation was then re-written
with DM as the independent variable. In a similar fashion multiplicative

modifying functions for b were successively incorporated into the model

2
accounting for the parameters: airflow rate, inlet air temperature,
initial moisture content and inlet grain temperature. Air humidity was
found not to be a significant factor in the model except as implicit I
in the equilibrium.moisture content.
Within the range:

inlet air temperature = 200.-500. deg F

initial m.c. = 0.20-0.60 d.b.

airflow rate = 60.-260. cfm/ft2

grain flow rate== 3.-9. bu/hr/ft2

inlet product temperature = 50.-lOO. deg F

the model was found to have an average error of approximately 0.5% d.b.

70

with a maximum error Of 1.6% d.b.

Concurrent dryer outlet air and grain temperatures and air humidity
models.-—- Using the midpoint value of each parameter as a base, the
grain temperature profile through the bed was generated, usingthe
CONCUR model. This profile was approximated, using GAUSHAUS, by a
function of the form: 9M= cl(Depth)c2; this will hereinafter be
designated the "standard profile".

. Next, the control variable airflow rate was varied from its mid-
point value over the intended range of the model being constructed,
holding all other parameters constant at thgig midpoint values. From
the profiles generated by this procedure, the deviation from.the
standard profile caused by the change in airflow rate was plotted as a
function of the value of airflow rate, for a particular depth in the
bed. The plot at each depth was found to be essentially linear. Thus
the product temperature at a particular bed depth could be predicted
by adding or subtracting the deviation from.the value of the standard
profile at that depth. In order to bring the parameter bed depth into
the model, it was then necessary to express the slopes of the lines
as a function of depth. GAUSHAUS was again used to provide the empirical
constants to the functional form: Slope 4= cl(Depth)cz.

The same procedure was used for the parameters inlet product teme
perature and inlet air temperature, giving three correction factors for
the standard profile, each based on the value of the parameter and
depth within the bed.

Variations in the inlet moisture content and absolute humidity
parameters also resulted in near-linear plots of deviation from.the

standard profile versus parameter value. These corrections, however,

71

were nearly independent of depth and hence required no depth factors in
their expressions. '

Finally, it was determined that the sum of the correction factors
for all parameters, when added to the value of product temperature taken
from the standard profile, provided a good approximation to the results
of the CONCUR model, using the same input data. Within the same ranges
as the preceding empirical moisture content model, this model produced
results generally within 5.0 deg F of the data of the CONCUR model, at
bed depths of one foot or greater.

The same technique was used by Roth(l97l) to produce empirical
models for the outlet air temperature and humidity of the concurrent
dryer. The results are: outlet humidity, maximum.deviation .0168 lb
water/lb dry air, maximum.percent error AO.%, mean percent error 18.%;
outlet air temperature, maximum deviation 52. deg F, maximum.percent
error 35.%, mean percent error 12.%. The four empirical models for the
concurrent dryer-outlet air temperature, humidity, grain temperature, and

moisture content-are programmed in Appendix B, SUBROUTINE DRXSIM.

Model of outlet air and product conditions for the countercurrent cooler.--
Roth(l97l) found that essentially the same procedure could be used
in formulating the empirical models for the countercurrent cooler, except
that a single additive depth correction factor could be used. That is,
instead of adding the depth dependent correction factors to the stored
profiles of the dependent variables, a linear depth correction factor plus
the sum of the other correction factors and a constant yielded a good
approximation to the data from.COUNT.
Significant nonlinearity of the correction factor for airflow rate
in the humidity and air temperature models required approximation of

the factor by a polynomial in the former case and by an exponential in

72

the latter. Only inlet air temperature and airflow rate were found to
significantly affect the outlet air temperature; inlet moisture content
and bed depth were the sole major determinants of outlet product moisture
content.
In testing the model over the ranges:
inlet air temperature, 35-60 deg F
inlet air humidity, .'ooz—.005 lb water/lb dry air
inlet product temperature, 110-160 deg F
inlet moisture content, 0.19—0.2A decimal, dry basis
airflow rate, 80.-l75. cfm
depth, 0.3-2.0 ft; ( 4‘ 1.0 ft for outlet product temperature)
Results were: outlet humidity, maximum deviation .00113 lb water/
lb dry air, mean percent error 12%; outlet air temperature, maximum.devi-
ation 3.9 deg F, mean percent error 1.0%; outlet product temperature,
maximum deviation 6.1 deg F, mean percent error 4.5%; outlet moisture
content, maximum deviation .0135 decimal dry basis, mean percent error
2.0%. The four empirical models for the cooler outlet air temperature,
humidity, grain temperature, and moisture content are found in Appendix
B, SUBROUTINE COOLSIM.
0.3;. Dynamic programming formulation and computer programming
With the preliminary information necessary for the solution now
available, the mechanics of the dynamic programming formulation will be
considered. Because the application of the dynamic programming princi-
ple to a practical problem is inseparable from the problems of computer
program coding and the characteristics of the machine being used, the
following discussion will detail the rationale for the program flow
pattern; the associated programs COOLONE, DRXONE, and RECONE with their

attendant subprograms are listed in Appendix B.

73

There are two variations of the dynamic programming principle which
can be applied to a process of this nature. In the first type, forward '
dynamic programming, the solution starts at a single state specified at
the beginning of the process and proceeds iteratively to the end of the
process, where the optimal solutions for a number of final states can be
explored without reworking the entire problem, For the other formulation,
backward dynamic programming, the solution begins at a single terminal
state and works iteratively backward to the beginning of the process; in
this case, only the last step must be repeated in order to investigate
alternate initial states.

For the present problem, the backward dynamic programming formulation
was chosen as the most useful. Corn received at an elevator in a single
day can vary widely in moisture content; the final moisture content
necessary to prevent deterioration in storage provides a relatively
invariant terminal state toward which the system.is directed. For a
single solution, no variation in ambient or economic conditions is per-
mitted; a parameter study of these factors using the present formulation
would require multiple trials.

In coding a d.p. computer program, there are two basic machine
limitations which must be considered: computer fast access memory and
available computer time. Of these, the first is the more serious problem;
a long program with many checks on constraints may be more efficient
computationally but may require too much memory to accomodate the large
arrays which are needed for in erpolation. 0n the other hand, there
are economic penalties for computationally inefficient programs, which
may be the overriding consideration. In practice, a balance must be
struck. I

The overall strategy employed for the solution was to write a

74

separate program for each stage in the process rather than to solve

the entire problem in a single program. This allowed the portion of

the memory allotted to program instruction storage to be kept to a
minimum" In addition, visual scanning of the intermediate output at
each stage permitted significant reduction of the size of the inter-
polating arrays for the subsequent stage, thus requiring only a moderate
amount of memory for the solution of large scale problems.

The general computational scheme for the cooler stage will now be
described; coding for the program is given in Appendix B, PROGRAM
COOLONE.

COOLONE first reads the user supplied information, including
ambient and economic conditions (fixed during the solution), the ranges
and increment sizes for the state variables, product moisture content
and temperature at the dryer-cooler interface, the range and increment
size to be used for the application of the control variables airflow and
bed depth, and the maximum.permissible values of outlet product moisture
content and temperature. In addition, a minimum permissible value for
moisture content is read; the use of this value will be discussed later.

The second phase of the program is a preliminary screening of the
grid sizes of both the state and control variables, in order to limit the
number of model evaluations necessary. This screening is accomplished
by using the constraints on maximum allowable outlet product temperature
and moisture content.

In the screening procedure, all parameters, except the one being.
checked, are held constant at that end of their respective ranges which
maximizes the probability of a feasible trial with respect to a given

constraint. The range of the remaining parameter is then searched until

75
the point of failure (if any) is found. Several such simple checks
using different constraints can reduce the problem size considerably
at the start. The initial screening can be done several times before
the iterations are begun in order to insure a near-Optimal starting.
grid size and a minimum.number of controls to be applied.

The third phase of the program is designed to evaluate the cooler
simulation model for the remaining node points of the inlet product
temperature-moisture content grid, using the remaining controls. The
order of these computations is arranged such that the desired outlet
conditions are most apt to be achieved at the beginning of this computa-
tional sequence; termination of the program occurs when no further
feasible solutions starting from.the remaining nodes are possible.- Thus,
the order of the D0 l00ps, outside to inside, is: (i) inlet product
moisture (starting at its lowest value), (ii) inlet product temperature
(starting at its lowest value), (iii) airflow rate (starting at its
upper bound), and finally (iv) depth (starting at its lower bound).

The constraints on outlet product properties and outlet air prop-
erties can be classified into two groups if all other parameters except
depth are temporarily held constant (as they are while the inner D0
100p is incrementing). A failure to meet the constraints on maximum
allowable outlet product temperature and moisture content indicates
that the bed is too shallow; excess bed length is indicated by the
conditions of air saturation, air-product equilibrium, water absorption
_by the product, or by excessive drying (indicated by the user-supplied
moisture constraint read in at the beginning of the program). These
checks are inserted within the inner DO 100p. A "bed too shallow" check

indicates that feasible solutions are still possible within the depth

76
100p but that no cost evaluation for the present iteration is necessary.
These procedures reduce considerably the number of process and economic
model evaluations required.

For each node of the inlet product temperature-moisture content
grid, the least cost feasible solution, if any, is determined by coma
parison with the costs incurred in feasible trials using alternate
controls; the indices of the grid, the minimal cost, and the corresponding
depth and airflow are then written sequentially on a low-speed storage
device. No arrays are used in this program and storage requirements~
are thus kept to a minimum.

The second program in the computer package for this concurrent
dryer-countercurrent cooler configuration, DRYONE, was designed to:

(i) evaluate the airflow and heating costs of the concurrent dryer using
all feasible controls, (ii) interpolate in the grid of partial costs
stored at the dryer-cooler interface, and (iii) perform a comparison

of the total costs to find the minimum.total operating cost for the
system. Having found the optimum, DRYONE then interpolates in the grid
of stored cooler controls at the dryer-cooler interface to recover the
Optimal depth and airflow for the cooler.

The general computational strategy is similar to that of COOLONE,
with several additional features. Recalling the values of cooler cost
and corresponding depth and airflow stored sequentially on the low
speed memory device, DRIONE uses the associated indices of product
temperature and moisture content to index the three two-dimensional
arrays needed for interpolation. Because many of the values of the array
would normally be infeasible if the original indices were retained,
DRIONE uses information supplied by the user (from.a scan of the COOLONE

output) to reduce the size of the arrays to a minimum. If memory

77

capacity is still a problem, the program can be rewritten so that the
control arrays need not be formed at this time, but may have the needed
values retrieved for interpolation after the minimum cost is determined.

User—supplied values for the economic and ambient conditions are
read, in addition to the bounds and increment sizes in the state and
control variables. Two new constraints for the dryer, maximum inlet
air temperature and maximum allowable pressure drop, may be specified
by the user to conform to the requirements of currently available
equipment.

The iterative scheme used is to initialize the control D0 loops
such that maximum.drying occurs, with the order of the 100ps (from outer
to inner) being: airflow rate, air temperature, and bed depth. All
available constraints are used to minimize the number of model and cost
function evaluations necessary. Next, all feasible control possibili—
ties are evaluated and the sums of all dryer and interpolated cooler
costs compared in turn to identify the minimum.suml The associated
controls for the cooler are then found by interpolation in the stored
grids. Finally a third program, RECONE, uses all optimal controls and
input data to identify all intermediate states for the process.

0.4. Procedure for use of the algorithm and an example

The input information necessary for the use of the non-air-recycle
process programs, COOLONE, DRYONE, and RECONE is summarized in this
section. Options available to the user are explained for each program.
Finally, an example of the use of the algorithm is given.

For program.COOLONE, the first of the three programs to be used, the
following information is to be supplied on data cards, using FORTRAN IV

Fl0.0 and 110 fields for real and integer values, respectively.

 

78

CARD 1

a. Maximum moisture content allowable at the entrance to the cooler,
decimal, dry basis (real).

b. Minimum.moisture content allowable at the entrance to the cooler,
decimal, dry basis (real).

c. Number of levels of moisture content to be included in the grid
between the maximum.and minimum values (integer). ' I
CARD 2

a. Maximum.product temperature allowable at the entrance to the
cooler, deg F (real).

b. Minimum.product temperature allowable at the entrance to the
cooler, deg F (real).

c. Number of levels of product temperature to be included in the
grid between the maximum and minimum values (integer).

CARD 3

a. Maximum.airflow rate to be considered as a cooler control
variable, lb dry air/hr/ft2 (real).

b. Minimum airflow rate to be considered as a cooler control
variable, lb dry air/hr/ft2 (real).

c. Number of discrete levels of airflow rate between the maximum
and minimum values (integer). V
CARD 4

a. Maximum cooler depth to be considered as a control variable,
feet (real).

b. Minimum cooler depth to be considered as a control variable,
feet (real).

0. Number of discrete levels of cooler depth between the maximum

79

and minimum values (integer).
CARD 5

a. Inlet air temperature to the cooler, deg F (real).

b. Inlet air humidity to the cooler, lb vapor/lb dry air (real).

c. Atmospheric pressure, psi (real).
CARD 6

a. Maximum allowable cooler outlet product temperature, deg F
(real).

b. Maximum allowable cooler outlet product moisture content,
decimal, dry basis (real).

c. Minimum allowable cooler outlet product moisture content,
decimal, dry basis (real).
CARD 7

a. Price of fan energy (for electrical energy, as programmed)
required, ¢/KWR (real). -

b. Time base over which the Optimization is taken, hr (real).

c. Fan efficiency, decimal (real).

d. Efficiency of fan power source, decimal (real).

In addition, the user has the Option of using the cooler model
supplied, SUBROUTINE COOLSIM, and the electrical-energy-based objective
function, SUBROUTINE CSTCOOL, or substituting his own subroutines. In
this as well as the following programs, constraints read in as data
values may be deactivated by setting them.to absurdly high or low'values
and allowing the method to find an Optimum.unconstrained in that variable.

The output format for the program COOLONE, which is written on a
storage device as well as printed, gives the following information:

grid index for moisture content, grid index for product temperature,

80

current optimal cost, airflow rate, and depth for the node.

In using COOLONE, some experience is required to be able to choose
an optimal sized grid. Recalling that the primary limitation in the
following program, DRYONE, is the size of the arrays which can be ac-
commodated by the fast access memory, the user is urged to carefully
consider the range and increment size in the product temperature and
'moisture content dimensions. Because only the Optimal controls of
depth and airflow rate will be stored, computer time for the COOLONE
program is the only restriction on the range and increment size of the
two controls. In initial trials using COOLONE, the use of a STOP card
after the initial screening process in the program is recommended; this
allows the program to use the constraints specified to initially adjust

the variable ranges to feasible size.

Before running the second program, DRYONE, the user should scan the
indices of the product moisture and temperature dimensions to note the
largest and smallest feasible values Of each found by the COOLONE simu-
lation; this information is used to reduce the size of the arrays needed
in DRYONE. For large scale problems the computer can be programmed to
do the scan automatically. In choosing ranges and increment sizes for
the control variables, the program is limited only by computer time,
not storage space.

A summary of the information needed for the program DRYONE follows;
values are read in FORTRAN IV Fl0.0 and 110 fields for real and integer
values, respectively.

CARD 1

a. Minimum.value of product moisture content to be considered in

 

81

the interpolating grid (found by scanning the COOLONE output), decimal,
dry basis (real).

b. Increment size in product moisture dimension (from.COOLONE),
decimal, dry basis (real). I

c. Lowest value of moisture content dimension used in COOLONE
program (real).

d. Number of feasible levels remaining in the moisture content
dimension (from the scan of the OOODONE output) (integer).
CARD 2

a. Minimum value of product temperature to be considered in the
interpolating grid (found by scanning the COOLONE output), deg F (real).

b. Increment size in product temperature dimension, deg F (real).

c. Lowest value of product temperature dimension used in COOLONE
program (real). 4

d. Number of feasible levels remaining in the moisture content
dimension (from the scan of COOLONE output), integer.
CARD 3

a. Lowest value of airflow rate to be used as a control for the
dryer, lb dry air/hr/ft2 (real).

b. Increment size of the airflow rate, lb dry air/hr/ft2 (real).

c. Number of levels of airflow rate to be used as controls (integer).
CARD 4 (These inputs do not include the temperature rise due to the fan.)

a. Lowest value Of inlet air temperature to be used as a control
for the dryer, deg F (real).

b. Increment size in inlet air temperature, deg F (real).

c. Number of levels of inlet air temperature to be used as controls

(integer).

CARD 5

a.

82

Lowest value of dryer bed depth to be used as a control for the

dryer, feet (real).

. b.
C.

CARD 6

CARD 7

CARD 8

a.

b.

Increment size in bed depth, feet (real).

Number of levels of bed depth to be used as controls (integer).

Inlet air temperature to heater, deg F (real).

Inlet air humidity to heater, lb vapor/lb dry air (real).
Atmospheric pressure, psia(real).

Inlet grain moisture content to dryer, decimal, dry basis (real).

Inlet product temperature to dryer, deg F (real).

Time base used for Optimization, hr (real).

Price of energy (electrical energy) for fan, ¢/KWH (real).
Fan efficiency, decimal (real).

Efficiency of fan power source, decimal (real).

Heat value of fuel used, Btu/gal (real).

Price of fuel used, ¢/gal (real).

Thermal efficiency of heater, decimal (real).

Maximum temperature constraint for dryer, deg F (real).

Maximum pressure drOp constraint for dryer, inches of water (real).

Output from.DRXONE includes the Optimal total process cost, Optimal

dryer depth, airflow rate, and inlet air temperature (without heat added

by the fan). In addition, the grain moisture content and temperature

at the dryer-cooler interface are specified. Interpolation performed

by DRYONE also identifies the optimal depth and airflow rate controls

83

for the cooler.

The remaining program, RECONE, uses the results obtained in DRIONE'
along with the component models to identify the remaining air and grain
conditions and to summarize the steady-state Operating conditions for
the process.

The required input information is specified in Fl0.0 fields as
follows:

CARD 1 (From DRYONE output)

a. Optimal dryer airflow rate, lb dry air/hr/ft2.

b. Optimal dryer inlet air temperature (without heat added by the
fan), deg F.

c. Inlet air humidity to dryer, lb water vapor per lb dry air.

d. Optimal dryer depth, ft.

e. Moisture content at dryer-cooler interface, decimal, dry basis.

f. Product temperature at dryer-cooler interface, deg F.
CARD 2

a. Inlet air temperature to heater, deg F.

b. Atmospheric pressure, psia.

c. Fan efficiency, decimal.

d. Motor efficiency, decimal.

e. Inlet grain moisture content, decimal, dry basis.

f. Inlet grain temperature, deg F.

CARD 3 (From.COOlONE input and DRXONE output)

a. Inlet air temperature, deg F.

b. Inlet air humidity, lb water vapor/lb dry air.

c. Optimal cooler depth, ft.

d. Optimal cooler airflow rate, lb dry air/hr/ftz.

 

'84
An example of the use of the program is next discussed; the con-
ditions for the example were arbitrarily chosen.
Input information for the gggle; program:
Air input conditions -- temperature, 52 deg F; absolute humidity,
.003 lb vapor/lb dry air; atmospheric pressure, 1A.3A psia.
Desired outlet conditions -— grain temperature, 80 deg F; product
moisture content, 0.19, dry basis.
State and control variable ranges and increment sizes (after initial
screening):
MOisture content: 0.203-O.l95 dry basis, increment, 0.001
Product inlet temperature: lOO:-lAO.deg F; increment, 5.0
Airflow rate: lOO.-AOO. lb dry air/ftZ/hr; increment 10.
Depth of bed: l.-2. ft; increment, 0.1
In the execution of this program, all remaining nodes were found to be
feasible; hence, in the setup procedure for running DRYONE, no adjust-
ments in the grid size were necessary. The range and increment size for
the dryer controls were chosen to be:
Airflow rate: A00.-850. lb dry air/ftZ/hr; increment, 50.
Inlet air temperature: 200.-500. deg F; increment, 50.
Bed depth: l.-3. ft; increment, 0.l
Inlet grain conditions were 0.25 moisture content, dry basis and
50. deg F.
Economic factors for both machine components were: electrical
price 2¢/KWH; LP gas price, l6.5¢/gal; fuel heat value, 91547. BTU/gal;

fan efficiency, 0.55; motor efficiency, 0.75; thermal efficiengy, 0.70. 9*5::’
WA ”......”—

‘

 

 

An optimization time base of one hour was used for each model; grain
flow rate was specified as 9.0 bu/hr/ftz. The maximum total air tem-

perature bound was 500. deg F; the maximum allowable pressure drop

85

constraint was deactivated. The objective function used reflected only
operating cost per square foot Of dryer cross-section: no capital costs
or product quality penalties were included. I
The Optimal conditions found were:
Cost: lS.7¢/ft2/hr (or l.7h¢/bu)
Dryer depth: 2.6 feet
Dryer airflow rate: 600. lb dry air/hr/ft2 (or 230.cfm/ft2)
Inlet fan dryer temperature: without fan component, 450. deg F;
with fan component, A69. deg F
Cooler depth: 2.0 feet
Cooler airflow rate: 163. lb dry air/hr/ft2 (or 35.cfm/ft2).
Using RECONE, the resulting intermediate steady state air and
grain conditions were identified. Outlet conditions for the air from
the cooler were: 13l. deg F and 0.0143 lb vapor/lb dry air; for the
dryer air: 128. deg F and 0.0155 lb vapor/1b dry air. Outlet grain
conditions for the cooler were: 79. deg F and 0.185 dry basis; for the
dryer, 132. deg F and 0.202, dry basis. The overdrying from.the
desired final moisture content is a result of the specified product
temperature constraining the solution.
From.these results, mass and energy balances were performed as
a check on the accuracy of the models used. For the cooler the moisture
content model indicated that the amount of water lost by the grain was
7.36 lb/hr/ft2 whereas the less accurate humidity model predicted a
gain of only 1.84 lb/hr/ft2 by the air. Using the former more reliable
figure and performing an energy balance on the cooler, an average figure
of 1100 Btu/1b water was obtained for latent heat of vaporization. This
value compares well with experimental data (Ha11,l957).

Similar computations for the dryer indicated a moisture loss of

86

20.8 1b/hr/ft2 from the grain versus a 7.5 lb/hr/ft2 gain by the air.
Again using the product moisture loss as a basis, the average latent
heat of vaporization was computed to be 1660 Btu/lb water. This value
is higher than would be expected, indicating poor predictions by one

or more of the dryer models. The foregoing balances are not efficiency
computations for the system. Their calculation serves only as a check
on the predictions of the component models used and hence provides no
measure of either component or system performance. If, in the judgement
of the user, the results of these checks are unacceptable, the component
models must be improved before fUrther use with the optimization
technique.

A summary of measured process parameters for a dryer-cooler combin-
ation of this type are next given (Anderson,1971). Because these data
are incomplete, no direct comparison can be made; however, they may
provide a useful benchmark in future process optimization work.

Cooler: depth, approximately 3 feet

average outlet grain temperature, 72.h deg F

average outlet grain moisture content, 16.5% wet basis

Dryer: average inlet air temperature, 530 deg F

average outlet air temperature, 130 deg F

depth, approximately 5.5 ft
average inlet grain flow rate, 725.A lb/min/lOOft2
average inlet moisture content, 21.9% wet basis

airflow rate, 5710 ft3/hr/100ft2

Ambient conditions:

average dry bulb temperature, 52.3 deg F

average relative humidity, 63.7%

87
D. Optimization of Configuration with Air Recycle
0.1. Dryer-cooler description ’

The second dryer-cooler configuration for which an optimization
technique was developed employed recycle of the warm.air leaving the
lcooler (Figure 10). This air, which has increased in humidity and
temperature during its passage through the cooler, is mixed with fresh
air, further heated, and forced through the dryer. Since the temperature
of the air at the cooler exit approaches that of the grain_from.the
dryer, there seems to be the potential for minimizing the heat which
must be added by the burner. This potential, however, will depend
strongly on the amount of moisture added to the air in the cooler and
on the ratio of cooler air to fresh air which enters the dryer. The fan,
which draws air through the cooler by suction also adds frictional heat
to the air prior to forcing it through the dryer.

Clearly, this is a much more difficult Optimization problem than
that for the process without recycle because of the added control
variables and the fact that the air properties must be accounted for in
the recycle process. In the deveIOpment of the Optimization procedure
which follows, the grain flow rate, as well as economic and ambient
conditions, will again be considered fixed. Thus, if the same models,
constraints, and objective function are used as in the previous con-
figuration, the Optimal results of the two solutions can be directly
compared.

D.2. Dynamic programming formulation and computer programming

The following development is in many respects an extension of the
non-recycle technique. Hence, to avoid repetition only the crucial
differences in the approach will be developed. Frequent reference to

the exposition of the former technique will be made.

88

 

GRAIN ‘*
INLET A'R
HEATER

l . I JHOT AIR

CONCURRENT FLOW

*L__l

 

 

 

 

 

 

 

 

 

 

DRYER
DRY, WARM EXHAUST
GRAIN ‘ ". AIR
WARM AIR
WITH SOME
ADDITIONAL
} MOISTURE
MAKEUP
AIR
COUNTERFLOW
COOLER

 

 

 

DRY, COOL coOL AIR
GRAIN INLET

Figure 10: Schematic Representation of Combination Dryer-Cooler
With Air Recycle

89

An overview of the system in terms of the dynamic programming
formulation reveals some significant differences from the preceding
problem. Unlike the process without air recycle, which could be called
a straight-through process (the air was discarded at each stage), the
process with recycle exhibits two streams--prOduct and air--flowing in
opposite directions between stages. In addition, there is a sidestream
of fresh air entering the system prior to the heater. Such a system
is termed a countercurrent flow process.

The dynamic programming ramifications for this type of system
have been treated extensively by Dranoff et al.(l96l). Basically, the
authors show that just as in the straight-through process the backward
dynamic programming recursion formula (Equation II.B.l) applies if each
stage can be shown to be independent of those preceding it in the product
stream. For the recycled air grain drying process, this condition is
met by dividing the system into three d.p. stages: (i) the dryer,
without fan or heater; (ii) the fan, heater, and mixing chamber for the
air; and (iii) the cooler. In the cooler, the last d.p. stage in the
system, the process of cooling from any given inlet moisture content and
product temperature will be the same, regardless of the use that is made
of the cooler exit air. The second stage (heater) model requires a
knowledge of the downstream air properties as well as of the properties
of the inlet air to achieve a specified set of outlet air conditions,
but needs no information about the dryer upstream. The first d.p. stage
is the dryer, which uses the air from the heater, but has no upstream
counterpart.

In determining which prOperties of the air and product to employ

as state variables, the product temperature and moisture content

 

9O

logically can be used to uniquely identify the state of the grain. It _

is also necessary to identify the state of the air between stages: This

is accomplished by the use of three additional state variables: air

humidity, temperature, and flow rate. The airflow rate, which ordinarily

would be designated a control variable, must become a state variable
because of the addition of outside air in the middle stage of the process
The resulting five-dimensional state space poses a critical test

for the computer implementation since a very large number of storage

locations are required even to accomodate a rough grid mesh. Dranoff

et al.,(l961), for example, consider the problem to be computationally

feasible only in the dimensionality is two or less. However, by taking

advantage of the properties and configuration of the process, the tech-

nique presented can give a good approximation to the Optimal costs and

controls for the system.
One of the difficulties which initially arose in the implementation

of the problem was due to the inability of the computer, the CDC 6500,

to work with arrays having a dimensionality greater than three. To

circumvent this problem, a subprogram (SUBROUTINE IENCODE, Appendix C)
was programmed to combine three of the indices into a single index, so
that the values could be stored in three dimensions.

The interpolation subroutine (SUBROUTINE INT'ERP, Appendix C)
was prograImned to do linear interpolation in uniltiple dimensions.

Other auxilliary subprograms which are required are: (i) DRYHEAT,

associated with the interpolation subroutine INTERP; (ii) the cooler
and heater models, 00015114 and DRYSIM; (iii) the cooler and heater

operating cost evaluations, CSTCOOL and DRYCOST; (iv) ZEROIN, a

91
search routine for the zeroes, or roots, of an unknown function (Dekker,
1967), and associated functions DEEPMNT, DEEPMNM, and HOWDEEP; and
(v) assorted psychrometric functions of the SYCHART package (Lerew,l97l).
To reduce the memory requirements for the program instructions,

the routines for the three stages were coded separately as programs

DRYER, HEATER, and COOLER. SiJmllation and cost evaluation for the

HEATER program were done internally; the program required no additional

sub progr ams .

The overall strategy for the computer implementation was to avoid
forming the cost interpolation array until the final dryer stage, thus

enabling the intermediate results to be stored in low-access-speed

storage devices. This was accomplished at the sacrifice of some com-

putational efficiency and extra lowespeed storage requirements. In the
interim, by employment of scanning techniques such as described earlier,
the size of the array created in the last stage could be substantially
reduced to manageable size.

In the first program of the sequence, COOLER, a grid of values of
each of the five dimensions at the cooler-heater interface is formed,
although the values are stored sequentially on the low-access-speed

storage device rather than forming an array. To accomplish this, an

initial feasibility screening of the specified input ranges takes
place, as in the previous algorithm, and user adjustments are made.
Next, for each pair of discrete (indexed) product temperatures and
moisture content values, a series of discrete decreasing airflow rates
is applied. For each airflow rate, the ZEROIN search routine is used
to fill the other two dimensions by searching for the bed depths where

discrete values of air humidity occur and noting the corresponding

92
temperature of the air at the cooler exit. The indices of the moisture
content, product temperature, airflow rate, and humidity are stored as

they are generated, along with aesociated nondiscrete values of air tem-

perature and cost. Various feasibility checks are made during the compu-
tations to minimize the number of model and cost evaluations necessary.

In the second program, HEATER, the Objective is to apply all avail-
able controls to each of the states represented by the nodal points at

the heater-cooler interface and to produce a discrete (indexed) five-

dimensional state space at the dryer-heater interface for interpolation

in cost and controls.
Because the grain is unaffected by the heater in this stage, only

the process costs and airflow conditions are altered. HEATER reads and

processes as a block all those state nodal values at the heater-cooler
interface which have repeated indices for product moisture content and

temperature; this is an efficient process since they had been generated

as a block in the COOLER program. Seven one-dimensional arrays are needed

for temporary storage and processing of a block; the minimum dimensions

needed for each is equal to the largest number of values to be read in

a block.
For each state in the block, all controls on heat and airflow add-

ed are applied in turn and the cmmllative cost computed. With the con-

trols available, it is only possible to get two of the three state vari-
ables affected by this stage, air temperature and humidity, into the

discrete, evenly spaced values necessary for simple interpolation in the

Therefore, an interpolation between the airflow rate values

DRIER stage.
resulting from successive controls is performed at this point, and the

resulting five state variable index values with associated costs and

heater controls are written on a second low-speed storage device.

 

93
Since no attempt is made at this point to determine which of the

values corresponding to” a given set of state indices is the optimal

cumulative cost, this procedure magr produce many redundant data sets.
However,

since the values are written in a block correSponding to the

indices of) moisture content and product temperature, they are stored
somewhat compactly on the memory device.

The third program, DRYER, using information supplied by the user
regarding the ranges of indices generated by the HEATER program, initi-
ally reads the indices and cost values from the low-speed memory device

and adjusts the indices to be used in the cost array to mininmm size.

By means of IENCODE, the indices in the airflow, air temperature, and

air humidity dimensions are combined into a single index.

This index,
with the remaining moisture content and product temperature indices,

is used to identify the location in the three dimensional array to be
filled with a cost value. By comparison of all costs having the same
set of indices, each position in the array is filled with the optimal

cumulative cost at the dryer-heater interface; gaps in the arrw are

filled by absurdly high cost values to prevent their use in the inter—
polation procedure.

Using the input air values indicated by the indices of the airflow
rate, air temperature, and humidity dimensions, and applying discrete
values of the only remaining control, dryer length, the operating costs

due to airflow in this stage are assessed and added to the interpolated

cost obtained from the dryer-heater-interface cost array.

Appropriate
feasibility checks and the program flow pattern are designated to limit

the number of evaluations necessary. The minimum of all costs found
by this procedure is the Optimal total Operating cost for the process.

 

9A

The remaining task is to recover all optimal controls for the 9
process . This is accomplished by an auxilliary program called RECOVRY,

which uses information supplied by the user from the preceding programs
to find optimal bed depths for both cooler and dryer, optimal airflows

through both components, and the Optimal amount of heat added by the

heater stage. RECOVRY also uses the simulation models to summarize the

conditions of operation between each of the stages and at both ends of

the process.

By the same interpolation procedure used to find Optimal cost, the

optimal amount of added air and heat in the heater stage are recovered.
Simple heat and mass balances, plus model evaluations recover all

intermediate air and grain conditions and remaining controls except

cooler depth. This is found by using the one-dimensional search tech-

nique, ZEROIN, to locate the depth at which the known cooler outlet
humidity occurs.

To facilitate use of the algorithm, a detailed description of

program inputs and intermediate steps follows. Information is to be

supplied on data cards using FORTRAN IV F10.0 and I10 fields for real

and integer values, respectively.

In preparation for using COOLER, the first of the four programs,
the desired outlet product moisture content and temperature, airflow

and depth ranges, and humidity increment must be chosen. Ambient and

economic conditions to be fixed for the duration of the solution are

needed. The following data cards are to be supplied by the user:
CARD 1

a. Maxilmlm moisture content allowable at the entrance to the

 

95

cooler, decimal, dry basis (real).
b. Mdnimum.moisture content allowable at the entrance to the

cooler, decimal, dry basis (real).
(L Number of levels of moisture content to be included in the grid

between the maximum.and minimum values (integer).

CARD 2
a. Maximum product temperature allowable at the entrance to the

cooler, deg F (real).
b. Minimum product temperature allowable at the entrance to the

cooler, deg F (real).
c. Number of levels of product temperature to be included in the

grid between the maximum and minimum values (integer).

CARD 3
a. Maximum.airflow rate through the cooler, 1b dry air/hr/ft2 (real).

b. Minimum airflow rate through the cooler, lb dry air/hr/ft2 (real).

c. Number of levels of airflow rate to be included in the grid
between the maximum.and minimum values (integer).

CARD 4
a. Maximum cooler depth to be considered as a control variable,

feet (real).

b. IMinimum cooler depth to be considered as a control variable,

feet (real).

CARD 5

a. Size of humidity increment desired for setting up the humidity

dimension at the cooler-heater interface, 1b water/1b dry air (real).

CARI)65
a. Inlet air temperature to the cooler, deg F (real).

 

96
b. Inlet air humidity to the cooler, lb water/1b dry air (real).
c. Atmospheric pressure, psia (real).
CARD 7

a. Constraint on maximum cooler outlet product temperature,
deg F (real).

b. Constraint on maximum.cooler outlet product moisture content,
decimal, dry basis (real). -

c. Constraint on minimum cooler outlet product moisture content,
decimal, dry basis (real). '

CARD 8

a. Price for fan energy (electrical energy, as programmed) required,
¢/KWH (real).

b. Time base over which optimization is taken, hr (real).

c. Fan efficiency, decimal (real).

d. Efficiency of fan power source, decimal (real).

The cooler model COOLSIM, and the electrical-energy-based objective
function, CSTCOOL, may be replaced by equivalent subprograms supplied
by the user, if desired. In the COOLER, HEATER, and DRIER programs,
constraints read in as data values may be deactivated by setting them
to absurdly low or high values.

The output format for COOLER, which is written onto a low access
speed storage device as well as printed, provides the following informa-
tion for each of the grid points at the cooler—heater interface: product
moisture content index, product temperature index, airflow rate index,
absolute humidity index, corresponding outlet air temperature and cor-

responding cost.

As explained previously, the initial part of COOLER is designed to

97

use feasibility checks for limiting the ranges of state and control
variables to be considered. Therefore, in initial trials, the use of
a STOP card immediately after the screening section is recommended.

The second program, HEATER, reads the information generated by
COOLER, then performs heat and mass balances calculating the amount'
of added fresh air and heat necessary to achieve discrete values of
outlet conditions; the following information is to be read from data -
cards:
CARD 1

a. Minimum.airflow rate at the cooler-heater interface as read
from.COOLER data, lb dry air/hr/ft2 (real).

b._ Airflow rate increment, as read from COOLER data, lb dry air/
hr/ft2 (real).

c. Minimum.air humidity at the cooler-heater interface, as read
from COOLER data, 1b water/lb dry air (real).

d. Air humidity increment, as read from COOLER data, lb water/
lb dry air (real).

e. Minimum.moisture content bound at heater-cooler interface,
as read from COOLER data, decimal, d.b. (real).

f. Maximum.moisture content bound at heater-cooler interface, as
read from.COOLER data, decimal, d.b. (real).

g. Minimum product temperature bound at heater-cooler interface,
as read from COOLER data, deg F (real).

h. Maximum.product temperature bound at heater-cooler interface,
as read from.COOLER data, deg F (real).
CARD2

a. Lower airflow rate bound at the heater-dryer interface,

98

lb dry air/hr/ft2 (real).

b. Upper airflow rate bound at the heater-dryer interface,
lb dry air/hr/rt2 (real)."

c. Maximum permissible ratio of cooler exhaust air to total air,
decimal (real).

d. Lower air temperature bound at the heater-dryer interface,
deg F (real).

e. Upper air temperature bound at the heater-dryer interface,
deg F (real).

f. Lower air humidity bound at the heater—dryer interface,
1b water/lb dry air (real).

g. Upper air humidity bound at the heater-dryer interface, 1b water/
lb dry air (real).
CARD 3

a. Number of levels of moisture content at heater-cooler interface,
from COOLER data (integer).

b. Number of levels of product temperature at heater—cooler inter—
face, from.COOLER data (integer).

c. Number of levels of airflow rate at dryer—heater interface
(integer).

d. Number of levels of air temperature at dryer-heater interface
(integer).

e. Number of levels of air humidity at dryer-heater interface
(integer).
CARD A

a. Inlet air temperature to heater, deg F (real).

b. Inlet air humidity to heater, lb water/lb dry air (real).

99

c. Atmospheric pressure, psi (real).

a. Heat value of fuel used, Btu/gal (real).

b. Price of fuel used, ¢/gal (real).

c. Heater efficiency, decimal (real).

d. Time base for Optimization, hours (real).

The size of each of the seven one-dimensional arrays in HEATER is
found by counting the maximum number of data sets to be read from.the
COOLER output which have the same indices of product moisture content
and temperature.

The output from.HEATER, which is both printed and written on a
low-access-speed storage device, consists of indices for the moisture
content, product temperature, air temperature, air humidity, and airflow
rate dimensions, plus values of cumulative cost, added heat, and added
air from the HEATER.model. To reduce the size of the cost array, to
be formed in the DRYER program, the range of indices for each state
variable is found from.a scan of the HEATER data and is input on the
data cards for DRYER; the size of the 3-dimensional cost array is given
by the.maximum.number of remaining feasible levels for (i) moisture
content and (ii) product temperature and (iii) by the product of the
remaining feasible levels of the other three state variables.

CARD 1

a. Number of feasible levels remaining in moisture content
dimension (integer).

b. Number of feasible levels remaining in product temperature
dimension (integer).

c. Number of feasible levels remaining in airflow rate dimension

(integer) .

100

d. Number of feasiblelevels remaining in air temperature dimen-
sion (integer).

e. Number of feasible levels remaining in humidity dimension
(integer). I
CARD 2

a. Lower feasible bound of moisture content at the heater-dryer
interface, decimal, d.b. (real).

b. Moisture content increment at heater-dryer interface, decimal,
d.b. (real).
CARD 3

a. Lower feasible bound of product temperature at the heater-dryer
interface, deg F (real).

b. Product temperature increment at heater-dryer interface, deg
F (real).
CARD 4 p

a. Lower feasible bound of airflow rate at the heater-dryer inter-
face, lb air/hr/rt2 (real).

b. Increment of airflow rate at the heater-dryer interface, lb air/
hr/ft2 (real).
CARD 5

a. Lower feasible bound of air temperature at the heater-dryer
interface, deg F (real).

b. Increment of air temperature at the heater-dryer interface,
deg F (real).
CARD 6

a. Lower feasible bound of humidity at the heater-dryer interface,

lb water/1b air (real).

101

b. Increment of humidity at the heater-dryer interface, 1b water/

lbah'heflj.

CARD 7

a. Lower
(real).

b. Lower

c. Lower
(real).

d. Lower

e. Lower
CARD 8

a. Lower

feet (real).
b. Upper
(real).

moisture content bound before adjustment, decimal, d.b.

product temperature bound before adjustment, deg F (real).

airflow rate bound before adjustment, lb dry air/hr/ft2

air temperature bound before adjustment, deg F (real).

humidity bound before adjustment, 1b water/lb air (real).
bound of dryer depth to be considered as a control,

bound of dryer depth to be considered as a control, feet

c. Number of levels of dryer depth between upper and lower bounds

to be considered (integer).

CARD 9

a. Moisture content of grain entering dryer, decimal, d.b. (real).

b. Temperature of grain entering dryer, deg F (real).

c. Time base used for Optimization, hours (real).

d. Price

Of energy (electrical energy) for fan, ¢/KWH (real).

e. Fan efficiency, decimal (real).

f. Efficiency Of fan power source, decimal (real).

g. Atmospheric pressure, psia (real).

a. Maximum.in1et air temperature to the dryer, deg F (real).

102

b. Maximum pressure drOp through dryer, inches of water (real).

The optimal total operating cost is given by DRYER, in addition to
the optimal dryer depth and the resulting grain temperature and moisture
content at the dryer exit .

The last program, RECOVRY, uses this, plus additional information
from COOLER and HEATER to recover by interpolation all intermediate
states and Optimal controls. The following data cards are required for
the use of RECOVRY; all input values are real and are read in Fl0.0
fields.

CARD 1 (from.DRYER output)

a. Optimal airflow rate, lb dry air/hr/ft2.

b. Optimal inlet air temperature, deg F.

0. Optimal inlet absolute humidity, 1b water vapor/lb dry air.

d. Optimal dryer depth, feet.

e. Resultant dryer outlet moisture content, decimal, dry basis.

f. Resultant dryer outlet product temperature, deg F.

Next, note the indices in the HEATER grid corresponding to the Op—
timal values of airflow rate, inlet air temperature, and inlet absolute
humidity from the Optimal DRYER results. For the resultant DRYER outlet
moisture content and outlet product temperature values, determine the
next lowest discretized values of both variables and their corresponding
indices in the HEATER grid. From the HEATER output, read the least
cost "added airflow" and "added heat" controls corresponding to the
five indices: (1) moisture content, (ii) product temperature, (iii)
airflow rate, (iv) air temperature, (v) absolute humidity.

CARD 2 (from.HEATER output)
a. Discretized moisture content corresponding to the index found

in the above procedure, decimal, dry basis.

103

b. Discretized product temperature corresponding to the index
found in the above procedure, deg F.

0. Least cost "added airflow" control corresponding to the five.
indices found in the above procedure, lb dry air/hr/ftz.

d. Least cost "added heat" control corresponding to the five
indices found in the above procedure, Btu/hr/ftz.

Now raise the moisture content index by one, holding the other four
constant, and read the least cost "added airflow" and "added heat"
controls in the HEATER output corresponding to the new set of indices.
CARD 3 (from HEATER output)

a. Increment size in moisture content dimension, decimal, dry basis.

b. Least cost "added airflow" control corresponding to the five
indices found by the preceding method, lb dry air/hr/ftz.

c. Least cost "added heat" control corresponding to the five
indices found by the preceding method, Btu/hr/ftz.

Next, return to the set of five indices used for Card 2 (i.e. return
the moisture content index to its original value). Then, raise the value
of the product temperature index by one, holding the other four constant,
and read the least cost "added airflow" and "added heat" controls in
the HEATER output corresponding to the new set of indices.

CARD 4 (from HEATER output)

a. Increment size in product temperature dimension, deg F.

b. Least cost "added airflow" control corresponding to the five
indices found by the preceding method, 1b dry air/hr/ftz.

c. Least cost "added heat" control corresponding to the five
indices found by the preceding method, Btu/hr/ftz.

The final three data cards are taken from the input parameters

104

supplied for the three prior programs.

CARD 5 (from DRYER input)

8.

b.

CARD 6

Inlet moisture content to dryer, decimal, dry basis.
Inlet product temperature to dryer, deg F.

(from HEATER input)
Temperature of inlet air to heater, deg F.

Absolute humidity of inlet air to heater, 1b water vapor/lb dry

Absolute humidity of inlet air to cooler, lb water vapor/lb dry

a.
b.

air.
c. Fan efficiency, decimal.
d. Motor efficiency, decimal.

CARD 7 (from.COOLER input)
a. Temperature of inlet air to cooler, deg F.
b.

air.
0. Maximum.allowable cooler depth, feet.
d. Minimum.allowable cooler depth, feet.
e. Atmospheric pressure, psia.

D.3. Example and diggussion of results

 

A second example problem was formulated, using ambient and economic

conditions (summarized on pp. 84-85) identical to those of the non-air-

recycle case, but employing the dryer configuration with air recycle.

In the COOLER program, after initial adjustments, the resulting

State variable ranges and increment sizes were:

product moisture content: 0.195-O.203 d.b.; increment, 0.002

product temperature: l30.-l60. deg F; increment, 5. deg F

airflow rate: 200.-500. lb dry air/hr/ftz; increment, 50. 1b dry

air/hr/rt2

105
humidity increment: 0.0001 lb vapor/1b dry air

The indices of these four state variables, the corresponding outlet
temperatures, and the associated costs were stored sequentially on the
low-access-speed storage device.

In the HEATER program, blocks of these cost data were processed
according to the indices of the first two state variables, product mois—
ture content and temperature. The size of the seven one-dimensional
arrays needed for block processing was chosen by scanning the COOLER
output data and locating that pair of product temperature and moisture
content indices which were most often repeated. The corresponding
(decimal) number of storage locations needed for each array was A9 for a
total of 3A3. HEATER then generated the indices for the five state
variables, the cumulative cost, and associated heater controls, which
were stored sequentially on a loweaccess-speed storage device. A constraint
'was applied which limited the maximum recycled air:total air ratio to
().5. Hence the number of feasible combinations was substantially
.reduced.

Prior to the running of the third program, DRYER, a scan of the
stored indices allowed adjustment of the ranges of the five state var-
iables to:

product moisture content: O.l95-0.203 d.b.; increment 0.002

product temperature: l30.-l60. deg F; increment 5.0 deg F

airflow rate: 650.-800. lb dry air/hr/ftz; increment 50.

air temperature: 200.-500. deg F; increment 50.

air humidity: .0055-.0069 lb vapor/1b dry air; increment .0002.

The resulting three-dimensional cost array size, using this rela—

tively coarse grid, was 5x6x22A or 6720 decimal storage locations.

106
Total execution time required for the three programs using a CDC 6500

computer was approximately 95 seconds. The optimal total process cost
was 15.8¢/ft2/hr or l.76¢/bu.

The final program, RECOVRY, identified all Optimal controls for
the process and produced values of the state variables at the inlet,
outlet, and intermediate stages of the process. These are

' dryer exit air temperature: 133.7 deg F

dryer exit air humidity: 0.0141 lb vapor/lb dry air

dryer inlet product temperature: 50. deg F

dryer inlet grain moisture content: 0.25 dry basis

optimal dryer length: 4.5 feet

Optimal airflow rate through dryer: 650. lb dry air/hr/ft2 (231.cfm/7

rt2

optimal inlet air temperature to dryer: 379.8 deg F.

dryer exit moisture content: 0.202 dry basis

dryer exit product temperature: 143.2 deg F.

heat added by burner: 42031. Iatu/hr/rt2

optimal fresh air added to system: 388.9 lb dry air/hr/ft2 (84.

cfm/ftz)

cooler exit air temperature: 142.1 deg F

optimal cooler depth: 1.47 feet

optimal cooler airflow rate: 261.1 lb dry air/hr/rt2 (56.4 cfm/ft2)
The outlet product conditions, as calculated by working backward using
the Optimal controls, are: 83.8 deg F and 18.9% d.b. The good agree-
ment with the specified outlet conditions of 80.0 deg F and 19.0% d.b.
indicates that even with the rough grid employed, a near-Optimal solu-
tion has been found and that little accuracy has been lost in the linear

interpolations performed. Because no Optimal controls or resulting

107

states lie on the bounds prescribed for the problem, a good choice of
ranges is indicated.

Mass and energy balances were performed on the dryer and cooler
components of the system; since the heater model was itself a mass and
energy balance, no information on adequacy of its model could be gained
by this procedure.

For the cooler, using the output data given, a mass balance indi—
cated a loss of 5.5 lb water/hr/ft2 from the corn versus a gain of 6.8
lb vapor/hr/ft2 by the air. Based on the change in moisture content Of
the grain (the more accurate model), a latent heat of vaporization of
1205. Btu/lb of water was calculated. Because the range of the cooler
model had been extrapolated beyond its proven depth limit for the pur-
poses of this example, better results can be expected using shallower
beds or more comprehensive empirical equations. Improvement in the
cooler model, which comes computationally first in the algorithm, will
reduce the error carried on in subsequent calculations. .

Similar calculations performed on the dryer output data indicated
a moisture input to the dryer of 112.6 lb moisture/ftZ/hr versus an
output of 96.6 lb moisture/ftZ/hr. This descrepancy was due primarily
to the model for outlet air humidity; in the solution process this model
was used only as a feasibility check and introduced no inherent error
into the subsequent calculations if the optimal predicted dryer conditions
were indeed feasible.

Presuming that the model for outlet air temperature is reasonable
(using the outlet product temperature as a basis) and adding all addi~
tional moisture to the air which is needed to balance the equation, the

resulting outlet air humidity of 0.0386 lb water/1b dry air is far

108'

below the saturation value of 0.1200 lb water/lb dry air at the pre-
dicted outlet air temperature, 133.7 deg F. Based on the more reason-
able moisture content difference, a latent heat of vaporization of
1041. Btu/lb water is calculated for the dryer.

Comparing the values obtained for latent heat with those given
by Hall(1957), the calculated values are reasonable; the drier grain
passing through the cooler requires additional heat, on the average,
to evaporate a pound of water than does that passing through the dryer.

The efficiency calculated for the system, based only on heat energy
reaching the dryer inlet, is 1940. Btu per pound of water evaporated.
The energy expended for airflow only is 0.063 HP-hours per pound of
water evaporated. If thermal, motor and fan efficiencies are included,
the figures become 2775. Btu and 0.153 HP-hours per pound of water
evaporated.
D,4. Comparison of;three example_programs

The Optimal controls and resulting states for the two previous
examples are, as might be expected, quite different; results of these
tests are summarized in Table 5. The slight superiority in per bushel
Operating cost for the dryer without recycled air is most likely at-
tributable to the favorable drying weather conditions chosen for the
example. In the recycling case, the amount of moisture picked up by
the air in its passage through the cooler makes it unprofitable to use
for recycle, despite the heat it has gained. Commercial dryers of the
recycle design are rated for Operating cost at 0 deg F (Anon.,l97l),
which suggests that under cold temperature conditions these dryers may
have the advantage of greater economy. It may be possible to develop
dryer designs which can be easily switched from one type to the other,

depending on weather conditions.

1W

 

 

 

Table 5: Summary of optimal conditions for three drying systems
Recycle Non-recycle Dryer
System System Alone
COOLER
Inlet air temp, deg F 52. 52. -
Inlet air humidity,
1b water vapor/lb dry air 0.003 0.003 -
Inlet grain temperature, deg F 143.2 131.8 -
Inlet grain moisture content, -
decimal, dry basis 0.202 O. 202 -
Optimal depth, feet 1.47 2.00 -
Optimal airflow rate, 1b/hr/ft2 261.1 163. -
Outlet air temperature, deg F 142. l 131. -
Outlet air humidity,
1b water vapor/lb dry air 0.0117 0.0143 -
Outlet grain temperature, deg F 83.8 79. -
Outlet grain moisture content,
decimal, dry basis 0.189 0.185 -
HEATER
Inlet air temperature, deg F 142.1 - -
52.
Inlet air humidity,
lb water vapor/1b dry air 0.0117 - -
2and 0.003
Optimal airflow rate, lb/hr/ft2 650. - -
Optimal heat input, Btu/hr/ft 42031. - -
Outlet air temperature, deg F 379.8 - -
Outlet air humidity,
1b water vapor/lb dry air 0.0065 - -
DRYER
Inlet air temperature, deg F 379.8 469. 491.
Inlet air humidity,
1b water vapor/lb dry air 0.0065 0.003 0.003
Inlet grain temperature, deg F 50. 50. 50.
Inlet grain moisture content,
decimal, dry basis 0.25 0.25 0.25
Optimal depth, feet 4.5 2.6 5.0
Optimal airflow rate, 1b/hr/ft2650. 600. 500.
Outlet air temperature, deg F 133.7 128. 142.
Outlet air humidity,
lb water vapor/lb dry air 0.0141 0.0155 0.0401
Outlet grain temperature, deg F 143.2 132. 119.9
Outlet grain moisture content,
decimal, dry basis 0.202 0.202 0.190
TOTAL COST, ¢/bu 1.76 1.71. 1.57

110

Because the optimal Operating conditions for the non-recycle sys-
tem lie on several bounds imposed by the model, it is possible that
additional savings in Operating cost may be gained by extension of the
model limits.

As expected, the amount of labor, computer memory, and time required
for running a program of the non~recycle design are minimal compared to
those for the more complex system. A single solution, using a fine
mesh in state and control variables, may be sufficient to achieve accept-
able results for the former algorithm, while the latter may require
several applications to narrow the limits of uncertainty.

A third program, utilizing the concurrent dryer only, was run in
order to compare operating costs with those of the cooler-dryer combin-
ations. For this case, the outlet grain moisture content desired was
again 0.19 dry basis, while the outlet grain temperature was uncon-
strained. A comparison Of the optimal results is shown in Table
Because no cooling was done, the cost per bushel is lower than fOr the

dryer-cooler combinations. The outlet grain temperature, however, is

119.9 deg F, which is unsuitable for storage without further cooling.

E. Summary and Conclusions

The primary goal of this study, two user-oriented algorithms for
design and analysis of competing grain dryer designs, has been achieved.
Instructions for the use and modification of the two programs have been
included. Both algorithms have been tested for performance, flexibility,
ease of use, and computational feasibility on an existing computer.

The empirical models which were necessary for the testing of the
two algorithms have shown deficiencies when subjected to heat and mass

balance checks; for successful design, some of these will require

 

111
improvement and/or extension of the ranges over which they apply; alter-
nate approaches to modeling the dryer and cooler components may be
required.

In the two comparable examples of algorithm use presented, the
dryer not using the air recycle was shown to have slightly superior
Operating cost performance under the constraints imposed and economic
and ambient conditions assumed. Optimal operating conditions for the

two dryer systems were quite different.

F. Suggestions for Further Study

The following suggestions for use and improvement of the algorithm
are made:

1. The primary requirement for further use of the algorithms is
the development of more accurate models for the system components.
Although the present models are sufficient to provide much information
about the nature of the optimal solution, they are inadequate for the
purpose of making precise design recommendations. For the cooler model,
in particular, an extension to greater depths is suggested.

2. User experience with the algorithms is required in order to
reduce the computational requirements for the solutions; parameter
studies of the effect of economic and ambient conditions will permit
a priori narrowing of the state and control ranges to be considered.

3. The algorithms, as presently programmed, are primarily written
for comprehensibility rather than computational efficiency. By recon-
sideration of the order in which evaluations are processed, by increased
efficiency in programming, and by devising new checks on the limits of
feasibility, computational speed and program memory requirements can be

further optimized.

112

4. Adaptation of these algorithms to other possible process var-
iations should be considered. For instance, the recycling solution
technique can be easily changed to investigate the possibility of re-
cycling only a fraction of the air from the cooler. Another design
worthy of interest is the insertion of a heat exchanger into the system.-
to utilize heat of the exhausted air from the dryer.

5. Two of the expressions used in the program require further
attention. The available formula for pressure drop through the dryer
and cooler is based on depth and an airflow rate measured in cfm.
Because the air volume is temperature dependent, this requires a cor-
rection in the solution procedure, in addition to the computational
inefficiency of conversion to lb dry air/hr, the basic units of the
airflow rate terms in the algorithms. An improvement would be an
expression for pressure drop based on mass flow rate and bed depth.

The second expression to be analyzed for reliability in the al-
gorithms is the term expressing the amount Of heat added to the air by
the fan and motor inefficiency. In the absence of a better equation,
this expression, developed from lowetemperature loweairflow'data in
Part I of the study, has been extrapolated far beyond its intended
limits. Because this inefficiency can make an appreciable contribution
to the heat added to the air under conditions of high airflow rates and
deep beds, some experimental checks on the adequacy of this estimate
are needed.

6. Medifications to the constraints and objective function may be
made, if desired. For example, fixed costs may be included in the ob-
jective function if they vary smoothly with respect to the control
variables; this requirement on smoothness prevents errors in interpola-

tion. In addition, compatible time scales must be chosen for fixed and

113

Operating cost in order to keep both equally represented in the objective
function.

7. In the event that premium prices are paid for quality grain,
attention should be given to the choice of an apprOpriate penalty term
for the Objective function which will accurately reflect the economic

cost of dryer Operation on grain quality.

REFERENCES

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design Of a moving bed grain dryer by dynamic programming. The
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American Society of Agricultural Engineers (1971). Agricultural
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Anderson, R. (1971). President of The Andersons Incorporated, Maumee,
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Anonymous (1958). Electric motors for farm use. Vocational Agri—
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Anonymous (1971). Carter—Day's HC dryer-~An exclusive new concept in
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Bakker-Arkema, F. W. and D. B. Brooker, Eds. (1971). Proceedings of the
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Bakker-Arkema, F. W. and L. E. Lerew (1972 in press). Simulation of
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Michigan.

Barre, H. J., G. R. Baughman, and M. Y. Hamdy (1970). Application of
the logarithmic model to deep bed drying of grain. Paper 70—327,
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Bellman, R. and R. Kalaba (1965). Dynamic Proggamming and MOdern Control
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Bloome, Peter Dale (1970). Simulation of low temperature drying of
shelled corn leading to Optimization. Thesis for the degree of
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Bussell, J. D. (1969). Problems of grain conditioning as viewed by the
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DeBoer, S. F. (1971). Graduate Research Associate, Department of

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115

Agricultural Engineering, Michigan State University, East Lansing,
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Dekker, T. J. (1967). Finding a zero by means of successive linear
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Dranoff, J. S., L. G. Mitten, W. F. Stevens and L. A. Wanninger Jr. (1961).
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Farmer, D. M. (1969). Department of Agricultural Engineering, Michigan
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drying tests.

Farmer, D. M. (1971). Materials handling system design for shelled
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Hamdy, M. Y. and H. J. Barre (1969). Theoretical analysis and simulation
of deep bed drying of grain. Paper 69-328, American Society Of
Agricultural Engineers, St. Joseph, Michigan.

Hamdy, M. Y. and H. J. Barre (1970). Analysis and hybrid simulation of
deep-bed drying of grain. Transactions of the American Society
of Agricultural Engineers:13,6, 752-757.

Harsh, S. B. and J. R. Black (1970). On the use of time-share computers
in extension education--the Michigan experience. Whatern Agri-
cultural Economics Association Proceedings.

Hukill, William.V. (1954). Drying of grain. Chapter 9 in J. A. Anderson
and A. W. Alcock, Storage of Cereal Grains and their Products.
American Association of Cereal Chemists, St. Paul, Minnesota.

Kirk, D. E. (1959). Column thickness for shelled corn dryers. Trans-
actions of the American Society of Agricultural Engineers:g,l,
42-3 0

Kline, C. M. and M. Best. (1971). Grain Drying Process and Apparatus.
U. S. Patent Number 3,373,503 Re. 27,030, January 12.

Larson, R. E. (1968). State Increment Dynamic Programming. American
Elsevier Publishing Company, Inc., New York. 256~pp.

Lerew, L. E. (1971). Guide to the use of SYCHART, a Fortran psychrometric
package. Unpublished report. Department of Agricultural Engineering,
Michigan State University, East Lansing, Michigan.

 

116

Maddex, R. L. (1971). Raising corn for profit. Michigan Field CrOps
short course, Lessons 5 and 6. Drying-Handling-Storing Shelled
Corn. Agricultural Extension Service, Department of Agricultural
Engineering, Michigan State University, East Lansing.

Meeter, D. A. (1965). Non-Linear Least Squares (OAUSHAUS). Michigan _
State University Computer Laboratory Program Documentation No. 87.

Mitten, L. G. and G. L. Nemhauser (1963). Multistage optimization.
Chemical Engineering Progress:29,1,52-60.

Morey, R. V. and R. M. Peart (1969). Optimization of a natural air corn
drying system. Paper 69-834, American Society of Agricultural
Engineers, St. Joseph, Michigan.

Nelson, G. L. (1960). A new analysis of batch grain dryer performance.
Transactions of the American Society of Agricultural Engineers:
3,2,81—88.

Perry, John H. (1963). Perryjs Chemical Engineers' Handbook, 4th Edition.
McGraweHill Book Company, New York.

 

Rosenbrock, H. H. (1960). An automatic method for finding the greatest
or least value Of a function. Computer Journal:3,175-184.
Algorithm programmed by D. M. Koenig, Department of Chemical
Engineering, Ohio State University, Columbus, Ohio.

Rosenbrock, H. H. and 0. Storey (1966). Computational Techniques for
Chemical Engineers.Pergamon Press, Oxford, Great Britain. 328 pp.

Roth, M. G. (1971). Research Associate, Department of Agricultural
Engineering, Michigan State University, East Lansing, Michigan.
Personal communication.

Schroeder, M. E. and R. M. Peart (1965). Dynamic programming method of
air allocation in a grain dryer. Paper 65—923, American Society
of Agricultural Engineers, St. Joseph, Michigan.

Scott, J. T.,Jr. (1969). The economics of corn conditioning and storage
alternatives for farmers. Agricultural Economics Research Report
98. Department of Agricultural Economics, Agricultural Experiment
Station, University of Illinois, Champaign, 22 pp.

Smith, C. L., R.'W. Pike, and P. w. Murill (1970). Formulation and
Optimization of Mathematical Models. International Textbook
Company, Scranton, Pennsylvania, 581 pp.

Thompson, Thomas-L. (1967). Predicted performances and optimal designs
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Threlkeld, J. L. (1962). Thermal Environmental Engineering. Prentice~
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117

Thygeson, J. R.,Jr. and E. D. Grossman (1971). Optimization of a
continuous through-circulation dryer. AIChE Journa1:l§,5,749-754.

‘Wilde, D. J. and C. S. Beightler (1967). Foundations of Optimization.
Prentice—Hall, Inc. Englewood Cliffs, New Jersey, 480 pp.

Wong, Peter J. (1967). Dynamic programming using shift vectors. Thesis
for the degree of Ph.D, Stanford University, Stanford, California.
(Unpublished).

APPENDICES

(Note:

APPENDICES

program name; the subprograms are listed alphabetically in Appendix C.)

Appendix
1 .

Appendix
1 .

Appendix

\OCDQO‘U’IJ-‘WND—J

10.

12:

A: Thesis Part I

Program CONTROL Page 120
Subprograms: COSTFUN, COSTRT, COVER, DIAG, EMC,

HADBRH, HAPV, HLDB, OFFDIAG, OPTWHIZ, PSDB, PVTG,

PVDBWB, PVHA, QUICK, SIDE, VSDBHA, WBDBHAS, WBL,

ZEROIN, ZEROINA

B: Thesis Part II

Program COOLONE Page 123
Subprograms: 00018134, CSTCOOL, EMC, PSDB, PVHA,

RHPSPV, VSDBHA

Program DRYONE Page 126
Subprograms: COOLERC, COOLERD, COOLERG, COSTONE,

DRYSIM, EMC, INTERP, PSDB, PVHA, RHPSPV, VSDBHA

Program RECONE Page 129
Subprogrmas: COOLSIM, DRYSIM, HOWDEEP, VSDHTA

Program COOLER Page 131
Subprograms: COOLSIM, CSTCOOL, DEEPMNM, DEEPMNT,

EMC, HOWDEEP, VSDBHA, RHPSPV, ZEROIN

Program.REATER Page 136
Program DRYER Page 139
Subprograms: DRYCOST, DRYHEAT, DRYSIM, EMC,

IENCODE, INTERP, PSDB, PVHA, RHPSPV, VSDBHA

Program RECOVRY Page 142
Subprograms: COOLSIM, DRYSIM, HOWDEEP, VSDBHA,

ZEROIN '

C: Subprograms

COOLERC Page 145 13. DRYCOST Page 148 25. PSDB Page
COOLERD Page 145 14. DRYHEAT Page 148 26. PVDBWB Page
COOLERG Page 145 15. DRYSIM Page 149 27. PVHA Page
COOLSIM Page 146 16. EMC Page 150 28. PVTG Page
COSTFUN Page 146 17. HADBRH Page 150 29. QUICK Page
COSTONE Page 147 18. HAPV Page 151 30. RHPSPV Page
COSTRT Page 147 19. HLDB Page 151 31. SIDE Page
COVER Page 147 20. HOWDEEP Page 151 32. VSDBHA Page
CSTCOOL Page 147 21. IENCODE Page 151 33. WBDBHAS Page
DEEPMNM Page 148 22. INTERP Page 151 34. WBL Page
DEEPMNT Page 148 23. OFFDIAG Page 153 35. ZEROIN Page
DIAG Page 148 24. OPTWHIZ Page 153 36. ZEROINA Page

119

The subprograms required for each main program are given after the

156
156
156
156
156
157
157
157
158
158
158
158

APPENDIX A

121

PROGRAM CONTROLCINPJT;OUTPUT)

c PROGRAM CONTROL HFADS DATA. CONVERTS UNITS.AN0 CALLS ROUTINBS
C DETERMINED P? OPTIONS CROStR BY THE USER.

10
13
2O

35
50
60

61

62
63

64

645
646

65
70
75
78
100
103
105

110
113

COHdON/PRESSIPATM/FCON/FANETApEHETA.THERETA; ELPHICEaFUPRICEoTAHa.F
1AcTuR ELCOST, IuLcUST/ICHECKIIFLAG/DT/DEPTH.TIRE/FINALIXHFINAL/STUF
zr/pzAGeLP. DMTN, THIN, uNAx, THAx/BDUNDIXMPND.BNDHIAMB/PSHB. HPrquTHBo
BXMZEROIOTHERIUFLX.PV/OT/OUE.TEE

EXTERNAL COVFR,SIDE

DATA IRASIS.IO}IHUHID.JHUHID.JCOST.J0PT.JBOTH.JHHICHoIPRESS.IFUEL/

110H RETbASIS.ianFRPERsorT. icH RETRULBoiofl AesHUH.lOH NO
icoST.iUH DPTIHIzb.1aH 80TH.loH INLETTEMP.10H PSI.10H
3 GALLON/ , ,

READ 10020310HFOK6.ICHECK7

PPINT 20001.!CHRCK6.ICHECK7

THE FOLLDRING SECTION SETS UP FOR QUICK

READ 10001;ISHFCKT;XHZERD.XMPI~AL.XH8N0

PRINT 20032.XHZbRu.XHFINAL.XM8N0.ICHECK1
IFIICHECK1.EO.IdASI5)io.13

XHZEROaXHZERO/I1.-xnzsR0) s xnrINALsXHFIRAL/tlg-XHFINALITXHBND-xnb
2ND/(1.-XM8ND)

JFIXHZERO.LE.0.4.ANDzXHZERO.CE.0.0IGO TO 20 S PRINT 1000!

STOP

READ luDOTZIrHEch;PATH

PRINT 20003.PATH.ICHECKA

IFIICHPcKA.EO.IPRrssaoo T0 35$PATH-u.491oPATM

READ 100012IPHPCR250.0MAX.GMIN

PRINT 20004.0.0HAX.0HI~.ICHECKZ

IFIICHECK2.EO.TU>oc.50
080~DEPTH11.PBTUMAx-DHAx-DEPTH/i.2STOMIN-0MIN-DEPTH11.25
IFIQ.0E.5..ANU.U.LE.50. )00 To 61 S PRINT 10003

STOP
IF![CHECK6.EO.J0PT.AND.(ICHECK7.NE.J80TH.AND.ICHECK7.NE;JHHICH))62
1.64

IFIUHAx.GT.5o..UR,nRrR.LT.5..DR.oHAx.LT,oHINI63.64

PRINT 10005

STOP

READ 1001010=PTH.TIRE.TAMBDB.TINLET.THAx.THINSATAHe-TAHHDB+459,59
PRINT ZCCOSJPEDTHpTIHEpIAHBDB

PRINT zooss.TIwLPT.THAx.THIN

TAHazTAHHDR
IFIICHECK6.E0.JUPI.AND.(ICHECK7.E0.JBOTH.OR;ICDECK7.FU.JHHICHII
1645,65

IFITMAx.GT.140..cR.THIN.LT.70..0R.THAx.LT.THINIo46.65

PRINT 10007

STOP

READ 10801}ICHFCK35HUHID

PRINT socco.uU~In.ICHECK3

Ir:{CHFcK3,en,THHMIU)70.90

IF(HUMID.GT.32.oAND.HUHlD.LT.100.)GO To 78

PRINT 10006

STOP

AHHAdasHUH 004'9. oesev:onRR8IATARe.AHeARaIsHseswAPVIPvITGO To 120
IF(ICHECK3.EO.JH1010)100.110

IFIHUHID. 07.3. 01.0ND.HU"ID.LT. 0.1)103.135

Pv= PVHA(HUHIP )THSP= HUHIUTGO T0 120

PRINT 10005

STOP

IF‘HUHIDOGTOCUCOARDONUHI00LT0100)1150113

PRINT 10006

STOP

122

115 HSP-HADBRH(ATA“8.HUHIO)SPVIPVHAIHSP)
129 FACTORa(u.?4?¢3cabPHSP)IVSDBHAIATAHBoHSP)TDELX'DEPTH/ZOS
c THE FOLLOHIRG SEPJIUN SETS UP FOR THE COST FUNCTIUNDIF 0550:
130 READ 100222FAR=TA.EHETA.THEHETA.ELPRICE.FUPRICt.HTVAL
PRINT 20007,rANtTA.EHETA.THERETA
PRINT 20075.PLPHICE.FUPRICE.HTVAL
IFIICHECK6.E0.JCO$TIGOT014DSPJPRICEsFUPRICE/HTVALSIFIICHECK0.E0.J0
1PTIGOT0150TCALL COSTFUNICOST.TINLET.0)SCOSTP80=COSTIOEPTH-1.25
PRINT 10023.5LCUSI}FULCUST0CUSIDCUSTPBU
140 CALL QUICK(AVEHCUB.TINLETpO)SAVEMCNB‘AVENCDB/I1o‘AVEMCOBISPHINT
110024,TINLeT,O,Av§PCDB.AVEMCHa

STOP
150 IFI1CchK7.ET.JaoTHIGO TO TOQIIFIICHECK7.ER;JRNICHI160:170
__,;7 c AIRFLOH IS FIXEU.0NLY INLET TEMP CAN VARY ‘

160 GUESSlzTHAXSCUE552=THINSJUE3050ALL ZEROINA‘GUE551PGUE55200.0000010
100VER)ITEE:(CUPSSI¢GUESSZIIZ.
105 CALL COSTFUNTCOSTOTEEGQUEISCDSTPBU‘COST/DEPTH’1325
AVEHCHRIXHFI”AL/(1.0XHFINALITPRINT 1002305LC05T0FULCUSToCOSToCOSTE
18U S PRINT 100220TFEoQUEoXMFINALoAVENCHB 5 STOP
w~f§C INLET IE"? IS PIREOpONLY AIRFLOH CAN VARY
170 TEE£TIVLET S GUES§130HAX$GUESS230HINSCALL ZEROINAIGUF55100UE552001
1000001,SIDEITUUh'IGJESSI*GUE552)/Zo100 T0 165
180 CALL OPTHHIZ
10001 FORMATIA1037F10007 '
10002 FORMATI3X.CMCISTURE CONTENT IS NOT IN THE NORMAL RANGE.)
10003 F0”MAT(3X,OAIRFLDH RATE INPUT IS NOT IN THE NURHAL RANGE.)
10004 FORMATI3X.OH=TRULH TEMPERATURE INPUT IS NOT IN NORMAL RANGE.)
10005 FORMATI3X,OA°SOLUTE HUHIDITY IS NOT IN NORMAL RANGE')
10006 I0RMATI3X,ORCLATIVE HUHIDITY INPUT IS INCORRbCT.)
10007 FORHATISX.'I”LFT AIR TEMP BOUNDS ARE INCORRECT.)
10008 FORMATI3X0'AIKFLUN RATE BOUNDS ARE INCORRECT.)
10010 FORMATIBF10.0)
10020 FORMATI2610)
10022 F0RHATI8F1000’
10023 FORMAT!3Xc°ELbCTeQONPONENT 3'0F7020*FUEL COHPUNENT 3.077620'T0TAL
' icUSToCTS/FTZ 390F7.2a' CTS/BU COST 3‘:F7.29
10024 FORMAT(3X.OIVLFT AIR TEMP, DES F athIOO 71*AIR FLOH RATE CFNIFTZ O
1'0F10.70'FINAL AVE MC; 088 .DF1003D.TINAL AVE "CONE 3 .0F1007’
20001 FORHATI1HO.3X,OTHE OPTION CHOSEN I5992A10)
20002 FORMATI1fiog3109INIT HC 3 th7. 4:3X:'DESIRED FINAL MC 3 OpF7. 4. 3X;
1'TOP LAYER MCISTURE BOUND 3 '0F7o 403x: 610’
20003 FORMATI1H0;3v,eATH PRESSURE x P.F7.4o3X.A10)
20004 FORMATIIHOI3YoOAIHFL0H INITIAL GUESS I *9F7a4oJXinPPER BOUND I Q,
1F7,4,3¥,9L0NCR BOUND ' 'F7O4PSXGA10) -
20005 FORMATI1H003Y0*UEU UEPTH,FT I 0pF7.493X,ODRYING TIHE,HR I O.F7.4,3
1xo*6M9IENT TPMPIF 3 ‘0F704’
20006 FOPRATI1H0;3¥.¢HUMI0ITY : c.F7.4.A10)
20007 FOPMATI1H003Y09hFEICIENCY 0F FAN I 'pF7.403Xo'UF HCTOH I -.F7.4.3X
1,.0F HRATEP : O.FI,4)
20055 FORMATI1ND'3700INLFT TEMP INITIAL GUESSpF ' '0F7o4o3Xo‘UPPEH BOUND
1 3 o.F7. 4.3X,'L0Nh” POUND : 0,F7. 4) ‘
20075 IURMATI1PO 3YpOthCTRIC PRICE; CTS/KHH- I 0, F7. 40 SX'.FUEL PRICE 0T3
1/GAL 0R /LH 8 ioF7.403xJ’FUEL HEAT VALUE: BTU/GAL OR ILR"0F7o4o//)
END

APPENDIX B

0000

(’10

(DC)

I)“

f) I)

(1

123

PROGRAM conLPNFIINPUI. OUTPUT. CHANaTAPEIIPCHANI
PROGRAM COOLONE Is THE INITIAL PROGRAM To PE RUN IN THE 3- -PRO0HAM SEQUENCE F05
OPTIMAL DESIGN Or A AON- -AIR- PECYCLE TYPE CONCURRENT DRYER-COUNTEPPLOH CUOLER
SYSTEM. ASSOCIATPD SUBPROORAMS REOUIRED-COOLSIM.CSTCOOL:FHCoPSDBoPVHAoRQPS°Vo
VSDBHA
COMMON/FIXED/TAHPAHAHB.ATAMB.SPVLCDN/INCOOL/XMUIoXTH.XGA.XDP/OUTC0
10L/HOUT.TOUTaTHUUT.XMOUT/PRICE/ELPRICE.TIME.FANETAoEMETA/PRESSIPAT
1M
PRINT 55555
READ MAX A“D MIN ALLOHABLE HCS AT THE ENTRANCE TO THE COOLER AND THE NUHPET OF
LEVELS USED. CALCULAfy INCREHENT SIZE.
READ locDOLXVMXINAXMMMIN.LEVELH
DELHPIYMMXIN-XVHNIN)/FLUAT(LEVELH'l)
PRINT 4cooc.xHMxIN.XHMNIN,LEVELM.DELM
READ MAX AND MIN ALLDHAULE PRODUCT TEMPERATURES AT THE ENTRANCE TO THE CDO.ER
AND THE NUHBFR OP LEVELS USED. CALCULATE INCREHENT SIZET
READ 1ucco;THMxINoTHHNIN.LEVELTH
DELTHaITHHxIM-THMNINI/FLOATILEVELTH-l)
PRINT 40001 .THMxIU.THMNIN.LEVELTH.DELTH
READ MAX AND MIN ALLUHABLE AIRFLDH RATES THRU THE COOLER AND THE NUMBER OF
LEVELS USED. CALCULATE INCREMENT SIZE.
READ 10000. G XOUT.GMNDUT.LEVELG
DELG- MMXQUT DMNOUTIIFLUATILEVELG- 1)
PRINT 4C303.CHXUUT.GMNOUT. LEVELG DELG
READ MAX AVE MIN ALLDHABLE BED DEPTHS OF THE COOLER AND THE NUHBER 0F LEVELS
USED. CALCJLATE INCHEMENT SIZE.
READ 19030}J°THHAX.UPTHHINALEVELD
DELD:(ngHHAx-PPTHMINIIFLOAT(LEVELD-l)
pRINT 78787."PTH"AX.OPTHMINALEVELDADELD
PRINT 40005
READ COOLING AIR IHLET TEMPERATURE AND HUMIDITY PLUS ATMOSPHERIC PRE$§UBE.
REA020003.TAMB, HAMB. PATM
PEAD MAX ALLOHABLE OUTLET PRODUCT TEMPERATURE AND MAx AND MIN ALLOWABLE OUTLET
Hcs,
READ zuccagTuouTHx.xHOUTMx.XMOUTMN
PRINT 7OUOJ.TAMH.HAHP.PATM
PRINT 70001.THOUTMX.XMOUTMX.XMDUTMN
ATAHHaTAM5o4S9.69
SPVLCDH:VSDBHAIATAM8.HAMR)/6O.
RHTN:RHDHHA(ATAWfioHAHB)$EUN=ENCTRHTN0TAH81
PEAD ELECTRIC PRICEo IIME OVER WHICH OPTIMIZATION IS TO BE PERFORMED. FAN AND
MOTOR EFFICIENCIPS,
READ 230:0;ELPPICE.TIME.FANETA.EMETA
pPINT 4020h0flLPHTCEIquCOFANETAOEMETA
QEGIN SEARCH FOR UPPER BOUND IN MC GRID DIMENSION. FIRST CHECK FEASIBILITT OF
PIN POSSIaLE OUTLET «g AGAINST CONSTRAINT.
PRINT 40307
XHOszIgHIMTYTHSTNVXINTXGASGHXUUTIXDPSDPTHMAX
CALL CDOLSIMIS)
PRINT 90:05.¥HOUT

[FIXMOIT.GT.YHGUTHX)STOP '
CHECK FEASTBILITY OF "IN POSSIBLE OUTLET PRODUCT TEMPERATURE AGAINST 000'
STRAINT,

XTHzTHWNIN
CALL CJCLSIHIII
PRINT 40008.THPUT
IFTTHOUT.GT.THPUTMX)STOP
USING HISECTIOH. FIND UPPER MOISTURE CONTENT BOUND;
"HIGHaLEVELHTMLOHa1
MUPTRv=LEVELM

10

15

20
25
30

35
A0

45

124

XMoxswaNxNoanATIMJPTRT-x)tDELHTXTHxTHMxIN
CALL COOLSIHT3)
IFTX"0UT.GT.XHCUTHX)35015
XTHnTHWNIN'

CALL COOLSTHT1)
TF(THOJT.GT.THOUTHX)35.20
MLOHBHUPTRY
IF‘(”HIGH-MLnH,OLL01‘2504°
MUPIHLOH

IF(MUP.EU.1)30,45

PRINT 90000

STOP

HHIGHIMUPTRY
MUPTRY:(HHIGH-PLOH)IQPHLOH
SOTOlO

FRTNT 400090WUP

C PEGIN SEARCH BY DISECTTON FOR LOHER BOUND TN AIRFLUH RATE GRID DIHENSIOuo
C USING MAX OUTLET PRODUCT TEMPERATURE CONSTRAINT.

46

50
55
57
60
62
65
70

PRINT 40010
KGHIGHSLEVELGSKGLON81TLONGTRYI1
XUAIGMYOUTSCALL COOLSIMT1)
PRINT 40008.THnUT
IFITHonT.GT.TH0UTMx)STOP
XGA¢RMN00T+FLOAT(LDdGTRT-1IPDELG$CALL COOLSIM(1I
IFITHoUT.GT.THOUTNXI55.60
KGLOHaLOHGTRY

IF ((KzHIGH-VGLUHI.Ec.1I62.57
LOHGTRY-(KGHIGH-KULONIIZ+KGLON
GOTO 50

KGHIGHsLOHGTRY
IFI(KGNInH-KzLOH),0T.1)OOT057
LONG-KRHIGH
IFTLONG.EQ.LEVFLGI65.70

PRINT 90001

LOHGsLEVPLG

PRINT 40011.LEVELG.LORG

C REGIN SEARCH BY OISECTTON FOR UPPER BOUND IN PRODUCT TEMP GRID DIMENSIOGA
C USING MAX OUTLET PRODUCT TEMPERATURE CONSTRAINT.

75
80
85
9O

95
100

105

PRINT 40012
xMO1=waN1wsvDPIPPTHMAXAXGABGNXOUT
xTHaTHwNINtCALL CuoLSIHI1I

PRINT 40008.THOUT
IFTTHOJT.GT.THnuer>5T0P ,
KTHHIGusLEVELTNIRTHLOU=1IIUPTATR=LEVELTH
xTH=THwN1N.FLUAT(IUPTHTR-1IPDELTHICALL COOLSINTII
IFTTHOUT.GT.THOUTDXI95.BD

KTHLOHSIUPTHTR ’
IFI(KTUHIGH-PTHLONI.LE.1)85.100
IUPTH24THLOH'

IFIIUPTH.EO.I)90.1OS

PRINT 90002

STOP

KTHHTGusIUPTHTR
IUPTHTRsTKTHHIRH-KTNLOHIYZoKTHLOH

GOTO 75

PRINT 40009.IUPTH

C RESIN SEARCH BY PISECTIOH FOR THE LONER BOUND 0F BED DEPTH.

PRINT 40313
XHOI=XNMN101¥0A=GHXOUTIXTH2THHN1N
KDHIGH=LEVELOIKULDUSiILDNDTRYa1

125

XDP-DPTHMAXSCALL COOLSIHT1)
PRINT 40C08.THnUT
IFITHOUT.GT.THOUTMXISTOP
110 xDPsDPTHMTN+rLOAT(LOHUTRY-1IvDELDSCALL COOLSIHT1’
IF!THOUT.GT.THOUTMXI115.120
115 KOLDHsLOHDTRY
IFI(KDNTGH-KnLnu),EO.1)122.117
117 L0R0TRv:(KDRTGH-KDLOH)/2+K0L0N
GOT0110
120 KDHIGHsLONDTRY
ITI(KnNIGH-KPLnNI.GT.1IGOT0117
122 LONDIKDHIGH
IF(LONO.EO.LFVFLOI125.130
125 PRINT 90099$STOP
130 PRINT 40011.LEvELD.LDNO
C.FIND OPTIMAL FEASIPLE DEPTH AND AIRFLOH RATE CONTROLS. FILLING THE PRODUCT "C“
C TEMPERATURE GRID NITH CORRESPONDING COSTS.
PRINT 40016
no 230 III=1.HUP
XHOItFLOATIIII-1IPDELN+XMMNIN
no 220 JJJ81.IUPTH
xTuernAT(JJJ-1ItDELTH¢THMNIN$RESTCST=1D.E25TBESTGA'O.OSBESTDPIOIO
DO 210 KKK=LON0.LEVELG
KKK1-LONG-KKK+LEVELG
XGAsFLOATIKKV1-1IPDELGOGHNOUT
DO 200 LlLIthPoLEVELD
LLLLILowb-LLL+LEVELD
XDPIFLOAT(LLL1-1IPDEL000PTHMTN
CALL COOLSIH¢4I
IFITHOUT.GT.THOUTHX.OR.XHOUT.GT.XMOUTHX)GOT021O
ATOuTaTOUT¢4S9.69
RNOUTanOBHAIATUUT.HOUTI
IFTRHOUT.GT.1.IGDTDZOO
IFII(XMOUT-EOMIIIAMOI-EUM)I.OT.1.IGOT0200
EQNQUTEEMC(RHUUT.TDUT)
IFIEOMnuT.GE.x“UUT)DOTOZOO
CALL CSTCOOLTCOST)
IFTCOST.LT.BPSTCSTI190.200
190 HESTCSTECOSTcassr§A:xGASREST0P.xpp
200 CONTINUE
210 ~CONTINUE
IFTRESTCST.LT.TD.E231211.220
211 HRITET11.33333)IIIbJJJ.6ESTCST.BESTGA.RESTDP
PRINT 33333.III.JUJ.DESTCST.BESTGA.BESTDP
220 CONTINUE
230 CONTINUE
ENDFILE'11
10000 FORMATTZFloocoTIC)
20000 FORMATIBF10.OI
33333 FORMAT(21533522.15)
40000 FORMATI1XPHAY “C‘PF5.3- MIN MCI-F5.3- NUMBER OF LEVELSIOISO MC
11NCREHENT30F6.4)
40001 FORMAT¢1H0.PRUR TEM”-MAX::F5.0¢ HINIPFS.C' NUMBER OF LEVELSEOISP
1 INCRPHFNT SIZEt'F5.1I
40003 FORMAT¢1H0vAIPPLON RATE'PAXIOF5.OP MINaor5.0- NUMBER OF LEVELsIo
1130 INCREPEPT EIZF=$F5.1.///)
40005 FORMATT1XtADOITIOHAL INFORMATION REOD FOP SOLUTIONPIIII
40006 FORHAT(1H00ELECTHICAL ancEsor4.2c TIME SCALE3'F4.2P FAN EFF-0F4
1.3. MOTOR EFT:tFA.3.///)
40007 FORMATI1XtSEARCH FDR FEASIBLE UPPER BOUND IN MC DIPENSION"

40008

40009
40010
40011
40012
40013
40016

55555
70000

70001
78787

90000
90001
90002
90005
90099

126

FORMATT1H0¢FPR THESE INPUTS. THE MINIMUM OUTLET PRODUCT TEMPERATUR
1E Is or7.2I
FoRHATI1HooARJU§TEO UPPER NOOE IS oISc.LOHER NUDE IS 1'///I
FURNAT(1X'3EOHCH FOR FEASIBLE LOHER BOUND IN AIRFLOH DIMENSION.)
FORMATI1H00UPPFH IDOE ISPISP.AUJUSTED LONER NONE ISPI5.///I
FORMATI1Y.SEARCH FOR FEASIBLE UPPER BOUND IN THETA DIMENSION!)
FORMATIIXPSEPHCH FOR FEASIBLE LOHER BOUND OF DEPTHPI
FORMATI1XERECII ITFRATION. PRINTING INDICES OF MC AND PRODUCT TEMP
1 AND ASSOCIATED COST.AIRFLOH.AND DEPTH.)

FORMATI1H1-SET UP GRID SYSTEM AT THE COOLER-DRYER INTERFACEP/III
roRMATc1x.INLET CONDITIONS-TEMPI-F7.3t ABS Hun..r5,5. ATM Passs-
1'F5.2I

FORMAT<1HocDESIREU OUTLET CONDITIONs-MAX GRAIN TEHP94F738-
1:.F7,4o MIN HC39E7. 4)
FURNATI1Hovn€PTH-MAX80F4,20
1CREMENT SIZE:PF5. 3. III)
FORMATesx..MCISTURE DIMENSION CONTAINS OALY CNE FEASIBLE LEVEL.)
FORMAT(3XQAIQFLU“ DIMENSION CONTAINS ONLY ONE FEASIHLE LEVEL'l/I
FORMATI3x.oTHETA UINENSICN CONTIANS ONLY ONE FFASIELE LEVELPI
FORMATI1H0.FCR THESE INPUTS. THE MINIMUM OUTLET HC IS «F5.4I
FOPHATI3XRDEPTH CONTROL CONTAINS ONLY ONE FEASIBLE LEVELPI

END

MAX MCI

HINSPF4.2* NUHPEH 0F LEVELSPPI3? IN

PROGRA“ DRYOI'hclNPUTo OUTPUT. TAPES?)

C PROGRAM DRYONE IS THE PND PROGRAN TO BE RUN IN THE 3-PPDGRAM SEQUENCE FUR 3PT-
c IHAL DESIGN or A NON-AIR‘RECYCLE TYPE COICURPENT GHYER- COUNTERFLOH GOUIPS
C SYSTEM. ASSOCIATEP SURPROGRAMS REOUIReon-cooLERr:.COOLER0.COOLERG.COSIONE.
c DRYSIH.EHC. INTERP. PSDB. PVHA.PHPSPV. VSDBHA.
DIMENSION INDEY(?I. ICECREHTZI
COMMON/CON/CthotUT'ZIVALUESIXMCIN.THIN.GP. SAocA. CP.CV/DOLLARITIHEp
1ELPRICF.FUELPhI. FUELHTOEHETAa FANETA:THERCTl/VECALL/UPTCOSTI o )0
lGCOOLI o I.DCOOL( . I. XNEN(2I. DXINVIZ). X“LUH DELM THLOHoDELTH/
2PRESS/PATH/IN/suwTENP.HAHa.GIN.DEPTH/GUT/XM.THETA.TOUT. HOUT/SINCO§
3T/O.TA48.TIN
EXTERNAL COOLERC.COOLERO.COOLERG
DATA PATH.RHOP;UPH.SA.CA.CP.CV/14.34.38.71.9..239...242..268..95/
NDTH82
BESTCST3100672
GP=RPH01.2444RHUP
CuklssA/IGPtCPISCUNztSA/CA
c REA0.MIN FPASIRLE Uc VALUE FROM COOLONE OUTPUT. INr;REHENT SIZE. MIN VALUE OF
C PC USED IN CUOL0“E PRUGRAN. AND NUMBER OF FEASIBLE LEVELS REMAINING IN COOLONE
C OUTPUT.
READ 10001.!”LON.DELH.XMNNIfloLEVELH
PRINT PCUO..YMLUH.DELH.XMMNIN.LEVELH
DXINV(1)=1./1tkn
XHHIGH=XHL0H0TLUATILEVEL"-1)PDELH
PTNQAH:ITIXIIX“LOH+OEL“/2.-XMMNINI/DELHI
c READ MIN FEASIJL‘ PH”U“CT TEMP VALUE FROM COOLONE OuTPUT. INCREMPNT SIZE. *IN
C VALUE OF PRODUCT TEHP USED IN COOLONE PROGRAN. Ana NJHRER Gr FEASIRLE Lgve;s
C REHAINING ID CO0LU“E UUTDJTG

REAQ 1JDG1.THL3H.DELTH.THMNINoLEVELTH

PRINT 203010THL040DELTHoTHflNINOLEVELTH
DxINV(?)=1./OELT4
THHIGHaTHLJNoILUATILEVELTH-lIPDELTH
HINDXTH=IFIXI(THLOuoDELTH/Z.-THMNINI/DELTHI

7
9
C

127

READ LOdEST AIRFLOU RATE.INCREHENT SIZE.AND NUMBER OF LEVELS TO BE USED.
READ iuooslGLUA.nELG.LEVELG
PRINT 2c0°2IanuODELGDLtVELG
READ LOREST INLET alR TEMP. INCREHENT SIZE. AND NUMBER OF LEVELS TO BE USED:
READ zuuuszTLUu.nELT.LEVELT
PRINT PODD3.TLCR.UELT.LEVELT
READ SHORTFST NED PEPTR.INCREHENT SIZE. AND NUMBER OF LEVELS TO BE OSED.
READ 1aoc3}DLOw.OtLO.LEVELD
PRINT accn4.PL0R.UPLO.LEVELD
READ INLET AIR TFHPERATJRE AND HJMIDITY T0 HEATER PLUS ATHOSPHERIC PRES§JRE
AND INLET GRAIN PC AND TEMPERATURE TO THE DRYER.
READ 13002.TAHP.HAHB.PATH,XHCIN.THIN
PRINT 2CD D5.TAPB. HAR8.PATH.XRCIN. THIN
READ TIME RASE FnR UPTIHIZATION. ELECTRIC PRICE. FAN EFFICIENCY.HOTOR EEFIC-
IEMCY. FUEL HEAT VALue. FUEL PRICE. AND THERMAL EFFICIENCY.
READ 1UCC?.TTHP. ELPRICE. FANETA. EMETA.FUELHT. FUELPRI. THERETA
PRINT 2: LD6.TIPE. tLPRICE. FANETA. EMETA
PPINT 2coo7.FUPLRT. FUELPRI. THERETA
READ MAXIHHH A.IR TEMP AND PRESSURE DROP CONSTRAINTS;
RElD iUCDZDTVAXopUPOPHX
PPINT Pcann.rmAx.PnR9an
ATAMQITAHBO4‘9.09
SPVLconuvsoauAIATAP8.HAHBI/6o.
RHIN:RRDRHA(ATAMR.HAHBI
EOMsENCIPHIN.TAHRI
INITIALIZE THE CPST INTERPDLATION GRID TO ABSURD VALUES.
DO 9 I-1.LEVFLP
DO 7 J-1.LEVFLTH
0PTcosTII.JI-13.E25
CONTINUE
CONTINUE
READ IN VALUES 0F COST AND CORRESPONDING AIRFLOH AND COOLER DEPTHS. BY

c COMPARISON. FILL THE OHIO HITH OPTIMAL VALUES.

11
15

C
C
20

C

C

READ (37.36030)I”.ITH.EXPENSE.GACOOL.DRCOOL
IFIEOFI37II25.15
IHtleHINDXM
ITHsITR-MINUXTH
OPTCOSTIIH3ITHI3EXPENSE
GCOOLIIN.ITHIIGACOOL
DCOCL(IHDITH)‘DPCUOL
GOT011
APPLY IN T'JRN ALL DRYER AIRFLOH RATE .AIR TEMPERATURE. AND BED DEPTH COUTRJLS
TO EACH NonAL POINT Of THE GRID. .
DO 180 16:1.LEVEL9
IGG:LEUELG-IC+1
GIN:GLOH¢FLOAT(IGG-lIODELG
no 170 IT:1.LEVELI
ITTzLEVELT-TT’l
TIN:TLnH¢FL0LT(III-1IPDELT
Do 160 10:1.LthLP
IDD:LEVELD-IO¢1
DEPTHanLoquLUAT!InD-1IADELD
CHECK MAXINUM AIP Tt“fERATURE AND PRESSURE DROP CUN§TRAINTS;
OzGINoNPVLCON
PDRoPsnEPTNtIDISR.I031.528
IFIPDROP.GT.PUPUPHXIGOTO160
DELTADCI.378‘4Rd9tPDR0P/FANETA/EHETA
SUHTEHPzTIhoRELTADn
IF(SUHTEHP.GT.THAX)GOTOlba
CALL DRYER SIMULATION TO PREDICT OUTLET GRAIN MC AND TEHPERATUPE AND CHECK Ir

128

C RESULT FALLS WITHIN GRID.
CALL DPYSIHI1I
IF(KM.GT.XNHIGH)GOT017O
Ircxw.LT.anfiu)soTo1¢0
IFITHETA.LT.THLUHIGOT0160
IFTTHETA.GT.THHIGH)GOT0170
APPLY SATUQATION, ABSORPTION , AND EQUILIBRIUM CHECKS TO OUTLET AIR AND
HUMIDITY PREDICTIONS 0r DRYER MODEL.
CALL DHYSIHI?)
ATOUTITOUT¢4S9,69
RHOUTaaHDaHA(ATOUT;HOUTI
IF‘RHOUT06T01036010160
IF(((XW-EOHI/IXHCIN-EOH)).GT.1.)GOT0160
EOMOUT-EHCIRHUUT.IOUTI
IF(EOHOUT.GE.XM)GUT0160
0 SET UP INTERPOLATION PROCEDURE AND INTERPOLATE.
XNENI1IIXHSXHEHC2)ITHETA
NSTOPEIO
DO 85 1:1.NOIH
85 ICECREMIIIIO
INDEXI1I8IFIXI(xn-XHLONI7DELN)¢1
{NEEx(2)aIFIX((THETA-THLOH)/DELTH)*1
vaLstLoquLOATTINDEXI1I-1IvOELH
IFIAHSIXVAL-YH).LI.10.E-8)90a100
90 NSTOREzNSTORF¢1$ICECR&M(NSTORE);1
100 XVALITNLON+FLOATIINDEXIZI-1I00ELTH
IFIAHSIXVAL-THETA).LT.10;E-B)110n120

110 NSTOREsNSTORF+1$IQECREHINSTUREIUZ

120 IFINSTDRE.EO.NOIH)130:140

130 PARCOSTaOPTCnSTTINDEXII).INDEXT2))

6070 145

140 CALL INTERPIPAPCOST.NDIH.INDEX.ICECREH.NSTORE.CO0LERO)

145 IFIPARCOST.LT.1.E-R.0R.FARCOST.GT.10.EZZIGOT0160

C EVALUATE COST OF DRYING AND ADD TO INTERPOLATED COST. COMPARE THE RE§ULT T0

C THE CURRENT OPTIMAL COST.

CALL COSToNEICnST,PDROP)
TRYCOSTaPAPCOSTPCUST
IFTTRYCOST.LT.PESTCST I130o160

C IF CURRENT TOTAL COST Is OPTIMAL. REPLACE THE PREVIOUS BEST COST AND INTER-

C POLATE TO FIND OPTIMAL COOLER DEPTH AND AIRFLOH RATE. PRINT ALL OPTIMAL

C RESULTS.

150 HESTCSTsTRYCDSTSHESTDPHtDEPTHTBESTGIGINSRESTTtTINSBESTHIXHSHESTTH'
1THETA ' .

PRINT 20013.9ESTCST,BESTH,BESTTH

PRINT 20314.0EST0PH.BESTG,BESTT

CALL INTERPICPTDCL INDIHDINDEXOICECREHONSTOREICUOLERD’
CALL 1NTERP(nPTGCL .NDIH.INDEXoICECREH.NSTOREoc00LERGI
PRINT 20015.0PTOCL;OPTGCL

160 CONTINUE

170 CONTINUE

160 CONTINUE

10001 FORMATTBF10.coI1oI

10002 FCPHATT8F10.0)

10003 FOPHATI2F10.00I10)

20000 FORMATI1H1¢ MIN [EASIBLE HC8tF5I30 MC INCREHENTzoFs,3.NxN "C IN
1COOLONE-GRlthr5.30 REMAINING FEASIBLE LEVELS"I4,//)

20001 FORMATIlHOo "IN IEASIBLE PRODUCT TEUP80F6.2' PRODUCT TEHP INCREH
1tNTIcF6,2a MIN PRODUCT TEMP IN COOLONE CRID‘*F602’ RFHAINING FEA
ZSIBLE LVL8-I3o/l)

20002 FORMATI1HO0LOHFR AIRPLON RATE BOUNDsOF632- INCKEHENTItF6.20 NUHB

on

129

15R OF LEVELSI'I4a/l)

20003 FORMAT¢1RooLnRsR AIR TEMP aouNuc.r0.2' INCRERENTa-F6.2o NUMeER 0

1F LEVELS'PIA.//I

20004 FORMATI1HOOLOHER BED DEPTH BOUND-OFQIZO INCRE"ENTIPF4.2* NUMBER

10? LEVELSI-I4.//I

20005 FORMATI1HOPHEATER INLET AIR TEHP3.F562. INLET HUMIDITYIOF654¢ AI

1" PRESSUREIPFO,20 GRAIN INLET HCIPF5.30 INLET GRAIN TEHP80F6.20(
1/2

20006 FORMATtiHORTIHF BASE FOR OPTIMIZATIONSPF4.2* ELECTRIC PRICE'PF4IZ

1' FAN EFFICIENCY-OFScSP MOTOR EFFICIENCY'PF5o30/l)

20007 FORMATI1HQOFUEL HEATIOF7.OP FUEL PRICEEPF502' THERMAL EFFICIENCT

1"F5030//0

20008 FORMATtlHOoHAX AIR TEMP cONsTRAI~T=.r7.2‘ MAX PRESSURE DROP c0~91

IRAINTIOF6.20//)

20013 FORMATI1HOPCURRENI OETIHAL COST'PF1005P OUTLET HC"?5.45 OUTLET

1 GRAIN TEHP'OFéoZI

20014 FORMATIIHOPCURRENI OPTIMAL DRYER DEPTRs.r4.2- AIRFLOH RATEs-r6.2-

1 AIR TEMP3096,2)

20015 FORMATIIHO-OPTIHAL COOLER DEPTH..r5.3. AIRrLOu RATEOPFOIZo/ll)
30000 FORMAT¢2ISISE22.15>

000

n 00 an on

END

PROGRAM RECONEIINPUT.OUTPUT)
PROGRAM RecoNE Is THE FINAL PROGRAM TO BE RUN IN THE s-PROGRAM SEQUENCE F0?
OPTIMAL DESIGN Or I NON-AIR-RECYCLE TYPE CONCURRENT DRYER-COUNTERFLOH COOLER
SYSTEM. ARsocIATED SURPROGRAMS REOUIRED'COOLS1M.DHYSIM.HOHDEEP.vsnaHA.
COMMON/PRESSIPaTM/IN/TREAT.BESTH.O.DESTDPH/OUTIZ.YZ.TOUTDRY.MOUTDM
1Y/INCOOL/BESTH.BESTTHoGIMHEAT.XDP/OUTCOOL/HOUToTOUTpTMOUTIXHOUT/FIX
2XED/TAMB.HAMO,ATAMR.SPVLCON/AITCM/MINCOOL
COMMON/VALUES/XMCIN.THIN.OP.SA¢CA:CP.CV
EXTERNAL HOHDEEP
DATA EPS/.01/
READ. FROM THE DPYONE OUTPUT. OPTIMAL DRYER AIRFLON RATE. INLET AIR TEMC.
INLET AIR HUMIDITY} DEPTH. PLUS RESULTING OUTLET MC AND PRODUCT TEIP;
READ 10000IBF5T0.BESTT.8ESTH.BESTDPH.BESTM.BESTTH
READ INLET AIR TEMP TU HEATER, ATMOSPHERIC PRESSURE. FAN AND MOTOR EFfiIQ-
IENCIES. AND DRYER INLET MC AND PRODUCT TEMP.
READ 1oooogTINHEAT,PATM.FANETA.EMETA.XMCIN.THIN
READ INLET AIR TEMP AND HUMIDITY T0 COOLER.0PTIMAL COOLER oePTH AND COOLER
AIRFLOH RATE.
READ 10000;TAMR.HAMD.XDP.OINREA1
CALCULATE ACTUAL AIM TEMP ENTERING DRYER.
ATARBaYAMB+459.69
O:RESTGoVSDBHA(ATAHB.HAfl6)/60.
POROP=RESTDPUPIUI58.I*01.528
THEATORESTTP.37854869PPUROP/FANETA/EHETA
SIMULATE DRYER Tn PREDICT OUTLET AIR TEMP AND HUMIDITY.
CALL DPYSIHI2>
PRINT 20000.TOUTDRY.H00TDRY.XMCIN.TRIN.3£STDPN
stLcoucvsnDuA(ATAMa,HAMa)/¢o,
DEPTstDPsCALL COULSIMIQ)
SIMULATE COOLER TO PREDICT COOLER OUTLET AIR TMEP AND HUMIDITY. PRODUCT TERP
AND Mc. '
CALL COOLSIMIAI
PRINT 50000.TOUT.HOUT.GIRHEAT
PRINT 55000.PE57"oRESTTH

130

PRINT 600000TAHB.HAHB¢DEPTH;XH0UT'THOUT

10000 FORMATIBII0OC’

20000 FORMATTINIPDDYFR §PECS'UUTLET AIR TEMPSPF6.2' HHHIDITYIPFbil? IN
ILET GRAIN NCB‘F5.3P PROD TEWP=OF501* DRYER DEPTHBOF4.20//l

5000 FORMATI1XGC00LER EXIT AIR CONDITIONS'AIR TEMPi'F6,2. HUNIDIT’I'FO
1.4. AIRFLOH RATE'.F602/’0

55000 FOPHATIIXODRYER EXIT GRAIN CONDITIONS-HCIOF4o3' PROD TERPIPF6.2.(
1”

60000 FORMATle'COnLFR §PEC5'INLET AIR TEHP3'F6.2* "UNIDITY3*7605' DE?
1THI0F6.20 OUTLET ”C30F6,4* OUTLET PROD TEHP3*F6021
END

OOOG

nnnn

131

PROGRAM COOLPRIINPUT. OUTPUTpsIN TAPEIIIGINI
PROGRAH COOLER IS THE INITIAL PROGRAM TO BE RUN IN THE A-PROGRAM SEQUENCE COR
OPTIMAL DESIGN 0r AN AIRPRECYCLE TYPE CONCURRENT DRYER- COUNTFRFLOH COOLER
SYSTEM, ASSOCIATED SURPROGRAMS REDUINED~~COOLSIMo CSTCOOL DEEPMNM. BEECHUTv ENC.
HOHDEEP.VSDBHA3RHPSPV.ZEROIN;
COMMON/FIXED/TAnquAMB.ATAMB.SPVLCON/INCDOL/XNOI:XTH.XGA.XDPIOUTCO
10L/HOUToTOUTaTNUUI}XMOUT/PRICE/ELPRICEoTIME.FANETA.EMETAIAITCH7HNO
2N/PRESS/PATH/TPIAL/THOUTMX.XMOUTMX
EXTERNAL HONDEEP.DEEPMNT.DEEPMNM
DATA EPS/.01/
PRINT 40004
READ MAXIMUM AND MINIMUM ALLOHABLE MCS AT THE ENTRANCE To THE COOLER AND TME
NUMBER or LEVELS USED, CALCULATE INCREMENT SIZE.
READ 10000}XMMXINoXMMNINgLEVELH
DELMuIXMMxIN-XMHNINT/FLOATILEVELH-lI
PRINT 40000.!MMxIN.XHHNIN.LEVELM.DELM
READ MAXIMUM ANU MINIMUM ALLOWABLE PRODUCT TEMPERATURES AT THE ENTRANCE TO THE
COOLER AND THE NNMRER OF LEVELS USED. CALCULATE INCREMENT SIZE;
READ 1ooao;TquIN.TMMNIN.LEVELTH
DELTRETTHMxIN-THMNINIIFLOATILEVELTH- 1)
PRINT 40031 ,TquIN.TMMNIN. LEVELTHabELTH
READ MAXIMUM AND MINIMUM ALLOHABLE AIRFLON RATES THRU THE COOLER AND THE
NUMBER OF LEVELS USED. CALCULATE INCREMENT SIZE.
READ 10000;GMXDUT.GMNOUT.LEVELG
DELG=IGMXOUT-OMNOUT)lFLOATILEVELG-i)
PRINT 40003.nMXOUT.GMNOUT.LEVELG.DELG.
PRINT 40005
PEAD MAXIMUM AND MINIMUM COOLER DEPTHS USED. THE INCREMENT SIZE IN HUMIDITY.
COOLING AIP INLET TEMPERATURE AND HUMIDITY. ATMOSPHERIC PRESSURE. MAXIMUM
ALLOWABLE OUTLET PRODUCT TEMPERATURE; AND MAXIMUM AND MINIMUM ALLONABLE
OUTLET MOISTURE CONTENTs,
READ ZDDODQDPTHMAX;DPTHMINSREAD 20000.0ELH
PEA020000.TAMB;HAMR.EATM
READ zsocnzTuODTMx;xMOUTMx.XMOUTMN
PRINT 7oono.TA~B.NAMB.PATM
PRINT 70001.THDUTMX.XNOUTMX.XMOUTMN
PRINT 7000200PT”"AX0DPTHHINoDELH
ATAMBOTAMHP469.69
svacoNsvsDBNATATAMB.HAMB)/bo.
RHIN=RHDBHA(ATAMBoNAMB)SEONSEMC(RHIN9TAMRI
READ ELECTRIC PRICE. TIME OVER MNICH OPTIMIZATION IS To RE PERFORMED: FAN AND
MOTOR EFFICIENCIPS.
READ ZDCDCIELPPICE.TIME.FANETA.EMETA
PRINT 40006.0LPRICE.TIME.FANETA.PMETA
RESIN SEARCH FOR UPPER aDUNo IN MC GRID DIMENSION. FIRST CHECK FEASIBILITY OF
MIN PossxaLE OUTLET M9 AGAINST CONSTRAINT.
PRINT 40007
XMOIszMNINSYTusTHMXINSXGAsGMXOUTIXDPsnPTHMAX
CALL COOLSIHISI
PRINT 90005.1HOUT
IFIxMOHT.G -T.YMOUTHX)GOT0250
CHECK FEASIBILITY OT MIN POSSIBLE OUTLET PRODUCT TEMPERATURE AGAINST CON-
STRAINT
XTuaTHININ
CALL COOLSIHT1I
PRINT 40008.THOUT
IFTTHOHT.GT.TMOUT"YIGOT0250
USING PISECTION. FIND UPPER POISTJHE CONTENT BOUND;
HHIGHaLEVELHTHLUHI1
MUPTRYzLEVPL"

132

10 XHOItxMMNIH+FLOAT("UPTRY-lI‘DELMIXTHBTHHXIN
CALL COOLSIHISI
IrcXMOUT.CT.XM0UTMxI55.15
15 XTHITHMNIN
CALL COOLSIHI1)
IF(THOUT.GT.THOUTMX)S5020

20 ”LONEHUPTHY
IFI(HHIGH-HLOH).L§.1)25090
25 MUPIMLOH
IFIMUP.EO.1)30}45
30 PRINT 90000
STOP
35 MHIGM=MUPTPY
40 "UPTRYI("HIGH-PLOHIIZOHLOH
007010

45 PRINT 40009oMUP
C BEGIN SEARCH BY RISECIION FOP LONER ROUND IN AIPFLOH RATE GRID DIMENSION:
c USING MAX nuTLET PPunUCT TEMPERATURE CONSTRAINT.
46 PRINT 40010
KCHIGH=LEVELG$VGLUH:ISLONGTRYI1
XCA-GMXOUT
CALL COOLSIM I1)
PfiINT 40000.TH0UT
IFITHOUT.GT.THPUTMXISTOP
so XCA=GHNOUT¢FLUAT(LOHGTRY-1IPDELG
CALL COOLSIMIII
IFITHouT.GT.TMOUTHx)55.60
55 KCLONILOMCTRY
IF ((KnHIGM-VGLOHI.EO.1)62.57
57 LONGTRvsIKRHICM-KCLONI/2+KOLOA

GOTO 50

so KGHIGHzLOHGTPY
Irc(ACHICH-KanwI,GT.1IGoT057

62 LONG=KOHIGH
IFcLowc.Eo.LPVFLCIas.70

65 PRINT OODOISSTDP

70 PRINT 40011.LEVELC.LouG

c PEGIN SEARCH By RISECTION FOR UPPER BOUND IN PRODUCT TEMP GRID DIMENSION.
C USING MAX ouTLET PRODUCT TEMPERATURE CONSTRAINTI

PRINT 40012

xMoI:XMMNIHsyuPsnPTMMAXSXGAssMXOUT

XTHsTHMNIN

CALL COOLSIHI1)

PRINT AcaflfipTHOUT

IF‘THOUTQGTOTHOUTMX)71072

71 PRINT AOOZUIPTOP
72 KTHMIcuzLEVELTusxTMLOuz1;IUPTRTR=LEVELTH
75 XTHsTHHNIH+FLUtTIIUPTHTHo1IPDELTH

CALL COOLSIHIII

80 KTHLON:IUPTHTH ,
IFIIKT4HIGu-MTHL3H).LE.1)85.100
85 IUPTH=KTML0H
IFIIUPTM.EO.1>90.105
90 PRINT 90002
STOP
95 KTHHIGHslUPTHTP
100 IUPTHTR=(KTMRIGH-KTMLORIIZoKTHLOR
GOTO 75

105 PRINT 4:009,IUPTM

133

o W.-
o: . 0‘ I.
c PEGIN SEARCH FOR LnNER SOUND OF BED DEPTH. FIRST CHECK FEASIBILITY OE DIN
C POSSIBLE OUTLET PRODUCT TEMPERATURE AGAINST CONSTRAINT.
PRINT 40013
XMOI:thNIPSKGAIGMXOUTSXTHITHMNIN
qusDPTMMAx
CALL COOLSIMIlI
PRINT JEUOB.THOUT
IFITMOUT.GT.TMOUTHXI106.107
106 PRINT 40020
STOP
C CHECK FEASIBILITV OI MIN POSSIBLE OUTLET MC AGAINST CONSTRAINT.
107 CALL COOLSIHISI
PRINT OCLOSoXHOUT
IFIxMOUT.GT.YhOUTHX,1030109
IOP PRINT 40023$STOP
100 KOUNTIU
XDPsDPTHNIN
CALL COOLSIHIII
IEITHouT.GT.TH0UTHX)1150120
C USING 1-0 SEARCH. FIND DEPIM AT NHICH PRODUCT TEMPERATURE CONSTRAINT IS
C SATISFIED.
115 GUE551IDPTHHINTEUESS£=DPTHHAX
CALL ZEROII(CUP581.GUESSZ.EPS.DEEPMNT)
DPTHMINIIGUESSITGQESSZIIZ,
KUUNTIi
C USING 1.0 SEARCH, FIND DEPTH AT HHICH MC CONSTRAINT IS SATISFIED.
129 YDFIDPTHMIHICAIL COOLSIMISI
IFIxMouT.LT.YMCUTMX.AND.KOUNT.EO.0)150.130
130 GUESS1EDPTPMINIGUESSP=CPTHMAX
CALL ZEROINIGUPSSl}GUE552,EPSoDEEPHNH)
IFIKOUNT.EG.0)135e14O
135 UPTHMIUIIGUESSIOGQFSSZI/Z.
GOT01SU
c COMPARE DEDTHS Arn CHOOSE THE MAXIMUM BECAUSE IT SATISFIES BOTH CONSTRAINTSI
140 DPTHMImsAMAXI(PPTHMIN.IIGUESSl’GUESSZIYZ.I,
150 PRINT 10014.0PTHMIN
PRINT 40014
nEFINE THE LIMITS PETNEEN HHICH HUMIDITY CAN VARY. THIS IS HELPFUL IN CNOOSING
THE INCREMENT 817E IN HUMIDITY.
XMOI:xMMNILSXTP:IHMNINIchcquOUTixDP=nPTNMINSCALL COOLSIMIZI
HLON=MOUT
XHCstMHNIH+FLDAT(“JP-1IODELH
XTHzTHMNINOFLUATIIUPTH'1IODELTH
XGA:FLCAT(LUwh-1)*PELG¢GPNOUT
qu=DPTHMAxsrALL COOLSIMIZ)
HHI=HOUT
PRINT 4oc15.HLCH.HPI
THE USE OF A STOP CARD AT THIS POINT Is RECOMMEMCEC FOR INITIAL GRID STITNSA
THE DU LOOP [Ték;T[uN§ TO FILL THE GPID WITH COSTS AND CORRESPONDING COBTRULS
RESIN.
PRINT 40016 -
OD 243 11"10MUP
PRODMEFLLATIIII-IIPDELMPXMMNIN
DO 230 JJJ:1.I“PTM
THETA:FLUAT(JJJ'I)ODEETMATHHNIN
:0 220 KKK=LOHCoLtVéLG
KKK1zLONG-VKVOLEVEL3
GA=FLOATIKKK1-1)oUFLGocMnouT
LSFARC~=0$ISPARCM=C
XMDIspacfiMIXTH:IHETAIXGA=GASXDP=DPTHHINICALL COOLSIMI4)

Of]

COG

134

IHSTARTleIXI(“OUT-HAHBIIDELHI‘Z
C PEGIN CHECKS ON PUDEL PREDICTIONS FOR HUMIDITY AT HINIMUM PED DEPTH.
C CHECK SATURATION OF AIR.
ATouTsTcUT.4s9.69
RHOuTanDaHAIATUUT.MOUTI
IF‘RHCHTOGTO1Q,GDIOZSG
c CHECK MOISTURE RITIU ION ABSORPTION.
IFII(XNOUT-EOMIIIPRODM-EOM)I.GT.1.)GOTO230
C CHECK FOR EOUILIPHUI“.
EDMDUT=EMCIRHDUT.TOUII
IFIEOMOUT.OE.XVOUTIGOTO 230
C CK OUTLET PRODUCT TEMPERATURE AND MC CONSTRAINTs.
IFIXMDUT.LT.YMOUTMNISDT0230
IFITHOUT.GT.THOUTHX.OR.XMOUT.GT.XH0UTHXIL$EARCH'1
C BEGIN CKS ON MODEL PREDICTICNS FOR HUMIDITY AT MAXIMUM BED DEPTH.
XDPBDPTHMAXSCALL COOLSIHI4)
IHTsPsIFIXI(ROOT-HAMNIIDELH)¢1
IFIIHSTAPT.EO.IIMTOPP1II151.153
C IF VARIATION OF HUMIDITY BETNEEN MAX AND MIN BED DEPTHS IS LT DELH. SKIB T3
C NEXT ITERATION Or INNER 00 LOOP.
151 PRINT 99900
GOTOZZD
C PERFORM MODEL FEASIBILITY CHECKS.
153 IFITHOUT.GT.THOUTMX.0R.XHOUT.GT.XMOUTMX)GOT0 230
ATOUTxTOUT+4S9.69
RMOJTaanaHAIATUUT.MOUTI
IFIRHOUT.GT.1.IGPT0155
IFI((XMOUT-EOHI/IVRQDH’EOH”oGTolo’GOTOlss
EONOUTaEMCIRMUuT.TOUTI
IFIEOMOUT.G&.X”UUT)155.154
154 IFIXMouT.LT.VMnUIMN)155.160
15$ ISEARCN=1
C IF NECESSARY BEGIN SEARCH FOR LONER FEASIBLE BOUND ON HUMIDITY DURING THIS 00
C LOOP ITEPATION.
160 IFILSEARCH.NF.1)GUT0181
165 LTPV=IIHTOP-IHSTART)/2+IHSTART
167 HMCN=FL0ATILTHY'1IPOELH+HAM8
GUESSl:DPTHHINTCUESSZSDPTHMAX
CALL ZEROINIOUPSSI.GUESSZ.EPS.HONDEEP)
XDP:(GJESS1+CUFSSZIl2.$CALL COOLSIHI4)
{FITHQUT.GT,THOUTHX.OH.XHOUT.GTo XMDUTMXI170v175
170 IHSTARTzLTRY
IF((IHTOP-IHSTLHT’oLEol)2300165'
17S LTRV=ILTPV-INSTARTI/20IMSTART
IFILTRV.EO.IPSTARI)180.167
18? IHSTART:LTNY¢1
181 IrIIsEARCH.En.1I185.205
C IF NECESSARY, HEOIN SEARCH FOR UPPER FEASIBLE BOUND ON HUMIDITY DURING {HIS OO
3 LOOP ITERATION.
185 ITPY=IIHTOP-IHSTAHTIIZPIHSTART
187 HNOuzFLOATIITHY-IIPDELHOHAMB
506551=DPTNMINIGugssz=DPTMMAx
CALL ZEROINIOUESSI.3UESSZ.EPS.HOHDEEP)
xaPztsIESSIoCUPSSZIIZ.TCALL COOLSIMI4)
ATDuTxTouT+4S9.69
RHOuT=°HU8HAIATUIT.HOUT)
IFIRMOHT.GT.1.)GDTU 190
IFIIIxHOUT-EONIIIPRUDM-EOM)).GT.1.)GOT0190
EDM00T=EMC<RNOHT.IOOTI
IF(EDMOUT.OE.XNOUT)190.188

135

188 IFIXMOUT.LT.XMOUTMN)190.195
190 IHTOP:ITPY
IFIIIHTOP-IMSTARTI.LT.1)230.185
195 ITRY=(IHT0P-ITPY)/2+ITRY
IFIITRY.EO.(IHTOP-I)I200.167
0 IHTUPaITRY
5 DEFPlsDPTHNIN
FIND THE DEPTH AT WHICH EACH INTERMEDIATE HUMIDITY OCCURS AND EVALUATE
CORRESPONDING AIP TEMPERATURE AND PROCESS COST.
DO 210 IhIIHSTAKTaI4T0P
HNOHtrLOATIIH'1IPDELH0HAHB
GUESS1-DEEP1TGUE5528OPTHMAX
CALL ZEROINIOUE581.GU6552.EPS.HOHDEEP)
qua(GUESSIoCUFS82)/2.ICALL COOLSIMI4I
GIGA'SDVLCUH
PDQOPIXDPG(0/5po,"10528
TOUTITOUTo.378548BOOPDROP/FANETAIENETA
CALL CSTCOOLICOSTI
HRITEI11.33333IIII.JJJ.KKK1.IH.TOUT.XDP.COST
PRINT 33333.III.J4J.KKK1.IM.TOUT.XOP.COST
DEFPszDP
210 CONTINUE
220 CONTINUE
230 CONTINUE
240 CONTINUE
250 CONTINUE
ENDFILE 11
10000 FORMATIZF10. 0 I10)
10014 FORMATIIMOvMININUM FEASIBLE BED DEPTH IS tFS. Sol/l)
20000 FORMATI8F10.2 I
05333 FDRMATIAIS.3F22.15)
40000 FORMATI1XCMAX MCI'F5.30 MIN MCItF5.39 NUMBER OF LEVELSEPISO MC
11NCREMENTI0F6.4)
40001 FOVMATI1Ho-PRUD TENP-HAX:0F5,00 MINI.F5,00 NUMBER OF LEVELSSfiISO
1 INCRPMENT SI7E=-F5.1I
40003 FORHATTIHO'AIHFLOH RATE-HAX80F5.00 MIN:¢F5.DP NUMBER OF LEVELSIQ
113' IMCHENENT SIZPtAF5.1.///I
40004 FORMAT<1H1.S:T UP GRID SYSTEM AT THE COOLER-HEATER INTERFACE*l//)
40005 FORMATI1x.ADOITIONAL INFORMATION HEDD FUR SOLUTIONoIIII
40006 FORMATI1H0-ELEPTRICAL PNICE3-r4.2' TIME SCALE3'F4.20 FAN EFFPPFQ
1.3. MOTOR EFF=0F4.5.///)
40007 FORMATI1x-SEAHCH FOR FEASIBLE UPPER BOUND IN MC DIMENSION.)
40008 FORMATI1H0~FOR THESE INPUTS. THE MINIMUM OUTLET PRODUCT TEMPERATUM
1E IS vF7.2)
40009 FORMATI1Ho-ADJUSTEO UPPER 0005 IS ‘15'oLOJER NUDE IS 1.///)
40010 FORMATIIXcSEaHCH FOR FEASIBLE LOHER BOUND IN AIRFLON DIMENSIONPI
40011 FORMATIIHavUPPEN PODE ISc15-.ADJUSTED LONEP NUDE IStIS.///I
40012 FOPMATIlXtSEAHCH FOR FEASIBLE UPPER ROUND IN TPETA DIMENSION.)
40013 FORMATI1x-SEAPCM FDR FEASIaLE LOHER ROUND OF DEPTH.)
40014 FORMATle'TU CHUUSF DELN. CHECK THE LIMITS OF UuTLET HUMIDITY?)
40015 FORMATI1H0-anru UOJND=-F7.5o UPPER BOUNO=vF7;5.///I
40016 FORMATI1X¢PEPIN ITPRATIUNS.PRINTINC INDICES 0F MC. PROD TENPoAIRFL
100.Aas Hun; AND ASSOCIATED AIR TEMP. DEPTH. CObTO)
40020 FORMATI1H0.STDPPEU-MAA DEPTH TOO SHORT TO REACH DESIRED OUTLET CON
1DITIONSO)
70000 FORMATI1X¢INLET CONDITIous-TEMPscF7.3o ABS HUS=*F8.5¢ ATM PRESS.
1~F5.2)
70001 FORMATI1H0o3=SIHED OUTLET CUNDITIONSPHAX GRAIN TEMP80F7;3' MAX MCI
1:.r7,4. MIN HC:¢I7.4I
70002 FORMATI1H0oMAx DEPTMc-F4.20 MIN DEPTHxaF4.2'I“C°EMENT SIZE IN A8§

136

1 HUH8'V8.5’

00000 FORMATIBX,0HHISYURF DIMENSION CONTAINS ONLY ONE FEASIBLE LEVEL.)
90001 FORMATI3X.tATHFL0d DINENSION CONTAINS ONLY ONE FEASIQLF LEVEL?)
90002 FORMATI3X,oTHhTA DINENSION CONTIANS ONLY ONE FtASIPLE LEVEL’)
90D05 FORMATI1HOPFPR THESE INPUTS. THE MININUH OUTLET M'C IS 0F594l
°9°00 FORMATISXPDIVFENENCE IN HUMIDITY THRU FED LT DELH*)

“curzn curnac1c0n

00(1

END

PROGRAW HEATCRIINPUT.OUTPUT.TAPe7.DOUT.TAPE11-OOUTI
PROGRAM HEATER IS THE 2ND PROGRAH IN THE 4-PPOGRAN SEQUENCE FOR OPTIMAL DESIGN
or AN AIR-PECYCLP TYPE CONCURRENT DRYER-COUNTERFLUH COOLER SYSTEM; ASSOCIATED
SURPROGRAHS HEOUIHPD-NONE.
HINIHUM DIMENSIOHS FOR THE ARRAYS ARE FOUND HY EXAMINATION OP COOLER UUIPUT.
EQUAL TO THE HAXIHUH NUNRER OF STATE NODAL VALUES HAVING REPEATED INDICFS ’09
PRODUCT ac AND TPHPERATURE.

DIMENSION IGSCPHI ).IHSCRH( ).TSCRACH( 9.CSTSCRHI I

DIMENSION GOHT( )IGAUDEDI I.DELHEAT( I

COMMON/PRESS/PATH

DATA CA.CV/.742o.45/
READ [NFJRWATION FROM COOLER OUTPUT-MIN AIRFLOH RATE AND INCREMENT. MIN HUNID-
ITY AND INCREHENT. MIN AND HAx MC. AND HIN ANO MAX PRODUCT TEMPERATURES‘
READ INFDRNAYION NEEDED TO SETUP GRID SYSTEM AT HEATER-DRYER INTERFAC§ AND
LIMITING RECYCLED AIR-FRESH AIR RATIO.

PRINT 50002

HEAD 200003GLONINoDELGIN.HL0HIN.DELHIN.XML0N.XMHIGH.THLOH,THHIGN

PRINT 60000

PRINT 30003.0LOHIN;OELGIN.HL041N.DELHIN

READ ZUOUD}XLUNG.XHIG,RAT101.XLOHT.XHIT.XLONH,XHIH

PRINT 30002,vNLUN.XNHIGH.THLDA.THHIGH.XL0NG.XHIG.XLOUT,xHIT.XLOHHp

1XHIH

PRINT 30005.0AT101
READ THE NUMPER 0F LEVELS OF Hc. PRODUCT TEMPERATURE: AIRFLOH RATE. AIR TE*P‘

FRATURE, I00 HUMIDITY TO as USED IN THE GRID. THEN CALCULATE INCREHEN! §IZES
IN EACH DIMENSION.

READ 200:1}LPVH.LEVTHoLEVG.LEVT;LEVH

”ELG=(XHIG~XLUVU)/FL0AY(LEVG'l’

DELT:(xHIT-XLUNTI/FLUAT(LEVT-1’

DELH=(XHIH-XLUUH)/FLOAT(LEVH-1)

DELH=IXMHIGH-XPLOH)lFLOATILEVH'l)

DELTA=ITHHIGH-THLUH)IFLUATILEVTH-l)

PRINT KUJasoLEVMALEVTHILEVGoLEVToLEVH.DELH:DELTHoDELG-DELToDELH
PEAD INLET AIR CDHFITIONS TO THE HEATER-TEMPERATURE. HUMIDITY. AN0 ATHO§°HERIC
PRESSURE.

PRINT 50301

pEAD ZUICOLTAHPoHAHB.PATF

PRINT 30001.TAHH.HARH,PATH
PEAD ECONOAIC FACTORS-FJEL HEAT VALUE. FUEL PRICE.HEATER EFFICIENCY, AND T46
TIME BASE USED FOR UPTINIZATION.

READ zgcga;rntLHr,FU€LPRI,THERETA.TIHE

PRINT 30004.7UELH13FUELPRI,THERETAoTIHE

PRINT 30003
pEAT) THE FIRST P019 00 ac AND PRODUCT TEMPERATURE INDICES AND BEGIN READINS
ALL GRID VALUES HAVINg THESE TRU INDICES.

IEND=O

PEA0c7I11111IIH.ITH

ITHOLD=ITHTI”ULU'I"

137

BACKSPACE 7

89 lSPRACHto

90 READ(7.10000)1VaITHoIG.IH.TACOSTCL
IFIEOFI7))10°.94

94 IFIITH,NE.ITNULU)95.100

95 BACKSPACE 7
ITHoLasxrasxwuLusgn
core 110

C STORE AIRFLOR AND HUHIDITY INDICES PLUS CORRESPONDING AIR TEMPERATURE AND COST
C VALUES FOR ALL DATA GROUPS HAVING THE SAHE MC AND PRODUCT TEMPERATURE INDICES.
100 ISCRACA:ISCRACH+1
IGSCRHIISCPACH)IIQ
IHSCRHIISCRACHI3Id
TSCRAC4(ISCRACH)IT
CSTSCRRIISCRACHIICOSTCL
GOTO 90
C APPLY ALL FEASIBLE CONTROLS (HEAT AND AIRFLOH ADDED) TO INLET AIR CONUIrovs:
109 IEPD=1TIM:IMOLD$ITHIITHULD
110 OD 140 JJJ:1.LEVT
TourngOHToFLOATIgJJ-1IPDELT
Do 130 KKK:1,LFVH
HOUT=XL0HH+FLUAT(KKK-1"DELH
IFIISCPACH.ED.1)GOTO 130
DO 120 LLL=1.ISCRACH
IFICSTSCRHILLL).GE.10.EZZ)115.116
115 GOUTILLL):-1,SGUTU 120
116 CINsoLnNIN¢FLOATIIGSCRHILLL)-1)-DELGIN
HINsHLOHINoFLDATIIHSCRHILLLI-1I-DELHIN
TINzTSCRACHILLL)
IFIAHSIHOUT-HAHH),LT.1.E-12)117.118
117 GOUTILLL):1000000000.
GOT0119
119 GOUT(LLL)IGINttHlN-HAfl8)/(HOUT'HAHB)
11¢ GADDEDILLL18nOUT(LLLI-GIN
IFI(GIT/GOUTILLL)I.LT.RATIOI)GOT01195
DELHEATILLLI=1.
GADDEDILLL)'-1.
GOTUIZU
1105 DELHEAT‘LLL)ITOUTPGOUT(LLLI'(CAéHOUT‘CV)‘TIN'GIN'(CA*HIN*CVI'TAHB'
1GADOEDILLL)'(CA*HAN5‘CV)
120 CONTINUE
HOLDING THE AIR TEPPPHATURE ANO HUMIDITY AND PRODUCT TEMPERATURE AND MC 01*-
ENSIONS coIsTAnT, INTEPPOLAIE IN THE AIRFLQR DIMENSION TD OBTAIN EVENLY SPACED
VALUES 0F CONTROLS AND COST.
DO 125 HMH32.ISCRACH
SONE:GGUT(HHV-1)
IFICONE.LT.XLOHG.0R.GON&.GT.KNIGIGOTO 126
IFICADCED(NM~-1).LT.0.3.OR.DELHEAT(HHH-1).LT.O.0)OOT012S
COSTONF:DELHEATI”H“'1)/FUELHY/THERETAGFUFLPR1'1I”E*CSTSC°H(HHH'1)
LNHEPE1zIFIXT(CONE-(LO~C)/OELGI+1
GTRY:X;OHG+FLO£TILHHFREl)oDkLG
IF(AHSIGONEPRTPY).Lleoh'ic)LHHEREl3LHHEQE1‘1
GTHO=GOUT(HM”)
IFIGTHJ.LT.XLUUB.U°.GTH0;GT.XHIG)GOT0 126
IFIGADCEDIMH“ ).LT.D.D.0R.UELHEAT(HHH ).LT.0.0)GOT012S
COSTTHO=DELH=ATI“HV)IFUtLHT/THERETAOFUELPRI'TINETCSTSCRHIHHH)
LNHeugzzlrIxc(CTonXLDRCI/DELGIo1
GTRY=XLOHG*FLUAT(LHHEHE2)cDELG
IFIABSIGTuu-CTPYI.LT.1.E-10)LRHERE2=LRHER52¢1
IFILHHERE1.EO.LNH§PEZ)GUTO 125

(3 Cl C)

138

IFIGTHO.LT.GONF)GUTO 125
NUH331

122 LHIPNcLHHERE1¢NUHB
GstOHSoFLOATILw1PN-1)oDELG
RATIOaIG-GHNEIIIGTPO-GONE)
PESTcsrsccnsrTwO-COSTONE).PATIOoCOSTONE
HESTGAPIIGAD‘E0(NH“)-GAUDED(NWHp1))IRATIO¢GADDPD(HMH-17
BESTOHT:(DELHEATIHNHI-OtLHEAT(HMM-l))vRATIO¢OELHEAT(MHH-1)
NRITE(11.40020)IfioITH.LH1PN.JJJ.KKK.BESTCST.HESTGAD.PESTDHT
PRINT 77777 ;IM.ITH.LR1PN.JJJ.KKK.8ESTCST,BESTGAD.RESTDHT
IF(LN1PN.EQ.LHHbR§2)GOT0 125 '
NUHSINUH891
GOTO 122

123 NuMC=1

124 LH2PN8LHHERE70NUH9
GstOH0.FLDATILH2PN-1)PDELG
PATIOIIG-GONFI/(GTHO-GONE)
RESTCST:(COSTUNE-COSTT40)0PATIO+COSTTHO
RESTGAD:(GADDEDINHY-1I-GADDEDIHRH))tRATIO¢GADDtD(HHH)
BESTDHTsIDELHEATIHHN.1)-BELHEAT(HRH))cRATIO+OELHEAT(MHH)
HRITEI11940020)I".ITHoLNZPN.JJJ.KKK.BESTCST.HESTGAD.PESTDHT
PRINT 77777 .I".ITH.LH2PN.JJJ.KKK.RESTCST.DESTGAD.RESTDHT
IFILH2°N.EO.LRHER§1)G0TO 125
NUHCINUHC¢1
GOTO 124

125 CONTINUE

130 CONTINUE

140 CONTINUE

150 CONTINUE

IFIIENO.E0.0)GOT0 89
ENDFILE 11

10000 FORMAT:AIS.E?Z,15.22x.E22.15)

11111 FORMATIZISI

20000 FORMATI8F10.0)

20001 FORMATtallc)

30000 FOPHAT(1XcAIRFLUH'LUNER BOUNDPPF7.2* INCREHENT SIZE=0F7;30 HUHID
llTY-LONER 300N0=.F7.5- INCREHENT sxze=~r7.5.1//>

30001 FORMAT(1X.AIP TEMPERATUREs-F7.2t Aas HUMIDITYI-F7.So ATH PRESSUN
1E:*F7.2.///)

30002 FORMATI1XoHC-LOHEP BND=PF5.4t UPPER BNOz'F5.4./lv PROD TEMP-LOHEB
1 BND=PF7.2- UPPER 6ND=OF7.2.//I AIRFLOH-LUHER BND:.F7,2. UPPER B
2ND=0F7.2.l/' AIH TFflP-LOHER BND!OF7.2* UPPER HNO=0F7.2.//* ABS Hg
SNIDITY.LODER BNU=PF7.S~ UPPER ENDc.F7.S.//l

30003 FORMATIIX.VA?IAst LEVELS-HC=oI4t PROD TEHP=PI4P AIRFLON=PI40 A
11R TEMo=-14n AUS HUN3014.//t INCREMENT SIZE'HC3'FS.4' PROD TEHP|
2.F5.1o AIRFLUU=°P5.1P AIR TEMPSPF5.1* ABS HUH3'F7.5:///’

30004 FORMATI1x.ECDNOMIC FACTORS-FJEL HEAT VALUE3-F9.0' FUEL PRICE80F91
120 THERMAL EFFICIENCY=OP4.3° TIRE SCALES'F4.1.///)

30005 FORMAT¢110PAX ALLUHASLE COOLER AIR/TOTAL AIR RAT1030P5.3.///7

40000 FCPHAT¢51533P22.15I

50000 FOPMATI1H1.HPAD INFORMATION FROM COOLER SOLUTIUNPIII

50001 FORMATI1XcPEID CONDITIONS OF ADDED AIR'III

50002 FORMATIlXtSET HP ORID SYSTEM AT THE HEATER-DRYER INTERFACE'IIII

50003 FOPHAT¢1X0PRINT INDICES OF FEASIRLE MC.PPOD TEMP.AIRFLOH.AIRTEHP.A
105 HUM; AND ASSUCIATED ACCUHULATED COST.ADDED AIR.ADDED HEAT9/7I

77777 FORMATISX.517.3b22.15I

END

0000

no

(In

nnn

139

PROGRAR DRYERIINPUT. OUTPUT. TAPES?)
PROGRAM DRYER IS THE 3RD PROGRAM TO BE RUN IN THE 4-PROGRAH SEQUENCE ION
nPTIHAL DES IGN 0. AN AIR- RECYCLE TYPE CONCURRENT DPYER- COUNTERFLUN cOULER
SYSTEM. ASSOCIATED SUPPRUGHAMS REOUIRED--0RYCOST. DRYHEAT. DRYSIH.EHO.IEOCOOE.
INTERP.PSDR.PVHA,HHPSPV.VSDUHA.
DIMENSION INOEXIS).ICECREH(3).INDISI.HAX(3)
COPNDN/CON/Cfifia.CUN2/VALUES/XACIN.THIN.GP.SA.CA.CP.CV/DOLLARITIHEo
1ELPRICE.ENETA.FANETA/HECALL/OPTCOST( o . )oxNENIZIoDXINV(2).XHL
20H.OELN.THLOU.D:L]H/PRESS/PATN/IN/SUHTEMP.HIN.GIN.DEPTfi/OUTIXH.THE
STA.TDUT.HOUT
EXTERNAL DRYHEAT
DATA PATH.RHOP.BPh.SA.CA.CP/14.34.36.71.9..239;..262..2607
BESTCST:10.E?2
GPIdPHo1,244oRH0P
CONI‘SA/IGPPPPISCUWZOSAICA
READ NUMBER OF FPASIPLE LEVELS REHAINING IN Mc. PRODUCT TEMP. AIRFLOH RATE.
AIR TEUPERATuRE. AND HUMIDIIY DIMENSIONS.
READ 10000.L=VPLM.LEVELIN.LEVELG.LEVELT.LEVELN
PRINT 20001.LEUELI4.LEVELTH.LEVELG.LEVELT.LEVELH
MAXI1)=LFVELN$HAX(7)2LEVELTSNDIH=3
READ LONER FEASIRLF PC BOUND AND INCREHENT SIZE AT HEATER-DRYER INTEREACE.
READ 100023X"LDH.DELH
DXINV(1)81./"ELH
XNHIGHzxMLOH¢FLOAIILEVELF-1I'DELH
PRINT 200 02.YNLUH.¥NHIGH. DELH
READ LONER FEASI"LE PRODUCT TEMPERATURE BOUND AND INCRENENT SIZE AT HEAIER-
DRYER INTERFACE
READ 10002.THLOH.DELTN
DXINV(?)I1./OELTH
THHIGHzTHLONoFLUAT(LEVELTH-iIOUELTH
PRINT 20003.THLUN.THHIGH.DELTH
READ LONER FEASIOLF AIRFLOA RATE BOUND AND INCRENENT SIZE AT HEATER-DRYER
INTERFACE,
READ 10002;GLUD.DELG
PRINT 20004.nLOH.DELG
READ LONER FEASIRLE AIR TEMPERATURE BOUND AND INCRENENT 5125 AT HEATER-DRYER
INTERFAce,
READ 130023TLUH.D§LT
PPINT R0005.TLGH.UFLT
95.0 LOHER FEASIDLF HUMIDITY BOUND AND INCREHENT SIZE AT HEATER-DRYER INTER-
FACE.
READ 10002.HLOU. DELH
PRINT 20006.HLON. DFLH
READ THE LORI R BOUNDS OF EACH DINENSION AS THEY ARE serene ADJUSTMENT-ans.
PRODUCT TEIP. AIRFLON RrTE.AIR TEMP.AND HUMIDITY.
READ 1ac021XUHNIN.TNHNIN.XLUAG.XLOHT.XLOHH
PRINT 2C007.YH“NIN.THHNIN.XL3~fioXLONT.XLOHH
quotzxrIxIIvaDNoOELM/2.-XNNNINT/DELHI
niugxru=xr1xc(THLU.¢DELIH/?.-THMNIN)/DELTH)
H1N3xfi:IFIY((GLUUOPELG/2.-XLDAG)(UELGI
NIUJXTEIIIxIITLuuonELT/2.-XL04IIIDELT)
MINuxN:1F1x((HLU.+DELH/2;-XLDAH)lDELH)
PEAa enUND 0F DRvEP DEPTH AND INcREHENT SIZE.
READ 99909.0L0“. UPICN.LEvELn
PRINT 20008.‘LOH.DHISH.LEVELD
DELD=(DHIGH-"LOHI/FLOATILEVELO'lI
READ DRYER INLET GRAIN HC AND TEN? PLUS ECONOMIC FACTORS--TIHE ON HRICH OPT-
IMIZATION IS aASPD,ELEcTRIc PRICE.H0TUR FFFICIENCY.FAN EFFICIENCY. AND AT-
MOSPHERIC PRESSURE.
READ 10002;XMCIN.THIN.TIHE.ELPHICE.EMETA.FANETA.PATM

140

PRINT PCCDOoYDCIN.THIN.PATM
PRINT 2001C.TI“E.ELPRICE.E*ETA.FANETA
C PEAD CDVSTRAINTS UN VAXIHUH AIR TESPERATURE INTO DRYER AND HAXIHUH PRESSURE
C DROP.
READ 200C12TrAx.PuRoPHx
PRINT 20311.THAX.PPROPHX
215 INDYHArsLEVELG-LEVFLT'LEVELH
c INITIALIZE TPE COST INTERPOLATION GRID TO ABSURD VALUES.
DO 9 IaloLEVFLM
DO 7 J31.LFVFLTH
0PTCnST(1.JoV)=151F25
CONTINUE
CONTINUE
c READ IN VALUES FROM HEATER OUTPUT AND BY COMPARISON. FILL THE GRID HIIH OPT-
C IHAL COSTS.
11 READ(37.30308)IHoITHaINDISIoINDIZIoINDtlIoEXPEWSE
IFIE0F(37))22015
15 INSIM-HINDXH
ITHleuunquxTH
INDISIEINDISI-VINDXG
IND(2):IND(2)-HI~DXT
IND(1)=IND(1)-HIND¥H
CALL IENCODETINDYoINnofiAX.NDIHI
IF(EXPFNSE.LT.OPTCOST(IH.ITH.INDY))16017
16 0PTCOSTIIM}ITH}INDYIPEKPENSE
17 GOTO 11
C APPLY IN TURN ALL AIRELDN RATE. INLET AIR TEMPERATURE. AND HUMIDITY COMBINv-
C ATIONS NITRIN THE OHIO. .
70 DO 1¢o 16:1.LEVELD
IGGILEVELG-IC*1$IHD‘SIIIGG
GINECLWR+FLOATIICE-II-DELG
IND(3):IGG
DO 180 IT=1oLEVtLI
ITTaLEVELT-IT*1
T1NaTLOu+FLOATIITT-lI'DELTSINDIZI8ITT
DO 170 IH=1aLEVELH
HINtHLDW*FL0AT(IH-1IPDELHSINDI1I'IH
ATIN=TIN*459.60
RHIN=RHDEHAIATIN.hINI
E0H=EMSIRHINoTINI
C APPLY ALL DISCRETE DEPTH CONTROLS IN TURN.
DO 160 IDEPTU=JaLEVELD .
IDH=LEVELD-IREPTH*1
DEPTHan0w+FLUATTIDD-lItDELD
ATEHP:TIN+450.69
CAPPHOX:GIHOVSFUHA(ATE”P.HIN)/6O,
PQRDP:DEPTN¢IOAPPROX/39.>001.528
CHECK HAquuM AIR TEMPERATURE AND PRESSURE DROP COVSTRAIRTSS
IFIPDRRP.GT.PD°UPMY)GOT0160
DELTAnng.378E4P690PDR0P/FANETA/EHETA
SUVTEH92T1u+DELTADD
IF(SUNTEMP,CT.THAXIGOT0160
ASLM=543TE1PO4'9,69
0:61N9VSDBNATASU”'”IN)/60.
CALL DRYER SIMULATION To PREDICT OUTLET GRAIN MC AND TFHRERATURE AND CHECK Ir
RESULT FALLS NITRIN GRID.
CALL DRYSIUII)
IF(xM,GT.XMHTCRIGOTOI7O
IF(XH.LT.XHLRH)0010160

'O‘JU!

()

GT)

141

IFITHETA.LT.THIORIGOT0160
IFITHETA.GT.THRICH)ODT0170
c APPLY SATURATION. ADSOPPTIDN. ANo EQUILIBRIUM CHECKS T0 OUTLET AIR TEMP AND
c HUMIDITY PREDIcTIONs UF DRYER MODEL.
CALL DRYSIN(9)
ATOUTITOUT+4R9.69
RHOUT:RHDBHA(ATUUT.HOUT)
IFIRHOUT.GT.1.)GOTO160
IFII(xn-EQHIIIYHCIN-EOHII.GT.1.TGOT0160
EONOUT:ENC¢RHUUI.TOUTI
IFIEOHOUT.GE.X“)OOTD160
c SET UP INTERPDLATIDN PROCEDURE AND INTERPOLATE.
CALL IFNCODEIINDY.INO.MAx.NDIM)
INDEX<1I=IFIYI(X“-XMLOH)/DELN)+1
1005xI2I-IFIYI(THETA-TRLDNIIDELTHI+1
INDEX<3I=INDY
XNENIIIEXHIXNENI2IITRETA
NSTOREso
DD 85 1:1.NOIM
85 ICECREW(I):O
XVAL:XML0H0FL0ATTINDEXT1I'1)PDELH
IFIAHSTXVAL'YH).LI.10.E*8)900100
9o NsToREsNSTOR=+1SICECRENINSTDREIII
100 vaLETNLoquLUATIINDEXIzI-iItDELTH
IFIAHSIXVAL-THFTA).LT.10.E-8)110o120
11n NsTORfiaNsTchogSICECREMINSTORE)32
123 NsToREENSTnRr+1$ICECREMINSTOREIIS
IFINSTORF,FO,HPIM)150o14D
130 PARCOST:0PTCOST(INDEXIII.INDEx<2).INDEX(3)9
GOTO 145
140 CALL INTERPIPAPCO§TANDIM.INDEX:ICECREW.NST0RE.DHYHEATI
14S IFIPARCOST.LT.1.E-R.0R.PARCDST.GT.10.E22)GDT0160
c EVALUATE COST OF DRYING AND ADD TO INTERPOLATED COST. COMPARE THE RE§ULT TO
c THE CURRENT OPTIMAL c051.
CALL DRYCOST!COST.PDROP.O)
TRYCOSTaPARCOSTOCUST
IrtTRYCOST. LT. RESTCST I150 160
C IF CURRENT TOTAL COST IS OPTIMAL. REPLACE THE PREVIOUS BEST COST AND PRINT THE
COPTIMAL cnsr AND CUPHE§PONOING DRYER INLET AIR AND DuTLET GRAIN CONDITIUUS PLUS
0 OPTIMAI DEPTH.
150 RESTCST=TRYCOST$RESTDPH‘DEPTHiflESTGtGINSDESTTPTINSPESTHsHINSBESTH'
1XM$BESTTH3THFTA
PRINT 20313.0EsrcST,eesrR,BESTTH
PRINT 70014JDESTDEHDBESTGDBEST1DRESTR
160 CONTINUE
170 CONTINUE
180 CONTINUE -
190 CONTINUE
10000 FORMATI8I13)
10002 FORMATIOFlJo J)
20001 FORMAT(1H1-FPACIHLE LEVELS REMAINING IN EACH DIHFNSION-MC3'13R PRO
10 TEWP:¢IS' alRILUN .iATE=013' AIR TEHP=RI3' EU"IDITY='13.//)
20002 FORMATIIHO'MOISTUHE CONTENT FEASIBLE BOULDS'LO”ER‘*F7v4' UPPER'TE
17 49 INCPEH:“T3‘t7o4o//)
20003 FORMATleo-PPUFUCT TEMP FEASIBLE BOUNOS'LOHE""F6o2 UPPER"F6.2'
1 INCREHENTzoFA. Zrll)
20004 FORMATIlhOPLCREH FEASIBLE AIRFLOH RATE Bourozoib.2. INCREHENTEAF¢
1.2. IACREIEHTzfifb. 2. II)
20005 FDRNAT¢1HocLOREH IEASIBLE AIR TEMP BOUNDsoFé. 2' INCPEMEN73'F6.29/
I/I

142

20006 FORMATI1HD-LOHFR [FASIBLE HUMIDITY BOUNDI-Fbu4' INCREMENT'PF6.40/

1/)

20007 FORMATTIHORLONER BOUNDS BEFORE ADJUSTMENT-PMC'F5.3* PRODUCT TEMP.

1cF5,2. AIRFLUC RATERRF5.2t AIR TEMPa-F5.2’ MUMIDITY=¢F5.4:I7)

20008 FORMATT1HDoDwYPR DEPTH--LONER HOUNDEAF5,3. UPPER poumnscF5I3. N9

lthR OF LEVEL58'I3o/l)

20009 FORMATIIHOEIULFT CONDITIONS T0 DRYER-~GRAIN HC=RF5,3t GRAIN TEHPI

10F6.49 ATMORPHERIC PRESQUREIOF6o4o//I

20010 F0RMAT(1H0RFCUFU“IC FACTORS'-TIME ON WHICH OPTIMIZATION IS BASEDPY

1F4.2' ELECTRIC PNICE:-F4,2c EFFICIENCIES-MUTUH:¢F4,30 FANaaF4;3
1.//)

20011 FORMAT<1H0.*CONSTRAINTS--MAX AIR TEMP8*F6.2* MAX PRESSURE DROP-org

1020],,

20013 FORMATT1Ho.choENT OPTIMAL COST:.F10.5- OUTLET HC=~PS.4* OUTLET

1GRAIN TEMPERATUREioF5.2I

20014 FORMATTIHOoCHHRENI OPTIMAL DEPTHavF4.2' AIR CONDITIONS-EFLON RATE

1:9F6.2¢ TEMPERATDPE8’F6,20 HUMIDITthFbg4a/(/)

30000 FORMATISI5SE°2.1DA44X)
99999 FORMATIZFICACoIIOI

non

on on

an

END

PROGRAM RECOVRYIINPJT.OUTPUT)
PROGRAM RECOVERY IS THE FINAL PROGRAM TO BE RUN IN THE A-PROGRAM SEQUENCE FOR
OPTIMAL DESIGN Or AN AIR-RECYCLE TYPE CONCURRENT DRYER-COUNTERFLOH COOLER
SYSTEM. ASSOCIATED SURPROCRAMS REUUIRED-COOLSIM.DHYSIH.HHDEEPR.VSDBHA.ZERDIN.

COMMON/PRESS/PATM/lN/THEATaBESTHADoRESTDPH/OUT/tvYZATOUTORYAHOUTOR

lY/INCDOL/RESTM.UESTTH.OINHEAT.XDP/OUTCUOLIHOUT.TOUTpTHOUToXMOUT/FI

ZXED/TAND.HAHR.ATAHP.SPVLCON/AITCH/MINCOOL

COMMON/VALUEC/YMCINoTHIN.GPaSAoCAACPpCV

EXTERNAL HONDEFP

DATA EPS/.01/
READ (FROM DRYER OUTPUT) OPTIMAL AIRFLDH RATE. INLET AIR TEMP: INLET HUMIDITY
DRYER DEPTR. PLO: CORRESPONDING OUTLET MC AND PRODUCT TEMP.

READ 10000;OFSTO.UESTT.dESTH.aESTDPH.RESTM.BEsTTH
READ (FROM HEATER OUTPUT) THE BASE VALUES 0F MC. PRODUCT TEMP. AIRFLON DATE.
Ann HEAT ADDED NEEDED FOR INTERPOLATION.

READ 100003RASFM.OASPTR.BASEDA0.OASE0HT
READ (FROM HEATER OUTPUT) THE VALUES OF Mc INcPEHENT. AIRFLON RATE. AND HEAT
ADDED NEEDED FOR INIEBPDLATION IN THE MC DIMENSION!

READ 10000;DFLM.PLUSGAM.PLUSRTH
READ (FROM HEATER OUTPUT) THE VALUES OF PRODUCT TEMP ILCREPENT. AIRFLUH RATE.
AND HEAT ADDED NFEOEO FOR INTERPDLATION IN THE PRODUCT TEMPERATURE DIMENSIJN;

READ 10000;DFLTH.PLJSGAT.PLUSTHT
READ INLET MC ANO PRODUCT TEMPERATURE 13 DRYER.

READ 100003X"CIN.THIN
READ TEMP ANO HU”I"ITY 3F INLET AIR TO HEATER. PLUS FAN AND MOTOR EFFICIENZIES

READ 10000$T1nuEAT;RINREAT.FANEIA.EMETA
READ TEMP Aun HUNIOITY 3F INLET AIR TO COOLER, PLUb MAX AND MIN POUND? UN
COOLER DEPTH. ANS ATMOSPHERIC PRESSURE.

READ 10000.TAMPAHA"8.UPTHMAX.OPTH“INAPATH
INTERPOLATE TO FIND OPTIMAL AMOUNTS OF ADDED AIR AND HEAT,

nHP1:8ASEH.DtLH1HTHP1=aASETHthLTH

GAOI=RASEGAD¢IPLUSHAN-RASEGADI'IEMPITBASEMIIDELM

CA02:8ASEGAD.(PLOSCAT-BASEGAD)t(aTHPl-RASETHI/DELTR

GADDEDI(GADIOGAD2’/2g

DHTl‘BASEDHT*(PLHSHTH-BASEDHTI'(BMP1“BASEHI/DELM

143

DHTZ=BASFDHToIPLU§TRT-BASEDHTI-(BTHPl-PASETHIIDELTH
DELHFATI(DHTI¢OHT2)/20
C CALCULATE AIRFLOU RATE THROUGH COOLER.
GINHEAT:PESTG-OADDED
C CALCULATE OUTLET ATR RUMIUITY FROM COOLER.
HIN000L=<BESTG.UE§TH-GAUDEooHINHEATIIGTNUEAT
CALCULATE OuTLET AIR TEMP FROM COOLER.
ABESTT=BESTT¢479.0Q
OAPPR0x39E3Tn0VSOBHA(A8ESTT.8ESTHI/60.
PDROPxQESTnPHt(UAPPROXISB.)P'1.528
TAP?HOX:HESTT¢,37&54889tPDROP/FANETA/EMETA
ANFNT:TAPPRO¥¢459,69
O=RE§Tq.vseauATA~EHT.BESTH)/6c.
PDPOP=RESTDPP'(U/53.I'01.528
THEATsRESTT*.37854809¢PUPOPIFANETAIEHETA
C SIMULATE DRYER T0 CALQULATE OUTLFT AIR TEMP AND HUMIDITY.
CALL DPYSIM(9)
PRIVT 700030TOUTnRYAHOUIDRYAXHCINOTHINDBFSTDPH
PRINT 30003.THEAT.RESTR.BEST3.BESTN.BESTTH
PRINT aoooa,nAnUED.DELHEAT.TINHEAT.HINHEAT
ATAMRaTAMBo4Sv,b9
SPVLcolzvsnewAtATAM3.HAMa)/6o.
C SEARCH FOR OPTIMAL COOLER DEPTH.
GUESS1:DPTHMIH1bUtSS?‘DPTHMAX
CALL ZFROIN(GUFSSI.GJESSZ.EPS.HOHDEEP)
DEPTH:IGUE381*:UE§SZI/2.
CALL CDOLSIH¢4T
PRInT 50003.TOUT.HOUT.GINHEAT.BESTH.BESTTH
PRINT 50000.TAMB.HAM3.0EPTH.XMOUT.TROUT
10000 FORMATTSF10.CI
20000 FORMATI1H1vDPYER SPECS-OUTLET AIR TEHPth6.2‘ HUMIDITYI'F6.49 IN
1LET GRAIN Mcsorb.éo PROD TEMP:oF5.1* DPYER DEPTR=¢F4.2.III
soooo FOHMAT¢1XODRYtP-HEATER INTERFACE AIR TEMP=*F6.2- HUMIDITY=PF6.4.
10AIRFLnu-.ro,2. MC=6F5.So PROD TEMP:*FS.1.//I
40000 FORMAT¢1A.HEATFR §PECS-ADDED AIR:.F¢,2. ADDED HEAT:.F10.20 TEMP
IINLET AIRB'F507' INLEI HUMIOIYY:.F604O/I)
50000 FORMATc1x.AEATcR-co0LER INTERFAcE AIR TEMP-~F¢.2c HUMIDIIY80F6,4-
1 AIRFLOH RATE20F6.2‘ MCsoF4.3- PROD TEMPI'F6;?.//)
60000 FORMAT¢1XOCOOLPH §PECS-INLET AIR TEMPs¢r6.2. HUMIDITY:¢F6,5Q DE?
1TH89F6,20 OUTLET MCa~F6.4t OUTLET PROD TEHPI’F6.21
END

C)

APPENDIX C

145

SURROUTINE COOLEHCIAJERAGE.KIPoLMNOP.INDEX,
C COOLERG IS CALLED FROM INTERP T0 INTERPOLATE COOLER COST.
DIMENSION INDEx(2).LMNOP(2)
COMMON/RECALL/OPTCCSTI . ).GCOOL( . I.DCOOL( . I.XNEhI2).DxI
1Nvt2).rMLOw.OtLH.THLOR.DELTH
STINTRn(x04E.ATwO.xx.xxx.ox)=x0NE¢(XTAOsXONE)ctxx-XXXI-D!
LMNQR(K1P):1HOFX(K1PI
xouezonTCOSTILHNOPTII.LMNOPIZII
LMNoP(KtP)-IHUFXTKIP)¢1
XTwoaopTCOSTTLMNOP(1IoLHNOP(2)I
GOT0(10020)0“IP
1o XVALDXHLoquLUATIINDEXCII'1IPDELM
AVERAGRaSTINTRPTxUNE.XTw0.XNENT1I.XVAL,Dx1Nv(1II
RETURN
20 vaLzTRL0u+FLUATTINDEXTZI-1)~DELTH
AvFRAcesSTINTRRIxUNE.XTRo,XNEH(2).XVAL.Dx1NVT2)I
RETuRN
END

SUBROUTINE COCLhRUtAVERAGEpKlppLMNOPaINDEX,
c COOLERD IS CALLE" FROM INTERP TO INTERPOLATE COOLER DEPTH.
DI“ENSION INOEYC?).LHNOPI2I
CO“MON/RECALL/OPTCOST( . ).GCOOL( o ).DCOOL( . I.XNENI2).DXI
INVIZI,XNLONAOELnoTHLONADELTH
STINTRRIxoME.XTHO.¥x.xxx.DXIcXONE¢TXTHo-XONEI-Ixx-XXXIOD!
LMNoPIKTP):IUUCXTMIPI
xonesDCOOL(LMNOPT1).LMNUP(2)I
LMNOP(KIP)II“UFX(KIPI¢1
XTWOIDCOOLTL“NOP(1).LHNUPIZ)I
GOTOIIUAZOIIKIP
10 vaLxquOquLDATTINOEXT1)-1I-DELH
AVERAGcssTTNTRPTxUNe.XTuo,x~Eu(1>.XVAL.OXINVI1T)
RETURN
20 XVAL:THLOH¢FLUAT(INDEX(2)'1)'DELTH
AVERAGE=STINTHP(XUNE.XTNO.XNEN(2I.XVAL,DXINVIZ)I
RETURN
END

SURROUTINE COOLERG(AVERAGE,KIP.LMNOP.INDEXI
C COOLERG IS CALLEO FROM INTERP T0 INTERPOLATE COOLEH AIRFLOU RATE.
DIMENSION INOE¥(2)}LMNOP(2)
COMMON/RECALL/OPTCOSTI . ).GCO0L( . ).DCCOL( . I.XNE~(2)oOXI
1NV¢2T,XMLOU.HELH.THLOH.UELTH
STINTRD<X0JE,XTHO.¥X.XXX.DX)8XONF*(XTHO~YONE)'(XX-XXXIPDX
LMUoP(<1P):IHDCXTKIP)
xoh =GCOOL¢LunnP(1).LMNOP(2)I
LMNOP(<1P):I“DFXTKIP)¢1
XTH0=GCOOLTLPNOPC1I.LMNOP(2)I
GOTOI1JAZO)I’I9
10 vaL=quDR+FL0ATTINDEXI1I-IIOOELH
AVERAGE:STINTRPIXUNE.XTRO,XNERI1).XVAL.DXINVI1II
RETURN

146

20 vaLaTRLOHoFLOATTINDEXTZI-1IOOELTH
AVERACEASTINTRRITONE.XTRO.XNEH<2).XVAL.0XINV‘2’I
RETURN
END

SUBROUTINE COOLSIMTIOO)

C COOLSIM PREDICTS OUTLET PRODUCT TEMPERATURE, HUMIDITY. AIR TEMP. AND HOASTJRE

C CONTENT FOR THE COUNTERFLON COOLER AS A FUNcTION 0. INLET CONDITIONS ANU 850

C DEPTH.
COMMON/FIXED/TAnspHAHB.ATANB.SPvLCDN/INCOOL/As.A6.XGA.Aa/0UTCOOL/A

11'A20A33A‘
DELTHIX)=x-140.
DELTIXI'X'450
DELH(XIIX'.0035
DELM(X,.X-.21
DELCFM(x>-x-100.
DELXTxaxx-,4
A7=xCA.stLCON
GOTOI10.15.20.101.IGO

C RRODUCT TEWPERATURF CALCJLATIONS,

10 DTH:,3RSS¢OELTR(A6)
DTI.6007'D&LT(TA‘B)
DH=264.9'DELH(HAHBI
UH:71.93'DELM(A5)
DCFM=-.44520OELCPH(A7)
DXLI-1R,98*DELX(A§)
A3:DTH¢DT+UH¢UM+OCFN*DXL*80o459
IF(IGO.NE.4)°ETURN

C HUMIDITY CALCULATIONS.

15 DTH8.00003667aDhLTH(A6)
DT=.COuclsoaoDFLTITAMBI
DH:.9179'DELH(PAHBI
DMx.02305o‘ELM(ASI
DCFM:5,474E-7tDELCFMTA7Iio2-6.1lsE-5*DELCFMIA7)
OXL=.00078930UFLX(A8)
A1=DTH+DT6DH+DP+DCFH¢DXLO.00772
IFTIGO.NE.4)RETURN

C AIR TEMPERATURE CALCULATIONS.
UTH:,9R2*DELTH(A6)
DcFH=2.035-<1.-EXP(.02190vDELCFM(A7))I
A2=DTHoOCFM+137.77

C "C CALCULATIONS.

20 DH=.9635*DEL“(A5)

DXL=-.u[8260D&LX(A8)
A4:DM.OxLo.20555
RETURN

END

SUBROUTINE COSTFUHICOST.TINLET.O)

c SUBROUTINE COSTFUM EVALUATES TRE PROCESS COST.
COMMON/ECOU/FAUETA.EHETA.THERETA.ELP9ICE.FUP9ICE.TAHR.FACT0R.ELC0§
lT.FULCOST{OT/DFPTN.TIHE

147

20 ”DROPBDEPTHPTQISB.I0‘1.528TDELT'.378548895899'PDROPIFANEYA/EHETA
ELCOST=ORPOROPcTIMEtELPRICEfii.1748031496062-4/EMETA/FANETA
FULCOST:FUFRICEOO'FACTORo(TINLET-TAHB-DELTI'TICE/THERETAPOOT
COSTIELCOST¢FULCO§TSRETURN
END

SURROUTINE cnsruugccosr,poqon)
c COSTONE EVALUATER THE COST or THE DRYING PROCESS.

COMMON/DOLLA°ITI“E.ELPRICE.FJELPRI.FUELHToEMETA.FANETAATHEREYA/SIH
1c05T/D.TAMR.TIN/IN/SUNTENP.HAND.DIN.DEPTH/VALUESIXHCIN.THIN.GP.5A.
2CA.CP,CV
ELCOSTxotPnfinpoTINE'ELPRICE'l.174803149606E-4/EMETAIFANETA
FULcosrsnIU.ICA+CVoRAnaIccTIN-TANDI/FUELRT/THEHETA-FUELPRItTIns
COST=ELCO$T+FULCO§T

RETURN

END

FUNCTION CDSTRTTOI

C FUNCTION COSTRT Is U§ED BY 0PTHRIZ To DETERMINE ISOCOST SLOPE.
COMMON/Ean/FANETA. EUETA. THERETApELPRICE. FUPRICE. TAMR.FACTOR. ELCO§
1T. FULCOSTZOT/DFPTH. TINE/CT/COSTLFT. TOFF
pDROPIOEPTHPTG/b".)P‘1.528TDELT3.378548695899GPDR0P/FANETA/EHETA
ELCOSTao‘PDRnPoTIME'ELPNICE'l.1748031496U65'4/tHETA/FANETA
FULCOST-FUPRICE'U'FACTDR0(TUFF PTAHBPDELTI'TIVE/THERETA06OA
COSTRTscosTLFT-ELCOST-FULCOST$RETURN
END

FUNCTION COVERTT)
c FUNCTION Coven IS-USEO av OPTHHIZ To FIND THE INTERSECTION or THE
c ISOSTERE AITH A TEMPERATURE BOUND.
COMMON/FINAL/XMIINAL/UT/DUEATEE
CALL OUICKIAVEPCOB)T.UUE)SC0VERBAVEHCDBBXHFINALSRETURN
END

\

sunanuTINE CSTCOOL(COST)
C CSTCOOL EVALUATER THE COST OF THE AIRFLON WHRU THE COUNTERFLOR COOLER-

CUHMON/FIXED/TA quHAHB.ATA“8. SPVLCON/PRICE/ELP”ICEoTIHE. FANETAoEMtTA
1TA/INCROL/PRCUM,THETA.GA.DEPTH

O=CAcsRvLc0N

RuuopzwsrTuoIU/Se.I-P1.526
COST=00PDRUPOIIMFOPLPHICE'Io748031496065'4/EHEIA/FANEIA

pETURN

END

148

FUNCTION DEEPHIMIUI
C DEEPHNH IS USE') IN A 1- 0 SEARCH TO FIND THE BED DEPTH AT HHICH THE HAxInUH

C ALLOHAHLE ouTLET MC UCCURS.
COMMON/TR!AL/THUUINA.XMOUTMXIINCOOL/XHOI.XTH.XGAoXDP/OUTCOOLIHOUTo
1TOUT.THOUT;XMUUT
xDP=D$CALL COOLSIHISI
DEEPHNwaxHOUI-xHDUTHx
RETURN
END

FUNCTION DEEPMNTTD)
C DEEPMNT Is USED IN A I-U SEARCH TO FIND THE OED DEPTH AT NHICH THE MAKINJN

c ALLowARLE nuTLET PDUDUCT TEMPERATURE OCCURS;
COMMON/TRIAL/THUUIMX.XHUUTWX/INCOOL/XMOI)XTHDXGAIXDP/OUTCUOL/HOUTO
1TOUTITHOUT;XNUUT
XDPEDQCALLCOOLSIH(1)
DEEPMNTsTHUUT-THUUTHX
RETURN
END

FUNCTION DIACIT)
FUNCTION DIAG IS USED BY OPTHHIZ TO LOCATE THE INTERSECTION OF THE

C ISDSTERE HITH THE DIAGJNAL OF THE FEASIBLE REGION.
COMMON/FINAL/XNFINAL/STUFF/DIAGSLP QHIH. THIN UNAX.THAX
UsnIACSLPoIT- THIN)+QMIN$CALL OUICKIAVEMCDBoT 0)
DIAOIAVEMCDB-thINALTRETURN

END

7')

SUHROUTINE DPYTUST(COST.PDRUP)

C DRYCOST EVALUAT89 THE COST OF THE DRYING PROCESS.
COMMON/DOLLAP/TIHEoELPRICEoEHETAoFANETA/SIMCUST/Q.ZZ.ZZZ
COST=QOPDROPOTINE'ELEHICF’I.174803149606E'4/EfltTA/FANETA

PETUPN
END

sUHRDUTINE DDYUE:TTAVERACE.KIP.LMN0PIINDEX)
C DRYHEAT Is CALLED rHoM INTERP T0 PERFORN A LINEAH INTEPPOLATION TN 0"? U’"'

C ENSION.
DIMENSION INDEx<3IILNHoP<3I
COMMON/RECALL/OPICOSTI . .

icDELTH
STINTRF(XONE.XTNUOXX3XXXpDX)3X0NE*(XYHO'XONE)"XX'XXY)*DX

LHNopcle):IUUFXIKIPI
X0NE=ODTCOST(LPNU?(1)pLNNOP(2)oLHNOP131)

).XNEHKZ)oDXINV(2)aXHLONoDELHoTHLOH

149

LHNOP(¥IP)IX“DEXTKIP)‘1
xTHUIOPTCOSTILNNOPTIJoLMNOPI2)oLHNOPT32)
GOTOTlO.20)onP

1o vaLsxwLOH¢FLUATTINOEX(1)-1)*DELH
AVERAGE-STINTRP(XUhE.XTN0,XNEH(1)'XVALeDXINVK13)
PETURN

20 XVALxTHLCH¢FLUATTINDEX(2)-1)'DELTH
AVERAGF=STINTHP(XUNE,XTNO,XNEH(2)oXVALoDXINV(23)
RETURN
END

SUBROUTINE DOTSIN<IGO>
C DRYSIH PREDICTS THE OUTLET GRAIN MOISTURE CONTENT AND TEMP PLUS THE OUTLET
C AIR TEMP AND HUMIDITY.
COMMON/CON/CON1.CONZ/VALUES/XHCIN.THIN.OP.SAuCA.CP.CVISIMCOST/AFaZ
12.ZZZ/IN/TIN.H1No§A.DEPTH/OUT/XnaTHETAoTOUToHOUT
GOT0(10.218).IGU
C OUTLET MC MODEL.
10 ch,69.cAc.,49
DF'HC-CONl
DM:1.-(HCosAoDFPTH/GA/CA7(10."(-1.531*AL051C(UF)*5.92)))*'((( 3:9
177-AL0310(UA))l1.n73)'(77.36/(UF‘111o9))*‘629.95'TIN)I428.5'(1.12f
2XMCIN¢,59I-(43?.Eb-THINIISB4.61)
C OUTLET PRODUCT TFMEE°ATUPE MODEL.
AT!N=T!N¢459.69 -
HHIN:RHDBHAIITIN.NIN1
XHE.EHC(HHIN.TINI
XMSDH0(XMC!N-XNh)‘XHE
THFTA3150.89C85'DEPTH'¢T-O.13458986)
IFITHIN.LE.7S.IGCTOZD
SLOPEIO.465366360DEPTH0'I'D-28550254)
GOTO4O
20 SLOPEIU.46493212'DEPTH**(-0.26856385)
4O DELTEHP=SLOPF*(THIN'75.)
THETAzTHETA+DELTENP
lF<TIN.LE.35Cn)GOT060
SLOpExu,197236?b¢UEPTHtt(-o.3155996)
GoToao
so SLOPE=D.25331944oUEPTH-nI-o.26230775>
80 DELTEHDESLOPCvITIN-SQO.)
THETAETHETAofitLTEHP
IFIAHCIN.LE.C.30)COT01OO
DELTEM98-1?8.bo(XHCIN-OoSO)
GO TO 122
100 DELTEHD:-183.0(XSOIN*O.30)
120 THETA:THETA*"ELTEHP
IFIHIN.LE.L.304)GUTOI40
DELTEHoz157.2727o(HIM-0.004)
GOTOlbU
140 UELTEME=205.O(Hl”-C.OO4)
160 THEThz'HETA‘OELTEHP
IFIAF.OT.150.)CUTO1UO
SLOPEsu.436715L10DFPTH-*(-0.28665312)
GOTOZOU
180 SLOPEaa.33196071«DEPTHo-I-o.16561c73I
200 DELTEHPzSLOPF'tAF'150.)

150

THETA-THETAvnELTEHP
DELTEMps-ls.7136H4'DEPTHODC“0.37998875’
THETAaTEETA+DELTENP

RETURN

C OUTLET AIR TEHPEPATURE MODEL.

210 Tourzl1c,2.A=.74¢Expt-o.5492o0EPTHI+37.39~(EXPT-3.351*!XHCIN-D.3)l-I.)*0;2
1-1.)00;2025*FXP(-.1275tDEPTH)t(TIN-SSC.)+,6$55'EXP('0.1348*DEPTH)!(141v.25
2(THIN-75.)‘o4050*FXPT'091575'DEPTH)'(AF'1SUI’

C OUTLET HUMIDITY “ODtLo

TEMPURY:0.31710ALUG(DEPTH/0.178)+(Oo0055‘DEPTH‘:41)*(X”CIN'0o3""
14,4965.6v(DERTH-4.E3x)¢t2+1.14SE-4)t<TIN-350.)‘1.208E-4RDEPTH9'035
24319(THIN-75.)+0.9CD‘(HlN-.OO35)

IFTAF.GT.1SO.)GUTO?20

Hoursrgnponvo,378E-4CIAF-150.I

RETURN

220 HOUT=TEMPORY6(-4.1SE-6-(DEPTH*1.155)"2#9.594E-5)*(AF-150o)

RETURN

END

FUNCTION EHCIHH.TI
C FUNCTION ENC IS USED TO DETERMINE EQUILIBRIUM HOISTURE CONTENT As A
C FUNCTION OF TENREpArURE AND RH.
IF(RH-.50)303o300o30°
300 F18-.0003922oT+.1$F2c-.0004353ET+.1328IF3:-.0005359tT+.164¢$S1:131
1835t(-9,*F1+6.0F2-731$52=13.BSBOT4.0F3v9,tF2*6.tF1)SBSRH-.17
IFTB>301o331.302
301 EMC =(510RH09H9RH/1.02¢(F1l.17'51*.02833)'RH)5HETURN
302 IFIRH-.34)303,303.304
303 A 9 .34-RH
EMC 2(81'A'A0A/1.02052'83B*B/1o02*TF2/.17-52'.92833)*8+(Fl/o17-51'
1. 02633) 'A)$PETURN

304 A8.51.WH
EHC I SZ’AOAOA/1.UZ*TFSI.17)*(RH'.34)*(F2/c17'52'n023333)’A5RETURN

309 TO: -.0005373oT+,16?4TF1=-.0007075vTo, 2075$F2=-;0007449tT+.2532
F33'0001C71.T*.5°51551813.8380(4.0F0w9..F1¢6.gp2-F3)
5? 8 13, B38 0‘4,VF3"9. OF296,0F1-FU)$83RH-.66$1F(3,305.305'306
305 A=RH-. 49
EMC: 51"'A"/1 0?‘(Fl/.17-Sl'.02833)‘A*(F07.17Io(.66..RH)5REIURN
306 IFTRH-. 83)307.307.3Da
307 A=.83- -RH
EMC: SlvoAwA/l, 0205258a838/1,02¢(F2/,17. .52..02H33).9.(;1/.17 51.
1. 028353)*A SRETURN
308 A:1.0.QH
EHC:52“‘A‘A/1.02*(F3/.17)0(RH'.83)+TF2/.17-320,029333)«A$REIUQN
END

FUNCTION HADRHHIOU.RHI

C HAOBPH IS USED TO FIND ABSOLUTE HUMIDITY.GIVEN THE DRY BULB TEMP AND RH,
HADBRH:HAPV(PH0PSDP(Dd)) SRETURN
END

151

FUNCTION HAPV(PV)

C HAPV IS USED TO VINO ABSOLUTE HUMIDITY. GIVEN THE VAPOR PRESSURE.

COHHON/PRESS/PATN
RAPv-.6219-PVIIRAIR-PVI : RETURN

END

FUNCTION HLDR (08)

C HLDB IS USED TO FIND LATENT HEAT. GIVEN THE DRY BULB TEHP.

501“)”

IFID9-491.69) 1.2.?
HLDB:1725.864-.05n77IIDa-459.69>5RETURN

IFIDR.609.69I 3.4.4
RLDR=IU75.aqss-.5ooes-<09.459.69I5RETURN

HLDB=SURTT13546739214’.91252755879DB‘DPIERETURN
END

FUNCTION HGHOEEPID)

C HOHDEEP IS USED IN A 1-0 SEARCH TO FIND THE BED DEPTH AT HHICH THE CURRENTLY
C SPECIFIED HUNIDITY OCCURS.

COHHON/OUTCODL/HDUToTOUToTHOUTpXHOUT/ATTCH/HNUW/INCOOL/PRODHoTHETA
1,6A,DEDTH

DEPTHaOSCALL CDULSTHTZI

HOHDEEP-HOUT-HNUH

RETURN

END

SUBROUTINE IENCUUETINDYDINDoHAonDIM)

C IENCOUE IS USED TO CONFINE THREE INDICES INTO A SINGLE INDEXS

30

DIMENSION TND‘S’oNAXTS)
1NDY:IUD(1)

MAXSIZE=HAX(1)

DO 30 V32; anv
INDY=IVDTN)‘“AXSIZE-MAXSTZE’INDY
NAXSIzE=NAx5Ich~AxINI

RETURN

END

SURROUTINE THTFHP‘VALUEDNDIMOINDEX:ICECREHONST”HEDDUHMY’

C INTERP Is USED TO SET HP INTERPOLATION IN FOUR DIMENSIONS OR LESS; FOR THIS
C DROORAN, T40 DIMENSIONS ARE INTERPOLATED.

126

DIMENSION INDEY(5)aICECREHCSIILMNOPISI
SUH30.J

DO 175 KIR=1.NnI~

DO 125 N=13N0IR

LNNORIN>=D

1?7
128

1285
129
130
131
132

1325
133
134
135

1355

1357

136
137

138

1385
139
140

141
142
143
144
145
146
147
148

149

150
151
152

1525

1527

153
154

155

1555
156
157

158
159
160

152

KI1

HATSTOPsMRKSTOP-LUKSTOPEO

D0 129 JJsloNDT"
lFtJJ-ICECRE”(K))129.127.129
TFTJJ-KIPIIZRoI7SI128

LHNOP(JJ)IINDEX(J4)
IFIK-NSTORE31295913001285

KIK¢1

CONTINUE

D0 174 1810?

IFII-2I1323135.132

MATTHEHIO

K81
IF(LMVOP(K).TNDh“K,,13301370133
lFIK-KIP)134,1370134

HATTHEUIK

LMNOPIHATTNEUIIINUEXTHATTHERI-I+2
IF(HATRTOP-1)136.13550136

CALL DNNMYIAVERAGE.KIP.LHNOP.INDEX)
IFCAVERAGE'lc0E23713570174017§
IFIAvEnAce-O.031745.174o1745
IFII-2I137;142.137
IFIK-NDINI13°.138o139

MATSTOPI1

CALL DWHHY(AVERAG§DKIP0LHN0P0INDEX)
TF(AVERAGE'IC9‘23)138501740174
IFIAVERACE-0.0I1745.174.1745
1F(HATTHEH-D)141014Dol41

KIK+1

GO TO 1325

KQK¢1

DO 173 11.102
IF(I-2I1443152}144
lFTlI-2)14501520145

MARK=0
IFTLHN0P(K)'TNDEXTK,315031470150
IFIN-NDINI14R:149I148

KIK¢1

GO TO 146

HATSTOPI1

GO TO 174

IFIK-KIP’151.154o151

MARKzK

LNNOP‘MARK)8]NPCK(HARK)-I102
IF(NRKSTOP-13156o15250153

CALL DHMNYIAVERAGE.KIP.LNNOP.INDEX)
IF(AVE?AGE-13.EZS)15270173.173
IFIAVFRAGi-C.C)1/45017501745
IF‘ll-2)15‘01590154
IFT‘-NDI”)156p1550155

HRKSTopzl

CALL DHMHY(AVERAGE.KIP.LHNOP,INDEX)

IF(AVERAGEPICogzs)155501730173
IFIAVERAGE-D.3,1745o17301745

IFTHARK-9)15“ol57(15fl

KIK¢1

GO TO 146

K8K¢1

DO 172 1118102

IF'I’2,1601016501601

1601
1602
1603
1605
161
162

163
164
1645
165

166
167

16R
169

171
172
173
174
1741
1742

1745
175

176

153

lFTll-ZI1632o16501602
IFTITI~ZI1603o105o1603

LUKEBO
IF(LHNOP(K)'TNDhXCK))1640161o16‘
IF(K-NOIH)16?91°3o162

K2K+1

GO TO 1605

MRKST0931

GO TO 173
1F(K-KIP)1645,166p1645

LUKEPK
LHNOP(LUKEI:INDEX(LUKE)'T11*2
IF‘LUKSTDP.1,166D1690166
lF(K-NDIN)1670155(162

KIK¢1

GO TO 1605

LUKSTOPcl

CALL DNMHY(AVEPAGE)KTPoLHNOPoINDEX)
IFTAVEQAGE'lc0523,17101723172
IFIAVERAGE-0.DI1745o172p1745
CONTINUE

CONTINUE

CONTINUE
IFIAVERAGE-0.031792.174lol742
VALUE'OOO

RETURN

VALUEaa.0

RETURN

SUM-SUMoAVERAGE

CONTINUE
VALUESSUHYFLOATTNDTH-NSTORE)
RETURN

END

FUNCTION OFFDIAOIO)

c FUNCTION orraIAc Is USED av OPTARIZ TO DETERMINE ISOSTEPE SLOPE.

COMMON/FINAL/XVFIHAL/ENUT/TONE
CALL OUICK‘AVEHCDabTONEDO)SUFFDIAG=XHFTNAL-AVEOCDBSRETURN

END

SUBROUTINE OPTNHIZ

SUBROUTINE OPTHHIZ CONTROLS Z'DIVEHSIONAL OPTIMIZATION

COHNON/ICHECV/IFLAC/DT/DEPTH.TINE/FINAL/XHFINAL/OT/OUE.TEE/STUFF/U
llAGSLP.ONIh.TNIN.UNAX.TNAX/CT/COSTLFT.TOFF/EOUwC/XNSND,BRON/ENOT/
2TON5/EcoN1FANETA,EHETA,THERETApELPRICE,FuPRICE.TAN8.FACTOR.ELCOSTo
4FULCOST

exTERNAL COVER,SIOE.DIAG.OFFDIAG.COSTRT$KOUNT=NSDIFFP1002iEPS=1.E'
17$EPszz1.Ee4

SURVEY POISTNRF CONTENTS AT EXTREME POINTS

CALL QUICKIAVENCDE.THIN.DNIN)ETMNONNzAvENCDBIPRINT 20001.THIN.ONIN
10AVEHCDB '1

CALL DOICK(AVEHCDS,THAX.OHIN)!THXOHN=AVEHCDBTPNINT 20001.THAX,OMIN

12
14
155
17

195
20

31
33
334
337

60
70

81
83
84

86

120
130

141
143

144

154

1.AVEHCDBTCALL OUICK(AVEHCDBoTHIN:OHAX)$T”NOHXIAVEHCDRSPR1NT 20001.
2THINoOWAXTAVEHCUHSCALL QUICKIAVEHCDB.TMAx.oMAXIITMxonxxAvEMcng
CHECK LH.TnP,NuoHOTTOH aOUNDS SEQUENTIALLY FOR FINAL AVEMC 1505153
PRINT 20001.TNAX.UHAX AVEMCDaTIFIXHFINAL. GT. TMNUHN, 0R.XNF1NAL LT: I
1MNOHXTGO T0 90 SGUESSlzurlNSSUEssztoHAxsTEEsTMINTCALL ZEROINA(GUE§
ZSLOGUESSZIEPcoSIRE)3025R03(GJE551*GUESSZ)/2.ICALL OUTCK(AVEHCDB TH
SIN 025:0)EKOUNT-KOUNT+1$DELTI.1
TEE-TH!N'DELTEGUE551:0%INSGUEssstMAXICALL ZEHU‘NA(GU6551OGUESSZ'§
lPSTSIDE)*IF(TFLAG.NE.1)GOT0 14SDELTsDEL7/2,scutu 12
00VE=<GUESSI*GUES§2>12.55L0PEN=<cows-CZEROIIDELTIDELTa.msCALLcosTE
1UN(COSTLFT. T”IN OZERO)

TOFFBTHIN+DELTthUhSSlzaflAXTG1E55230MIN$CALL ZEKOINATGUESS105UESSZO
1EPS, COSTRT)$IF(1FLAG.NE. 1)GOT0 17IDELT:DELT/2 $5010 155
ORIGHTEIGUESS1+GUESSZTl2.$$L0PECt(uRIGHT-QZEROT/DELT

PRINT 20002.7MIN.92ER0

PRINT 3C002.9LOPEH.SLOPEC.COSTLFT

IFTSLOPEC.GT.SL0P§H>60 TO ZOSIF(BNDH,LE;XH8ND)60 To 195$RRINT 2001
10,8NDM;XHBND

COSTPBUECOSTLFT/DEPTH'I.assPRINT 20004.c05TLFT.COSTPau.TnIN.ozERo
GO TO 70

IF!XMFINAL.GT.THNUNX.0R.XHFINAL.LT.TMXONXIGC T0 70%GUESS18THIN%GUE
15528TMAX$QUExuwAX$CALL ZERO!NA(GUESSIoGUESSZoEPS:COVFR)STZEROI(GU:
2551¢GUFSSZTI°.TCALL OUICK(AVEHCDB.TZERO.CUE)*KUUNT:KOJNT¢1$DELQI;1
OUEIQMAX‘DELC$GUE$51=TWINSGUESSZITMAX$CALL ZERUINA(GUE551IGUESSZ'E
1PSOCOVER7$IFTlrL‘u.NE.1,GOTU 33!DEL0:DELQ72.$GUT0 31
TONE!(GUESSloGUESS?)/2.SSLOPEMODELO/(TZERO-TUNh)SDELTa,1sCALL COST
1FUNIC09TLFTaTZER0.0HAX)
TOFF'TZENOoURLTSGUF531'dHAXSGUESSZ'QHTNSCALLZEHUINA(GUESSlteUESSZO
1EPS COSTRT)$TF(1FLAG. NE. 1)SUT0337!DELTIDELT/2 $6070 334
ORIGHTaIGUFS<1+GUh552)/2. TSLOPECI(ORIGHT-QMAX)/DELTSPRINT 20002.TZ
1ERO. OHAKTPRTHTsocnz.SLOPEH.SL0PEC COSTLFTSIEISLOPEC. GT. SLOPEH)GO
1T0 70

IFTBNDH.LE.XMBHD)GO T0 60$PRINT ZCOIOoBNDHoXMBNDSSTOP
COSTpanscosTLFT/HEPTn-1.25$PRINT 20004.COSTLFToCOSTPnU.T!ER0p0NAl
IFtXMFINAL.GT.THXQHN.0R.XHFINAL.LT.TMXNMX)GO TH 13n$GUESSIBUM1N
GUESSZ:OMAXITEE=TMAXSCALL ZEROINA(GUESS1oGUESSZoEPSoSIDE)SOLEBOI
1(GUESSI*GUE592)/2,TCALL OU!CK(AVEMCDB.TEE.QZERU)TKOUNT=KOUNT¢1
GUF581=OMINIC UFSSZRJHAXSDELTs. 1

TEE8THAX'DFLT CT;ALL ZERQINAIGuess1.cuessz.EPs.sIuE>sxrIIFLAG. NE. 1)
160 To 93 s DELTstLT/Z. T 60 To 81
GONE=<GUESSI¢GUES§2>/2.TSLOPEN=IozERo-OONET/DELTIDELT.,1

CALL COSTFUNTCOSTLFT.THAX.OZERO)
TOFFzTNAX+DELTCGUQSSI=OHAX$GUESSZ=ONINTCALL ZEHUINATGUF531oGUESSZv
1EPS.COSTRTI$TF(IFLAG,NE.1)GUTO 86IDELTtnELT/2.$GOTO 84
ORIGHTs(GUFS?1¢GUESS?)/2.ISLOPEC=(BRIGHT-OZEROIIDELT

PRINT 20002.Tnax.uzeao

PRINT 3000203LCPEH55L0PCC.COSTLFT

IFISLOPEC.LT.SLUPEM)60 To ISQIIFIBNDM.LE.XH5N0IGO T0 120

PRINT 2cuas,:NfiH.XWENUTSTOP

COSTPBU=COSTLFT/ULPTA.1,25

PRINT 20004.?USTLTT.COSTP8U.Twa.ozEROTsTop
IF(XHFINAL.GT,THNNMN.OR.XHFINAL.LT.TMxOHNTGO Tn 19g3nupsglstq1~
GUEssstHAXIOchnhINTCALL ZEROINATGUE581.GUE852.EPS COVERTTTZERos
1<GHE551onu -S¢2I/?. TCALL 0U!CK(AVEHCDB TZERO.NUEITxounI-KOUNT+;
DELo=.1 IVRTNT ?03 n2 TZE R0 GWIN

DUE=°HIN*UEL°1°UF§SI= T INIGUE552:THAXICALL ZEROINAIGUE551.cuEssz,
IEPS.C0JER)IIV(T‘LAG.NE.1)GOT3 14SIDELQ=DEL0/2.TGDTQ 141
Toua=(auESS1.GuE582I/2.£SL0PEN=DEL0/ITONE-TZEHNTTDEL13,1

CALL CWSTFUNTCWSTLFT:TZtR0.0WIN)
TOFF=TZERO-DFLTSGNE551=UMAXISUESSZ=OMTNTCALL ZhRO!NA(GUESS1.GUES$2

155

1aEPS.COSTRT)TIFIIELAG.NE.1TGOTO 146$OELT80£LT123$00T0144
146 OLEFT:IGUESS1onUF§SZIIZ.SSLOPECBIQHIN-DLEFTIIDELT
PRINT 30002.9L0PEH;SLOPEC.COSTLFT!IFISLOPECTLTT SL9
1PEH)GO T0 19CSIFIBNDH.LE.XHBND)GO TO 180IPRINT 20003.8NOHTXHBNO
STOP
180 COSTPBU:COSTLFT/PEPTH01.25TPRINT 20004.005TLFTTCOSTPRU.TlEROIONIN
STOP
190 IFIKOUNT.E0.0)200g210
200 PRINT 20005
STOP
210 IFIKOUUT.NEo?)?20¢23O
220 PRINT 30003$STOP
C THERE IS AN INTERIOR OPTIHUW. BEGIN SEARCH FOR IT.
230 GUFSS12TMINSOUFSSZITHAKSOIAGSLP:(OMAx-OMINIIITMAX-THIN)
CALL ZEROINAIGUES§1aGUESSZ.&PSTDIAG)SIF(IFLAG.NE.1)GO T0 ZSOZSSTOE
2302 TZEROsIGU5551+CUE$$2)/2.$OZERO=DIAGSLP*(TZERO-THINIOOMINSDELT8,1
CALL OHICKIAVEMCOB;TZERO.OZERO)SIFIBNDM;L5.xH3N0)00To 231
CHIN=OZEROITNAXITZFROSGOT0230
231 GuESS1:OHINSOUFSSZIOMAXSTONE:TZERO oDELTSCALL ZEROINAIGUESS1.
1GUFssz;EPs;oerIAOI
IFIIFLAO.NE.1)00 IO 233£OELT80ELT/2.SGO To 231
233 OOME"GUESS10GUES§ZIl2.£$L0PEH'(OONE'OZEROI/CTUNE'TZEROISCALL COST
1ru~¢aosrLrT,TZFHO.CZER3)TOELTaoi
234 TOFFITZEHOODFLTi89253180HAXIGUESSZ'OHINSCALL ZEROINAIGUESS1oGUESSZ
1IEPS.COSTRT)TIrIIILAG.NE.1IGOTO 237SDELT:DELT/25550TO 234
237 NHIGHT:(GUES°1+UUESS2IIZISSLDPEC'(ORIGHT-OZEHOIIDELT
IFISLOPEC-SLOPPH)272.275.270
270 PRINT 20008,TZFHO,OZERO.SLOPEHISLOPEC.COSTLET
THstrzeHQIONAXIOZEROSGO T0 274
272 PRINT 20009.TZEHo.ozERo.SLOPEMTSLOPEC.COSTLET
THAX=TZER0$O“I“IO£ERO
274 CURDIFFIABSISLOPEC-SLOPEH)TIFIABS(CURDIFF-DIFFI:GT.EP52)GO TO 278
275 COSTPaUxCOSTLFT/OEPTflci.25TP9INT 20011oTZEROoQ£ERO
PRINT 30011oCUSTLET.ELCUST.FULCOST.COSTPBU
STOP
270 DIFFscnPDIFFTunTozsn
20001 FORMATI3X.¢I”LFT AIRTLHP.OEGF I fiaF6.4.3x.0AIRTLOH RATEOCFHIFTZ 8!
1,F7,4,3X30F1”AL AVEoMcooDECoDoBo 3'0F703’
20002 FORMATISXo’FINLL AVE 0c ISOSTERE caosses BNO AT INLET TEMP . 9,;51
14,3x.oAIRFLON DATt :6,F7.4)
20003 FORMATtsxl'Tut no OF THE TOP LAYERT'TF7.4o2xotHAS HOT REACHED THE
1MAX. PFRMISSIELE HC.5.F7.4I
20004 FORMAT(3X,.A“ DPTIHUH CUST.0,F7.4.2X,ECTS/FT2 URO.F7.4,2X,¢CT§)BU
10ccuPs AT INLET T§HP=*.F8.4.2xnfiAIRFLOH PATE "oF8.4o'CFH/FT2f)
20005 FORHATI3X,GIT IS IMPOSSIBLE TO REACH THE DESIRED nc NITHIN THE STA
1TED BOUNDS.)
20008 FORMATI3X,OTHE UPIIHUH UCCURS AT T.GT.O,F8.4I*OQ.LT.90F8.4D‘.HC SQ
1OPE 8..FR.400.FUST SLOPE :0,F8.4,t.COST :0.F8.4)
20009 FORMAT(3X,0THE UPTINUH OCCURS AT T.LT.*IF8.4ovoO.GT.*:F8.4»P.HC SL
10PE x-.F8.4,o.CUST SLOPh :-.F5.4.-.COST c 9.78.4)
20010 FORMATI3X.0TNE no or THE TOP LAYERaPaF7;4,2XotCANNOT REACH THE MAX
1. PERMISSISL' VC.*IF7.4.6HITHIN THE STATED BOUNDS.)
20011 FORMATISXoOTVE OPTIMUM OCCJRS AT T 3'0r8040'tu 8':F8.4.'CFH/FT2.?I
30002 FORMATI3x.oISUSTFHE SLOPE s c,F7.4,2x.oISOCOST SLOPE 3.,r7,4,2x,
1-ISOCOST VAL“t.CT$ 8 ..r7,4)
30003 FORAATI3X,0THE IINAL Hc ISOSTEHE CROSSES OTHER THAN 2 BOUNDS-1THE
IMETHOO FAILS.)
30011 FOPHATI3X.°TCTAL GUST/F128 '.F8.4o0 ELECT.CUHPUNELT8 ’0F894030 L
1UEL COMPONENT oaf§.4 p 0 COST/BU 8':F8.4. POTS.)
END

156

FUNCTION PSDR (DB)
C PSDB IS USED TO FIND THE SATURATION VAPOR PRESSURE: GIVEN THE DRY BULB QEH’:
DATA RIAoBQC.D}toTiG/.3286182232E04,~.274055258361426505..54189607
A6328951E32;o.4§137r384112655E-1p.215321191636394E-4.-.462026656819
B952E-5,.2416127209874501..1215465167060555'2/
IFIDB-491.69I 10202
1 PSDBSEXP(23.39P4-11286.6489/D8‘.46057OILOGTDH)TSRETURN
2 PSDB'R'EXPI(APDUPIP'DB'(CODBO(D*DB'E)III/(DB'IF-GDDB)IISRETURN
END

FUNCTION PVDPNRIDBIARI
c onawa IS USED To FIND THE VAPOR PRESSURE. GIVEN THE DRY BULB AND NET PvLa
c TEMPS.

COMMON/PRESSIPATM

AcPSDBINEI S 82.62194PHLDBIHBI'PATH : c:.2405-TA-PATP)-<wB-DB>

PvDaHRsIAnn-C'PATHIIIB+.155770CJ SRETURN

END

FUNCTION PVHA (H"
c PVHA IS USED TO FIND THE VAPOR PRESSURE. GIVEN THE A85. HUMIDITY.
COMMON/FPESS/PAT”
PVHAquoPATH/(,6219¢HAI SRETURN
END

FUNCTION PVTDITHYPVI
c PVTG Is USED To LUCATE THE VAPOR PRESSURE AT HHICH EQUILIBRIUM no OF THE Gath
0 000098.

COMMON/PRESS/PATM/ANN/PSNB.HPFG.ATNBTXMZERO

TEMP3ATkH-(PShn-TRYPV)tHPFG'?.586029106029/IPSWB'PATH)/(1o*0.15577

1.Tnypv/pATu)(TnvuthRYPVIPSDPITEMPIsTEHPrzTEHP-4S9.69

vagszZERO-PHCITNYRH.TEHPFISRETURN

END

SUBROUTINE QUICKIAVEHCDE.TINLET.O)
C SUBROUTINE QUICK SIMULATES THE BATCH-IN-BIN DRYING PROCESS
DINEMSICN XMCOPI21)
COPMON/PPESS/PAT“/ICHECK/IFLAG/DT/DEPTH.TIHE/BOUND/XMBND.BNUM/AHBI
1PSH8.HDFG,AT“H}XMZER0/OTHERIDELX.PV
EXTERNAL PVTG
ATINLETtTINLET¢459oo9$PSAT3950H(ATINLETI'RH'PV/PSATSHARstHADBRHIA].
1INLET,RH)$ATVB:HPUP4A$(ATINLETOHABSo49soo55Ooo001’
TEHgATue-459.59 TPSNH:PSDBIATH6) S HPFGsHLDBIATNB)
IFIEHCI1..TRPI.LE,anER0)1,2
1 T0=Twa s 00 TO 3

25
26

65
650
10001

157

GUESS1:.1SGUPSSZIPSNRSCALL ZEROINIGUE581AGUESSZ.1.6-eanT0)
IFIIFLAGAEQI1’25026

PRINT 10001

STOP

PVATTG:(GUESS1OOUESSZI/29
TGITHB-(PSHB-PVATIG)‘HPFG'Z.5860291060297CP$W8'PATHI/(13'0o15577

1'PVATTG/PATH)

XNEzenctRHzTINLET)
DELM=XWZERO-XHE :v:(1094.-o.57-TINLETT*(1.*4.39*EXPI~1412.5PDELH>I
VONE-v.(TINLFT-70.)0I1.*0.2/DELM)
XKI-.0456453°*.0055836759rPSAT'0.33804141'0'°o940613420

stx-TIHE T x-9.o T 00 65 KKI1021
DD-VONP.xR-DELN-x/O/ITINLET-TC)¢22.19316276535
RATIOanPIDDI/IEXPTDDToExPIYT-1.I 1 XHCDFIKKIBHATIOODELfioxHE
x:x.DELngNDN=XNCUUI21ISSUH=TXHCDBI1I+XMCOP(21I)/2.$00850LL?2.20
SUN-SUW+XMCDGILL)SAVENCUH=SUHPUELX/DEPTHTRETUHN

FDRNATISX.-TNE HODEL CANNOT HANDLE THIS CASE')

END

FUNCTION RHPSPVID1302)

C RHPSPV IS USED TO FIND THE RH, GIVEN THE PARTIAL AND SATURATION VAPOR PRESSURE

(I C)

”(I

A-D1 s B=02 c 00 10 x
ENTRv QHDBHA

AIPSDBIDII s BaPVHA(02)
RHPSPVIB/A

RETURN

END

FUNCTION SIDFIO)

FUNCTION SIDE IS USED BY OPTHHIZ TO FIND THE INTERSECTION OF THE
ISOSTERE WITH A TFHPERATURE BOUND,

CONMON/FINAL/XNFINAL/UT/OUEATEE
CALL QUICKIAVENCDUATEEAU)SSIDE'AVEHCDB'XHFINAL$RETURN
END

FUNCTION VSDPHA IDB.HA)

VSDBHA IS USED Tn FI~Q THE SPECIFIC VOLUME; GIVEN THE DRYBULB TEMP AND 595.
HUMIDITY.

COMMON [PRESS/PATH
VSDBHA=53.SSOUP'(96219¢HA)/144./.6219/PATH$RETURN
END

158

FUNCTION HEDDHASIofioflA.Gl.GZoEPS)

C NBDBHAS IS USED TO FIND THE NET BULB TEMP,, GIVEN THE DRY BULB TEMP. ANU AQS;
C HUMIDITY.

C uBL

EXTERNAL NFL
COMMON ISPECIAL/PVETB
AIGITBIGZDTBIUFSPVIPVHA‘HAIDCALL ZEROIN(A;B:EPboNBLISHBDBHAS8‘A09

1)/2.$RETURN

END

‘UNCTION HRLITNBI

IS THE FUNCTION USED TO DESCRIBE THE NET BULB LINE.
COMMON/PRESS/PaTM/SPEC!AL/Pvaoa

PHH:PSOB(THBI
DBL-Tuu.na-(pwr-PV)/T.2405~(PHD-PATH)«T1.+;15577*PVIPATM))*l,§2195

1-HLDR(THB)) TRFTURN

END

SUBROUTINE ZFHfiINIAa9;EPS.FUNC)

C SUBROUTINE ZEROIN 15 THE 1-OIMENSTONAL zefio-FINDING ROUTINE USED BY
C 30TH PARTS OF THE THE§IS.

POOVU'IbvaNF-‘b
C
o

C

12

COPMON/ICHECK/TFLAG

REAL I."

FAtFUNCCA) s FDIFUNCTB> S FC-FA 5 CIA S {FLAGIO
IFTSIDN<1.;F°>.Eu,stD~(1..FC)>4oo.1

IFLAGI1 S RETURN

IFIADSTFCI'ADSIFFII 2.3.3

CIR $ PaA $ AIC 3 FCIFB S FBIFA S FAOFC
IFTAHsrc-a)-?.vtP$) 12.12.4

I:(B-A)cFa/(Fa-FA) S J-LEGVARTI) S H=(C+B)/2. 5 IFTJ-O) 7:507
Is-I*8 S CHIMTIIH'II‘IH'I) I IF(CHINT) 50807

13M

IF(ABS¢8-I)-FPS) 9.10.10

I=SIGN(1..(C-H))-EPS*B

A38 % Q21 T FA=FH T FbIFUNC(9I
IFI5IG”(1.;F°).$IGNI190FCII 1.11:1

CxA$ FC8FA 5 GO TO 1

A:(coB)/2. 1 FA8FUNCTA)
1r(SIcu(1.;rn>.Ea,510~(1..r8)T B=c S RETURN
END

SUHROUTINE ZFROINA (ADBDEPSDFUNCI

C SUBROUTINE ZEROINA I§ THE 1-DINEN5IONAL ZERO-FINDING ROUTINE USED BY

400

0PTHHIZ AND CONTROL.

COHHON/ICHECV/IFLAG

"EAL I)"

FA=FUNC(A) S FD=FQNCIBI I FC=FA S C=A $ IFLAG'O ,
IF(SIGT(1.;F“),tU!SIGN(1;,FC))400,1

IFLAGI1 S RETURN

O

POCDVm-bMNP
O

h.»
an»

159

IF(ABS(FC)-Aner9)) 2.3.3

C38 S Q=A S A=C T FC:FB t FBsFA S FA'FC
IF(‘BSIC’B,'90'bP§’ 12012.‘

I:(B-A)vFB/(FU-fAI i J=LEGVARIII S M=(C+B)12. 3 IF<J~OT 7.5.7
18-I¢B s CHIMT=(R-I)‘IN'I) s IFICHINTI 8.8.2

IIM

IFIABSIB'II'CPSI 9.10.10

I-SXDNT1..¢c-b))ogps.e

A38 S Oil $ FAOFH T FBBFUNCIBI
IF(SIGU(1.2Fn)-SIO~(1..FC)) 1.11.1

ccAs FezfA 5 GO TO 1

Ascc.a,/2. s FAsFDHCTA)
IF<SIGN(1.}FA>.hO,SIGN(1..FB)) 88C 5 RETURN
END

“000000