A CALCULATION MODEL OF FIXED
BED WHEAT AERATION
By
SUKIT NITINAI
Bachelor of Engineering
King Mongkut ' s Institute of Technology
North Bangkok, Thailand
1982
Master of Science
Oklahoma State University
Stillwater, Oklahoma
1991
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial fulfillment of
the requirements for
the degree of
DOCTOR OF PHILOSOPHY
December, 1997
A CALCULATION MODEL OF FIXED
BED WHEAT AERATION
Thesis Approved:
11
•
ACIOJOWLEDGMENTS
The author wishes to express sincere appreciation to
Dr. Bobby L. Clary for his invaluable technical expertise,
encouragement and criticism as the author's dissertation
advisor. Thanks is also extended to chair of the advisory
committee, Dr. Bennett L. Basore; and members of the
advisory committee, Dr. Glenn A. Kranzler and Dr. John P.
Chandler, for their guidance and critical rev~ews.
A special thanks is extended to Rajamangala Institute
of Technology, Thailand for financial support during my
graduate study program at Oklahoma State University.
Finally, I would like to express my appreciation to my
wife, Tuk, for her encouragement, understanding, and
patience to take care of our beloved children, Nan and Nat,
during my graduate study program.
~l~
TABLE OF CONTENTS
Chapter
I. INTRODUCTION
Purpose of the Study
II. LITERATURE REVIEW . . .
Effects of Aeration .
Deep Bed Simulation Models
State of the Art
Grain Properties
III. METHODOLOGY .....
Mathematical Models
IV. COMPUTER PROGRAM DEVELOPMENT.
Main Program . . . .
Constraint Subroutine Programs.
V. RESULTS AND DISCUSSIONS .
Grain Temperatures ..
Grain Moisture Contents
VI. CONCLUSIONS
LITERATURE CITED
APPENDIXES ..
APPENDIX A  RESULTS OF EXPERI MENTAL DATA.
APPENDIX B  SIMULATION RESULTS
APPENDIX C  PROGRAM CODE
APPENDIX D  PHYSICAL PROPERTIES OF GRAIN,
AIR, AND WATER. . . . . . . .
lV
Page
1
6
8
8
10
18
20
22
24
28
28
29
41
42
59
67
70
73
74
76
81
. . . . . 87

LIST OF TABLES
Table Page
I. Equilibrium Moisture Content of Red Wheat .
II. Effect of Time, Airflow Rate, and Bed position
on Grain Temperatures During Aeration as
Predicted by the Simulation Model
III. Correlation of Predicted and Measured
Temperature at Various Airflow Rates
IV. Anova of Predicted and Measured Temperatures
· 36
· 42
· 52
at Various Airflow Rates . 54
V. Average Absolute Error of Temperature Prediction
Various Time Periods and Airflow Rates . . 56
VI. Predicted and Measured Final Moisture Contents
at Difference Airflow Rates by Location.. . 60
VII. Correlation of Predicted and Measured
Final Moisture Contents .
VIII. Anova of Predicted and Measured Final
· 64
Moisture Contents at Various Airflow Rates. 64
IX. Experiment Data at Q = 0.67, 1.34 and 2.68 1/ (sm3 ) . 74
X. Experiment Data at Q = 5.36, 8.04 and 10.7 1/ (sm3 ) . 75
XI. Properties of Grain (hard red winter wheat) 87
XII. Properties of Water 88
XIII. Properties of Air 88
v

LIST' OF FIGURES
Figure
1. Damaged by Grain Heating
2. Damaged by Insects.
3. Loss of Germination and Baking Quality
4. Damaged by Mites .
5. Safe Storage Conditions
6. A Typical Grain Aeration
7. Effect of Aeration on Insect Population
in Oklahoma Wheat Storage with
Application of Malathion at Binning
8. Effect of Aeration on Red Flour Beetles
in Oklahoma Wheat Storage
9. Flow Chart of Main Program
10 . Flow Chart of T, H Calculation Subroutine.
1I. Predicted and Measured Grain Temperature
Profiles at Q = 0.67 1/ (sm3 ) .
12. Predicted and Measured Grain Temperature
Profiles at Q = 1. 34 1/ (sm3 ) .
13. Predicted and Measured Grain Temperature
Profiles at Q = 2.68 1/(sm3 ).
14 . Predicted and Measured Grain Temperature
Profiles at Q = 5.36 1/ (sm3 ) .
15. Predicted and Measured Grain Temperature
Profiles at Q = 8.04 1/ (sm3 ) •
Vl
Page
2
2
3
3
4
9
. . . 11
12
30
34
· .. 44
· . . 46
· .. 47
. . . . 49
· . . 50
Figure
16. Predicted and Measured Grain Temperature
Profiles at Q = 10.7 11 (sm3 ). • ••
17. Scatter Plot of Predicted Temperature
at Q = 0.67 1/(sm3 ) •••.••.
18. Scatter Plot of Predicted Temperature
Page
· .. 51
· . . 53
at Q = 2.68 1/(sm3 ) ••••••• 53
19. Plotting of Average Absolute Error
Over different Time Periods
at Q = 0.67, 1.34 and 2.68 11 (sm3 ) .
20. Plotting of Average Absolute Error
Over Different Time Periods
· . . 57
at Q = 5.36,8.04 and 10.72 11 (sm3 ) •••••• 57
21. Predicted and Measured Final Moisture
Contents at Q = 0.67 1/ (sm3 } • · ·
22. Predicted and Measured Final Moisture
Contents at Q = 2.68 1/ (sm3 ) • · ·
23. Predicted and Measured Final Moisture
Contents at Q = 8.04 1/ (sm3 ) • · ·
24. Average Absolute Error of Final Moisture
Content by Various Airflow Rates
at Q = 0.67, 2.68 and 8.04 1/(sm3 ).
25. Scatter Plot of Predicted Moisture
Content at Q = 0.67 11 (sm3 )
Vl.l.
. . . . 61
· . . 61
· . . 62
· . . 63
. . 63
LIST OF SYMBOLS
A Constant in Equilibrium Relative Humidity equation
C Constant in Equilibrium Relative Humidity equation
m Rate of mass transfer
'¥ Rate of energy transfer
h Coefficient of convective heat transfer, W/(m2 K)
k Thermal conductivity, W/(m K)
E Specific internal energy, kJ/kg
e Base energy of internal energy
T Air temperature, °c
e Grain temperature, °c
H Specific humidity of dry air, kg/kg
M Moisture content of product, %db
Va Velocity of air, m/s
vp Velocity of grain, m/s
Pa Density of air, kg/m3
pp Density of grain, kg/m3
E Void ratio
c Specifi c heat, kJ/(kg K)
w Water content in grain, kg/kg
wb Wet basis
db Dry basis
as subscript:
a Dry air
v Water vapor
p Product or grain
w Water in grain
viii
CHAPTER I
INTRODUCTION
Upon harvesting, wheat moisture content should be
reduced by drying to preserve grain quality and to extend
the storage time. In addition, decreasing the moisture
content controls fungi and insect growth. Grain aeration is
also performed to decrease grain temperature and moisture
content for proper storing. The main purposes of aeration
are to prevent moisture migration and to cool the grain.
This process also reduces mold growth, insect activity, and
undesirable odors.
Information in Figures 1 through 5 (McLean, 1989) shows
the general effects of grain moisture and temperature on
grain storage life with regard to grain heating, insects,
germination, baking quality, and mites. Information given
in each figure shows that the grain storage regions are
affected by environment. For instance, Figure 1 shows the
effect of temperature and moisture content on grain heating.
Figure 2 and Figure 4 show the effect of temperature and
moisture content on insects and mites.
1
p
.Q..) .::.l
40
30
~ 20
c..
.E,
I
10
10 15 20 25
Moisture cont~nl %
Figure 1. Damaged by grain heating
p
Q)
3
;;
30
~ 20
c..
E
Q)
I
10 Safe
10 15 20
M,olsture content %
Figure 2. Damaged by insects
25
2
.u
10 15 20 25
Moisture content %
Figure 3. Loss of germination and
baking quality
40
30
.U
.II.>. ::I ;;
~ 20
Co
E
II>
I
10
10 15 20
Moisture cont,enl %
Figure 4. Damaged by mites
25
3
P
<11
3
iii
40
30
;; 20
Co
E
<II
I
10
10 15 20 25
Moisiure contenl %
Figure 5. Safe Storage Conditions
Note: Figures 1 through 5 adapted from
Mclean (1989)
4
5
Information in Figures 1 through 4 is combined in
Figure 5 and shows the moisture content and temperature
conditions under which wheat can be safely stored. The safe
temperature range is between DoC and 16°e, and the safe
moisture content range is between 8 percent and 12 percent
wet basis.
Oklahoma has periodic cold fronts in October and
November, wi thambient temperatures below 10oe. This
weather can be utilized to increase the performance of wheat
aeration systems in this region.
Although the aeration process can extend the storage
time and preserve wheat quality, the factors involved such
as air flow rate, temperature, and moisture content must be
identified and defined. To find an optimum aeration time,
experimental testing must be conducted which will require a
lot of effort, time, and cost.
The concept of mass and heat transfer simulation,
however, can be used to help reduce the time of actual
experi mental testing of wheat aeration systems. In
addition, it is difficult to conduct experiments in the
field or laboratory due to weather conditions and/or size of
storage, so computer simulation i s needed.
By developing a computer model of mass and heat
transfer, where temperatures and humidity can be easily
changed, length of aeration time can be obtained without
actual experimental testings.
Purpose of The Study
6
The purpose of the study was to develop a personal
computer calculation model which simulates wheat aeration of
a fixed bed bin. A personal computer simulation program
will allow farmers and others to easily determine the
appropriate air flow rate for their fixed bed aeration,
which will results in high quality grain and optimum storage
life.
Objectives of the Study
To meet the purpose of the study, the follow objectives
were set forth:
1. To develop a calculation model for cooling wheat
1n a standard fixed bed bin, in order to predict
the effects of airflow rate, air temperature, and
alr humidity on cooling time. The partial
differential equation model will be solved by the
implicit finitedifference method.
2. To develop a simulation procedure by utilizing a
structured programming language.
7
3. To evaluate simulation performance of the aeration
system in a fixed bed wheat storage bin with
varying inlet air conditions.
CHAPTER II
LITERATURE REVIEW
Effects of Aeration
Brooker et a l. (1974) stated that despite the time that
high grai n quality can be preserved, which is most important
for farmers and others who store grain, the quality of gra~n
will decrease due to the growth of fungi and insects. To
prevent the development of insects and fungi, grain
temperature and moisture content can be controlled by using
aeration.
Aeration (as shown in Figure 6} is usually
accomplished by ventilati ng stored grain with a low a1.r flow
rate. The process accomplishes the following:
 Prevents moisture migration by maintaining a uni form
temperature through the grain.
 Cool s the grain to reduce mold growth and insect
activity.
 Helps to eliminate gra1.n odors.
8
9
GRAf'N
Air direction
Plenum Chamber
Blower
Figure 6. A Typical Grain Aeration
10
Many experiments on aeration have been conducted to
study the preserving of grain. Epperly et al. (1987)
conducted an aeration experiment in Oklahoma. They
investigated the insect growth rate in two grain bins. Both
bins were of the same size and condition, except the first
bin used conventional aeration while the second bin used
specific (forced) air flow rate aeration. The results
showed insect infestation of the second bin to be
significantly less than that of the first one (Figs. 7 and
8). It was found that grain temperature had to be kept at
about lOce to 13°e to maintain the ideal range of
temperature.
Deep Bed Simulation Model
Several investigations of deep bed calculation models
have been conducted to determine suitable models for each
particular case such as cooling, heating, and drying.
Schuman (1929) may have been the earliest to develop a
mathematical model that predicted heating or cooling with a
mass of crushed material with fluid flow. His model had
many simplifying assumptions such as constant air
temperature and air velocity in order to fit the Bessel
function form. A reason was that at that time (1929), the
modern digital computing machine had not been developed, and
W
ID o
0:: n.
25,,_,
UNAERATED
AERATED
204~~~~
0::
~154~~~
III
Io
W
In
~
1L10 4+"
o
0::
w
m
::!:
:::>
z 54_~
01~r_._,~~_.,_.__,
J A S o N
MONTHS
D F M
Figure 7. Effect of Aeration on Insect Population in
Oklahoma Wheat Storage with Appli cation of
Malathion at Binning
(adapted from Epperl y et al., 1987)
11
W
III
 UIlUAil.TED
 ~1l1UUD
~3+~~r+~
Ii.
It:
W
Ii.
o~
~2++~~!
'\
1 \ " 1 \ \
\
\
\
\
\
\
\
0' f=::::::=:;:::::==~.'r~r===r=:..::......~,
JUN JUl AUG SEPT OCT NOV DEC JAN FEB MAR APR
M,ONTHS
Figure 8. Effect of Aeration on Red Flour Beetles in
Oklahoma Wheat Storage.
(adapted from Epperly et al., 1987)
12
~
I
13
it was not possible to easily find the solution of a
mathematical model. Regarding the Bessel function, for
which we already know the solution, Schuman's model 1S ln
analytical form rather than in numerical form. There is no
val idation of Schuman's model. However, his work encouraged
further investigation of a drying calculation model.
After Schuman ' s presentation, the methods used to
develop mathematical models of deep bed simulation can be
classified in three different models. They are:
1. Graphical or logarithmic model
2. Heat and mass balance model
3. Partial differential equation model
Logarithmic Model
Hukill (1954) developed a mathematical model of deep
bed grain drying. He used a fully exposed equation of grain
drying (eq. 1) and a diffusion differential equation which
assumed the rate of moisture lost at some given depth x,
after time t , and is proportional to the rate of decrease in
alr temperature at (x,t) (eq. 2).
M  Me = (Mo  Me) ekt
dMR
dt
= C dT
dx
By using the similitude technique with boundary and
ini t i a l condition of deep bed storage, a series of
(1 }
(2 )
14
dimensionless curves was obtained from his equation (Hukill,
1954) .
MR = M Me
Mo  Me
= 2X (3 )
2X + 2t  1
where MR = rate of moisture lost
M = grain moisture content
Mo (initial) , Me (equilibrium)
X = dimensionless depth units
t = dimensionless time units
C, k = constant values
This equation provided a simple method for predicting
the moisture distribution in a grain drying bin. However,
his model underestimated the time required to dry grain to
specific moisture contents.
Because of the model's simplicity and computational
economy, some investigators were interested in modifying the
logarithmic models to fit their experiments. However, the
accuracy was acceptable only at low temperatures and low air
flow rates.
Heat and Mass Balance Model
The availability and speed of present day digital
computing machines can reduce time and effort in performing
more complex calculations. Several investigators,
therefore, have developed and presented the deep bed
15
calculation model utilizing heat and mass balance equations
rather than the logarithmic method. Some of the
investigations were as follows:
Boyce (1965) conducted barley drying in a 12inch deep
bed. He developed a layerbylayer technique which was
based on thin layer theory to calculate the temperature and
moisture content of air and grain. The results showed that
the drying times by his calculation were too l ong. He
recorrunended that more experiments be conducted to determine
a better procedure.
BakkerArkema et al. (1967) conducted an experiment to
cool high moisture content cherry pits in a deep bed. His
numerical model, which was based on heat and mass balance
conditions, showed good agreement with the experimental
results. However, his model was limited to the cherry pit
products.
Thompson et al. (1968) conducted corn drying in a deep
bed and reported his process as a series of thin layers in
which heat and mass balance models were applied (similar to
Boyce's model). The results showed an effective model, but
limited to certain inlet air conditions.
Henderson and Henderson (1968) used a thin layer
calculation model in their experiment on deep bed grain
drying. Their numerical results showed that the calculation
16
time for drying was too long when compared with the actual
results.
The calculation models of Bloome (1972), Sutherland et
al. (1971), and Ingram (1979) are based on heat and mass
balance equations. Consequently, the limitations of these
models are their accuracy and range of applicability. In
each case, they present some modification to improve the
models for their particular conditions. However, all of
their suggestions demonstrate the need for further
investigation of drying simulation models.
One of the disadvantages of heat and mass balance
models is accuracy. This is because the models use the
assumption of equilibrium moisture content conditions.
M(x,t) = Me(x,t)
T(x,t) = 8(x,t)
M = grain moisture content (dry basis)
Me = grain moisture content at equilibrium
T = air temperature, °C
8 = grain temperature,. °C
(4 )
( 5 )
In a practical situation, equilibrium conditions of
grain drying or cooling are not likely to occur, unless the
temperature and moisture differences are very low.
Therefore, the outcomes of some investigations show
predicted time to be longer than actual time. However, the
advantage of simplified equations is that they make the
model easy to compute and take less time than the partial
differential equation model. The partial differential
equation models are more complex and requires a lot of
computation time ..
17
In the last decade, the development of digital
computing machines has made it easier to handle some partial
differential equation models. Investigators continue to
develop suitable partial differential equation models
utilizing the much faster and more accurate digital
computing machines.
Partial Differential Equation Model
During the years from 1960 to 1980, only a few
investigators used partial differential equation models ~n
drying simulation, because so much computing time was
required and only a few computers were available. However,
the partial differential equation models showed a tendency
to be more accurate and have a wider range of inlet air
conditions.
Morey et a 1.. (1978) stated that beginning in 1966 I
Michigan State University conducted a series of theoretical
analyses to develop a full model of heat and mass transfer
simulation. In 1974, Brooker et al. presented drying models
that included physical characteristics such as convective
heat and mass transfer coefficients, and air and grain
densities, for single kernel drying.
18
These models used four partial differential equations
to predict air temperature, air humidity, grain temperature,
and moisture.. The finite difference method (F.D.M.) with a
timestep calculation was used in the program. However, the
performance and accuracy was not high, because an explicit
finite difference method was used which d i d not include an
error control technique.
State of The Art
The calculating power of digital computers since 1980
has grown at an exponential rate. These advanced machines
have been utilized by researchers in fix bed aeration
models. Investigations conducted by Parry (1985 ) and Costa
and Figueiredo (1993) are among the l atest.
Parry (1985) presented a mathematical model and
described the model as a general mathematical framework of
heat and mass transfer in drying particular solids. The
assumptions in the model are onedimensional (lD) mass flow
rates and cons t ant densities for both air and grain,
together with neglect of conductive and radiative he.ating
effects, so that convective grain drying can be represented
19
by a general system of partial differential equations of the
form
( 6)
Where U = property matrix vector
A = air velocity matrix vector
B = grain velocity matrix vector
b = energy and mass transfer matrix vector
(Details of the equation are given in Chapter III)
With certain simplifying assumptions, Parry stated that
other models appearing ln the literature coul d be considered
as particular versions of his general models under further
appropriate assumptions. For the fixedbed condition, the
model was less complicated. It is believed that Parry's
model is the most compl ete to date.
Costa and Figueiredo (1993) developed a numerical
technique for solving the partial differential equations
model for a fix bed dryer based on Parry's general model.
They proposed a technique to solve the set of partial
differential equations by using the characteristic method or
by the more conventional finitedifference discretization
method. Numerical results from their proposed model were
compared with experimental resul ts of Boyce (1965) and
showed good agreement.
20
Costa and Figueiredo's model is not as accurate as
Parry's model, but it has the fo l lowing characteristics:
generality, simplicity, reasonable accounting for the
essentials of the involved phenomena, and less computing
time and storage. Costa and Figueiredo suggest t hat their
proposed model can be used with the dynamic simulation of a
fixed bed dryer or other related processes.
Nevertheless, the above investigations of s i mulation
have not been conducted to work wi th the aeration process or
applied to the now universally used personal computers.
Grain Properties
One of the important parts of computer simulation is
the accuracy of grain properties. The major properties are
thermal conductivity (kpl, specific heat (Cpl, bul k density
(Ph)' particle density (Ps)' and porosity (Pl.
Epperly (1989) proposed grain properties, used in his
aeration experiment, in the following equations.
Thermal conductivity adapted from Chuma e t al. (1981).
~(W/m K) = 0 .144 + 0.0006 Mw ( 7 )
Specific heat adapted from Mohensin (1980) .
Cp (kJ /kg K) = 1. 258 + 0.0113 1 Mw ( 8 )
Mw = grain moisture content % wb
21
Bulk density is the weight of a mass of intact
individual units of the material packed in a given volume
(inc l uding pore space) by a specific method. Epperly used
750 kg/m3 for soft white wheat for moisture content of 0% to
10%. For each 1% increase in moisture content above 10%,
the density decreases by 3.7 kg/m3 •
Porosity (P) can be calculated from bulk density and
particle density as:
P = 1  ( Pb / Ps } ( 9 )
Where Pb = Bulk density
Ps = Particl e density
CHAPTER III
METHODOLOGY
Models developed by using partial differential
equations (P.D.E.) of energy and mass transfer have the
advantage of greater accuracy and parametric flexibility.
However, it is more complicated and time consuming than the
heat and mass balance method.
At the present, due to the highspeed central
processing unit (CPU) of digital computing machines,
including personal computers, the partial differential
equation can be numerically solved with relatively short
,
computation time. The problem encountered in the past,
utilizing the partial differential equation models, has been
eliminated. In addition, new investigations on parameters,
for instance, convective heat transfer coefficient, have
been conducted and the conclusions revised so that it can be
easily updated in the simulation program.
The simulation model developed by this researcher was
based on Parry's (1985) general model (eq. 6).
Modifications were made on the assumption that heat and mass
22
transfer during aeration is slower than that of the drying
process. Partial differential equations can be set in
vector form and solved by the appropriate methods (CrankNicolson
or Runge Kutta, Costa and Figueiredo, 1993).
The procedure used for developing the computer
simulation program was as follows:
1. Developed a simulation model based on Parry's general
models, with the equation modified for fixed bed aeration.
The boundary and initial conditions were:
t = 0, To = T(x),
t = 00, T(x) = To,
Ho = M(x)
MIx) = Ho
2. Simplified the model with the assumption proposed by
Costa and Figueiredo (1993). In the fixed bed, heat and
mass transfer during aeration were likely to be at
equilibrium moisture content state. The assumptions were:
 Air flow in one dimension is at a constant
velocity;
 Lateral dryer walls are adiabatic and impermeable;
23
 Conduction of heat and moisture between particles is
negligible;
 No shrinkage of bed occurs;
 Total air pressure is constant through the bed;
24
 Physical properties (density and specific heat)
of dry air and solids, liquid water, and water
vapor are constants;
Longitudinal mass and heat dispersion in the alr flow
are negligible;
 No initial gradient of moisture and temperature
exists within the solid particles;
The mass transfer rate from the solid to the al'r is
described by the drying kinetics equation {or thin
layer drying rate equation, aM/at = k (M  Me)ll) .
3. Used numerical techniques as in Costa and Figueiredo's
proposed model. The numerical solutions were obtained by
using semiimplicit finitedifference methods (Crank
Nicolson) .
Mathematical Models
The program mathematical model was developed based on
Parry's general models (eq. 6).
Where
aU+AOu+BoU=E
at ax oy
U = [H,M,T,6F
(10)
A = Va 0 0 0 B = 0 0 0 0
0 0 0 0 0 vp 0 0
0 0 Va 0 0 0 0 0
0 0 0 0 0 0 0 v P 1
b = bI = m/epa
b2 m/epa
b3 {'II  mEv (T) } / {ePa (ca + cvE) }
b4 { If' + mEw ( e) ) / { Pp (cp + cvM) )
...J
where: Ev ( T ) = ev + Cv T
Ew ( 8 ) = ew + cw8
m = rate of mass transfer, kg/s
'¥ = rate of energy transfer from
If' = h(T8 } mEv(8v), J/s
h = heat transfer coefficient, J/kg DC
E = specific internal energy, J/kg
e = base energy of internal energy, J/kg
T = air temperature, °c
8 = grain temperature, °C
H = specific humidity of dry air, kg/kg
M = moisture content of product, %db
Va' Vp = velocity of moist air and product, m/ s
Pa , Pp = density of air and product, kg/m3
e = void ratio
25
26
c = specific heat, kJ/kg K
w = water in grain, kg/kg
suffix a = dry air, v = water vapor
p = product, w = water in grain
Applied to the fixed bed case, with one dimension and
grain velocity = 0, equation (1) reduces to
(11)
Because of low temperature differences and airflow
rates, near equilibrium conditions were produced between air
and grain during aeration. Since the Biot Number (hl/k) is
low and airflow rates are low, grain temperature and air
temperature were assumed to be equal at equilibrium moisture
content conditions.
hence, M(x,t) = Me(x,t)
T(x,t) = 8(x,t)
Me = grain moisture content at equilibrium
(x,t) = space and time
the final vector matrix will be
(12)
Where
U = [H,TF
27
A = EPa + Pp OMe Pp BMe I
BH aT
Ppv BMe e:Pac1 + pp (c2 + BMe)
aH aT
B = l €P~V. 0
E:Pav acl 1
where v = cwe  cvT  hEg
c2 = cp + cwM
hfg = latent heat, kJ/kg K
The equation can be solved by using the semiimplicit scheme
by CrankNicolson.
CHAPTER IV
COMPUTER PROGRAM DEVELOPMENT
The computer program was developed as a main program
and subroutines. C language, a structured programming
langu age, was used to code the program. Two characteristics
of a structured programming language is that; one (l), it
runs from top to bottom and, two(2) as a subroutine, it has
only one specifi c output for any given inputs . These
characteristics make the program easy to read, correct
( debug) I and modi fy .
The overall program development is discussed under
main program and subroutine programs.
Main Program
The purposes of the maln program are to control and
cal l the subroutine programs. In addition, the main program
will perform the following t asks:
Initialize parameters, physical constants of grain air and
water.
Get input conditions, such as rate of alr flow and
temperatures.
28
29
 Call a subroutine to calculate nextstep conditions.
 Check prediction errors and make calculations (iteration
loop) .
Print out results.
Listing of the program is presented in Appendix C.
Figure 9 shows the flow chart of the main program.
Constraint Subroutine Programs
Temperature and Humidity Computer Model
The major part of the computer program is a temperature
and humidity prediction subroutine. A computer model was
developed from the system of partial differential equation
(eg. 12). The Crank Nicolson's method (semiimplicit finite
difference) was used to solve the computer model for
temperature and humidity prediction. Equation (12) can be
expanded as follows;
aH + ( ppaMe )aT + ep"Ya aH = 0 (13 )
at aT at ax
ppyaMe )aH +[ ePaCl + (C2 + yaMe )]aT + ePavaC1BT = 0 (14)
aH at aT at ax
from eg. (13) and (14), rearranging to (15) and (16),
+ 2pp al'1e = ePava 3H (15)
at ax
r
Initialize Data
(constants, variables)
L____________ ,, ____________ _ ~
~
Begin 1st time step
', ' +
I Begin 1st node ~~~
~late new temp
~~ at next time) .... l
L '
'~T  I Is ~ Nol
~Pred temp  New Tem~~J
_____________ < E ~~
~ I
I pre~.=p ; New Temp I JI
__ No
<~ sit the last nodp..:.?'
~ ... 
Yes£
I'~~p r i n t resul~ts l
~_~ old temp = New temp I
~:~>NO
~~~
Yes I ~~(
End ')
/
Figure 9. Flow Chart of Main Program
30
.J
f
(epaCl + ppC2)QT + 2 PE.y ~
at at
= ePavaCl aT.
ax
31
(16)
In order to find a numerical solution, the system of
partial differential equation (eg. 15 and 16) were
transformed to finite difference form by assuming constants
as follows;
A = ePa B = 2pp C =
Where m is the starting point (known value), n 1S the
time at point fi, and the symbol 1 is for the next time step.
At point m, time n, the finite difference equation that
represents eg. (15) becomes:
A (H.n 1  fIu,) + B (Mem 1  Mem ) = c (Hm,  ~1) (17)
At At AX
At point m, time n+l, the finite difference equation
that represents eq. (15) becomes:
A(H.n 1  H",) + B (Mem 1  Mem) = c (H".'  F!,..l I' ) (lB)
At At AX
Summation of eq. (17) and eq. (18) is the be semiimplicit
finite difference form, or CrankNicolson form.
AX
l et At/AX = r3,
r
32
To find the absolute humidity of aeration at point m
and time n+1, the equations were arranged to form the final
equation:
H.n' = Ii,..  ( B ) ( Hm1 + H' m1) (19)
2A/C (3
For grain temperatures, eq. (16) was used by
the same method to develop a finite difference of Crank
Nicolson form.
At point m, time n, the finite difference equation that
represents eq. (16) becomes:
= (20 )
At point ill, time n+1, the finite difference equation
that represents eq. (16) becomes:
E (MelD'  Me",)
At
= F (T", ,_ TID  1 ")
b.X
(21 )
Surrunation of eq. (20) and eq. (21) is the semiimplicit
finite difference form, or CrankNicolson form.
2D(TID '  Tm} + 2E(Me",' Me",) = F b.t (Tm'  T'ml + Tm  Tm1 )
b.X
let b.t/ b.X = ~ I
To find the absolute humidity of aeration at point m
and time n+1, the equation were arranged to form the final
33
equation:
Till' = Till  ( B ) ( Tm 1 + T' m1) E (Melli'  Me",) (22)
2D/F ~
Equation (19) and (22) were the simulation models which
were computer coded to the subroutine program, 'calc_T_H'.
Figure 10 shows a flow chart of the subroutine program.
Equilibrium moisture content model
Pfost et al. (1976) presented the equilibrium moisture
content (emc) equation for wheat as follows.
ERR = 1  exp( A( T + ClemcN
Where T = Temp, OF (dry bulb)
erne = equilibrium moisture content, % dry basis
ERR = equilibrium re.lative humidity, decimal unit
For hard red winter wheat,
C = 55.815 OF
N = 2.2857
so,
ERR = 1  exp(O.000023008( T + 55.815)emc2.2857) (23)
Hence, ernc can be defined as,
ernc (2 .2857) = In ( 1. ERHl (24)
O.000023008*(T + 55.815)

Cal.
culate T, H ')
r " l In~tialiize Data
(constants, variables)
,
Calc. air,
Icalc.E M C
grain entha~
I I
~a lCU_l_a_t_e,n_e_ _w_ '_ T_, . H l
I
y
I
Adjust T, H
~_ r
~ ~~ . NO I
~ s RH < ~. 99? o J
._,..
YES I
!
/ _ L_____ .
( Return
.. '~
34
Figure 10. Flow Chart of T, H Calculation Subroutine
35
Pfost et al. (1976) stated that the standard error of
ernc  0.0071. Table 1 shows results of equation (24) within
the temperature range of 40 of to 120 of.
Equation (23) was the computer model to calculate
equilibrium relative humidity (ERH) , and equation {24} was
the comput.er model to calculate equilibrium moisture
content (emc) . The models were coded in C programming
language. The subroutine programs of both models are 'ERH'
and 'EMC' which can be shown as follows:
double ERH(double mdb, double TF)
{
/******************************************************/
/* This sub. calc. equilibrium RH in decimal unit */
/* RH Relative Humidity [decimal unit] */
/* TF Temp, deg F (dry bulb) * /
/* TC Temp, deg C (dry bulb ) */
/* mdb EMC dry basis [decimal unit) */
/* Use only for red wheat */
/******************************************************/
double TC,RHi
TC = (TF32.1/ 1 .8;
RH = 1. exp{2.3008e5*(TC+55.815)\
*pow(mdb*100.,2.2857});
return (RH) ;
}
double EMC{double H, double T)
{
}
/******************************************************/
/* This sub. return EMC, decimal unit db */
/* H humidity ratio, lbllb dry air */
/* T Temp, deg F (dry bulb) */
/* TC Temp, deg C (dry bulb) * /
/* mdb EMC % dry basis */
/* Rh Relative Humidity [decimal unit] */
/* Use only for red wheat */
/******************************************************/
doubl e mdb,Rh, TCi
Rh=RH(H,T) ;
TC= (T32.)/1.8i
mdb =pow (log ( 1 .  Rh) / {2 . 300 8e5 * (TC+ 5 5 . 815) ) , (1. /2 . 2857) ) ;
return (mdb*.Ol);
36
TABLE I
EQUILIBRIUM MOISTURE CONTENT OF RED WHEAT I %db
Temp F 40 50 60 70 80 90 100 120
C 4.4 10.0 15.6 21.1 26.7 32.2 37.8 48.9
RH [%]
5 4.9 4.7 4.5 4.4 4.2 4.1 4.0 3.8
10 6.7 6.4 6.2 6.0 5.8 5.6 5.5 5.2
15 8.0 7.7 7.5 7.2 7.0 6.8 6 .. 6 6 .. 3
20 9.2 8.9 8 . 6 8.3 8.0 7.8 7.6 7.3
25 10.3 9.9 9.6 9.3 9.0 8.7 8.5 8. 1
30 11.3 10.9 10.5 10.2 9.9 9.6 9.4 8.9
35 12.3 11.8 11. 4 11.1 10.7 10.4 10.2 9.7
40 13.3 12.8 12.3 11. 9 11. 6 11 .. 2 10.9 10.4
45 14.2 13.7 13.2 12.8 12.4 12.0 11. 7 11. 2
50 15.2 14.6 14.1 13.6 13.2 12.8 12.5 11. 9
55 16.1 15.5 15.0 14.5 14.1 13.7 13.3 12.7
60 17.1 16.5 15.9 15.4 14.9 14.5 14.1 13.5
65 18.2 17.5 16.9 16.3 15.8 15.4 15.0 14.3
70 19.3 18.6 17.9 17.3 16.8 16.4 15.9 15.2
75 20.5 19.8 19.1 18.5 17.9 17 .. 4 16.9 16.1
80 21.9 21.1 20.4 19.7 19.1 18.6 18.1 17.2
85 23.6 22.7 21. 9 21.2 20.5 20.0 19.4 18.5
90 25.6 24.7 23 .. 8 23.0 22.3 21.7 21.1 20. 1
95 28.8 27.7 26.7 25.9 25.1 24.4 23.7 22.6
Equil ibrium Moisture Content of Red Wheat [%db]
Note: Calculated from equation (24)
37
Enthalpy of Water Vaporization Model (hfg)
This model calculates enthalpy of wate.r vaporization
(hfg) from a given temperature. The model is from
Thermodynamic Properties of Water at Saturation, ASHRAE 1985
Fundamental Handbook. Linear interpolation of latent heat
of vaporization hfg [Btu/lb] and temperature [OF] were
calculated for the model from a temperature of 32°F to
hfg [Btu/lb] = 1093.3136  0.56761364 * T[OF]
The hfg model was computer coded to the subroutine
program which can be shown as follows;
double hfg(double TF)
{
double BTU;
BTU = 1093.3136  0.56761364*TF;
return (BTU)
Vapor Pressure Model
(25)
The model is adapted from Thermodynamic Properties of
Water at Saturation, ASHRAE 1985 Fundamental Handbook. The
saturation pressure over liquid water for the temperature
range of 32 of to 392 of is given by:
In(Pvs) = A/T + B + CT + DT2 + ET3 + Fln(T) (26)
where
Pvs Saturation vapor pressure, psia
T Absolute temp., oR (OF + 459.67)
A = 10440~4, B = 11.2946669
c = 0.02700133, D = 0.1289706e4
E = 0.2478068e8, F = 6.5459673
The computer code of the model is shown as follows;
double Pvs (double T)
{
}
double TRK;
TRK = T + 459.67;
return (exp (10440. 4/TRK 11.2946669  0.0270013 3 *TRK
+1. 28970 6e5*TRK*TRK 2.4 78068e9*TRK*TRK*TRK
+6.5459673*log(TRK)));
Relative Humidity Model (RH)
From basic thermodynamics, absolute humidity (H) is
38
defined as the ratio of the mass of water vapor to the mass
of dry air contained in the sample (H= Mw / Ma). Mw and Ma
are in gas form so that they can be changed in mole fraction
form by using their molecular mass ratio.
H = 0.62198 * mole fraction of vapor
mole fraction of a~r
0.62198 is the ratio of molecular mass of vapor
(18.01534) to molecular mass of air (28.9645). Because
vapor pressure of any gases relate directly to their mole
fraction, absolute humidity will be proportional to vapor
pressure ratio.
39
let Pv = Vapor pressure
and Patm = Atmospheric pressure
H = 0.62198 * JX.... (27 )
Pair
H 0.62198 * Pv (28 )
(PatIn  Pv)
Hence Pv = H*Pgtm (29)
(O.62198+H)
So, relative humidity (RH) can be derived from the
ratio of vapor pressure (Pv) to saturated vapor pressure
(Ps), RH=Pv/Pvs, for a given temperature (T) and humidity
ratio (H). H can be found by using equation (28) and a
glven RH and temperature.
computer coding of the model can be shown as follows:
double RH{double H,double T)
/*************************************************/
/* This sub. cal c, Relative Humid. [decimal] */
/* from Humid. ratio and Temp [deg F) */
/* Pv ; vapor pressure */
/* Pvs: sat. vapor pressure */
/*************************************************/
double P,Pv,Ps,RH;
P = Pabs;
Ps = Pvs(T);
Pv H*P/(O.62198+H);
RH Pv/Ps;
return (RH);
double H(double RH,double T)
{
/*************************************************/
/*
/*
/*
/*
This
from
Patm
Ps
s ub. calc. Humidity Ratio [lb/lb]
Relative Humidity and Temp [deg FJ
: atmospheric pressure [psia]
: sat. vapor pressure [psia]
*/
*/
*/
*/
/*************************************************/
,.
"
I
I:
I'
II !,
,.;1,
\,
double H,P,Pv;
P=Patm;
Pv = Pvs(T)*RH;
H =(O.62198)*Pv/(PPv) ;
return (H) ;
40
CHAPTER IV
RESULTS AND DISCUSSIONS
A calculation program was developed based on
mathematical models (presented in Chapter III). A computer
simulation program was then written, based on a calculation
program in C computer language and compiled by a Ccompiler
(Turbo C 2.0 version). Aeration simulations were run by the
computer simulation program, and the results, grain
temperatures and moisture contents,were obtained. The
simulations were made utilizing the same initial parameters
(temperature, humidity, and moisture content) and airflow
rates which were used in the experiment. The results of the
simulation, grain temperature, and moisture content, were
then graphi cally compared with the experimental results.
Experimental aeration was tested and the resu lts
obtained by Epperly (1989). The experiment can be described
as fol l ows. Aeration was tested by using a 6ft diameter
bin filled to a height of 9 ft with hard red winter wheat.
Six different airflow rates(Q), from 0.67 to 10.72 1/(sm3 ),
were chosen for testing. At each airflow rate, grain
41
42
temperatures and final moisture contents were recorded
versus time by using a data logger machine.
Grain Temperatures
The computer simulation results (Appendix B) were
printed at 1 hr and 5 hr time periods. Time intervals that
were consistent with the time intervals use by Epperly in
his experimental testing are shown in Table II.
TABLE II
EFFECT OF TIME, AIRFLOW RATE, AND BED POSITION
ON GRAIN TEMPERATURE DURING AERATION AS
PREDICTED BY THE SIMULATION MODEL
Airflow rate (Q){ Litre/(sm3 )
Temperature, °c
X/L : Location of graJ.n over total height of grain, ft/ft
X/L
Q=0.67
25h
SOh
150h
Q=1.34
0.0
14.4
14.4
14.4
10h 14.4
25h 14.4
55h 14.4
10Gh 14.4
Q=2.68
10h
20h
40h
70h
14.4
14.4
14.4
14.4
0.1
23.6
14.7
14.4
26.8
14.7
14.4
14.4
18.8
15.4
14.9
14.6
0.2
33.2
23.9
14.5
35.6
24.1
14.4
14.4
27.8
17 .1
14.9
14.6
0.3
36.4
31.2
14.5
37.9
32.0
14.4
14.4
33.9
22.0
15.9
15.1
0.4
37.1
35.3
15.3
38.3
36.3
19 .. 4
14.5
36.4
28.7
15.9
15.1
0.5
37.2
36.7
15.7
38.3
37.8
25.7
14.5
37.0
33.3
18.0
15.9
0.6
37.2
37.1
21.4
38.3
38.2
31.8
14.9
37.2
35.8
20.7
15.9
0.7
37.2
37.2
26.4
38.3
38.3
35.4
14.9
37.2
36.8
24.4
16.3
0.8
37.2
37.2
31.4
38.3
38.3
37.2
19.3
37.2
37 . 1
29.3
17.3
0.9
37.2
37.2
34.3
38.3
38.3
38.0
23.8
37.2
37.2
32.8
17.3
1.0
37.2
37.2
36.0
38.3
38.3
38.2
29.9
37.2
37.2
35.2
20.2
43
(Table II concluded)
X/L 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Q=5.36
2h 18.3 29.S 35.3 36.5 36.6 36.7 36.7 36.7 36.7 36.7 36.7
6h 18.3 19.3 26.4 32.2 3S.3 36.3 36.6 36.7 36.7 36.7 36.7
12h 18.3 18.3 18.3 20.0 25. 4 30.3 33.9 3S.6 36.3 36.6 36 . 6
24h 18.3 18.3 18.3 18.3 18.3 18.3 18 . 7 19.6 23.5 27.2 31.4
Q=8.04
2h 17.8 26.S 34.9 37.6 38.2 38.3 38.3 38.3 38.3 38.3 38.3
Sh 17.8 18.0 23.0 29.9 35.0 37.2 38.1 38.3 38 .. 3 38.3 38.3
12h 17.8 17.8 17.8 17.8 18.1 20 . 0 24.7 29.5 33.7 36.1 37.4
20h 17.8 17.8 17.3 17 . 8 17.8 17.8 17.8 17. 8 19.1 19.1 24.8
Q=10.7
1h 18.9 28.7 33.8 34.8 35.0 35.0 3S.0 3S.0 35.0 35 . 0 35.0
2h 18.9 23.3 30.4 33.7 34.7 35.0 35.0 35.0 3S.0 35.0 35.0
4h 18 . 9 18.9 22.3 27.5 31. 8 33.9 34.7 34.9 35.0 35.0 35.0
8h 18.9 18.9 18.9 18.9 20.3 23 . 1 27.5 30.9 33.2 34.2 34.7
14h 18.9 18.9 18.9 18.9 18.9 18.9 19.0 19.0 20.4 21.5 26.3
Data from simulation results (Table I I ) and
experimental results (Appendix A) were then plotted and
shown in graphical illustrations (Figure 11 through 16).
Information given in Figures 11 through 16 shows the
graphical comparison of grain temperature between computer
prediction results and experimental results at the different
a i rflow rates.
Information in Figure 11 shows graln temperature
profiles from computer simulation and actual measurements at
Q = 0.67 1/ (sm3 ). The initial grain temperature was 37°C,
and the inlet air temperature was 14°C. In Figure II,
40.0
35.0 
30.0
o
cE:i. 25.0
~
20.0
I
I
I
I
I
I
I
I
15.0 ~=W I:s . ~
/
;f
/
/
/
/
",'"
/$.'"
",IS
, .... ..~
10.0 +1r4+;r+~r__+~~
o 0.1 0.2 0.3 0.4 0.5
Haight, XIL
0.6 0.7 0.8 0.9
025htest
m SOhtest
.. 150h·test
 4  2Shcalc
 8  50hcalc
 .6.  160hcalc
Figure 11. Predicted and Measured Grain Temperature Profiles at Q = 0.67 1/(sm3 )
Ill>
If!>.
45
temperature profiles of the simulation are shown at duration
of 25, 50, and 150 hours.
At the first 25hr profile time period, simulation
results (grain temperatures) were a little higher than
measured temperatures throughout the bin. From the next
25hr time period (50hr profile), most of the calculation
temperatures were higher than measured temperatures, except
at X/L = 0.1, where the calculation temperature was lower
than measured temperatures. At the lSOhr profile,
calculated temperatures at the bottom to the middle of bin
were less than measured temperatures.
Data in Figure 12 shows the results of grain
temperature profiles at Q = 1. 34 1/ (sm3 ). Initial grain
temperature was 38°C, and inlet air temperature was 14°C.
Simulation time was chosen at the 10hr, 25hr, 55hr, and
100hr intervals. At the lOhr profile, both grain
temperatures, calculated and measured, were closer than the
others. However, at the 2Shr, 55hr, and 100hr intervals,
the difference between calculated and measured temperature
was much wider than that at the 10hr profile.
Information in Figure 13 shows the simulation results
at Q = 2.68 1/(sm3 ). Initial grain temperature was 37°C and
inlet air temperature was 14°C. Simulation time was chosen
at lOhr, 20hr, 40hr, and 70hr intervals. At the time
40
35
30
u
Eci. 25
(I)
I
20
,1>   0 
~ III,' ,;' 0'
/
tf
/
/
/
I
I
I
I
I
I
~; "
/1'
Jf/
151· 1~1"" ........ ·:...;=i·w=• =__=__' _~r ,~ // ~ ___ ~ ___ x_~K
/
.ll' 
X
/
/
/
/
/ .
10+1+1~_4~~_4~_4~~
o 0.1 0.2 0.3 0.4 0.6
Height, XlL
0.6 0.7 0.8 0.9
~ 10h·test
I II 25htest
A 55htest
)( 1OOhtest
 "¢  10hcaJo
 0  25hcalc
 l1  55hcalc
 ~  100hcaJo
Figure 12. Predicted and Measured Grain Temperature Profi les at Q = 1.34 1/(srn3 )
~
0'\
40.0
35.0
30.0
u
cEi. 25.0
~
20.0
,.,.. :a /
/
/
/
15.0 j('E;:: ;g: ~
I
I
I
P
I
I
I

,. ,. ,.
)<
10.0+1~r__+~~~_r~~_+;_~
0.0 0.1 02 0.3 0.4 0.5
Height, XIL
0.6 0.7 0.8 0.9 1.0
..10h·lasl
m20h·tast
ts 40htest
)( 70htest
 ~  10hcalc
 0  2Ohcalc
 :6  40hcalc
 i<  70hcalc
Figure 13. Predicted and Measured Grain Temperature Profiles at Q = 2.68 1/(sm3 )
If».
...J
48
periods of 10 hr and 20 hr, predicted temperatures were very
close to the actual ones. However, at the 40hr and 70hr
intervals, both calculated temperature profiles were lower
than measured temperatures.
Figures 14 and 15 show grain temperature profiles of
simulation and actual measurement at Q = 5.36 11 (sm3 ) and
8.04 1/{sm3 ). Initial grain temperature was 38°C, and inlet
air temperature was 18°C. Periods of prediction at 2hr,
6hr, 12hr, and 24hr intervals were chosen for Q = 5.36
11 (sr:n3 ) • Twohr, 5hr, l2hr, and 20hr intervals were
chosen for Q = 8.04 1/(sm3 ). Both simulations show more
difference in calculated temperature and measured
temperature than those of previous simulations.
The final simulation set airflow rate at 10.7 1/(sm3 ).
Initial grain temperature was 3SoC and inlet air temperature
was 19°C. Grain temperatures were plotted and shown in
Figure 16 at time periods of I hr, 2 hr, 4 hr, 8 hr, and 14
hr. From the beginning of simulation to the Ihr period,
results of temperature calcul ation coincided with measured
temperatures. At the 2hr period, temperature calculation
differed more than at the lhr period. As simul ation time
continued (at 4hr and 8hr periods), temperature
calculation differed even more. However, error of
temperature calculation tended to decrease as simulation
40
35 
30
u
Eci. 25
~
20 ~.
15 .
/
/
/
/
r_.;A: : /  
/
r;I
/
/
/
/
/
)f
/
/
/
/
)!t" ....
;;
;,;,'"
4;'
   ~**~
~
X
;' '"
;' '"
;'
;' X '"
'" '"
;,X
'"
10~ I ~r_+~~~
o 0.1 0.2 0.3 0.4 0.5
Height, XlL
O.S 0.7 O.S 0.9
"2htest
m 6h·test
:6 12htast
)( 24htest
 + 2hcalc
 8  Shcalc
"11 12hcalc
 +< ~ 24hcarc
Figure 14. Predicted and Measured Grain Temperature Profiles at Q = 5.36 1 / (sm3 )
IP>
to
40
35
30
()
Eci. 25
III
~
20
15
I
I
I
I
,;
,,.;' ,;,;
I
I
I
/
I
I
/
il
/
/
/
p'"
~.,.,.;'
/
/Ji
/
/
/
/
/
/
.A'
/
/
/
/
., "","'"
   ..   == =tt.  ",*""   ~   )("
,;,;
Ii,.;'
__ 4
.,.K ....
I
/
/
/
/
X~
'2htest
mShIesl
A 12htest
)( 20htsst
 ...  2hcalc
 i3  5h·ca/c
 A  12hca.lc
 oK  20hcalc
10+1~~_+~_4~r__++_~
o
Figure 15.
0.1 0.2 0.3 0.4 0.5
Height, XlL
0.6 0.7 0.8 0.9
Predicted and Measured Grain Temperature Profiles at Q 3 = 8 _ 04 1/ (sm )
lJ1
o
o
ci.
E
~
40.0
35.0
30.0
25.0
20.0
15.0
/
L.
o 0.1 0.2
/
/J!
/
0.3
/
/
/
/
/
.A""/
0.4
A .... ,
//
0.5
Height. XlL
X
/
//
. /
X
0.6 0.7
//
)(//
0,8 0.9
0 . 1 hIesl
B!I 2htest
' 4htest
~ Bhtest
I
)I( 14htest
;f.  ..  1hcalc I
 B  2hcalc
 ~ 4hcalc
 i(  8hcalc
 iIC  14hcalc
Figure 16_ Pr edicted and Measured Grain Temperature Profiles at Q = 10.7 1/(sm3 )
VI
I'
52
time reached the final 14hr period of aeration.
To examine whether predicted temperatures were
correlated with measured temperatures, a scatter plot was
developed for each airflow rate, and linear regression of
data was calculated and analyz.ed. Figures 17 and 18 show
samples of scatter plots at airflow rates 0.67 and 1.34
1/(sm3 ). Table III shows a summary of correlation of
predicted and measured temperatures at various airf l ow
rates. Table IV shows a summary of analysis of variance of
predicted and measured temperatures.
TABLE III
CORRELATION OF PREDICTED AND MEASURED TEMPERATURES
AT VARIOUS AIRFLOW RATES
Q, 1/ (sm3 ) 0.67 1. 34 2.68 5.36 8.04 10.72
Correlation, R 0 .. 9812 0.9879 0.9868 0.9576 0.9563 0.9670
R2 0.9627 0.9760 0.9738 0.9170 0.9146 0.9350
Standard error 1.7665 1. 4850 1. 6895 2.2628 2.6947 1. 7709
observations 30 40 40 40 40 50
Note: R2 is coefficient of determination.
40
35
30
o
!If 25
~
(J)
0
E 20
.2
"'0
.2
o 15
]
a:
10
5
O+~~I+I~I ~I~I+~I
o 5 10 15 20 25 30 35 40
Measured temperature, C
Figure 17. Scatter plot of Predicted Temperature
at Q = 0.67 II (sm3 )
40
35
30
(J)
:; 25
"@
(J)
0
f: 20
.2
"'0
(J)
~ 15
~
0..
10
5
O++ ~I ~1r11~1+1+1~1
o 5 10 15 20 25 30 35 40
Measured temperature, C
Figure 18. Scatter Plot of Predicted Temperature
at Q = 2.68 11 {sm3 }
53
TABLE IV
ANOVA OF PREDICTED AND MEASURED TEMPERATURES
AT VARIOUS AIRFLOW RATES
Q = 0.67 I/(smA3)
Regresslion
Residual
Total
Intercept
X Variable 1
df S8
2256.664919
28 87.37808142
29 2344 ... 043
Coefficients Standard Error
1.15052202.8 1.208540523
1.065323878 0.039615976
Q = 1.34If(sm" 3)
df SS
MS
2256 .. 664919
3.120645765
I Stat
0.951992926
26.89126931
MS
F
723.1403653
Pvalue Lower 95% Upper 95%
0.349246807 3.626107845 1.32506379
1.52157E21 0.984 174139 1.146473618
F
Regression
Residual
Total
4023.907407 402.3.907407 1409.767806
38
39
108.463593 2.854305078
4132.371
Coeffi cients Standard Error t Stat Pvalue Lower 95% Upper 95%
Intercept 2.233366769 0.866464821 2.577561968 0.013953733 3.987433157
0.479300381
X Variable 1 1.085336376 0.028906171 37.54687478 1.19477E31 1.026818891 1.143853862
Q '" 2.68 J/(smA3)
df S8 M8 F
Regression
Residual
Total
3406.724697 3406.724697 1544.828348
Intercept
X Variable 1
38
39
1 .728609405
1.027675943
Q = 5.36 11(sm"3)
df
83.79930278
3490.524
Standard Error
0.734365466
0.026146651
8S
Regression 2152.027671
ResilduaJ 38 194.5713285
Tolal 39 2346.599
Coefficients Standard Error
Intercept 0.639041533 1.469320728
X Variable 1 1.030091087 0.050245749
2.20524481
t Stal
2.353881665
39.30430445
MS
2152.027671
5.12029812
I Stat
0.434923105
20.5010593
Pvalue Lower 95% Upper 95%
0.023851483 3.2152546 19 0.24196419
2.19203E32 0.9747448131 .080607073
F
420.2934324
Pvalue Lower 95% Upper 95%
0.666078372 2.335442878 3.613525943
3.81215E22 0.928373882 1.131808293
54
55
(Table IV concluded)
a '" 8.04 If(smA3)
df S8 MS F
Regression 2953.955315 2.953 .. 955315 406.7858881
Residual 38 275.9444348 7.261695652
Total 39 3229.89975
Coefficients Standard Error t Stat Pv.alue Lower 95% Upper95%
Intercep\ 1.980104042 1.572422727 1.259269538 0.215613464 5.163307545 1.203099'461
X Variable 1 1.101880673 0.054632569 20.16893374 6.73899E22 0.991.2B2814 1.212478531
Q = 10.72 1/(sm"3)
dl SS MS F
Regressi'on 1 21£6.111281 2166.111281 690.6833361
Residual 48 150.5369191 3.136185814
Total 49 2316.6482
Coefficients Standard Error t Stat Pvalue Lower95')!. Upper 95%
Intercept 0.00320229 1.127715081 0.002839627 0.997746078 2.270624099 2.264219519
X Variable 1 1.026908397 0.039074391 26.28085494 3.80046E30 0.9483441151 .105472679
Information in Table III, shows a very strong
correl ation between predic t ed and measured temperature a l l
a i rflow rates (greater than 95%). At low airflow rates
{O.67, 1.3 4 , and 2.68 1/(sm3 )), the degrees of c orrelation,
98_12%, 98.79% and 98.68%, are a little more than that of
higher airflow rates, 95.76%, 95.63% and 96.70%.
Hypothesis testing of correlation can be made by
examining each of the Fva l ues included in Table IV. All
Fvalues are much more than FO. 01 ,1,28 (7 . 64) which means there
is a high confidence in correlation of predicted and
measured temperatures.
56
Information presented In Table IV also shows the
deviation of prediction by the slope of regression. There
is no deviation, if the slope is 1. O. Slope of regression
can be found in Table IV, in the Coefficients column and X
Variabl e row. The range of slopes is 1. 0269 to 1..102.
Accuracy of the temperature prediction can be
determined by observing the temperature error; difference
between prediction and measurement. At each prediction,
average absolute error was calculated from summation of
grain temperature difference (at the same time period and
locations) between simulation results and testing results at
various airflow rate(Q) conditions. A summary of the grain
temperature error at various airflow rates is shown in Table
V. Graphical comparisons of Table V are shown in Figures 19
and 20.
TABLE V
AVERAGE ABSOLUTE ERROR OF TEMPERATURE PREDICTION AT
VARIOUS TIME PERIODS AND AIRFLOW RATES
Unit : Degree Celsius
1. 34 2.68 5.36 8.04 10.72
Profile #1 .74 .31 .37 .58 .55 .15
Profile #2 1. 08 .87 .81 1. 85 2.60 .68
Profile #3 2.19 1. 87 1. 70 3.43 3.39 1. 74
Profile #4 2.05 1. 83 1. 37 1. 15 2.61
Profile #5 .45
Maximum 2.19 2.05 1.83 3.43 3.39 2.61
4
o 3
....
g
CD
.$
~ g 2
.0
(1j
CD
OJ
~
Ql
> « 1
o++~~
o 50 100 150
Time, hr
57
oQ = 0.67
DQ= 1.34
~Q=2.68
Figure 19. Pl otting of Average Absolute Error Over Different
Time Periods at Q = 0.67, 1.34 and 2.68 1/(sm3 j
3
o
....
g
<D
El
:::> g 2
.0
(1j
<D
OJ
~
CD
~
¢O=5.36
00= 8.04
lxO= 10.7
O +++++~
o 5 10 15 20 25
Time, hr
Figure 20. Plotting of Average Absolute Error Over Different
Time Periods at Q = 5.36, 8.04 and 10.7 l/(s~)
From information presented l.n Table V,
concluded:
it can be
 Maximum average absolute error occurs at the middle
of simul ation and tends to have more error at the higher
58
airflow rate. At high airflow rate, heat and mass transfer
rates between the grain kernel and air will be high enough
to cause error in the initial and boundary conditions of
equilibrium between grain and air. Additionally, the higher
airflow rate causes an increase in the convective heat
transfer coefficient causing the Biot number to increase.
 When aeration time is close to the final stage, for
example at Q = 5.36, 8.04, and 10.7 1/(srn3 ); average
absolute errors decrease to 1.37, 1.15 and 0.45°C
 At every airflow rate simulation, sunrrnation of error
is increased as the simulation time continues until
approximate by the middle of the simulation. After that,
the error trends to decrease as the simulation time
approaches the final state; therefore, prediction time and
temperatures at the final state of simulation will have
sufficient accuracy to predict the aeration time needed for
hard red winter wheat.
59
Grain Moisture Contents
Grain moisture contents were measured at the end of the
aeration experiment. The measured grain moisture content
and prediction results are shown in Table VI. The
difference from both results also are shown in Table VI.
InformatIon presented in Table VI is plotted ~n Figures
21 through 23 and shows the graphical comparison of grain
moisture content between the computer prediction results and
the experimental results at different airflow rates. Three
airflow rates {0.67, 2.68 and 8.04 11 (sm3 ) ) were chosen to
compare with the calculation results.
Information presented in Figure 21 shows the moisture
content profile of simulation at Q = 0,67 11 (sm3 ) • The
profile shows that the final moisture content was 12.2 %wb
and the average error was 0.62 %wb (Table VI).
Information in Figure 22, shows the final moisture
content result at Q = 2.68 11 (sm3 ) • The result was 12.2%wb.
The average error is 0.64 %wb as shown in Table VI. At
the next simulation, Figure 23 (Q = 8.04 I/sm3 ) , the
prediction of final moisture content is a little less than
the actual measurement, so the error ~s higher than those of
lower airflow rate. The final moisture content was 10.3 %wb
and the average error was 1.88 %wb.
TABLE VI
PREDICTED AND MEASURED FINAL MOISTURE CONTENTS
AT DIFFERENT AIRFLOW RATES BY LOCATION
Q : Airflow rate, Litre/ (sm3 )
Moisture content, %wb
60
Q = 0.67 Q 2.68 Q = 8.04
L (ft) test calc cliff test calc cliff test calc diff
0.5 12.81 12.20 0 .. 61 13. 00 12.20 0.80 12.00 10.31 1. 69
1.0 13.13 12.20 0.93 12.67 12.20 0.47 12.11 10.31 1. 80
1.5 13.34 12.20 1.14 12.77 12.20 0.57 12.29 10.31 1. 98
2.0 13.03 12.20 0.83 11. 83 12.20 0 .. 37 12.16 10.31 1. 85
2.5 12.93 12.20 0.73 11.98 12.20 0.22 12.27 10.31 1. 96
3.0 12.73 12.20 0.53 11.80 12.20 0.40 12.42 10.31 2.11
3.5 12.66 12.20 0.42 11.91 12.20 0.29 12.46 10.31 2.15
4.0 12.42 12.20 0.22 11.72 12.20 0.48 12.50 10.31 2.19
4.5 12.11 12.20 0.09 11.85 12 . 20 0.35 12.37 10.31 2.06
5.0 11.92 12.20 0.28 11.44 12.20 0.76 12.35 10.31 2.04
5.5 11. 84 12.20 0.36 11.38 12.20 0.82 12.30 10 . 31 1.. 99
6.0 11. 75 12.28 0.53 11. 36 12.28 0.92 12 . 22 10.31 1. 91
6.5 11.70 12.28 0.58 11. 25 12.28 1. 0] 12.18 10.47 1.71
7.0 11.75 12.28 0.53 11.24 12.28 1. 04 12.17 10.47 1. 7 a
7.5 11.68 12.28 0.60 11.29 12.28 0.99 12.25 10.47 1. 78
8.0 11.34 12.28 0.94 11.49 12.28 0.79 12.12 10.47 1. 65
8.5 11.0] 12.28 1. 25 11.74 12.28 0.54 12.02 10.47 1. 55
Average 0.62 0.64 1. 88
Note: Testing data from Epperly (1989)
15
14
13
..c 12
0 .o fO C Q 00
~
<f.
E 11
~ c 0 10
0
~
B 9
(/)
'0
:::2: 8
7
6
5 I I I I I I I I I I I I I I I I
0 0.5 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9
Length, ft
Figure 21. Predicted and Measured Final Moisture
Content at Q = 0.67 1/(sm3 )
15
14
13
~
12
cf
C 11
.$
:5 10 o
~ B 9
III
'0
~ 8
7
6
~~oo.cO·oo·Q·o
o 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9
Length, It
Figure 22. Predicted and Measured Final Moisture
Content at Q = 2.68 l/(s~)
61
15
14
13
~ 12
'#
"t 11
.! 0 0 .. 0 0 [J 0" 0 0 0 01 [J .. Ct ·0 iJ 0" o[] l5 10
o
~ 9
.2
fIJ
'0
:::E 8
7
6
5 I I I I I I I I
o 0.5 1 1.5 2 2.5 3 3..5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9
Length, ft
Figure 23. Predicted and Measured Final Moisture
Content at Q = 8.04 l/(s~)
62
Figure 24 shows the average absolute error of predicted
moisture content at various airflow rates. The error tends
to increase when airfl ow rate is increased.
Sample scatter plot of predicted and measured moisture
contents at Q = 0.67 1/(sm3 ) is shown in Figure 25.
Correlation of predicted and measured moisture content was
calculated and shown in Table VII. Information presented in
Table VIII, shows analysis of variance.

3
2.5
..c
3:
~0 2 ..: e
Qi:
.$
~ 1.5 (5
!/) ..c
<tI
CD
Ol
<tI
W
«>
0.5
0 I
0 2 3 4 5 6 7 8 9
Airflow rate, I/(sm'B)
Figure 24 . Average Absolute Error of Final Moisture
Contents by Various Airflow Rates
at Q = 0.67, 2.68 and 8.04 l/(s~)
14
12
j
~ 0
C 10
QJ
C
0
()
~ 8
2
·'0"
E 6
"0
QJ
t5
"B ~ 4 u..
2
0
0 5 10 15
Measured moisture content, %wb
Figure 25. Scatter Plot of Predicted Moisture
Cont ent at Q = 0.67 1/(sm3 )
63
TABLE VII
CORRELATION OF PREDICTED AND MEASURED
FINAL MOISTURE CONTENTS
0.67 2.68 8.04
Correlation, R 0.7696 0.5619 0.4253
0 .. 5924 0.3160 0.1809
Standard error 0.0252 0.0327 0.0690
observations 17 17 17
Note: R2 is coefficient of determination.
TABLE VIII
ANOVA OF PREDICTED AND MEASURED FINAL MOISTURE
CONTENTS AT VARIOUS AIRFLOW RATES
Unit : Q , 1/ (sm3 )
At Q = 0.67
Regression
Residual
Total
Intercept
X Variable 1
dt
15
16
Coefficien ts
12.7548382
·0.04297686
SS
0.01388833
0.00955709
0.02344542
Standard Error
0.11263258
0.00920507
MS F
0.Q1388833 21.7979359
0.00063714
t Stat Pvalue Lower 95% Upper 95%
113.242881 2.0587E23 1.2.5147674 12.994909
4.66882596 0.00030279 0 .06259701 0.02335671
64
(Table VIII concluded)
At Q = 2. 68
Regression
Hesidual
Total
Intercept
X Variable 1
df
15
16
Coefficients
12.7021863
0.04001738
At Q = 8. 04
Regression
Residual
Total
Intercept
X Variable 1
df
15
16
Coefficients
12.8961139
0.20720069
S8
0.00740162
0 .. 0160438
0.02344542
Standard Error
0.17976607
0.01521224
SS
0.01576041
0.07135232
0.08711274
Standard Error
1.39387754
0.11383239
65
MS F
0.00740162 6.92007323
0.001106959
t Stat PvallJe Lower 95% Upp,er 95%
70.6595313 2.4021E20 12.3190237 13.0853488
2.63060321 0.01890975 0 . .07244153 0.00759323
MS F
0.01576041 3.3132233
0.00475682
t Stat Pvalue Lower 95% Upper 95%
9.25197051 1.3759E07 9.92513243 15.8670954
·1.82022617 0.08873459 0.44982884 0.03542745
From information presented in Figure 25 and Table VII,
there is a fair degree of correlation, 77% at Q = 0.67
1/(sm3 ), and decreases to weak correlation when the airflow
rate is increased, 56% at Q = 2.68 1/(sm3 ) and 42% at Q =
The analysis of variance, Table VIII, also
shows weak correlation, because the regression slopes
(0.04, 0.04 and 0.21) are low approaching the value of
zero.
The explanation is that ln the practical situation,
physical properties of grain and moisture content may not
match the initial assumption. For example, no gradient of
66
temperature and moisture content of a bin cross section, and
bin wall is in adiabatic condition. These results also show
the effect of airflow rates leading to lower correlation and
larger average absolute error at high airflow rates.
The predicted and measured results were not identical,
but at the low airflow rates, average absolute errors were
within 1%, 0.62% at 0.67 1/(sm3 ), and 0.64% at 2.68 1/(sm3 ).
This result suggests that the simulation model may be used
for moisture content prediction where 1% moisture content
accuracy is acceptable.
To improve accuracy of calculation, a number of node
(20) were applied, however, the error of prediction were not
significantly different from the present results. A longer
time step (2x) may be used to reduce simulation time, in
case of aeration with very low airf l ow rate.
CHAPTER VI
CONCLUSIONS
Information presented in Chapter I shows that good
aeration is an important feature in maintaining high quality
grain. In order to achieve good aeration, aeration time
needs to be determined . It would take a lot of effort to
develop and test the actual aeration to find an aeration
time, because i t is affected by many parameters such as
temperature and moisture content. A computer simulation of
aeration could determine the aeration time.
The purpose of this study was to develop a personal
computer calculation model which simulates wheat aeration of }
',.
ii!
" a fixed bed bin. An aeration simulation program was
developed to find the appropriate time for aeration of hard
red winter wheat. To simulate a fixed bed aeration, a
calculat i on model was developed based upon Parry's general
model (eg. 6 ) . Particular conditions of a fixed bed
aeration were applied to t he cal culation model (objective
1 ). After the model was developed , a computer program
(objective 2) was implemented to simulate hard red winter
67
wheat aeration. From information obtained in this study,
the evaluation of the simulation program (objective 3) can
be summarized as follows:
68
 At every simulation, temperature prediction error
increased until the simulation reached the middle of
aeration time.. After that the prediction error decreased to
a minimum at the final time of aeration.
 prediction of grain moisture content does not show a
correlation with the measured moisture content, but at low
airflow rates (less than 4.0 1/(sm3 ), Figure 24), average
absolute error is within 1%.
 Higher volumetric airflow rate has an effect on
increasing the average absolute error of grain temperature
and moisture content.
All of the above findings are the result of the initial
assumptions of the calculation model which have been made to
reduce the complexity of the model. In the actual testing,
some initial assumptions were found not to be precisely
correct; for instance, one assumption made earlier in the
study is the slow heat and mass transfer between grain and
medium fluid which does not occur at the higher airflow
rate.
The calculation program, because of the above results,
is not a good predictor (withou t further modification) to
predict grain temperatures at the middle of the aeration
time at high airflow rate (greater than 8.04 l/(sml )).
However, the prediction at the final stage is acceptable to
predict the aeration time. Therefore, fixed bed aeration
time of hard red winter wheat can be predicted by this
computer simulation program.
69
I
, I
I
I
I
!
1 I
i
.'.1 i,
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APPENDIXES
73
1 J I
,i I
1
I
L
11
'I I
, I
j ,
\ ' j I
I I
,j
:I
'I
i\
I i
i , '
APPENDIX A
RESULTS OF EXPERIMENTAL DATA
This following tables show the experimental grain
temperature results (Epperly, 1989) of wheat aeration:
TABLE IX 'I
EXPERIMENTAL DATA AT Q = 0.67, 1. 34 AND 2.68 II (sm3 )
Q: unit airflow rate [Li trel (sm3 ) ]
Temperature : °c
X/L 0.0 0 .1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Q=0.67
25h 14.4 22.8 31.0 35.0 36.0 36.2 36.6 36.9 37.2 37.2 37.2
SOh 14.4 16.8 22.0 29.0 33.8 36.0 36.5 36.6 36.8 36.8 36.8
150h 14.4 1 4 . 4 14.5 15.1 17.0 19.2 22.0 25.0 27.6 30.0 30.0
Q=1. 34
10h 14.4 26.9 34.9 37.0 37.4 37.8 38.3 38.3 38.3 38.3 38.3
2Sh 14.4 15.6 22.3 29.3 34.1 36.9 38.0 38.3 38.3 38.3 38.3
55h 14.4 14.4 14.8 17.0 20.2 24.3 28.1 31.7 34.4 36.2 36.7
100h 14.4 14.4 14 . 8 15.5 16.2 16.8 18.2 20.0 21.2 24 . 7 26.1
Q=2 . 68
10h 14.4 17.7 28.9 34.9 36.8 37.2 37.2 37.2 37.2 37.2 37.2
20h 14.4 14.6 18.4 24.6 30.0 34.1 36.6 37.2 37.2 37.2 37.2
40h 14.4 14.4 14.9 15.9 18.4 21.4 24.9 28.1 30.7 33.2 34.3
70h 14.4 14.4 14.4 15.4 15.9 16 .8 17.6 18.6 20.5 22 . 1 24.1
74
75
TABLE X
EXPERIMENTAL DATA AT Q = 5.36, 8. 04 AND 10.7 1/ (sm3 )
Q: unit airflow rate [Litre/(sm3 )]
Temperature : DC
X/L 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Q=5.36
2h 18.3 28.7 33.4 35.7 35.8 35.8 36.1 36.7 36.7 36.7 36.7
6h 18.3 19.7 22.9 27.6 30.8 33.2 35.0 36.2 36.4 36.7 36.7
12h 18.3 18.9 18.8 20.3 22.0 23.4 26.8 29.7 31. 8 33.8 34.3
24h 18.3 18.3 18.7 19.0 19.0 19.0 19.7 20.6 21. 6 23.3 28.0 I
!
Q=8.04
:,
t
i!
2h 17.8 27.6 32.7 35.7 37.9 38.3 38.3 38.3 38.3 38.3 38.3 'J
5h 17.8 21. 0 23.5 25.5 29.1 31.8 34.1 36.2 37.6 38.3 38.3
12h 17.8 18.0 19.5 19.3 20.0 21.2 22.6 24.6 26.8 28.9 30.1
20h 17.8 18.0 18.3 18.7 18.8 19.0 19.2 19.7 20.6 21.5 25.3
Q=10.72
1h 18.9 30.0 33.8 35.0 35.0 35.0 35.0 35.0 35.0 35.0 35.0
2h 18.9 25.0 29.1 31. 6 33.5 34.5 35.0 35.0 35.0 35.0 35.0
4h 18.9 19.8 23.0 25.2 27.6 29.8 32.0 33.4 34.2 34.8 35.0
8h 18.9 18.9 19.8 20.2 21.0 22.2 23.8 25.8 27.6 29.4 31.6
14h 18.9 18.9 18.9 18.9 18.9 18.9 19.3 19.8 20.6 21.8 23.4
APPENDIX B
SIMULATION RESULTS
'I
The following data show the simulation printout of
wheat aeration Q = 0.67, 1. 34, 2.68, and 10.72 1/(sm3 ):
Desired Time ;:: 27S.08 hrs
AIR VELOCITY 0.36 ft/m.in
AIR VOLUME O.OS CFM/bush or 0 . 67 l / s/cu.m
No. Node ;:: 1 0 Length = 9.00 [ft]
dx= 0.90 [ft]
dt = O.OSO [min ] BM ;:: 0.01
Temp 0.0 0.1 0.2 0.3 0.4 O.S 0.6 0.7 0.8 0.9 1.0
Temp 14.4 37.2 37.2 37.2 37.2 37.2 37.2 37.2 37.2 37.2 37.2
Mdb 13.9 14 .3 14.3 14.3 141 .3 14 .3 14.3 14 .3 14.3 14.3 14 .3
Shr T = 34.8 37 . 1 37.2 37.2 37.2 37.2 37.2 37.2 37.2 37.2
Shr M = 14 .3 14.3 14.3 14.3 14.3 14.3 14.3 14 . 3 14.3 14.3
10hr T = 31.8 36.6 37.2 37.2 37.2 37.2 37.2 37.2 37 .2 37.2
10hr M ;:: 14.2 14.3 14.3 14.3 14.3 14 .3 14.3 14.3 14 .3 14.3
lShr T ;:: 28.9 3S.8 37.0 37.2 37.2 37.2 37.2 37.2 37.2 37.2
lShr M 14.2 14.3 14.3 14 .3 14 .3 14.3 14.3 14.3 14 . 3 14.3
20hr T ;:: 26.2 34.6 36.8 37.2 37.2 37.2 37.2 37.2 37.2 37.2
20hr M = 14 .2 14.3 14 . 3 14.3 14.3 14.3 14.3 14.3 14.3 14.3
2Shr T = 23.6 33.2 36 . 4 37.1 37.2 37.2 37.2 37.2 37 .2 37.2
25hr M 14.1 14.2 14.3 14.3 14 .3 14.3 14.3 14.3 14.3 14 .3
30hr T 21.3 31.S 3S.7 37.0 37.2 37.2 37.2 37.2 37.2 37.2
3Dhr M 14. 1 14.2 1 4.3 14 .3 14.3 14.3 14.3 14.3 14.3 14.3
35hr T 19. 1 29.7 34.9 36.7 37.1 37.2 37.2 37.2 37.2 37.2
35hr M 14.0 14.2 14 .3 14.3 14.3 14.3 14.3 14.3 14.3 14.3
4Dhr T 17 .3 27.8 33.9 36.4 37.0 37.2 37.2 37.2 37.2 37.2
40hr M 14.0 14.2 14.3 14.3 14.3 14.3 14.3 14.3 14.3 1 4 . 3
45hr T 15.8 25.8 32.7 35.9 36.9 37.2 37.2 37.2 37.2 37.2
45hr M 13 .9 14 .2 14.2 14.3 14.3 14.3 14.3 14.3 14 .3 14 .3
SOhr T 14.7 23.9 31. 2 35.3 36.7 37.1 37.2 37.2 37.2 37.2
SOhr M ;:: 13 .9 14.1 14.2 14.3 14.3 14.3 14 .3 14.3 14.3 14.3
S5hr T 14.4 22.1 29.7 34.6 36.4 37.0 37.2 37.2 37.2 37.2
55hr M = 13 .9 14.1 14.2 14 .3 14 .3 14 .3 14.3 14.3 14.3 14.3
60hr T ;:: 14 .4 20.S 28 . 0 33.7 36.1 36.9 37.2 37.2 37.2 37 . 2
60hr M ;:: 13.9 14 .1 14.2 14.3 14 .3 14.3 14.3 14 .3 14.3 14.3
6Shr T ;:: 14.4 19.1 26.2 32.6 35.6 36 . 8 37.1 37.2 37 . 2 37.2
65hr M ;:: 13.9 14 . 0 14.2 14.2 1 4 .3 14.3 14.3 14.3 14.3 14.3
70hr T 14.4 17.9 24.4 31.4 3S.0 36.6 37.0 37.2 37 .2 37.2
70hr M 13.9 14.0 14.1 14.2 14 . 3 14.3 14.3 14.3 14.3 1 4.3
7Shr T 14 .4 17.0 22.7 30.1 34.2 36.3 36.9 37.2 37.2 37.2
75hr M 13 .9 14 .0 14.1 14.2 14 .3 14.3 14.3 14.3 14.3 14.3
80hr T 14 .4 16.2 21. 0 28.7 33.3 35.9 36.8 37.1 37.2 37.2
80hr M 13 .9 14.0 14.1 14.2 14 .2 14.3 14.3 14.3 14.3 14.3
85hr T ;:: 14.4 1 5.6 19.4 27.2 32.3 35.4 36.6 37.1 37.2 37.2
85hr M 13.9 13 . 9 1 4 .0 14 .2 14.2 14.3 14.3 14.3 14.3 14.3
76
77
90hr T 14.4 IS.2 18.0 2S.7 31.2 34.8 36.4 37.0 37.2 37.2
90hr M 13.9 13.9 14.0 14.2 14.2 14.3 14.3 14.3 14 .3 14.3
95hr T 14.4 14.9 16.7 24.1 29.9 34.1 36.0 36.9 37.1 37.2
95hr M 13.9 13.9 14 .0 14.1 14.2 14.3 14.3 14.3 14 .3 14.3
100hr T 14 .4 14.7 1S.5 22.6 28.6 33.3 35.6 36.7 37.1 37.2
100hr M 13 .9 13.9 13.9 14.1 14.2 14.2 14.3 14.3 14 .3 14 . 3
10Shr T 14.4 14.7 14 .7 21.2 27.1 32.4 35.2 36.5 37.0 37 .2
10Shr M = 13 . 9 13.9 13.9 14.1 14.2 1.4.2 14.3 14.3 14.3 14.3
110hr T 14.4 14.6 14 .6 19.9 25.6 31.4 34 . 6 36.3 36.9 37.1
110hr M 13.9 13.9 13.9 14 .0 14 .2 14.2 14 .3 14.3 14.3 14.3
115hr T = 14.4 14.6 14 .6 18.8 24.1 30.3 33.9 35.9 36.8 37.1
l1Shr M 13 .9 13.9 13 .9 14.0 14.1 14.2 14.3 14.3 14 .3 14 . 3
120hr T 14.4 14.6 14.6 17.9 22.6 29 . 1 33.1 3S . S 36.6 37.0
120hr M 13 .9 13.9 13 .9 14.0 14 .. 1 14.2 14 .. 2 14.3 14.3 14.3
125hr T 14.4 14.6 14.6 17.1 21.1 27.8 32.2 3S.1 36.4 36.9
125hr M 13.9 13.9 13 .9 14.0 14.1 14.2 14 .2 14 .3 14.3 14.3
130hr T 14.4 14.6 14.6 16.5 19.8 26.5 31.2 34.5 36.1 36.8
BOhr M 13 .9 13.9 13 .9 14.0 14.0 14.2 14 .2 14.3 14.3 14.3
135hr T 14.4 14 . 6 14.6 16.0 18.S 25.2 30.1 33.9 35 .. 8 36.7
135hr M = 13.9 13 . 9 13.9 14.0 14.0 14.1 14 .2 14.3 14.3 14.3
140hr T = 14.4 14.5 14.5 IS.6 17 .4 23.9 28.9 33.2 35.4 36.S
140hr M = 13 . 9' 13.9 13 .9 13.9 14.0 14.1 14 .2 14 .2 14.3 14.3
14Shr T = 14.4 14.5 14.S 15.4 16.4 22.6 27.7 32.3 34.9 36.3
145hr M 13.9 13.9 13.9 13.9 14.0 14.1 14 .2 14 .2 14.3 14.3
150hr T 14.4 14.5 14.S 15.3 15.7 21. 4 26.4 31.4 34.3 36.0
150hr M 13 .9 13.9 13.9 13.9 13.9 14.1 14.2 14.2 14.3 14.3
15Shr T 14.4 14.5 14. S 15 . 2 15.2 20.3 25.0 30.4 33.7 35 . 7
15Shr M 13 .9 13.9 13 .9 13.9 13 .9 14.1 14.1 14.2 14.3 14.3
160hr T 14.4 14.5 14.5 15.1 15.1 19.3 23.7 29.4 32.9 35.3
160hr M 13.9 13.9 13 . 9 13.9 13.9 14.0 14.1 14 .2 14.2 14.3
165hr T 14.4 14.5 14.5 15.0 15.0 18.5 22.4 28.3 32.1 34.9
165hr M 13.9 13.9 13.9 13.9 13 .9 14.0 14.1 14 .. 2 14.2 14 . 3
170hr T 14 .4 14.5 14 .5 15.0 15.0 17.8 21.1 27.1 31.2 34.4
170hr M = 13.9 13.9 13 .9 13.9 13 .9 14 . 0 14 .1 14.2 14 .2 14.3
175hr T 14.4 14.5 14 .5 14 .. 9 14.9 17.2 19.9 26.0 30.2 33.8
175hr M 13 .9 13.9 13 .9 13 .. 9 13.9 14 .0 14 .0 14 . 2 14 .2 14.3
180hr T 14.4 14.S 14.5 14 .9 14.9 16.7 18 .8 24.8 29.1 33.1
180hr M = 13 .9 13.9 13 .9 13 .9 13.9 14.0 14 .0 14 .1 14 .2 14.2
185hr T = 14.4 14.5 14.5 14.9 14.9 16.4 17.8 23.6 28.0 32.4
185hr M 13.9 13.9 13.9 13.9 13 .. 9 14.0 14.0 14.1 14 .2 14.2
190hr T = 14.4 14.S 14.5 14.8 14.8 16.1 16.9 22.5 26.8 31.7
190hr M 13.9 13.9 13.9 13.9 13 . 9 14.0 14.0 14 .1 14.2 14 . 2
195hr T 14.4 14.S 14.5 14.8 14.8 15.9 16.1 21.4 25.6 30 . 8
195hr M = 13.9 13.9 13.9 13.9 13 .9 14.0 14.0 14 .1 14 .1 14.2
200hr T = 14.4 14.5 14.5 14.8 14.8 15.8 15.8 20.4 24.3 30.0
200hr M = 13 .9 13 .9 13.9 13.9 13.9 13.9 13.9 14.1 14.1 14.2
205hr T = 14.4 14 .5 14.5 14.7 14.7 15.7 15.7 19.6 23.1 29.1
205hr M = 13 .9 13.9 13.9 13.9 13.9 13 .9 13.9 14.0 14.1 14.2
210hr T = 14 . 4 14.5 14.5 14.7 14.7 IS.6 15.6 18.9 21.8 28.1
210hr M = 13 .9 13 .9 13.9 13.9 13.9 13 .9 13.9 14.0 14.1 14.2
21Shr T 14 .4 14.5 14.5 14.7 14.7 15.5 15.5 18.3 20.6 27.2
21Shr M 13.9 13.9 13.9 13.9 13.9 13.9 13.9 14.0 14.1 14 .2
220hr T 14 .4 14.5 14.5 14.7 14.7 15.5 lS.5 17.8 19.5 26.4
220hr M 13.9 13.9 13 .9 13.9 13.9 13 .9 13 . 9 14 . 0 14.0 14.2
22Shr T 14 .4 14.5 14.S 14.6 14.6 15.4 15.4 17 .4 18.4 25.5
225hr M 13 .. 9 13.9 13 .9 13.9 13.9 13 .. 9 13.9 141.0 14.0 14.1
230hr T 14.4 14.5 14.5 14.6 14.6 15.3 15.3 17.2 17.4 24.7
230hr M 13 .9 13.9 13 .9 13.9 13.9 13 .9 13.9 14.0 14.0 14.1
23Shr T 14.4 14.5 14 .S 14.6 14 .6 15.2 15.2 17.0 17.0 23.9
235hr M 13.9 13.9 13.9 13 .9 13.9 13 .9 13 .9 14 .0 14 .0 14.1
240hr T 14.4 14.5 14.5 14.6 14.6 15.2 15.2 16.8 16.8 23.1
240hr M 13 .9 13.9 13 .9 13.9 13 .9 13 .9 13 .9 14.0 14.0 14.1
245hr T 14.4 14.5 14.S 14.6 14.6 15.1 15.1 16.7 16.7 22.S
78
245hr M == 13 . 9 13.9 13.9 13.9 13.9 13.9 13.9 14.0 14.0 14. 1
250hr T = 14 .4 14 .5 14 .5 14.6 14 .6 15.1 15.1 16.5 16.5 21. 8
250hr M == 13.9 13 .9 13 .9 13.9 13.9 13.9 13 .9 14 .0 14 . 0 14.1
255hr T = 14.4 14.5 14.5 14.6 14 . 6 15.0 15.0 16.4 16.4 21.3
255hr M == 13 .9 13.9 13.9 13.9 13.9 13.9 13 .9 14.0 141 . 0 14.1
260hr T == 14 . 4 14.5 14.5 14.5 141 .5 15.0 15.0 16.2 16.2 20.8
260hr M = 13.9 13.9 13.9 13.9 13 .9 13.9 13 .9 14 .0 14.0 14 . 1
265hr T = 14.4 14.5 14 .5 14 . 5 14.5 14.9 14.9 16.]. 16 . 1 20 . 3
26Shr M = 13 .9 13.9 13 .9 13.9 13.9 13 .9 13 .9 14 .0 14 .0 1 4.1
270hr T 14.4 14.5 14 .5 14 .5 14.5 14.9 14.9 16.0 16.0 19.9 270hr M 13 .9 13.9 13.9 13.9 13.9 13 .9 13.9 14.0 14.0 14.0 I
275hr T 14.4 14.5 14.5 14 .5 14.5 14 . 9 14 .9 15.9 15.9 19.5
275hr M 13 . 9 13.9 13 . 9 13.9 13.9 13 .9 13 .9 14.0 14.0 14 .. 0
Desired Time 135.08 hrs
AIR VELOCITY 0.72 ftlmin
.AIR VOLUME 0.10 CFMlbush or 1.34 l/s/cu.m
No. Node = 10 Length = 9.00 [ ft]
dx= 0.90 [ft]
dt = 0.050 [min] BM == 0.01
Temp 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Temp 14.4 38.3 38.3 38.3 38.3 38.3 38.3 38.3 38.3 38.3 38.3
Mdb 13.9 14.3 14.3 14.3 14 . 3 14.3 14.3 14.3 14 .3 141 .3 14.3
Shr T 32.6 37.7 38.3 38.3 38.3 38.3 38.3 38.3 38.3 38.3
Shr M 14.2 14.3 14.3 14.3 14.3 14.3 14.3 14.3 14 .3 14.3
10hr T = 26.8 35.6 37.9 38.3 38.3 38.3 38.3 38.3 38.3 38 . 3
10hr M = 14.2 14.3 14.3 14.3 14.3 14.3 14.3 14.3 14.3 14.3
IShr T = 21. 6 32.3 36.8 38.0 38.3 38.3 38 . 3 38.3 38.3 38.3
IShr M = 14.1 14.2 14.3 1 4 .3 14 .3 14 .3 141 . 3 14.3 14.3 14.3
20hr T = 17.4 28.3 34.8 37.4 38.1 38.3 38 .3 38.3 38.3 38.3
20hr M = 14.0 14 .2 14.3 14.3 14.3 14.3 14 .3 14 .3 14.3 14.3
25hr T = 14.7 24.1 32 . 0 36 . 3 37.8 38.2 38 . 3 38.3 38.3 38.3
25hr M = 13.9 14.1 14 .2 14 .3 14 .3 14.3 14 .3 14 .. 3 14 .3 14.3
30hr T = 14.4 20.4 28.5 34.5 37.1 38.0 38.3 38 .. 3 38.3 38.3
30hr M = 13.9 14.1 14.2 14.3 14.3 14 .. 3 14.3 14 .3 14 .3 14 . 3
3Shr T = 14.4 17.6 24.7 32.0 35.9 37 . 6 38.1 38.3 38.3 38 . 3
3Shr M == 13.9 14.0 14.1 14 .. 2 14 .3 1 4 . 3 14.3 14.3 14 .3 14 . 3
40hr T = 14.4 IS.7 21.0 29.0 34.1 36.8 37.9 38.2 38.3 38.3
40hr M 13 .9 13.9 14.1 14.2 14.2 14.3 14.3 14 .3 14.3 14.3
4Shr T 141 .4 14.6 17.8 2S.7 31.8 35 . 7 37.4 38.0 38.3 38.3
45hr M 13.9 13.9 14.0 14.1 14.2 14.3 14.3 14..3 14 . 3 14.3
SOhr T 14.4 1 4 .. 4 lS.3 22.4 28.9 34.0 36.6 37.7 38 . 2 38.3
SOhr M 13.9 13 .. 9 13 .9 14 .1 14.2 14.2 14.3 14.3 14.3 14.3
55hr T 14.4 14.4 14.4 1 9 . 4 25 . 7 31.8 35.4 37.2 38.0 38.2
55hr M 13.9 13.9 13.9 14 .0 14.1 14.2 1 4 .3 14.3 14.3 14.3
60hr T 14.4 14.4 14.4 1 7.2 22.S 29.2 33.7 36.4 37.6 38.1
60hr M 13.9 13.9 13.9 14.0 14.1 14.2 14.2 14.3 14.3 14.3
6Shr T 14.4 14.4 14 . 4 1S.8 19.5 26.4 31.6 3S.2 37.0 37.9
6Shr M 13 . 9 13 .9 13.9 13.9 14.0 14.2 14.2 14.3 14.3 14.3
70hr T 14.4 14.4 14.4 14.9 17.0 23.5 29.1 33.7 36.2 37.5
70hr M = 13.9 13.9 13.9 13.9 14 .0 14.1 14.2 14.2 14 . 3 14 . 3
7Shr T = 14.4 14.4 14.4 14.6 lS . 0 20 . 7 26.3 31 . 7 3S.0 36.9
75hr M = 13.9 13.9 13.9 13.9 13.9 14 . 1 14.2 1 4.2 14 .3 14.3
BOhr T = 14 .4 14.4 14 .. 4 14.6 14 .6 18.4 23.4 29.4 33.4 36.1
BOhr M = 13 .9 13.9 13.9 13.9 13.9 14.0 14.1 l4 .2 14.2 14.3
8Shr T = 14.4 14.4 14.4 14.S 14 .S 16 . 7 20 . 6 26.8 31. 5 35.0
8Shr M 13 .9 13 .9 13 . 9 13.9 13 .9 14.0 14.1 14.2 14 . 2 14 .3
90hr T 14 .4 14.4 14.4 l4.5 14 .5 15.6 18.1 24.2 29.1 33.6
90hr M 13 . 9 13.9 13 .9 13.9 13.9 13.9 14.0 14.1 14.2 14.2
95hr T 14 .4 14.4 14.4 14.5 14. S 15.1 16.1 21. 6 26.S 31. 8
9Shr M 13 .9 13 .9 13.9 13.9 13 .9 13.9 14.0 14.1 14.2 14.2
100hI' T 14 .4 14.4 14.4 14. S 14.S 14.9 14.9 19.3 23.8 29.9
79
100hr M 13.9 13.9 13 . 9 13.9 13.9 13.9 13 .9 14 .0 14 .1 14 . 2
10Shr T 14.4 14.4 14 .4 14 .5 14 . 5 14.8 14 . 8 17.6 21.0 27.8
1 0Shr M 13 . 9 13.9 13 .9 13.9 13 . 9 13 .9 13 .9 14.0 14.1 14.2
110hr T 14.4 14.4 14 .4 14.5 14.5 14.7 14.7 16 . 5 18.5 25.8
1l0hr M 13.9 13.9 13.9 13.9 13.9 13 .9 13.9 14 .0 14 . 0 14.1
115hr T 14.4 14.4 14 .4 14 . 5 14.5 14.7 1 4.7 15.9 16.2 24.0
l1Shr M 13.9 13.9 13.9 13.9 13.9 13.9 13 .9 14.0 14 . 0 14. 1
120hr T 14 .4 14.4 14.4 1 4.5 14 . 5 14.6 14.6 1 5.7 15.7 22.3
120hr M 13 . 9 13 .9 13.9 13.9 13 .9 13.9 13.9 1 3 . 9 13 .9 14.1
125hr T 14 .4 14.4 14.4 14 .5 14.5 14.6 14.6 15.5 15 . 5 20.9
125hr M = 13 .9 13.9 13 . 9 13.9 13 . 9 13.9 13 . 9 13 . 9 13 .9 14.1
130hr T = 14.4 14 . 4 14 .4 14 .5 14.5 14.6 14.6 15.3 15.3 19.7
DOhr M 13.9 13.9 13 . 9 13 .9 13 . 9 13 .9 13.9 13.9 13.9 14 . 0
13Shr T 14.4 1 4.4 14.4 14.5 14 .5 14.6 14.6 15.2 15.2 1 8.8
135hr M = 13.9 13.9 13 . 9 13.9 13.9 13.9 13 .9 13.9 13 .9 1 4.0
Desired Time B8.08 h r s
AIR VELOCITY 1. 45 ft/min
AIR VOLUME 0.20 CFM/bush or 2 . 68 l/s/cu.m
No. Node = 10 Length = 9.00 [ftl
dx= 0.90 [ft]
dt = 0.050 [min] BM = 0.01
Temp 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 O.S 0.9 1.0
Temp 14.4 37.2 37.2 37.2 37.2 37.2 37 . 2 37.2 37.2 37.2 37.2
Mdb 13 .9 14.3 14.3 14 .3 14.3 14 . . 3 14 . 3 14.3 14 .3 14.3 14.3
5hr T = 26.2 34.6 36.S 37.2 37.2 37.2 37.2 37.2 37 .2 37 . 2
5hr M = 14.2 14.3 1 4 . 3 14.3 14.3 14.3 14.3 14.3 14 .3 14 .. 3
10hr T 18.8 27.8 33.9 36.4 37 . 0 37.2 37.2 37.2 37.2 37 . 2
1 0hr M 14.0 14.2 14 .3 14.3 14.3 14.3 14.3 14.3 14.3 14.3
15hr T 16.3 21.3 2B.2 33.6 36.0 36.9 37.2 37.2 37.2 37.2
15hr M 14.0 14 .1 14 .2 14 .3 14.3 14.3 14 . 3 14.3 14.3 14 . 3
20hr T 15 . 4 17.1 22.0 28 .7 33.3 3S . 8 36.8 37 .1 37 . 2 37.2
20hr M = 13.9 14.0 14.1 14 .2 14 .2 14.3 14.3 1 4 .3 14.3 14.3
2Shr T = IS.1 lS.l 18.S 23.4 28.7 33.2 35.6 36.7 37.1 37.2
2Shr M = 13 . 9 13.9 14.0 14 .1 14 .2 14.2 1 4 .3 14.3 14.3 14.3
30hr T = lS . 0 IS.0 16.9 19.4 23 . S 29.1 33.0 35.5 36.5 37 . 0
30hr M 13 . 9 13 .9 14.0 14.0 14.1 14.2 14.2 14.3 14 .3 14.3
35hr T = 14 . 9 14.9 16.2 17. 1 19.9 24.S 29 . 0 33.0 35.3 36.5
35hr M = 13 .9 13.9 14.0 14.0 14 .0 14.1 14 . 2 14.2 14.3 14.3
40hr T = 14.9 14.9 lS . 9 lS.9 18.0 20.7 24 . 4 29.3 32 . 8 35.2
40hr M 13 .9 13 . 9 14 . 0 14. 0 14 .0 14.1 HI.1 14.2 14 .2 14.3
45hr T 14.8 14.S 15 . 7 15.7 17.1 18.S 20 . 7 2S.1 29.2 33.0
45hr M 13 .9 13 .9 13 .9 13.9 14.0 14.0 14.1 14 .1 14 .2 14 .2
SOhr T 14.8 14.8 15.6 lS.6 16.6 17.2 18.7 21. 6 24.B 29.9
50hr M 13 . 9 13 .9 13.9 13.9 14.0 14.0 14.0 14.1 14 .. 1 14 . 2
5Shr T 14.7 14.7 15.4 15 . 4 16.3 16.4 17.7 19 . 5 21.0 26 . 7
5Shr M = 13 . 9 13.9 13.9 13.9 14.0 141.0 14.0 14 . 0 14.1 14.2
60hr T 14.7 14.7 15.3 IS . 3 16 .1 16 .1 17 .1 lS.4 18.6 24 . 0
60hr M 13 .9 13 . 9 13.9 13 . 9 14 . 0 14.0 14.0 14.0 14.0 14 .1
65hr T 14.6 14 . 6 15 .2 1 5.2 16.0 16.0 16.6 17.S 17.S 21. S
65hr M 13 .9 13.9 13 . 9 13.9 14 . 0 14.0 14 . 0 14.0 14 .0 14. 1
70hr T 14.6 14.6 15.1 15.1 15.9 15.9 16.3 17.3 17 .3 20.2
70hr M 13 .9 13. 9 13.9 13 .9 14.0 14.0 14 .0 14 . 0 14.0 14 .1
7S!hr T 14.6 14 . 6 15.1 15.1 15.8 15.8 16.1 16.9 16 . 9 19.1
75hr M 13.9 13 .9 13.9 13.9 13.9 13.9 14.0 1 4 . 0 14 . 0 14.0
SOhr T 14.6 14.6 15 . 0 15.0 15.7 15.7 15.9 16 . 6 16.6 1 8.2
80hr M 13 . 9 13.9 13.9 13.9 13 .9 13.9 14 . 0 14.0 14 .0 14.0
85hr T 14.6 1 4.6 15.0 lS . 0 IS.6 15.6 15.8 16 . 3 16.3 17.6
85hr M 13.9 13 .9 13. 9 13.9 13.9 13.9 13 .9 14.0 14.0 14.0
80
Desired Time 21.02 hrs
AIR VELOCITY 5.79 ft/min
AIR VOLUME 0.80 CFM/bush or 10.72 l/s/cu.m
No. Node = 10 Length 9.00 [ft]
dx= 0.90 [ ft]
dt = 0.050 [min] BM 0.01
Temp 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Temp 18.9 35.0 35.0 35.0 35.0 35.0 35.0 35.0 35 . 0 35.0 35.0
Mdb 11.5 14.3 14.3 14.3 14.3 14.3 14.3 14 . 3 14.3 14.3 14.3
1hr T 28.7 33.8 34.8 35.0 35.0 35.0 35.0 35.0 35.0 35.0
1hr M 13.5 14 .1 14.3 14.3 14.3 14.3 14.3 14.3 14 .3 14.3
2hr T = 23.3 30.4 33.7 34.7 35.0 35.0 35.0 35.0 35.0 35.0
2hr M 12.5 13.7 14.1 14.3 14.3 14.3 14,.3 14.3 14.3 14.3
3hr T 20.2 26.1 31.2 33.8 34.7 34.9 35.0 35.0 35.0 35.0
3hr M 11.8 13.0 13.8 14.1 14.3 14.3 14.3 14.3 14.3 14.3
4hr T 18.9 22.3 27.5 31.8 33.9 34.7 34.9 35.0 35.0 35.0
4hr M 11.5 12.3 13 .3 13.9 14.2 14.3 14.3 14.3 14.3 14.3
shr T 18.9 19.9 23.5 28.8 32.2 34.0 34.7 34.9 35.0 35.0
Shr M 11.5 11.7 12.5 13 .5 13.9 14.2 14.3 14.3 14.3 14 .3
6hr T 18.9 18.9 20.8 25.4 29.5 32.6 34.1 34 .. 7 34.9 35.0
6hr M 11.5 11.5 12.0 12.9 13.6 14.0 14.2 14 .3 14.3 14 .3
7hr T 18.9 18.9 19.5 22.4 26.3 30.4 32.9 34.2 34.7 34.9
7hr M 11. S 11.5 11.6 12.3 13 .1 13.7 14.0 14 .2 14.3 14 .3
8hr T 18.9 18.9 18.9 20.3 23.1 27.5 30.9 33.2 34.2 34.7
8hr M 11.S 11.5 11.5 11.8 12.5 13.3 13.8 14.1 14.2 14.3
9hr T 18.9 18.9 18.9 19.2 20.8 24.6 28.2 31.S 33.3 34 ... 3
9hr M 11.5 11.5 11.5 11.6 12.0 12.7 13.4 13 .8 14.1 14.2
10hr T 18.9 18.9 18.9 18.9 19.6 22.1 25.2 29.1 31. 8 33.6
10hr M 11.5 11.5 11.5 11.5 11.7 12.2 12.9 13.5 13.9 14.1
11hr T 18.9 18.9 18.9 18.9 19.0 20.4 22.4 26.4 29.6 32.3
I1hr M 11.5 11.5 11.5 11.5 11.5 11.9 12.3 13 .1 13.6 14.0
12hr T 18.9 18.9 18.9 18.9 18.9 19.5 20.5 23.8 26.9 30.5
12hr M 11.5 11.5 11.5 11. 5 11.5 11.6 11.9 12.6 13 .2 13.7
13hr T 18.9 18.9 18.9 18.9 18.9 19. i 19.4 21.7 24.0 28.4
13hr M 11.5 11.5 11.5 11.5 11.5 11.5 11.6 12.2 12 .. 6 13.4
14hr T 18.9 18.9 18.9 18.9 18.9 19.0 19.0 20.4 21.5 26.3
14hr M 11.5 1l.5 11.5 11.5 11.5 11.5 11.5 11.9 12.1 13 .1
15hr T 18.9 18.9 18.9 18.9 18.9 19.0 19.0 19.8 19.8 24.4
15hr M 11.5 11.5 11 .. 5 11.5 11.5 11.5 11. S 11.7 11.7 12.7
16hr T 18.9 18.9 18.9 18.9 18.9 18.9 18.9 19.5 19.5 22.9
16hr M 11.5 11.5 11.5 11.5 11.5 11.5 11.5 11.6 11.6 12.4
17hr T 18.9 18.9 18.9 18.9 18.9 18.9 18.9 19.4 19.4 21. 8
17hr M 11.5 11. 5 11.5 11.5 11.5 11.5 11.5 11. 6 11.6 12.2
18hr T 18.9 18.9 18.9 18.9 18.9 18.9 18.9 19.2 19.2 21.0
18hr M = 11.5 11.5 11.5 11.5 11.5 11.5 11.5 11 . 6 11.6 12.0
19hr T 18.9 18.9 18.9 18 . 9 18.9 18.9 18.9 19.1 19.1 20.5
19hr M 11.5 11.5 11.5 11.5 11.5 11.5 11.5 11.5 11.5 11.9
20hr T 18.9 18.9 18.9 18.9 18.9 18.9 18.9 19.1 19.1 20.0
20hr M 11.5 11.5 11.5 11. 5 11.5 11.5 11 .. 5 11.5 11.5 11.8
21hr T 18.9 18.9 18.9 18.9 18.9 18.9 18.9 19.0 19.0 19.7
21hr M 11.5 11.5 11.5 11.5 11.5 11. 5 11.5 11.5 11.5 11.7
APPENDIX C
PROGRAM CODE
The calculation program was code by C language.
Program editor and compiler are Turbo C 2.0 software program
from Borland company.
/*****************************************************************.****/
/* AERATION RESEARCH Project */
/* by using CRANKNICOLSON impl icit method */
/* Sukit NITINAI May 9, 97 * /
/* */
/* Hard red winter wheat Hi = 0.129 wb (14.3% db) */
/**********************************************************************/
#include<stdio.h>
#include<math.h>
#define node
#define EPSILON
:fIdefine P_ATM
10
1e3
14.696
/*
/*
/*
No. of node in xaxis */
Iteration criterion per unit */
Atsmospheric pressure [psia] */
void calc_T_H();
double EMC() ,ERH(),dM() ,M();
doubl e hf g ( ) , Pvs () , RH () ,H () ;
double Ti,Hi,DX,HO,HN,BM,R,CMT,CMHi
double Told[node+5] ,Tnew[node+5];
double Hold [node+5] ,Hnew[node+5];
double Mold [node+5] ,Mnew[node+5];
double VR,ro_a,rop,v_a,ca,cv,cw,cp,Tai,Tgi,dt;
double EMCi,Hai,Hgi;
main ()
(
long int istep,nstep;
int i,j,ic:ount,iprn,st.atus,N;
double sumerr,L,endtirne;
doubl e Tnewp[node+5] ,Hnewp [node+5];
doubl e CFM_BU;
clrscr () ;
/****** initialized data and constant ************/
= O.B ;
0.44 B ;
0.077775;
/*
/*
/*
Volume of air
void ratio
air density
81
[cu. it/bush] */
*/
Ib/cu ft (50F) */
r
rop 49.78; /* product density lb/cu ft */
v_a = CFM_BU*7.23182; 1* air velocity ft/min *1
ca 0.2397 /* specific heat of air Btu! lbF * I
cp == 0.42 1* specific heat of product Btu/lbF */
cv 0.4472; 1* specific. heat of vapor Btu/lbF */
cw 1. 1* specific heat of water in grain Btu/lbF*I
L 9.0; /* Length of BED, feet *1
dt .05; 1* time step, min * I
iprn = 1200;
endtime = 1261.;
1* print every 1200 step */
/* total time, min *1
Tai 66. /* T air enter, deg 1" *1
Hai == 0.005;
Tgi == 95.
EMCi== 0 . 14286;
/* entering Humidity ratio, Ib/lb *1
1* Ti grain, deg F * /
1* decimal unit mdb *1
Hgi == H(ERH(EMCi,Tgi) ,Tgi}; /* Hin grain, Ib/lb */
DX == L/node;
BM == 0.25*dt/DX;
nstep = (long int)
/* canst
(endtime/dt) + 1;
[min/tt]
printf ( '\n Desired Time % 5. 2f hrs ", endtime/60.);
printt("\n AIR VELOCITY %5.2f ft/min" ,v_a};
printf ("\n AIR VOLUME %5. 2f CFM/bush" , CFM_BU) ;
printf(" or %5.2f l/s/eu.m" ,CFM_BU*13.4);
*/
printf ( "\n No. Node = %d Length = %6.2 f [tt]", node, L) ;
printf { "\n dx %6.2f [tt]" ,DX);
printf ( "\n dt %6.3f [min] BM =%5.2f",dt,BM);
Told [0]
Hold[O]
Tnew[D)
Hnew[D]
Tai;
Hai;
N = node + 2;
for{i==1;i<==N;i++} (
Told[i]
Hold[i]
)
/* add tictieius nodes */
printf(" \n Temp');
Tgi;
Hgi;
for(i==O;i<==node;i++) printf("%5.1f",i*0.1);
printf (" \n Temp");
for(i==O;i<=node;i++) printf("%5.1f" , (Told[i]32.)/l.8);
printf(" \n Mdb ");
for(l=O;i<=node;i++) printf{'%5.1f",100*EMC(Hold[1] ,Told[i]»;
/************* STARTING POINT TIME = 0.0 ******************1
for(istep==l;istep<=nstep;istep++)
(
sumerr = 5.;
ieount = 0;
cm 0.0;
CM!! = 1. 0;
Tnew[O]
Hnew[O]
1* Initial for enter loop */
Told [OJ
== Hold[O]
1* GUESS TEMP. */
for(i=l;i<=N;i++) {
82
Tnew[i] = Told[i] ,
Hnew[i] = Hold[i] i
TnewJl[i] 0.;
HnewJl [i 1 = 0.;
}
/* ITERATION LOOP until unchanged T and H */
while (sumerr > EPSILON
{
sumerr=O. ;
if (icount > 10) break,
for(i=I,i<N;i+t)
(
if(i == N) {Told[N] =Tnew[N]=Told [Nl];}
calc_T_H (i) ,
surnerr +=
surnerr +=
fabs«Tnew(i]Tnewp[i])/Tnew[ij),
fabs«Hnew[i]HnewJl[i ) /Hnew[ij);
HnewJl[i]
TnewJl[i ]
} 1* end sweep
++icount,
/* end if icount
} 1* end while A
surnerr 2.;
icount = 0;
eMT = 1. 0;
CMH = 0.0;
Hnew[i] ;
Tnew[i],
i = 1 to N */
*1
*/
while( sumerr > EPSILON
{
sumerr=O. ;
if (icount > 10) break;
for (i=1; i<N; i++)
(
if(i == N) (Told[Nl=Tnew[N]=Told[Nl];)
calc_T_H(i) ;
surnerr +=
surnerr +=
fabs ( (Tnew[i) TnewJl [i] ) ITnew[i] ) ;
fabs«Hnew[i]HnewJl[i]l /Hnew[i]l;
HnewJl[i]
TnewJl[il
Hnew[i],
Tnew[i] ,
) /* end sweep i = 1 to N */
++icount;
/* end if icount */
) /* end while * /
for(i=l;l<=N;i++)
for(i=l;i<=N,i++)
for(i=l;i<=N;i++)
Hold[i]
Told [i]
Mold [i]
Hnew[i] ;
Tnew[i] ;
Mnew[ij ;
1************** PRINT RESULTS **********************/
if (istep%iprn==O
printf("\n%4.0fhr T =",dt*istep/60);
83
for(i=1;i<=node;i++)
printf { .. %5 .1f U , (Tnew [ i] 32) 11. 8) ;
printf (" \n%4. Ofhr M =', dt*istep/60 . ) ;
for (i=l ;i<=node;i++)
printf( "%5.1f",lOO.*M(i» ) ;
1* goto next time step */
1* end istep *1
1********* FINISH at to endtirne ****************1
/**************~****
void calc_T_H(int i)
(
END OF THE PROGRAM **********************/
double del_M,TF,H,DT,DH;
double A,B,C,D,E,F,G,Ml,M2;
Ml EMC(Hold[il,Told[i ] );
M2 dM(i) + Ml;
if(Ml< 0 .04) M1 0.04;
if(Ml>0.25) Ml 0.25;
if(M2<0.04) M2 0.04;
if(M2>0.25) M2 0.25;
G = 6.;
TF Tol d [i] ;
H Hold [i];
A 1. *v_a;
B 2. *ro....:p/ (VR*ro_a) ;
D VR*ro_a*G*(ca+cv*H) +ro....:p* (cp+cw*M1) ;
E 2.*ro....:p* «cwcv)*(TF32.) hfg(TF));
F VR*ro_a*G*v_a*(ca+cv*H);
DT BM*F/D*(Told[i+l]Told[il]+Tnew[i+l]Tnew [i1])+E/D*del_M*CMT;
DR BM*A*(Hold[i+l]Hold[i1]+Hnew[i+l ]Hnew[il ) ) + B*del_M*CMH
Tnew[i]
Hnew [i]
Mnew[i]
if (Tnew[i ]
if (Tnew[iJ
if (Hnew[i]
if (Hnew [i]
Told[i] +DT;
Hold[i]+DH;
M2;
> Told[i]) Tnew[ i )=Told[i);
< Tnew(il ] ) Tnew[i]=0 . 5*(Tnew[i1]+TF);
> 0.03) Hnew[i]=O.03;
< O. 0) Hnew [ i] = 0 . 0 005 ;
double hfg(double TF)
{
/**~**************** * ************************I
/* This fuction calc. hfg [BTU/lb] * /
1* of vapor from temp . 32 F to 120 F *1
/ ********************************************/
double BTU;
84
BTU = l093 .31360.56761364*TF;
return (BTU) ;
double EMC(double H, double T)
{
/******************************************************/
/* This sub. return EMC, decimal unit db * /
/* H humidity rat.io, llblllb dry air */
1 * T Temp, deg F (dry bulb) ,. /
1* TC Temp, deg C (dry bulb) * /
/* mdb EMC % dry lbasis * I
/* Rh Relative Humidity [decimal unit ] * 1
/* Use only for red wheat */
/******************************************************/
double mdb,Rh,TC;
Rh=RH (H, T) ;
TC= (T32.)/1.B;
mdb =pow( log (1. Rh) / (2. 30 08e 5* (TC+55. B1 5)) , (1. 12 . 2857))
return (mdb*.Ol);
double H(double RH,double T}
(
/*************************************************/
/* This sub. calc . Humidity Ratio [lb/ l b] */
/* from Relative Humidity and Temp [deg F] */
/* Patm: atmospheric pressure [psia] */
1* Ps : sat. vapor pressure [psia] *1
/*************************************************1
double H, Patm,Pv;
Patm P_ATM;
Pv Pvs (T) *RH;
H (0.6219B)*Pv/(PatmPv);
return (HI;
double Pvs(double T)
{
/******************************~******************/
/* This sub. calc. Sat vapor pressure [psiaj */
/* from Temp [deg F] */
/* 32 F < T < 180 F * /
/* TRK: Temp rankine [R] */
1 *************************************************/
double TRK;
TRK = T + 459.67;
return (exp(10440.4/TRK 11.2946669 O.02700133*TRK \
+1.289706e5*TRK*TRK 2.47B068e9 *TRK*TRK*TRK \
+6.5459673* l og(TRK)));
double RH(double H,double T)
(
/*************************************************/
/* This sub. calc. Relatitive Humid . [decimal] */
1* from Humid. ratio and Temp, deg F */
/ * Pv: vapor pressure, psia * /
/ * Pvs: sat. vapor pressure, psia * /
/*************************************************/
85
}
double Patm,Pv,Ps,RH;
Patm
Ps
Pv
RH
if(RH
P_ATM;
Pvs (T) i
H*Patm/(O.62198+H) ;
Pv/Ps;
>1.} RH =0 . 99;
return (RH) ;
double ERH(double mdb, double TF)
{
}
/******************************************************/
/* This sub. calc. equilibrium RH i n decimal unit */
/* RH Relative Humidity [decimal unit] * I
/* TF Temp, deg F (dry bulb) * /
/ * TC Temp, deg C (dry bulb) */
/* mdb EMC dry basis [decimal unit ] */
/* Use only for red wheat * /
1******************************************************/
double TC,RHi
TC = (TF32.)/1.8;
RH = 1. exp(2.3008e5*( TC+55.815 }*pow(mdb*100.,2.2857)};
return(RH) ;
double M( lnt l)
double a,b,m;
a EMC(Hold[O] ,Tai);
b EMC{Hold[O] ,Tgi);
m a+(aEMC(Hold[O] ,Tnew[i]»)*(EMCia)/(ab) ;
return (m) ;
double dM(int i)
{
double dM,k,TK,M , Me;
TK (Told[i]32.)/ 1 .8 + 273 .;
k 2.4e8*exp(6144.0/TK) i
M Mold Ii ] ;
Me EMC(Hold[il],O.5*(Told[il ] +Tol d[i]);
if (M > Me) return (0.);
aM= k*(MMe) *dt/60.;
return (dM ) ;
/***** ************* END Subroutine ****************/.
86
APPENDIX D
PHYSICAL. PROPERTIES OF GRAIN, AIR, AND WATER
The physical properties that were used in the
calculation model was from the actual experiment of Epperly,
D. R., (1990) and the thermodynamics property table of air
and water, ASHARE (1985).
TABLE XI
PROPERTIES OF GRAIN(HARD RED WINTER WHEAT)
Property, unit
Bulk density, kg/rn3
Particle density, kg/m3
Specific heat, J/kgOC
Themal conductivity, W/moC
87
Value
793.3
1440
1758
0.1461
TABLE XII
PROPERTIES OF WATER (AT 20°C)
Property, unit
Densi ty, kg 1m3
Specific heat,J/kgOC
Themal conductivity, W/moC
Value
999.3
1758
0.1461
TABLE XII I
PROPERTIES OF AIR (AT 20°C)
Property, unit
Dens i ty, kg 1m3
Specific heat, J IkgOC
Value
l. 29
1. 007
88
I
VITA
Sukit NITINAI
Candidate for the Degree of
Doctor of Philosophy
Thesis: A CALCULATION MODEL OF FIXED BED WHEAT AERATION
Major field: General Engineering
Biographical:
Personal Data: Born in Cholburi, Thailand, September
24, 1959, the son of Kin and Meung Nitinai.
Education: Received the degree of Bachelor of
Engineering in 1982 from King Mongkut's Institute
of Technology, North Bangkok, Thailand; received
the degree of Master of Science (Agricultural
Engineering) in May, 1991 from Oklahoma State
university, Stillwater, Oklahoma; completed
requirements for the degree of Doctor of
Philosophy at Oklahoma State University in August,
1997.
Profession: From 1982 to present, Instructor at College
of Engineering, Rajamangala Institute of
Technology, Bangkok, Thailand.