A mathematical simulation of grain drying

J. agric. Engng Res. (I 969) 14 (3) 226-235

A Mathematical

Simulation

of Grain

Drying

H. B. SPENCER* A mathematical simulation of the grain-drying process is presented with the aim of providing a method of producing design data for driers. Comparison of the model’s predictions with independently published experimental data is made for bed depths of 0.5 ft and 2 ft with initial moisture contents (d.b.) in the range 0.25 to 0.47. Quite good agreement was obtained between the experimental results and the model predictions. 1.

Introduction

Attempts to formulate mathematical models of the grain drying process have appeared only recently. Some of these methods such as the work of Boyce,’ and Henderson,* are based on et a/.,‘, 4 is far more rigorous, The work of Bakker-Arkema rather empirical foundations. although their work4 did not show very good agreement between experiment and theory. The approach in the present paper is similar to the approach of Bakker-Arkema but the equation describing the drying rate of a particular point in the bed is significantly different. 2.

Derivation of equations

In obtaining the following set of differential following assumptions were made.

equations

describing

the drying

(1) Thermal properties of grain (dry), moisture, and air are constant temperatures under consideration. (2) Conduction heat transfer within the bed is negligible. (3) The effect of condensation within the bed is negligible. (4) The problem is one dimensional. 2.1 Consider

within

range of air

Equation describing air temperature

the flow into an elemental

The sensible

of deep beds the

heat flow into the volume

volume

of unit area and depth dx.

in time dt is: G, Ca T dt.

The sensible

heat of the air flowing out of the volume G, CacT+z

The change and (2) i.e. :

in sensible

in time dt is: dx) dt.

heat with respect to x within the bed is the difference

... between

Eqns (1)

G, Ca$dx

dt.

. . . (3)

Within the elemental volume there is air of an amount amount of air with respect to time is:

EP~

dx. The change of sensible heat of this

r Epa dx Cagdt. *N.I.A.E. Scottish Station, Penicuik, Midlothian 226

. (4)

H.

B.

227

SPENCER

~:

KEY TO SYMBOLS air flow rate, lb/min/ft2

-= =

specific heat of air at constant air temperature, “R

t

=

time

& Pu

7 void ratio =m density of dry air

X

:

co-ordinate:

4 h

=

heat flow rate, Btu/unit

=

heat transfer

8

:=

grain temperature,

CCi

=

specific heat of grain, Btu/lb/“R

C, a

=

specific heat of moisture

=

moisture

hv

-=

latent heat of evaporation,

H

=

specific humidity

Pr a,

=

density

==

initial moisture

a,

=

surface of grain moisture

S -

-=

ratio of surface area of grain berry/volume

( 1 DV

=

diffusion

a, b,,f”(o)

=

constants

L), t

=-

diffusion

=

energy of activation

R

=

gas constant

PI.5

:

PJ Pi.0

=

density of water vapour at saturation saturation vapour pressure

= =

density of water vapour in inlet air Air Velocity ft/min=G,/P,

c,, Ca T

b’ll

measured

pressure

from bottom

Btu/lb/“R

of bed of grain

time

rate, Btu/unit

time/ft3/‘R

“R

content,

within grain. Btu/lb/“R

d.b.

of water vapour

Btu/lb in air

content content of grain berry

coefficient (see Eqns 19, 20) constant

(76.8 sec/cm2) (12.20 kcal/mole)

The total change in sensible heat of the air in time dt is therefore: Ca dx dt (G.g+ As the air moves through

the bed a convective

heat exchange

q=h (O--T) Equating

(5)

EP,$).

dx dt.

occurs that can be expressed

as:

. . . (6)

Eqns (5) and (6) gives: (7)

2.2. Equation describing grain temperature In describing the temperature changes in the grain due account must be taken of the fact that the grain is a composite material consisting of dry grain and moisture. C

228

A

MATHEMATICAL

SIMULATION

If in time dt the grain temperature changes by an amount losing moisture by evaporation at the rate of &/at then: <

at3 -= ,&

\,-h

2.3. The amount

dt (C,+C,a)

dx

(t3-T)+-hv$pG (1 -E) I dx

-h

I

I heat balance

DRYING

‘-1

J is the differential

GRAIN

dO, and the grain is at the same time

poz

(I-E)

OF

equation

giving:

(er)-ll$p,

(I -a))

Equation describing

of water convected

. (8)

dr

(PC (I-

~1 G

+ Cd4

(9) I

change in vapour density in the air

into the elemental

volume in time dt is:

G, H dt. The amount

of water convected

out of the volume

is:

G, H+gdx ( Thus change of specific humidity

of water vapour

dt.

i

(II)

with respect to x is: G, g

The change

(IO)

within

dx dr.

the elemental

(12)

volume

with respect to time is:

aH

&puar dt dx. Now changes in specific humidity grain. In time dt an amount:

are brought

about

,.. (13) by the evaporation

of the water

from the

pc (I -- E)‘$dt dx of water vapour

is transferred

to the air passing

through

(14) the elemental

volume.

Thus: aH aH Ggvq=Pd-d~. This equation

can be written

in terms of the vapour

au

density

pDassuming

. (15) p(, is constant:

(16) 2.4 At any position

Expressions

jtir the drying rate aalat

x in the bed the grain dries according I $=.f(po.

to the equation

T, 0, a, a,,, . . .)

.

(17)

The form of the function f (p”, T, 8, a, a,, . . . ) can be determined from experiment or by theory. Unfortunately, to date no really comprehensive study of the form off(p,,, . . . ) has been

H.

B.

239

SPENCER

undertaken. However Becker5 has published a study in whichf(p,. . , . ) can be obtained subject to the restriction that no external resistance to mass transfer occurs, and that rates of change of environmental conditions do not influence the value of f(pu, . . ). In fact Becker’s form of f(p,. . . . ) is very much simpler than the general form proposed in Eqn (17), it reduces to: a, a,, t ).

$=f(6,

(18)

Becker gives .c’,Dt

Dt

for

and

a--a,V -=gexp(--bZ(ij’ a,--c(,

DtI

(20)

for

For a wide range of moisture Differentiating

contents,

Becker obtained

experimentally

a,,=O.l 1.

Eqns (19) and (20) with respect to time gives:

for

and

g--b2

2 D (a -a,)

(22)

for D= D, expl -E/RT

where The application of a further condition air is saturated, i.e.

g-0 The actual condition

applied

is also required

Thus Eqns (7), (9), (I 6) and (18) represent in beds.

(23,

to ensure no drying takes place when the

when p,,=-pus.

in the computations g-0

.

).

(24)

was:

when p,.>,O.99 P,.~. the system of equations

(25) describing

the drying

of grain

230

A MATHEMATICAL

2.5. The boundary

conditions

applicable

SIMULATION

OF

GRAIN

I)R\I’INC;

Boundary conditions to the drying problem At x=0,

iiT T= T,, ; z==O,

At x

are:

t-:0

>o;

Tr,==drying

air temperature

t=o

T- To 0=8,

or 0=f(x)

if required

a=a,

or a=f(x)

if required

At x=0;

t >0

T=T,)

3.

or T=,f(t)

if required

pO=pI-O or p,=,f(t)

if required

Method of solution

Eqns (7), (9), (10) and (18) as they stand represent a formidable problem in solution. However certain simplifying approximations can be made which considerably reduce their complexity. Eqns (7) and (9) contain both space and time derivatives of their dependent variables T, and pU. Examination of the experimental results of Reference 8 reveal that time derivatives are of the order of 2 ‘A of the space derivatives for high temperature flows (about 200” F). We shall therefore V a~,> make the quasi steady approximation that - and at are negligible in comparison to their space at derivatives g and 2. This leaves the equations

to be solved as: aT h (0-T) -=~ ax G, Ca (I-E) %” --axwPG Va

ae

. aa

. . . (27)

at

1

(l-s), at==-h(e-T)+h$i$, ii PC (I-E)

(CC tC,a)

. . (28)

~=.f(% a, a,, t). As these equations are non-linear using a finite difference technique.

the only way to produce

(26)

. . . (29) solutions

is by numerical

means

3.1. Finite diflerence scheme The basis of the finite difference scheme is shown in Fig. 1. The bed is split up into equal elements of depth Ax. The problem is solved by determining the values of the air temperature T,

kl.

B.

SPENCER

=n 9” ,T:f(x)at

T ,

I,,,,

_--I

./

f

I---

0 l,,,,,,

//,/~1~/1111~

i

Insulatedwalls Fig. I. Finite difference scheme

and vapour density given time t, i.e.

pi’ at the mid point of each of these elements

using Eqns (26) and (27) for a

Ax (30) AxThe method of Runge and Kutta is then used to advance the solutions of Eqns (28) and (29) by one time interval. An Atlas autocode programme has been developed to carry out the solution by the methods indicated.

Comparison of results with experiment

4.

Before numerical solutions can be obtained expressions for the heat transfer obtained, and values ascribed to Ca, pc, E, CG, C mr hv and p_.. The expression

rate, 11, must be

for h used was that given by Boyce’ 0.6011

(31)

The following

values of the physical

Specific heat of air, Cu, at constant Density

(solid) of dry grain=76*5

Void ratio

constants

were used.

pressure=0.241

Btu/lb/‘R

lb/ft”

&=-O-43.

Specific heat of dry grain=0.3 Specific heat of moisture

in grain=0.67

Latent heat of evaporation=

1032 Btu/lb

I j

from Reference

6

232

A

MATHEMATICAL

SIMULATION

OF

GRAIN

DRYING

In the case of wheat Beeker5 gives:,f”‘(0)=0.588 a-0.862,

D=D, Expressions

for pus can be obtained

j exp,-

b2-

1.301

11080 cm2/sec. 8

from Reference

7;

p,.,%==[O.6219ps/(14*7-ps)]

. (32)

p‘,

where ps=exp

(.54*6329- 1230.1688, T-5.16923

log, (T)l

In order to test the predictions of the model, comparisons were made with the experimental work of Woodforde and Lawton’ and Clark and Lamond.9T lo These comparisons are shown in Figs 2, 3, 4, 5, Experimental conditions relating to each test are shown on each graph. I.0

0.22

ri?

0.9

0.8 ~

o-7 -

0,16a r' ': 0 0.14-

i i

0.6-

E o.5;

\\

Theory

/>

0.4 Expt.(Ref.

8)

0.3 -

110

0.2 HO

r

0

0

I

I

I

I

I

I

I 20

I 40

I 60

I 80

I 100

I I20

r

70-

u 0

,

I

20

40

I 60

I 80

I

100

I 120

I 140

Time,

601 0

1 I40

mn

x 10m4 Ib/ft3, bed depth 0.5 ft. Comparison of: Fig. 2. Drying air temperature IIO’F, Go=34 Iblmin, p ..=5.98 (Top left) average moisture content (wet basis); (Topright) exit air relative humidity; (Bottom left) exit air, dry bulb, temperature, “F; (Bottom right) moisture content of exit air, grains/lb of dry air

H.

B.

SPENCER

Expt.( Ref.8)

Theory

Theory

60

Time,

min

-5.59 x 10m4 Ib/ft3, bed depth 0.5 ft. Comparison of: (TOP, Fig. 3. Drying air temperature 15O”F, G,=3.8 Iblmin, pVO-left) average moisture content (wet basis); (Top right) exit air relative humidity; (Bottom left) exit air, dry bulb temperature, “F; (Bottom right) moisture content of exit air; grains/lb of dry air

As can be seen very reasonable agreement is obtained with the average moisture contents, but the comparison of the exit air dry bulb temperatures are not so good. It is possible that condensation could account for these discrepancies, however the results of experiments by Clarke and Lamond9T lo indicated that no significant condensation occurs, at least on the surface layer of the grain bed. 5.

Conclusions

A mathematical model of the grain drying process has been developed which prediction of the drying rates from both high and low initial moisture contents.

gives good

234

A MATHEMATICAL

0.04

SIMULATION

OF

GRAIN

I)RYIY<;

c

,.,2: 14or

Time,

min

Fig. 4. Drying air temperature 19O”F, G,=3.8 Ib/min, p,,=5.19 x 10e4 lb/ft3, bed depth O.Sft. Comparison of; (Top left) average moisture content (wet basis); (Top right) exit air relative humidity; (Bottom left) exit air, dry bulb, temperature, “F; (Bottom right) moisturr content of exit air, grains/lb of dry air

The model developed is very simple and it may be that further refinements, such as description of condensation phases may produce even better agreement. However condensation and the subsequent absorption or partial absorption of moisture by the grain has yet to be described experimentally. Similarly, the description of the desorption rates of drying grain having partially absorbed condensed vapour is likely to prove very difficult. This may restrict the applicability of the present model to beds of a given depth. However at the present no indication is available as to what this depth might be.

ti.

R.

SPENCER

Time,

h

Fig. 5. Drying air temperature 167”F, G,=3.02 lb/ min p,,=4.55 x lo-” Ib/ft3, bed depth 2 ft. Comparison of: (Top left) average moisture content (dry basis); (Top right) surface layer moisture content (dry basis); (Bottom right) exit air dry bulb, temperature, “C; (Bottom left) exit air relative humidity REFERENCES

Boyce, D. A.

Grain moisture and temperature

changes with position and time during through drying.

J. agric. Engng Res., 1965, 10 (4) 333. Henderson, J. M.; Henderson, S. M. A computational procedure for deep bed drying analysis.

J. agric. Engng Res., 1968, 13 (2) 87 Bakker-Arkema, F. W.; Bickert, W. G.; Patterson, R. J. Simultaneous heat and mass transfer during the cooling of a deep bed of biological products under varying inlet conditions. J. agric. Engng Res., 1967, 12 (4) 297 Bakker-Arkema, F. W.; Bickert, W. G.; Morey, R. V. Gekoppelter Warme-und Stoflaustausch Wohrend des Trocknungsvorgangs in einem Behalter mit Getreide. Landtech. Forsch., 1967, 17 (7) 175 5 Becker, H. A. A study of diffusion in solids of arbitrary shape with application to the drying of the wheat kernel. J. appl. Polym. Sci., 1959, 1 (2) 212 6 Farm grain drying and storage. Min. of Agric. Fish. and Food (H.M.S.O.) Bulletin No. 149. 7 Brooker, D. B. Mathematical model of thepsychrometric chart. Trans. A.S.A.E., 1967, 10 (4) 558 a Woodforde, J.; Lawton, P. J. Drying cerealgrain in beds six inches deep. J. agric. Engng Res., 1965, 10 (2) 146 ’ Clark, R. G.; Lamond, W. J. Drying wheat in 2 ft beds. 1. J. agric. Engng Res., 1968, 13 (2) 141 ‘O Clark, R. G.; Lamond, W. J. Drying wheat in 2 ft beds. III. J. agric. Engng Res., 1969, 13 (4) 323