Moisture transport caused by natural convection in grain stores

J. agric. Engng Res. (1990) 47, 23-34

Moisture Transport Caused by Natural Convection in Grain Stores E. A. SMITH;* S. SOKHANSANJt * PaisleyCollege of Technology, High Street, Paisley,PA1 2BE, UK. I’ University of Saskatchewan,Saskatoon,Saskatchewan,S7NOWO, Canada. (Received

12 May

1989; accepted

in revised

form

6 January

1990)

Observations suggest that natural convection in grain stores produces moisture movement. The process is very slow but the moisture content of the grain can change enough during months of storage to influence the quality of the grain. The equations which describe this process are presented in this paper. The fact that the movement of moisture is very slow is used to simplify the equations of heat and moisture transfer. The simplified equations have the same form as the equations used to describe heat transfer alone. Standard methods for solving heat transfer equations were used to solve the simplified equations. The method is shown to be reasonably accurate by comparing the results with experimental data. Also, the equations were used to simulate the movement of moisture in a grain bin where some data were recorded. Overall, the simulation was reasonably accurate; in particular it showed that the moisture was moved to the top-centre of the bin. But there was not enough information (such as weather data and packing density) to give a very accurate simulation. Earlier work suggests that heat is normally transported by conduction in grain stores. In this paper, it is shown that natural convection can significantly affect heat transfer in the presence of moisture movement. This is achieved by using an approximate analysis of the heat and mass transfer equations which shows that conduction normally dominates the process. However, convection is important if the resistance to airflow is low enough and if the radius of the storage bin is approximately equal to the height of the bin. 1. Introduction Observations made by Schmidt’ and discussed by Hall’ suggest that natural convection occurs in grain stores. The resulting heat and mass transfer leads to a build up of moisture in some regions of the store. The quality of the grain in these regions is at risk. From a study of only the heat transfer processes, Smith and Sokhansanj3 have shown that conduction is the main form of heat flow in grain stores. This agrees with the work of

several authors4*5 who show that only conduction is required to model the temperature in grain stores. However, in some circumstances it is possible for convection to dominate the process of heat transfer. These circumstances are determined by the size of the Rayleigh number and the shape of the store. Natural convection is an important method of heat flow if the resistance to airflow is low and if the shape of the cross-section of the store is approximately square. This result was derived from a study3 of heat transfer alone, but it is shown in the present paper that the result is similar when both heat and moisture transfer are important. Natural convection increases the rate at which heat flows through the store. Because of this the average temperature of the grain bed is closer to the value of temperature outside the store. It has been shown by Beukema, Bruin and Schenk6 that it is an important process during the cooling of some agricultural produce. 23

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Notation

amplitude of temperature variation in Eqn (12), “C heat capacity of air, J me3 heat capacity of the mixturt-’ of air and grain J mheat capacity of the rm:u:e of water and glass spheres, J mm3 K’ heat capacity of water, (qAv J me3 K-’ d in Eqn (11); d = L@p, dH/dT, J mm3 K-’ D height of the cylinder used in Buretta and Berman’s experiment, m g gravitational acceleration, m/s2 H humidity of air, kg water/kg dry air HS,, humidity of saturated air, kg water/kg dry air k thermal conductivity of the mixture of air and grain, Wm-‘K-l k* in Eqn (11); k* = k$ I(&,, L M Nu

+ 4, m*/s

Height of the cylindrical bin of grain, m moisture content of grain, dry basis Nusselt number defined by Nu = QD/k( T2 - &)

air pressure, Pa average value of the vertical heat flux in Buretta and Berman’s experiment, W/m* R radius of the cylindrical bin of grain, m Ra Rayleigh number when moisture flow is ignored Ra = vagSALl(c+) Ra* Rayleigh number when moisture flow is included Ra* = Ra(1 + H) i?

x [I+

Ra’

(dHldT)(l

- E)/(E/~)]

Rayleigh number in Buretta

and Berman’s experiment Ra’ = gK’p’( T2 - T)DILY’Y’ r radial coordinate in cylindrical coordinate system, m T temperature of the mixture of fluid and porous material, “C TA average air temperature in Eqn (12), “C T, ambient air temperature in Eqn (18), “C G, K temperatures in Buretta and Berman’s experiment. Measured at the top and bottom of the cylinder as shown in Fig. 1 t time, s time in Eqns (12) and (18), ld days time when T = TA in Eqn to (1% days V superficial velocity of the air, m/s velocity v in the radial V, direction, m/s velocity v in the vertical VZ direction, m/s Z vertical coordinate in the cylindrical coordinate system, m thermal diffusivity of the mixture of air and grain; (Y= kl(C,),, n-?/s thermal diffusivity of the lx’ mixture of water and glass spheres; (Y’ = k’/(C& m2/s thermal expansion coefficient B of air; with sufficient accuracy /3 = l/(T + 273.15), K-r time scale over which temperatures change significantly in approximate analysis, s AT approximate value of the temperature difference between the ambient temperature and the

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SMITH;

E

8 K

A p v’ p pa pP

o

S.

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SOKHANSANJ

temperature at the centre of the storage bin, “C ratio of the molecular weight of water and air E = O-622 azimuthal coordinate in the cylindrical coordinate system, rad permeability of the porous medium, m latent heat of vapourization of water, J/kg dynamic viscosity, kg m-’ s-l kinematic viscosity of water, m*/s density of humid air, kg/m3 density of dry air, kg/m3 bulk density of grain, kg/m3 in Eqn (ll), u = [(C,), + d] 4wc, + 4

$ x V

porosity of the porous medium stream function which is used to simplify the equations b replacing the velocity r~, m Y/s Gradient; a a ‘la

V=~e,+;~e,+~e,

V*

Laplacp; v*=$+g+f-$+$ when the superscript (‘) is used, the value is calculated for the mixture of water and glass spheres, used in Buretta and Berman’s experiment, rather than for the mixture and air and grain in a silo

Although conduction is the main form of heat flow in grain beds, the process of diffusion, the equivalent method of moisture transfer, is not important. Davidson7 has shown that while natural convection produces very low air velocities, they are not as low as the velocities produced by diffusion. Thus, mass transfer by diffusion can be ignored. Simulations”o which assume that convection is the only form of mass flow have produced reasonable results. These simulations calculate the temperature throughout the bin using a conduction model. The temperature variations are then used to estimate the air velocity. Gough” has shown that this approach can give reasonable estimates of the changes in moisture content even if the bin does not have a rectangular cross-section. Nguyen” has developed a numerical solution of all of the equations which describe the transport of heat and mass by natural convection. Several simulations are presented in the paper, including the case where there is an air space above the grain bed. Although this detailed model is successful, the approximate models of Gough” and others*” gave reasonable results and this suggests that an approximate model may be adequate. Work by Close” suggests a way of simplifying the model of heat and mass transfer. Close’* studied the problem of the natural convection of saturated air in a porous medium. Using the fact that the temperature of the saturated air is related to its humidity, the equations were simplified so that they were in the same form as the equations of heat transfer alone. These equations have been studied in detail13*14 and the numerical methods for solving the equations have known properties of accuracy and stability. To take advantage of this result of simplification it is necessary to relate the humidity of the air in grain stores to the temperature of the air. In grain stores the air is not saturated so the same technique cannot be directly applied. In the present work a modified technique is developed. This paper describes the equations which model heat and moisture flow in grain stores. An approximate analysis of the equations shows the conditions when heat transfer by natural convection is important. The equations are solved numerically and the method is tested by comparison with experimental results. Finally the model is used to simulate the slow moisture transfer which occurs during the storage of wheat. The completed results are similar to those observed by Schmidt.’

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2. Model of heat and mass transfer

When grain is stored for prolonged periods, large temperature gradients are produced in the store because of variations in the ambient temperature. These thermal gradients lead to natural convection and the resulting airflow moves moisture from one area to another. One of the equations which describes natural convection is a variation of Darcy’s law which takes into account the influence of gravity6 V =

-(VP

-

gp)K/p

0)

where v is the superficial velocity. This velocity also satisfies v.v=o

(2)

if p is constant. This can be written

since this is a two-dimensional problem where u B = 0 and there is no dependence on 8. With this equation the stream function x can be used. In cylindrical coordinates15 this is

u,=--1 ax and v,=--- 1 ax r i3z

(3)

r &

By using x rather than v, Eqn (2) is automatically satisfied. Taking the curl of Eqn (1) and replacing v by x gives

The density of air is a function of temperature

T and humidity

H so

The density of dry air, pa and the thermal expansion coefficient, /3, are assumed to be constant. The ratio of the molecular weights of water and air is E = O-622. Following the work of Close,” the term aH/ar is replaced by aH -=--

dHaT

dr

dT dr

In the problem studied by Close,” the air was saturated so the humidity H was only a function of temperature T. To follow the method used by Close it is necessary to formulate H as a function of T for air in grain stores. For this air the humidity and temperature are related by the equation of equilibrium moisture content. If it is assumed that the moisture content is constant then an equation is formed in which H only depends on T. In grain stores the moisture content changes much less than the temperature so it is reasonable to treat moisture content as a constant; observations’ show a change of approximately 1% wet basis in a period of 30 months while temperature changes by 30°C. With M = 0.15 dry basis in Chung’s” equation of equilibrium moisture content we get H = H,,,(T)

exp [ -37.73/(

T + 51)]

(7) Eqn (7) is used to calculate a representative value for dH/dT. As with the values of /I and pa it is assumed that dH/dT is a constant in the calculations. For the example problem

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described later the value of dH/dT was 3.1 x 10e4 K-‘. The values of dH/dT range from 1.8 x 10e4 K-’ when T = 5°C to 4.7 X 10e4 K-’ when T = 2O”C, so for typical values of grain temperature dH/dT is approximately constant so long as H is given as a function of T as in Eqn (7). Eqn (4) becomes

once Eqns (5) and (6) are used to replace dp/dr. The modified Rayleigh number, Ra*, for this problem is a constant. It is almost equal to the Rayleigh number Ra; the effect of H and dH/dT creates a small difference between Ra and Ra*. The term A is a representative value of the temperature difference across the grain store; it is later equated to the amplitude of the ambient temperature variation in Eqn (12). The energy equation’2*‘3 is (9) As with Eqn (4), the velocity v has been replaced by the stream function x using Eqn (3). The heat capacity of the air in the porous medium”’ is C, = 1230 J me3 K-l. The heat capacity of the mixture of air and grain16 is (C,), = 1.5 x lo6 J mP3 K-’ and the thermal conductivity of the air and grain mixture16 is k = 0.159 W m-i K-l. As the air flows round the grain store it exchanges moisture with the grain and this process is described by

(10) In this equation the derivatives G’H/dt, dH/dr and aH/dz were replaced equivalent temperature derivative multiplied by dH/dT, as in Eqn (6). Using Eqn (10) to replace dM/at in Eqn (9) gives

a7- laXaT

oar=;arz----+k

iaxazaz ar

r

by their

(11)

where (Y= [(C,), + d]@/($C, + d), k* = k@/($C, + d) and d = @pa dH/dT. Eqns (8) and (11) can now be solved for T and x. The air velocity v can then be obtained from x using Eqn (3). The moisture content throughout the grain store can be calculated from the drying rate aM/dt, from Eqn (lo), together with the initial moisture content. To solve Eqns (8) and (11) it is necessary to know the boundary conditions for T and x. Typical conditions for grain stores, would be that the temperature of the air and grain at the top and sides of the store is given by T = TA + A sin (td - to)

(12)

Here it is assumed that the temperature varies sinusoidally over a year, with A equal to the amplitude of the sine wave. This gives a reasonably accurate summary of the temperature variation in the problem studied in Section 5. At the bottom of the grain store the temperature gradient, normal to the ground, is zero so there is no heat exchange with the ground. Also, the air velocity is zero on any solid surface so x = 0 on the walls, top and bottom of the store.

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MOiSTURE

3. Criterion

TRANSPORT

for natural convection

Eqn (11) shows that the temperature throughout the grain store is influenced by conduction and natural convection. If the air velocity is slow enough then conduction controls the flow of heat. But as the velocity increases, natural convection becomes a more important method of heat transport. To estimate the condition for convection to be important, an approximate analysis was undertaken. In this analysis the values used are typical over the whole bin. The variables r and z are replaced by length scales R and L. Also the derivatives were approximated as follows g

becomes T

where AT is an estimate of following calculation it was temperature variation in Eqn but in this case Ax = x. With these approximations

and

g

becomes $T

the temperature difference across the grain store. For the assumed that AT = A where A is the amplitude of the (12). Similar approximations are made for the x derivative Eqn (8) becomes x=RaRn/(k+F)

(13)

where Ra* is approximated by Ra because (1 + H) [I+ (dH/dT)(l - E)/(E/?)] is approximately 1.6 under most circumstances [it equals 1.06 when T = i?‘C, h = O-005, dH/dT = 3.1 x 10m4 K-l, p = 3.5 x 1O-3 K-’ and E = O-6221. In Eqn (11) the term describing the transport of energy by natural convection is 1dxdT ---_---r i3r az

laxaT r a.2 &

2 -xAT R2L

The constant 2 is only a very approximate value in keeping with the approximate nature of the analysis. The true value of the constant could be zero or as large as 10 or as low as -5. Similarly the conduction term in Eqn (11) is

If the convection term is larger than the conduction

term then

which with Eqn (13) becomes (14) This is the criterion for natural convection to play a significant part in heat transfer. Eqn (14) becomes Ra> 1.4 x lo3 when L/R = 1, k* = 7.32 x lop5 m2/s and cy= 1.1 X lo-’ m2/s (using16 k= O-159 W m-l K-l, @ = 0.4, C, = 1230 J me3 K-l, A = 2.5 x lo6 J/kg, pa = 1.23 kg/m3 and dH/dT = 3.1 x 1O-4 K-l). When the effect of humidity is ignored in Eqn (14) the constant k* is replaced by k/C,. The ratio k*/(k/C,) -0.56. Thus the effect of humidity in Eqn (14) is not large.

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The shape of the grain store, as represented by the term (L/R + R/L)*, has an effect on the criterion for natural convection [Eqn (14)]. The smallest value occurs when L/R = 1. For other values of this ratio, the Rayleigh number Ra has to be larger for natural convection to be a major form of heat transfer. 4. Numerical

solution

The equations of heat and mass transfer have been converted into the standard form13 [Eqns (8) and (ll)] for problems involving natural convection. Thus the numerical solution of these equations can use any of the many techniques’3”4 for this problem. For this paper the finite element method was used for the space variables and the Crank-Nicolson’7 method for time. The area was divided into triangular elements with a cubic basis function. By reducing the time step and increasing the number of elements the accuracy of the result was increased. This process was continued until there was a temperature difference of less than O*OYC between one solution and the next most accurate solution. For the problem discussed later, which simulates the movement of moisture during grain storage, a time step of 1 h was required to achieve acceptable accuracy over a period of 30 months. To ensure that the computer program was operating correctly the results of numerical simulations were compared with experimental results obtained by Buretta and Berman.‘* In these experiments, a cylindrical bed of glass balls was filled with water, heated from below and left to reach a steady state. The sides of the cylinder were thermally insulated so the flow of heat was mainly vertical. The bottom of the bed was held at a constant temperature T2 and the temperature at the top of the bed was T, (Fig. 1). The diameter of the bed was 0.305 m and a variety of bed depths from 0.0428 to O-081 m were used as well as a range of glass ball diameters from 0.003 to 0.0143 m. The experiments established that the Rayleigh number Ra’ was related to the Nusselt number Nu by Nu=l when Ra’ < 40 (15) log,, (Nu f 0.076) = 1.154 log,, Ra’ - 1.823 when 40 < Ra’ < 100

Fig. 1. Bed of porous material relationship between the Rayleigh

used by Buretta and Berman to measure experimentally number and Nusselt number. A range of bed depths (D) 04428 to O-081 m was used

the

from

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The Nusselt number in this experiment is Nu = QD/[k’(T, - T,)] where Q is the average value of the rate of vertical heat flow across a unit cross-sectional area (average heat flux). Nu is the ratio of Q to the rate at which heat is conducted through the bed. When Nu = 1 all of the heat flows by conduction. But when natural convection is important the heat is transported more rapidly so Q increases and Nu becomes larger. The experimental results which are summarized by Eqn (15) show that when Ra’ < 40 heat transfer is by conduction but when Ra’ > 40 natural convection plays an important part in heat transfer. The physical processes involved in the experiments are modelled by the following equations13 O=$(i$)

+$(is)

+(61i)DRa’g

(16)

laxaT laxaT at r ar a2 r a2 ar

do

These are the same as Eqns (8) and (11) except that the constants have values for a mixture of water and glass spheres rather than grain and air. The values used for the constants” were cd = k’/(C& with k’ = 0.6105 J s-i m-i K-l, (C,)w = 4 x lo6 J mm3 K-’ and cr’ = (C&,,,.,/(C& with (C,),, = 9.67 x 10’ Jmd3 K-‘. Eqns (16) and (17) were solved using the method described earlier in this Section. For these experiments the boundary conditions for x and T are that x = 0 on the top, bottom and sides of the cylindrical bed. Also dT/dr = 0 on the vertical, insulated sides and T = Tl on the top surface while T = T2 on the bottom surface. The simulations were left to run until a steady state was reached. It was decided that this state was reached when the results were constant for 24 h of simulated time t. Once this state was reached the average value of the vertical heat flux Q, was calculated. The vertical heat flux at any point is (C,),u,T

- k’ $

Ra’

Fig. 2. The graph of Nusselt number (Nu) against Rayieigh number (Ra ‘) for Buretta and Berman’s experiment. The results of the experiment () are compared with the computer simulation(- - -)

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The average value of this was obtained by integration over the horizontal cross-section of the porous bed. The Nusselt number Nu should be independent of height z but numerical errors resulted in Nu varying a little with height. The average value of Nu over five heights was calculated. This process was repeated for several values of Rayleigh number Ra’ to give the results shown in Fig. 2. The results are compared with the experimental values from Eqn (15). There is reasonable agreement between the two sets of results but the values from simulation are a little lower than the experimental results for large values of Ra’. It is possible that numerical errors built up in the simulations with large values of Ra’ because these simulations took a long time to reach a steady state. 5. Moisture

movement

in silos

The reason for developing the mode1 of natural convection was to predict the changes in moisture content which can occur in grain silos. Data which shows that this mass transfer occurs, were collected by Schmidt.’ This simulation was carried out to illustrate the results achieved in a typical situation where this mode1 would be used. As usual, there is a lack of recorded data so the question is whether the simulated results are sufficiently close to the observed values to be acceptable. In this example, is the observed moisture content close enough to the calculated values? In cylindrical steel bins, which contained 35 .24 m3 (1000 bushels) of wheat, the temperature and moisture content distribution was measured on a number of occasions over a period of 30 months. Figs 3 and 4 show examples of these data. Fig. 3 shows the

25’C

I-

(a)

(b)

Fig. 3. The temperature (“C) of grain throughout a cylindrical bin after 23 months shows the experimental results and 36 shows the results of the simulation

of storage.

3a

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(a)

(b)

of moisture content (dry basis) in a cylindrical bin after 30 months of storage. Fig. 4. The distribution 4a shows the experimental results and 46 shows the results of the simulation

temperature distribution in the summer after 23 months of storage and Fig. 4 shows the moisture content after 30 months of storage. To calculate the moisture content as a function of time, Eqns (S), (10) and (11) were solved numerically as described in Section 4. The ambient temperature of the air was T, = 13.5 + 14 sin (t,, - 137) (18) which fitted the recorded data reasonably well. The temperature of the grain and air at the top and sides of the bin was equated to T,. At the bottom of the bin the vertical heat flux was assumed to be zero so the isotherms were vertical at the ground. On the top, bottom and sides of the bin the air velocity was assumed to be zero so these surfaces formed the streamline x = 0. The simulation was continued for 30 months starting from July 1941 [(td = 198 in Eqn (18)] with a moisture content of 0.149 dry basis and a uniform temperature Ta given by Eqn (18). The diameter of the bin was 4.267 m and its height was 2.438 m. The values of the constants which appear in Eqns (S), (10) and (11) are as follows: Ra* aI = 1.06 x 10m5 m s-r ‘C--l, o = 680.97, k* = 7.32 x 10e5 m*/s and pP = 772.32 kg/m3. These constants were evaluated using T = 13~5°C and A = 14°C from Eqn (18) also,‘6*‘s dH/dT = 3.07 x 1O-4 K-‘, H = O-005, /3 = 3.66 x 10M5 K-l, C, = 1230 J me3 K-‘, E = 0.622, k = 0.159 W m-r K-‘. $I = 0.4, (Y= 1.08 x lo-’ m*/s, P/K = 4 x lo3 kg mm3 s-l, 1= 2.49 x lo6 J/kg and g = 9.81 m/s*. The results of the simulation are given in Figs 3 and 4 beside the recorded data. The simulation is approximately correct, in that the right amount of moisture is moved to the observed position and the temperature distribution is approximately correct. The reason for the difference between the simulation and the observed values is that there was not

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enough data to simulate all of the physical processes. In particular the solar radiation level, wind speed and rate of heat exchange with the ground were not recorded. In the model it was assumed that the air velocity was zero on the wall of the bin. This means that the moisture content should stay constant at 0.149 if diffusion can be ignored. The experimental results show some change in moisture content at the edge of the bin so it may be that there was some flow of air through the walls or that diffusion is important. The model assumed that there was no heat exchange with the ground. The experimental results show that some isotherms are not at right angles to the ground so some heat flow occurred. Lack of data on the ground conditions prevented this heat flow being modelled. For this experiment the criterion for natural convection [Eqn (14)] shows that natural convection is important if Ra > l-4 x lo3 For the experimental data Ra = 3.3 x 103 which suggests that natural convection will have some influence on the temperature, but this influence is likely to be small. Earlier work’ suggests that Ra needs to be substantially above the critical value (l-4 x 103) for the temperature to be significantly affected by convection. 6. Discussion

Natural convection in wheat stores results in a very small airflow which slowly redistributes the moisture throughout the store. This low rate of mass flow suggests that it is not necessary to solve the full set of heat and moisture flow equations. A simplified set of equations should give reasonably accurate results. In previous work simplified equations were used to estimate the moisture flow”” but not to calculate how this moisture would be distributed throughout the store. The method described in this paper calculates this distribution of moisture. If the rate of moisture flow is much larger than that recorded by Schmidt’ in wheat stores, then more accurate methods would be required. The air velocity which produces the moisture flow is increased if the Rayleigh number Ra is larger, and this happens if the size of the grain seeds is larger. Thus the air velocity will be larger for shelled maize and even larger for whole cobs of maize. With a program which simulates airflow in grain stores it will be possible to study ways of preventing this flow from producing a significant movement of moisture. One way may be to introduce artificial ventilation to disrupt the flow caused by natural convection. Another method may be to have vents in the side or top of the bin which can be opened or closed to alter the flow. 7. Conclusions

This paper derives the equations which describe the process of heat and moisture transfer involving natural convection in grain stores. It is shown how to simplify these equations so that they are in the standard form which is normally used to model heat transfer alone. Methods for solving this standard form of equation have been widely studied. The accuracy of the program used to solve these equations was tested by comparing results with those from Buretta and Berman’s experiments.‘* To reduce the equations to the standard form it was assumed that the rate of moisture flow was small, and this is observed in grain stores. The simplified equations work reasonably well when used to simulate experiments by Schmidt.’ These experiments recorded the change in moisture content, throughout a bin of wheat, caused by natural convection.

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An approximate analysis of the simplified equations showed that natural convection is a significant form of heat transfer if the Rayleigh number Ra is larger than a critical value Ra>

(

k+E

1

‘k*l(2a)

This is almost the same as the criterion when mass transfer is ignored;3 the only difference is that k* is replaced by k/C,. For values used in this paper the critical value of Ra was O-56 times the value when mass transfer was ignored (k*C,/k = O-56). References

J. L. Wheat storage research at Hutchinson, Kansas and Jamestown, North Dakota. United States Department of Agriculture Technical Bulletin 1113, 1955 * Hall, C. W. Drying and Storage of Agricultural Crops. Westport: AVI Publishing Company, ’ Schmidt,

1980 E. A.; Sokhansanj, S. Natural convection and the temperature of stored produce-a theoretical analysis. Canadian Agricultural Engineering 1990, 32: 91-97 l Yaciuk, G.; Muir, W. E.; Sinha, R. N. A simulation model of temperatures in stored grain. Journal of Agricultural Engineering Research 1975, 20: 245-258 ’ Jiang, S.; Jofriet, J. C. Finite element prediction of silage temperatures in tower silos. Transactions of the American Society of Agricultural Engineering 1987, 30: 1744-1750 ’ Beukema, K. J.; Bruin, S.; Schenk, J. Three-dimensional natural convection in a confined porous medium with internal heat generation. International Journal of Heat Mass Transfer ’ Smith

1983, 26: 451-458 M. R. Natural convection of gas/vapour mixtures in a porous medium. International Journal of Heat Mass Transfer 1986, 29: 1371-1381 a Muir, W. E.; Fraser, B. M.; Sinha, R. N. Simulation model of two-dimensional heat transfer in controlled-atmosphere bins. In: Controlled Atmosphere Storage of Grains (Shejbal, J., Ed.). New York: Elsevier Publishing Company, 1980 ’ Lo, K. M.; Chen, C. S.; Clayton, J. T.; Adrian, D. D. Simulation of temperature and moisture content in wheat stores due to weather variability. Journal of Agricultural Engineering Research 1975, 20: 47-53 lo Gough, M. C. Physical changes in large scale hermetic grain storage. Journal of Agricultural Engineering Research 1985, 31: 55-65 ” Ngnyen, T. V. Natural convection effects in stored grains-a simulation study. Drying Technology 1987, 5: 541-560 ‘* Close, D. J. Natural convection with coupled mass transfer in porous media. International Communications in Heat Mass Transfer 1983, 10: 465-476 ” Combarnous, M. A.; Bories, S. A. Hydrothermal convection in saturated porous media. Advances in Hydroscience 1975, 10: 231-307 l4 Bejan, A. Convection Heat Transfer. New York: John Wiley & Sons, 1984 ” Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena. New York: John Wiley & Sons, 1960 ” American Society of Agricultural Engineering Yearbook (33rd Edn). St. Joseph MI 49085, 1986 ” Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes. Cambridge University Press, 1988 ‘* Buretta, R. J.; Berman, A. S. Convective heat transfer in a liquid saturated porous layer. Journal of Applied Mechanics 1976, 43: 249-253 ” Kaye, G. W. C.; Laby, T. H. Tables of Physical and Chemical Constants (14th Edn). London: Longman Group Ltd, 1973 ’ Davidson,