Numerical analysis of natural convection heat transfer in stored high moisture corn

J. agric. Engng Res. (1988) 40,275-284

Numerical

Analysis of Natural Convection Heat Transfer Stored High Moisture Corn

in

C. L. G. DONA; W. E. STEWARTJR*

A numerical study was performed on the simulated natural convection heat transfer occurring inside a grain bin which contains stored corn. The simulations were intended to reveal the effects of flow inertia, container aspect ratio, and temperature dependent thertnophysical properties on grain temperatures. These simulations utilized a uniformly heat-generating porous medium in a short, vertical circular cylinder with an adiabatic lower end and an isothermal cylinder top and side. The thermophysical properties of air and corn were used for the porous medium. Modified Darcy momentum equations, which include inertia terms, were used. Streamline and isotherm patterns were developed for different cylinder height to radius ratios, boundary temperatures, and heat-generation rates. Results show that unicellular flows exist for large height to diameter bin

aspect ratios and multicellular flows develop for small aspect ratios. Maximum temperatures are greater for simulations using equations which included temperature dependent property terms. The location of the maximum temperature moves up from the bottom of the cylinder as storage time increases. In general, maximum temperatures at steady-state decrease as the cylinder height increases, at constant radius. Isotherms developed are very similar to previously published experimental data.

1. Introduction This investigation considers the numerical solution of the transient and steady-state streamlines and isotherms formed by the convective and conductive heat transfer in an air-corn porous medium in an enclosed, short, vertical circular cylinder similar to a storage bin. The corn is considered to be generating heat uniformly throughout its volume. A number of experimental and numerical studies have been done on beds of biological products. Burton et al. ’ investigated the temperature of unventilated stacks of potatoes stored in rectangular bins. They found that the maximum temperatures in six and twelve foot rectangular potato stacks occurred near the stack top on the vertical centreline. They attributed the upward displacement of the maximum temperature from the centre of mass to the convective air circulation. Holman and Carter’ determined moisture and temperature distributions for soybeans stored in farm bins for up to six and a half years. Schmidt3 conducted similar research on wheat storage, determining moisture and temperature distributions for various wheat storage times and ambient conditions. Beukema and Bruin4 developed a two-dimensional model for the prediction of the temperature and moisture distributions in a small parallelepiped (0.76 m x 0.76 m x 0.5 m) containing agricultural food products. They compared the results of the model with the experimentally measured temperatures of potatoes and a synthetic model material that simulated a biological product with a constant rate of heat generation and no moisture loss. Excellent agreement between the experimental and simulated results was found for the model material. In potatoes, the mathematical model predicted temperatures close to the experimentally measured ones at the bottom of the cylinder but underestimated the temperatures at the cylinder top. * Energy Research Laboratory, Department of Mechanical and Aerospace Engineering, University of MissouriKansas City, Truman Campus, Independence, Missouri 64050, USA Received 1 September 1987; accepted in revised form 23 April 1988

275 002l-8634/88/08027S+

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Notation CP

d 9

H

k, k,

P

4 r

x Ra

bulk grain specific heat diameter of corn gravitational acceleration height of cylinder bulk grain thermal conductivity Ergun product constant pressure uniform heat generation rate per unit volume dimensionless radius coordinate direction dimensional radius radius of cylinder modified Rayleigh number, gR2W4pcph vfk,f,T

Ra,

reference modified Rayleigh number, Ra T/T, dimensionless time t dimensional time i T temperature TCC isothermal boundary temperature dimensionless radial velocity 4 dimensional radial velocity v, dimensionless axial velocity 0, dimensional axial velocity ts, Z dimensionless height

i o! Y ; K

dimensional knl(w,)

height

f

specific weight, pg porosity dimensionless temperature, (T- TJWWk,) bulk permeability, E3d2

150(1_E)Zk, P Vr P ; II/

I$

dynamic viscosity kinematic viscosity of air, p/p density of air (Pc,)ln/(Pcp) r volume averaged temperature difference, rayg- T, dimensionless stream function dimensional stream function

Subscripts

f m CC

air bulk grain value evaluated at boundary temperature, T,

Superscripts

-+

vector quantity

In a later paper, Beukema et ~1.’compared the experimental results of the synthetic model material with the simulated results from a different mathematical model where three-dimensional natural convection in a confined porous medium with internal heat generation was modelled with the assumptions that the inertial forces in the momentum equation and the viscous dissipation of the air in the energy equation were negligible. A single energy equation was derived from assuming a one-phase porous medium. From a comparison of the simulations modelling natural convection and those modelling pure conduction, they concluded that natural convection in unventilated food products was sizeable and resulted in lower average food product temperatures. Stewart and Dona’ investigated numerically the natural convection of an internally heat generating porous medium in a vertical cylinder. For cylinders with volume equal to those of small grain bins (50 m3) and heat generation rates typical of high-moisture corn (15 to 25%), they found that the air velocities resulting from convective heat transfer were sufficient for sizeable inertial effects to occur. The situation considered here is a vertical cylinder containing an internally heat-generating porous media. The cylinder side and top are isothermal and at the same temperature. The bottom surface of the cylinder is adiabatic. These boundary conditions simulate those occurring in a food storage bin with its bottom insulated by the ground and the top and side remaining at the ambient temperature. For the fluid, the thermophysical properties of air are used and

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Insulated

Fig. 1. Schematic of numerical model

for the porous medium, a range of thermophysical properties typical of corn (Thompson et al.,’ Sinha and Muir,’ Patterson et al.*) are used. A range of heat generation rates representative of et af.’ A modified form of Darcy’s high moisture corn was based on the data of Thompson law was used to include inertial effects. 2. Numerical model The system modelled is a vertical right circular cylinder of radius R and height H, as represented in Fig. 1. The cylinder was assumed to be filled with spheres having bulk thermophysical properties typical of corn. Moisture content and heat generation rates were assumed constant for a given initial moisture content and ambient temperature. Assuming that the fluid and solid were in thermal equilibrium and that temperatures varied only in the radial and azimuthal directions, the temperature distribution was represented in cylindrical coordinates by a single energy equation

(1) The momentum

equation

for generalized

flow in a porous

medium

(Irmay”)

is

V Z+ $ = -(a+bu)G ( 1 where a and h are constants and y = pg. In component form in cylindrical coordinates,

dp -= dz

dp -_= dr Equations ductivity),

(3) and

(4) reduce

this becomes

-y(a+b(v,Z+u,2)“‘)VZ-y.

-y(a+h($+ui)“‘)V,-y.

to Darcy’s

dp -= dz

equations

-fV,-w

for b = 0, l/a = yK/p

(hydraulic

con-

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dp -=--

ANALYSIS

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pv K’

dr

Equations (3) and (4) are similar in form, except for the buoyancy term pg, to the modified Ergun equation for one-dimensional air flow in granular biological material developed by Patterson et a[.’ BP = mu+nu* h

where AP = pressure differential, u = superficial air velocity, h = bed height,

150Ml

m=

--E2)

and

d2e3

1.75k,p( 1 -E)

n=

de3

Comparison of Eqns (3) and (4) with Eqn (7) yields the coefficients a and b in terms of the Ergun constant, k,, values of which are available for corn9 (a = -m/y, b = -n/y). The continuity equation is satisfied by the stream function, $, in cylindrical coordinates as,

1a+ &=--_. 1a$ fir=-;-, F aZ v ar Non-dimensionalizing

63)

Eqn (1) with r = J/R,

z = F/R,

0 = (T-

T,)/(4RH/k,)

results in a single non-dimensional

Ic/= $l(uR),

and

t = afIR*

energy equation:

ae ia+ ae i a+ ae i a r_ae +-i+_, a*8 r ar a2 r ar (1ar az

“at+--_----=_r aZ ar

R H

(9)

The pressure terms in Eqns (3) and (4) are eliminated by taking cross derivatives. Also by using the Boussinesq approximation p = pa[l - (T- T,)/Tj, ignoring third order and higher terms, and non-dimensionalizing the resultant single momentum equation it becomes

(l+;;oT)(;g-$$ _c2)+~;,I,(Ek)(~Et) X

where

i (

a** r azar

1 r*

a$ i a$ aZ -4 r ar

a$

i

r* i3r

r

i

a't+b+;;!?;2 ar* 1) T

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The coefficient

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of a0jaY in Eqn (10) is the Rayleigh Ra

number

SR2fQ4PC,)f

=

vfk ; T

which varies with temperature

(11)

The boundary conditions assumed were an isothermal cylinder side and top and an adiabatic cylinder bottom. Stream function values at the cylinder surfaces and axial centreline were set equal to zero, resulting in a model with (1) impermeable boundaries at the cylinder side and top (a, = 0 at the top and t’, = 0 at the side), (2) symmetry at the axial centreline (0, = 0), and (3) slip flow at all surfaces (u, non-zero at the side and v, non-zero at the top). Thus, the boundary

conditions

for Eqn (9) are &l,Z,

t) = 0

d(r, H/R, t) = 0 M(r, 0, aqo,

with the initial condition

t)jaz

= 0

Z, t)/ar

= 0

of Qr, z, 0) = 0

and the boundary

conditions

for Eqn (10) are IC/(r,0, t) = 0 $(r, H/R, t) = 0 ti(l,z,

t) = 0

$(O,z, t) = 0. Equations (9) and (10) were expressed in transient, finite difference form where $ and 0 were solved for explicitly. The thermophysical properties of corn and the heat capacity of the air were assumed constant with temperature and set equal to their values at the boundary temperature, T,. Temperature varying values of pr and vf, p = p,(l -(TT,)/T) and v = v,(l+2(TT,)/T,), were used. Equation (9) was used to obtain B’s, which were then substituted in Eqn (10) to obtain new $ values. This process was continued until the tI and tj values reached equilibrium. Equilibrium was defined as the point where the temporal change in 0 was less than lo- 5 times the value of 8. The upwind finite difference scheme (Patankar”) was used to express the mixed Ic/and 0 derivatives in the momentum equation. A grid size was chosen by increasing the number of grid points until the T- T, results changed less than 1% with a doubling of the grid size. Boundary effects were negligible because of the relatively large bin dimensions.6

3. Results and discussion The transient development of isotherms and streamlines was followed for constant ambient and grain conditions. Constant values of outside temperature (155”C), corn moisture content (20%) and heat generation rate (20.3 W/m’) were assumed.7 A grain bin of radius 2m and height 4 m was used. Figs 2 and 3 show the predominately conductive isotherm and streamline patterns after approximately 12 days of storage, the isotherms being symmetric around the vertical centreline and the streamlines being symmetric around the horizontal centreline.

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Dimensionless rodws f

Fig. 2. Transient isotherm results after 12 days (t = 75) for 4 = 20.3 W/m’, z = 2 (H = 4m), r = 1 (R = 2m). 0 = dimensionless temperature

and

Dimensionless radius r

Fig. 3. Transient streamline results after 12 days (t = 75) Jar 4 = 20.3 W/m3, z = 2 lH = 4 mi, and r=I[R=Zm)

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0 Dlmenslonless

Fig. 4. Steady-state

isotherms

Fig. 5. Steadv-state

streamlines

radius r

at 46 days (t = 450) for 4 = 20.3 W/m’, z = 2 (H (R = 2 m) 0 = dimensionless temperature

= 4m),

and r = I

at 46 days (t = 450) .fbr 4 = 20.3 W/m3, z = 2 (H = 4m), (R = 2m)

and y = I

Dtmenslon

Ies

radius r

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Dimensionless

Fig. 6. Steady-state

Fig. 7. Steady-state

isotherms for 4 =

streamlines

ANALYSIS

OF

HEAT

TRANSFER

radius r

20.3 W/m’, z = 1 (H = I m), and r = 4 (R = 4~). dimensionless temperature

I

I

2 Dimensionless

3

(j =

radius I

with multicellular$owfr 4 = 20.3 Wjm3, z = I (H = I m), and r = 4 (R = 4m)

Arrows indicate the direction of air flow. The maximum temperature has remained on the vertical centreline but has moved off the bottom of the bin, indicating the initial effects of convection. At 46 days (Figs 4 and 5), the bin is within 0.5% of equilibrium. The maximum temperature is now near the top of the bin (83% of H) and the streamline centre has moved upward, indicating the upward movement of air flow in the bin. The streamlines are compressed near the bin side, evidence of a relatively large downward velocity as compared to the slower, upward velocities nearer the centre of the bin. Although the study was simplified by assuming constant heat generation rate and constant moisture content, the simulated temperature distribution in Fig. 4 closely resembles the temperature distributions found experimentally in grain bins with ambient conditions similar to those modelled here.z*3 Figs 6 and 7 show the equilibrium isotherms and the streamlines for a short, wide bin (R = 4, H = 1) with the same ambient and grain conditions as above. The isotherm patterns are still centred around the vertical centreline but the maximum temperature is located off the vertical centreline on the bin bottom. The isotherms move upward slightly in the radial direction, indicating the larger convective-heat transfer taking place at the upward and outward part of the bin surface. The streamline pattern in Fig. 7 shows a main convection cell centred toward the outward bin surface with relatively high velocities. In addition, a recirculation cell near the bin centre occurs as the colder, heavier air near the bin top and centre sinks toward the bin bottom and overcomes the relatively weak convective forces near the bin centre.

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Fig. 8. Average and maximum temperatures with corn jtir a range of bin height H and radius R (heat generation rate, 4 = 20-3 W/m’). Maximum temperatures: ?? , R = 2m, A. R = 3m, ?? . R = 4m. Average temperatures: 0, R = 2m, A, R = 3 m, 0, R = 4m

2o01 175 150 b-’ +.- 125e 2 i IOO5 E 0

7550 25-

1 OO

I

I

I

IO00

I500

2000

Reference

modified

Rayleigh

I

2500

3c

number Ra,

Fig. 9. Volume averaged and maximum temperatures, which include inertia effects, with and without thermophysical properties variation (R = 2m, H = 4 m) 0, average temperatures, p, v constant; ?? , average temperatures, p, v variable; a, maximum temperatures, p. v constant; A, maximum temperatures, p, v variable

Fig. 8 shows the volume-averaged and maximum temperatures at equilibrium obtained by varying the bin radius and height with the same ambient and grain conditions used earlier. In general, the maximum temperature increases with volume. However, at low volumes, the low height-to-radius bin (R = 4m) displays lower maximum temperature for volumes less than 187~m3 than for the bins with higher height-to-radius ratios (R = 2 m and 3 m). The average bin

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temperatures show similar dependency on bin geometry and volume. Average temperatures, though, exhibit a more gradual increase in temperature with bin volume. For 20% moisture content corn, a range of ambient air temperature and heat generation rates from Thompson7 was used to compare the maximum and volume-averaged equilibrium temperatures obtained by including variable thermophysical property effects with inertia effects. For plotting purposes, a temperature independent Rayleigh number (Ra_J was used although the temperature dependent Rayleigh number defined in Eqn (11) was used in the computer simulations. Fig. 9 shows the resultant curves. Both the maximum and average temperatures resulting from the inclusion of property effects are higher than their counterparts obtained by ignoring property effects. The differences in temperatures are significant, especially at the higher Rayleigh numbers corresponding to ambient temperatures above 18°C (Ra, = 1500). The effect of inertia terms is small, resulting in temperature variations less than 1%. 4. Conclusions Natural convection heat transfer in a short vertical cylinder, similar to grain storage bins, has been numerically investigated for a uniformly heat generating porous medium of air and corn. Constant heat generation rate and moisture content were assumed. Average and maximum temperatures calculated from inclusion of inertial effects were significantly higher than temperatures calculated using Darcy’s law without inertial terms. The numerical results show unicellular flows for large H/R aspect ratios and multicellular flows for some low aspect ratios. In general, grain temperatures increase with increasing heat generation rate (increasing Rayleigh number) and increasing cylinder (bin) volumes. However, short, wide bins may exhibit lower maximum temperatures than tall, narrow bins of the same volume. Isotherm results for corn under steady-state conditions are quite similar to previously published experimental results for wheat and soybeans. References ’ Burton W. G.; Mann, G.; Wager, H. G. The storage of ware potatoes in permanent

buildings II. The temperature of unventilated stacks of potatoes. Journal of Agricultural Science 195546: 150-163 ’ Holman, L. E.; Carter, D. G. Soybean storage in farm type bins. Illinois Agricultural Experiment Station Bulletin 553, 1952, pp. 451495 3 Schmidt. J. L. Wheat storage research at Hutchinson. Kansas and Jamestown, North Dakota. USDA Technical Bulletin 1113,“1955, p. 98 4 Beukema, K. J.; Bruin, S. Heat and mass transfer during cooling and storage of agricultural products. Chemical Engineering Science 1982, 37(2): 291-298 5 Beukema, K. J.; Bruin, S.; Schenk, J. Three-dimensional natural convection in a confined porous medium with internal heat generation. International Journal Heat Mass Transfer 1983,26(3): 451L 458 a Stewart W. E., Jr; Dona, C. L. G. Free convection in a heat generating porous medium in a finite vertical cylinder. Proceedings ASME/JSME Thermal Engineering Joint Conference, Honolulu, HA, 1987, Vol. II, pp. 353-358 7 Thompson, T. L.; Villa, L. G.; Cross, 0. E. Simulated and experimental performance of temperature control system for chilled high moisture grain storage. Transactions of the American Society Agricultural Engineering 1971, pp. 554559 8 Sinha, R. N.; Muir, W. F. eds. Grain Storage: Part of a System. Westport, Connecticut: AVI, 1973 9 Patterson, R. J.; Bakker-Arkema, F. W.; Bicker& W. G. Static pressure airflow relationships in packed beds of granular biological materials such as grain-II. Transactions of the American Society Agricultural Engineering 1971, 14: 172-174, 178 10 Irmay, S. On the theoretical derivation of Darcy and Forchheimer formulae. Transactions of the American Geophysical Union 1958,39: 702-706 11 Patankar, S. V. Numerical Heat Transfer and Fluid Flow. New York: Hemisphere, 1981