TEMPERATURES IN GRAIN BINS

INTRODUCTION

Western Canadian farmers are nowstoring most of their harvested grainfor one or more years. Losses in storedgrain can be caused by the growthof insects, mites and fungi. Theselosses can be reduced by lowering thetemperature of the stored grain(Sinha, 1964; Sinha, 1963; Sinha, etal, 1962; Wallace, et al, 1962). todesign and evaluate methods of reducing the temperature in grain binsit is necessary to predict the temperatures in typical farm grain bins.

Babbitt (1945) predicted the temperatures in a semi-infinite grain bulkwith one-dimensional heat flow usingan analytical method. Some of the assumptions required in this methodwere: (1) the grain bulk is initiallyat a constant temperature throughout, (2) the outside temperature is aharmonic function of time, and (3)thermal properties of the grain areconstant. To overcome the limitationsimposed by these assumptions, thepossibility of using the method offinite differences to predict graintemperatures was investigated. Usingthe method of Dusinberre (1961)prediction equations were developedfor a cylindrical grain bin, one of themost common bin shapes on WesternCanadian farms. Predicted temperatures are compared to temperaturesmeasured hourly for eight months ina 450 bushel plywood grain bin.

DEVELOPMENT OF

PREDICTION EQUATIONS

Insects, mites and fungi usually develop initially near the centre of agrain bulk where the grain cools theleast during the winter. Therefore,the prediction equations were developed for the horizontal planethrough the centre of a cylindricalgrain bin. Based on the measurements of Babbitt (1945) convectiveheat transfer was assumed negligiblecompared to conductive heat transfer.It was assumed that the temperaturesthroughout the bin were symmetricalabout the vertical axis and that heat

RECEIVED FOR PUBLICATIONSEPTEMBER 9. 1969

CANADIAN AGRICULTURAL ENGINEERING, VOL. 12, No. 1, MAY 1970

W. E. Muir

Member CSAE

Department of Agricultural EngineeringUniversity of Manitoba

flow was only in the radial direction.The grain was assumed to be free ofinsects and fungal growth. Since insects and fungi are the main sourcesof heat generation in grain bulks(Christensen, et al, 1969) heat generation was assumed to be negligible.

A sector of the cylindrical bin isconsidered, which has an angle between the radii of one radian and adepth of one unit (Figure 1). Thesector is (Jivided into a finite numberof concentric partial rings of equalradial distance, Ar, except for thecentre and outside rings which areone-half the radial distance.

The conductive heat transfer between any two spatial increments iscalculated by the general equation:

K A (el- (1)

where K = thermal conductivity ofthe grain, Btu — ft/hr_ ft2 _ °p,

A = cross-sectionial area, perpendicular to the direction of heat flow, ft2,

AT = temperature gradientAr along the direction of

heat flow, °F/ft,

Centre of

Grain Bin

q = rate of heat flow,Btu/hr.

Substituting in equation 1, the totalrate of heat flow qn into a spatialincrement n, is:

qn =K(nAr +^|) A6Az

At

fTn+1 - TnlI Air—J +

K(nAr- *f) AGAz f^1^ T") (2)

where n = number of spatial increment,

Ar = length of radius increment, ft.

A0 = angle between radii ofsector = 1 radian

Az = depth of sector, measured along the verticalaxis of the cylinder =1 ft.

The rate of change in heat energycontained in spatial increment, n, iscalculated by the equation:

Q = Vpc (Tn - Tn)(3)

where Q == Change in heat energyin the increment,Btu/hr,

OutsideAir

Outside Wallof Grain Bin

Figure 1. A sector of the grain bin

21

V

p

c

At

(Tn-1 " T„)

volume of element, ft3,

density of grain, lb./ft3,

specific heat of grain,Btu/lb — °F,

time increment betweensuccessive temperaturecalculations, hours,

Tn = temperature of spatialincrement n, at time t,°F,

Tn' = temperature of spatialincrement n, at timeT + At, °F

The heat balance equation for spatialincrement n, can now be written bysetting the total rate of heat flow intothe increment, equation 2, equal tothe rate of change in heat energy contained in the increment, equation 3,and simplifying:

K(n + j) (Tn+1 " Tn) + K(n - h)nAr2pc

At(4)

Equation 4 is solved for Tn:

qc = h(NArA6Az) (Ta - TN) — (8)

where qc = rate of convective heattransfer to the surfaceof the bin, Btu/hr,

h = convective heat transfer coefficient, Btu/hr_ ft2 _ °F,

N = radius of bin dividedby size of spatial increment,

Ta = temperature of ambientair outside bin,v °F,

TN =- temperature of spatialincrement, N, °F.

Writing the heat balance equation forspatial increment N, and solving forthe temperature at time t -f- At:

tn =8NB

(4N-1)M

8NB 8N-4

T + I 8N-4 |a T I(4N-1)M N-l

i - gi"a _ aara i T (9)(4N-1)M (4N-DHJ iN

where B is the dimsensionless Biotnumber:

is released by the freezing of themoisture in the grain at some giventemperature range, then an apparentspecific heat is used in the predictionequation for each spatial increment asit passes through that temperaturerange. The apparent specific heat iscalculated by:

when

c'I

AT'

SL =

apparent specific heat,Btu/lb. — °F,

latent heat of fusion,Btu/lb. of grain,

temperature rangethrough which latentheat of fusion is released, °F,

true specific heat of thegrain through the temperature range, AT'.

AT' =

c =

ln { 2nM J T"+l + { 2nM J Tn-1 B * — (10)

EXPERIMENTAL GRAIN BIN

Temperatures were measured, with24-gauge copper-constantan thermocouples, at 36 locations in a circularplywood bin, from December 2, 1968to July 21, 1969. The bin, which hada wooden floor, was 9.5 feet in diameter and 8 feet high. Bin temperatures reported in this paper were recorded hourly by a Honeywell strip-chart recorder (accuracy ± 1°F) andwere taken along the east-west diameter of the bin, four feet above thefloor. Surface temperatures were measured using thermocouples located atthe interface between the grain andthe plywood wall. Except for thesingle thermocouple at the centre ofthe bin reported temperatures are anaverage of the two thermocoupleslocated one on either side of the bin.Outside temperatures were obtainedfrom a Canada Department of Transport Meteorological Station eightmiles from the grain bin.

The bin contained 450 bushels ofNo. 4 Northern grade wheat. Grainsamples were taken at ten locationsthroughout the bin when the bin wasunloaded in August, 1969. Averagedensity was 45 Ib./ff (56 lb/bu) andaverage moisture content was 11.4%(wet basis). Fungal counts indicatedthat there was very little fungalgrowth in the grain which might becontributing to heat generation in thebin.

i-i T„—- (5)

where the dimensionless modulus, M,is defined by:„ _ cp(Ar) 2

KAt(6)

It is seen from equation 5 that thetemperature Tn at time t -f- At isequal to a weighted average of theinitial temperatures of the space increment, n, and the adjacent spaceincrements.

For the central space increment, 0,the development of the temperatureprediction equation is similar to thatabove except there is heat flowthrough only the outer face of thespatial increment. The equation is:

•s - (*) '"J (7)

In developing the equation for theoutside surface increment it was assumed that the bin wall had negligible heat capacity and infinite thermal conductivity. The heat flowacross the outside surface of the binwas considered to be only convectiveand was calculated by:

22

The temperature at the end of eachfinite time increment is calculated forevery spatial increment using thetemperatures calculated for the endof the previous time increment. Rateof heat flow is assumed constantthroughout any given time increment,that is, temperatures of the incrementchange by a step function at the endof each time increment. The error issmall if either the time increment orthe rate of change of the temperatureis small. Size of the time and spaceincrements must also be chosen sothat the coefficients of equations 5, 7and 9 are greater than zero, or elsethe equations become unstable. Calculations are best done with a digitalcomputer because of the large number of separate calculations that mustbe done.

In the results presented in thispaper, the thermal properties of thegrain were assumed to be constant.If thermal properties vary with temperature or location in the bin, then,for each time increment, differentvalues can be substituted into the prediction equations for each spatial increment according to its location andtemperature. If latent heat of fusion

CANADIAN AGRICULTURAL ENGINEERING, VOL. 12, No. 1, MAY 1970

RESULTS AND DISCUSSIONPreliminary temperature predic

tions using the equations developedabove indicated that increasing thesize of the spatial increments from0.24 feet to 0.95 feet and increasingthe time increment from 0.05 hours to1.0 hour had negligible effect on predicted temperatures but reduced computer time to one-sixth. Long termtemperature predictions, except forthe surface, were the same usingeither hourly outside temperatures orthe daily mean outside temperature(average of daily maximum and minimum temperature).Predicted temperatures were in closeagreement with measured temperatures at all distances from the wallexcept at the wall (Figures 2, 3 and4). At the centre of the bin predictedand measured minimum temperatureswere within 3°F and occurred at thesame time, two months after the outside temperature reached a minimum.The grain bulk warmed up faster inthe spring than predicted. Duringmid-afternoon measured surface temperatures were often higher thaneither the outside temperature or thepredicted surface temperature. Thesedifferences were probably caused bysolar radiation which was not takeninto account in the prediction equations. Surface temperatures follow thediurnal variation of outside temperatures. At 0.5 feet from the wall thediurnal variation was approximately1°F (Figure 4).

Except where otherwise noted thethermal properties used in the prediction equations are those given byMoote (1952) for wheat with moisture content of 10.9% and densityof 51.2 lb/ft3. These values are thermal conductivity — 0.0918 Btu —ft/hr — ft — °F and specific heat —0.44 Btu/lb — °F. Predicted temperatures were affected only slightly bysubstituting the thermal propertiesfound by Babbitt (1945) (thermalconductivity = 0.0872 Btu — ft/hr_ ft2 _ °p, specific heat _ 037Btu/lb — °F) (Figure 3). Sinceboth Moote and Babbitt conductedtheir tests at temperatures above80°F the thermal properties of wheatat low temperatures may differ somewhat from those used in the prediction equations.

There appears to be no temperatureat which latent heat of fusion is released by freezing of the water inthe grain. In all the curves of mea-

UJ

CEUlQ.

UJ

90

70

50

30

10

-10

-•—•

o

J_

OUTSIDE AIR TEMPERATURE

MEASURED TEMPERATURE

PREDICTED TEMPERATURE

PREDICTED TEMPERATURELatent heat of fusion at 32 °F

/'WV„' :v°

J_ JL-30

DEC'68 JAN.'69 FEB. MAR. APR.

TIME

MAY

/ *A

A/•a*

JUN. JUL.

Figure 2. Predicted and measured temperatures at centre of grain bin

90

70 -

uj 50<z

I-

uj 30a.

10

-10

-301

-• —

O

OUTSIDE AIR TEMPERATURE

MEASURED TEMPERATURE(Average of two thermocouples)PREDICTED TEMPERATUREThermal properties from Moote (1952)

PREDICTED TEMPERATUREThermal properties from Babbitt (1945)

_L J_ JL

DEC.'68 JAN/69 FEB. MAR. APR.

TIME

MAY JUN.

I I

JUL.

Figure 3. Predicted and measured temperatures 1.9 ft. from wall of grain bii

CANADIAN AGRICULTURAL ENGINEERING, VOL. 12, No. 1, MAY 197023

sured temperatures there is no dis-cernable leveling out of the curve ata freezing temperature. If the moisture in the grain froze at 32°F releasing latent heat equal to 16.4Btu/lb of grain, then the predictedtemperature curve shown in Figure2 would result which is significantlydifferent from the measured temperature curve. Work is now in progressto determine the thermal propertiesof wheat and other grains at low temperatures.

The prediction equations developed in this paper could be used formost cylindrical bins containing drygrain. For grain bins which have ahigh diameter to height ratio, the prediction equations probably will not beapplicable since vertical heat flowwould become important. Furtherwork is now in progress to developprediction equations for two-dimensional heat flow. Temperatures insteel bins probably will follow thepredicted temperatures more closelythan temperatures in wooden binssince the solar radiation absorptivityfor steel is less than for wood. Whenapplying the prediction equations tocylindrical concrete bins, the size ofthe surface spatial increments can bechosen to correspond to the thicknessof the concrete wall. Specific heatand thermal conductivity of the concrete can be substituted into the prediction equations for the outside spatial increments.

CONCLUSIONS

Measured temperatures at thecentre of a 450 bushel grain bin werein close agreement with those predicted by equations developed usingthe method of finite differences. Itappears that these equations can beused to predict temperatures in circular grain bins of other sizes providedthe grain is dry. The method has thefollowing advantages:

(a) initial temperatures in the bin donot have to be assumed uniformfrom the wall to the centre of thebin, since measured temperaturesacross the bin can be used in thecalculations.

(b) Outside temperature can varyaccording to any function oftime.

(c) Thermal properties of the graincan be any function of temperature and location in the bin.

24

40

OUTSIDE AIR TEMPERATURE

PREDICTED SURFACE TEMPERATUREMEASURED SURFACE TEMPERATURE

PREDICTED TEMPERATURE 6" FROM WALLMEASURED TEMPERATURE 6" FROM WALL

PREDICTED TEMPERATURE 12" FROM WALLMEASURED TEMPERATURE 12" FROM WALL

30

20

UJor

orUJo.

UJ10

DEC. 2 DEC. 3 DEC. 4 DEC. 5 DEC. 6 DEC. 7 DEC. 8

TIME

Figure 4. Predicted and measured temperatures showing diurnal variation

(d) Latent heat of fusion can betaken into account in the calculations.

The prediction equations presentedhere do not take into account solarradiation, vertical heat flow in thebin and heat generated by insects,mites and fungi.

SUMMARY

Equations based on the method offinite differences are developed forpredicting the temperature at thecentre of a cylindrical grain bin. Heatflow was assumed to occur only in theradial direction. Predicted temperatures were in close agreement withthose measured in a 450 bushel plywood grain bin over an eight monthperiod. The predicted minimum temperature at the centre of the bin waswithin 3°F of the measured value andoccurred at the same time. Themethod takes into account variationsin the initial grain temperatures, theoutside temperature, and the thermalproperties of the grain.

ACKNOWLEDGEMENTS

The author wishes to acknowledgethe financial support of the CanadaDepartment of Agriculture and theassistance of Dr. R. N. Sinha and Mr.H. A. H. Wallace of the Canada Department of Agriculture, Winnipeg.

REFERENCES

1. Babbitt, J. D. 1945. The thermalproperties of wheat in bulk. Can.J. Research, F. 23: 388-401.

2. Christensen, C. M. and H. H. Kaufman. 1969. Grain storage, the roleof fungi in quality loss. Universityof Minnesota Press, Minneopolis.

3. Dusinberre, G. M. 1961. Heattransfer calculations by finite differences. International TextbookCo., Scranton, Penn.

4. Moote, Irene. 1953. The effect ofmoisture on the thermal propertiesof wheat. Can J. Tech. 31: 57-62.

5. Sinha, R. N., E.A.R. Liscombe andH. A. H. Wallace. 1962. Infestationof mites, insects and microorganisms in a large wheat bulk afterprolonged storage. Can. Ent. 94:542-555.

6. Sinha, R. N. 1963. Suitability ofclimatic areas of Canada for infestation of some major storedgrain insects on farms. Proc. Ento-mol. Soc. Manitoba. 19: 31-39.

7. Sinha, R. N. 1964. Effect of lowtemperature on the survival ofsome stored products mites. Acaro-logia 6: 336-341.

8. Wallace, H. A. H. and R. N. Sinha.1962. Fungi associated with hotspots in farm stored grain. Can. J.Plant Sci. 42: 130-141.

CANADIAN AGRICULTURAL ENGINEERING, VOL. 12, No. 1, MAY 1970