TRANSIENT HEAT FLOW IN PIGGERY FLOORSTRANSIENT HEAT FLOW IN PIGGERY FLOORS by F. V. MacHardy Member...

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TRANSIENT HEAT FLOW IN PIGGERY FLOORS by F. V. MacHardy Member C.S.A.E. Agricultural Engineering Department, University of Alberta, Edmonton, Alta. Introduction would be most accurately described by the second possibility. Interest is increasing in cleaning J L piggery floors by means of high pres sure water jets. For most successful operation of this system, bedding should be eliminated. The question then arises as to the most suitable flooring material. With this overall problem in mind, research was undertaken at the Uni versity of Alberta to determine the nature of the heat transfer taking place from pigs lying on unheated floors into and through the flooring material. Theory Fourier's Law for unidirectional conduction of heat states that the instantaneous rate of heat flow dQ/d(9 is dependent upon the three factors, area A, temperature gradient dt/dx and a factor called the thermal con ductivity k. Expressed mathematic ally: A9: dO - kA dt "dx~ (1) Equation (1) is general and may be applied both when the temperature at any point remains constant, and when the temperature at a point varies with time. The process of heat flow in a case in which temperature varies with both time and position is called unsteady state conduction, and problems dealing with this state are referred to as problems in tran sient flow. The problem at hand, that of a pig lying down on an unheated sleep ing floor, is clearly, initially at least, a problem in transient flow. In calculations dealing with tran sient flow in slabs, there are two gen eral possibilities. The first is that the surface temperature of the slab im mediately assumes the temperature of the contacting heat source, in other words, the surface coefficient of heat transfer h has an infinitely large value. The second possibility is that the surface coefficient is a controlling factor, and the surface temperature does not immediately assume the temperature of the contacting heat source. It was anticipated that heat transfer from the pigs into floor slabs o a.5 l.o 1.5 2.0 2.5 3.0 3.5 U 6 x - k6 - pcQ Cpr*B, rZB Figure I. A complete treatment of the dif ferential equations for transient heat flow, together with their analytical solutions, are contained in McAdams (1). A number of writers have plotted solutions to transient flow problems making use of dimensionless ratios. This approach enables problems to be solved without becoming involved with the intricacies of the mathe matical manipulations required for analytical solutions. Figure 1 illus trates a Hottel (2) Chart for obtain ing the surface temperature of large slabs in terms of three dimensionless ratios: 1.) An unaccomplished tempera- , .. (ta - ts) ture change, Y 4 A— 6 (ta - tb) 2.) A relative time, X = ^0 = k 0 rni2 /ocprm2 3.) A resistance ratio, m = Rs = k Rm rmh where A Area through which heat flows at right angles, square feet cp Specific heat of solid, Btu/ (lb) (deg F) h Coefficient of heat transfer be tween surroundings at ta and surface at ts, Btu/ (hr) (sq ft) (deg F) k Thermal conductivity of solid, Btu/(hr)(sq ft) (deg F per ft) 13 rm Normal distance from midplane to surface in feet. For the case of a slab heated from one side rm is the total thickness of the slab in feet. ta Temperature of surroundings, deg F. ts Temperature of surface of slab, deg F. tb Original uniform (base) tem perature of solid, deg F. </> Time, from start of heating or cooling, hours pep p Density of solid, pounds per cubic foot Proposition The direct substitution of known values for the physical properties of any slab and a known value for the surface coefficient into the dimen sionless ratios of Figure 1 will yield slab surface temperatures. This re search project utilized the conjunct relationship: by measuring surface temperatures, and combining these with the known physical properties of the floor slab, it was anticipated that the nature of the heat source would be revealed. Specifically, it was hoped that a value for surface coefficient would be obtained. This would in turn enable engineers to consider pigs as simply being an iso thermal heat source exhibiting a known surface coefficient. Test Apparatus and Procedure The Department of Animal Sci ence, University of Alberta, generous ly made available three pens in one of their piggeries for an investigation of this heat flow problem. Sleeping platforms, 5' x 8' x 3" thick were built into the pens. Ma terials were: sand and gravel concrete, concrete with expanded shale aggre gate, and, concrete consisting of 1:1 mixture of Portland cement and ver- miculate. Thermocouples were in stalled in the sleeping platforms at the time of construction. Four ther-

Transcript of TRANSIENT HEAT FLOW IN PIGGERY FLOORSTRANSIENT HEAT FLOW IN PIGGERY FLOORS by F. V. MacHardy Member...

Page 1: TRANSIENT HEAT FLOW IN PIGGERY FLOORSTRANSIENT HEAT FLOW IN PIGGERY FLOORS by F. V. MacHardy Member C.S.A.E. Agricultural Engineering Department, University of Alberta, Edmonton, Alta.

TRANSIENT HEAT FLOW IN PIGGERY FLOORSby

F. V. MacHardyMember C.S.A.E.

Agricultural Engineering Department, University of Alberta, Edmonton, Alta.

Introduction would be most accurately describedby the second possibility.

Interest is increasing in cleaning J Lpiggery floors by means of high pressure water jets. For most successfuloperation of this system, beddingshould be eliminated. The questionthen arises as to the most suitable

flooring material.

With this overall problem in mind,research was undertaken at the Uni

versity of Alberta to determine thenature of the heat transfer takingplace from pigs lying on unheatedfloors into and through the flooringmaterial.

Theory

Fourier's Law for unidirectional

conduction of heat states that the

instantaneous rate of heat flow dQ/d(9is dependent upon the three factors,area A, temperature gradient dt/dxand a factor called the thermal con

ductivity k. Expressed mathematically:

A9:dO

- kAdt

"dx~ (1)

Equation (1) is general and may beapplied both when the temperatureat any point remains constant, andwhen the temperature at a pointvaries with time. The process of heatflow in a case in which temperaturevaries with both time and positionis called unsteady state conduction,and problems dealing with this stateare referred to as problems in transient flow.

The problem at hand, that of apig lying down on an unheated sleeping floor, is clearly, initially at least,a problem in transient flow.

In calculations dealing with transient flow in slabs, there are two general possibilities. The first is that thesurface temperature of the slab immediately assumes the temperatureof the contacting heat source, in otherwords, the surface coefficient of heattransfer h has an infinitely largevalue. The second possibility is thatthe surface coefficient is a controllingfactor, and the surface temperaturedoes not immediately assume thetemperature of the contacting heatsource. It was anticipated that heattransfer from the pigs into floor slabs

o a.5 l.o 1.5 2.0 2.5 3.0 3.5 U 6

x - k6 - pcQ

/© Cpr*B, rZB

Figure I.

A complete treatment of the differential equations for transient heatflow, together with their analyticalsolutions, are contained in McAdams(1). A number of writers have plotted

solutions to transient flow problemsmaking use of dimensionless ratios.This approach enables problems to besolved without becoming involvedwith the intricacies of the mathe

matical manipulations required foranalytical solutions. Figure 1 illustrates a Hottel (2) Chart for obtaining the surface temperature of largeslabs in terms of three dimensionless

ratios:

1.) An unaccomplished tempera-, .. (ta - ts)

ture change, Y — 4 A—6 (ta - tb)

2.) A relative time,X = ^0 = k 0

rni2 /ocprm2

3.) A resistance ratio,m = Rs = k

Rm rmh

where

A Area through which heat flowsat right angles, square feet

cp Specific heat of solid, Btu/ (lb)(deg F)

h Coefficient of heat transfer be

tween surroundings at ta andsurface at ts, Btu/ (hr) (sq ft)(deg F)

k Thermal conductivity of solid,Btu/(hr)(sq ft) (deg F per ft)

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rm Normal distance from midplaneto surface in feet. For the case

of a slab heated from one siderm is the total thickness of the

slab in feet.

ta Temperature of surroundings,deg F.

ts Temperature of surface of slab,deg F.

tb Original uniform (base) temperature of solid, deg F.

</> Time, from start of heating orcooling, hours

pep

p Density of solid, pounds percubic foot

Proposition

The direct substitution of known

values for the physical properties ofany slab and a known value for thesurface coefficient into the dimen

sionless ratios of Figure 1 will yieldslab surface temperatures. This research project utilized the conjunctrelationship: by measuring surfacetemperatures, and combining thesewith the known physical propertiesof the floor slab, it was anticipatedthat the nature of the heat source

would be revealed. Specifically, itwas hoped that a value for surfacecoefficient would be obtained. This

would in turn enable engineers toconsider pigs as simply being an isothermal heat source exhibiting aknown surface coefficient.

Test Apparatus and Procedure

The Department of Animal Science, University of Alberta, generously made available three pens in oneof their piggeries for an investigationof this heat flow problem.

Sleeping platforms, 5' x 8' x 3"thick were built into the pens. Materials were: sand and gravel concrete,concrete with expanded shale aggregate, and, concrete consisting of 1:1mixture of Portland cement and ver-

miculate. Thermocouples were installed in the sleeping platforms atthe time of construction. Four ther-

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mocouples were located flush withthe concrete surface and two werelocated at the bottom of each slab. Itwas considered that the thermo

couples at the surface would measurethe temperature of the junction between the slab and a pig whenever apig lay directly on one of the couples.The leads from the thermocoupleswere connected to a variable-span 12point recorder.

Two series of tests were made, onerecording surface and base temperatures of two slabs simultaneously,while for the other test surface tem

peratures of all three slabs were recorded simultaneously. Throughoutthe tests no bedding was used on thefloor slabs and from three to five

hogs, each approaching marketweight, were kept in the pens. Thisnumber of hogs together with thelimited size of the sleeping platforms,insured a reasonable incidence of con

tact between the hogs and the thermi-couples.

Results

Time In Hours

Figure 2.

Figure 2 shows a curve of surfacetemperature obtained from the recorder when a pig laid down on athermocouple located on the surfaceof the 1:1 vermiculate slab.

From this figure it is obvious thatthe surface temperature did not immediately assume the normal bodytemperature of the pig. The rise fromthe slab base temperature over aperiod of time indicates a heat flowcondition where boundary layer is animportant factor. The second state oftransient flow must then apply: thatwith surface resistance controlling.Data from this temperature curvewere combined with the physicalcharacteristics of the slab, and thetheoretical relationships plotted inFigure 1, to reveal the nature of heatflow from the pig into the sleepingfloor. Referring to Figure 1 the valueof X was obtained by combining timevalues from Figure 2 with the physi

cal properties of the slab. The valueof Y was obtained by combining thetemperature data from Figure 2 withthe normal body temperature of hogs.With values of X and Y available, intercepts on Figure 1 gave a value form. From m in turn, a value for h thesurface coefficient was derived. The

appendix shows the calculationswhich were carried out to arrive at a

numerical value for h. The value ob

tained was:

h 1.5 btu/fr" hr °F

Conclusions

The method developed here mayserve to reduce the physiological phenomena associated with heat dissipation from animals to terms more

readily handled by engineering methods.

Further investigation is required todetermine the value of surface coef

ficients over a range of ambient temperatures, and to attempt to fix thesevalues accurately in certain ambienttemperature ranges.

Flaving done this, a suitable standard for flooring materials could beestablished. A suitable standard would

appear to be that heat loss to theflooring material per unit contactingarea should not exceed heat loss to

the surroundings from a standing animal under conditions of thermoneu-

trality.

APPENDIX

To obtain h

Vermiculite concrete, 1:1 mixrm = 14 ft.k = 0.167 btu/hr ft °FcP = 0.20 btu/lb °Ft

P = 75 lb/ft.a

Substituting for X in Figure 1

0T67 X 1/275 X0.20X(i4)s= 0.089

X(0 = 14) — 0.045X(6 — i/8) = 0.025

From recorder chart, Figure 2

Y(* = K) , I01"82

x (e = 14)

Y(6 ---. y4Y(* = V8)

From FigureX and Y

m = 0.45

from which

h = 1.5

Theoretical time vs. surface temperature curve for sand and gravelconcrete

14)

100 - 64

= 0.53

= 0.61

= 0.75

for there values of

14

Using h = 1.5

m = 2.56

and rm = 1/4 ftk = 1.0 btu/i't hr °F

cp = 0.20 btu/lb °F

p = 150 lb/ft.3

from which values of X may be obtained, and, from X and m, valuesof Y, solving:

ts (9 =3 0) = 64 °Fts (9 == i/8)= 68.5 °FU (9 == 14)= 70.5 °Ft: (9 == i/9)= 72.5 °F

List of References

1. McAdams, William H., "HeatTransmission", McGraw Hill,New York, 1954.

2. Hottel, H. C, from (1).

Continued from page 6

the material play an important part.These properties seem not to havebeen extensively investigated thoughacurate knowledge of them is essential for systematic design of hoppersand mechanical conveyors, and ifsimilarity principles, which are so important in such fields as fluid flowand heat transfer, are to be applicablein solid-particle transport. Thus, properties including particle dimension,densities, a variety of friction factorsand drag coefficients, should be evaluated with attention to the effect ofsuch variables as moisture content.

BIBLIOGRAPHY

1. Vogt, E. G., White, R. R., Friction in the flow of suspensions,Ind. Eng. Chem. 40, 1731, 1948.

2. Henderson, S. M., Perry, R. L.,Agricultural Process Engineering,Wiley, 1955.

3. Carlson, H. M., Frazier, P. M.,Engdahl, R. B., Meter for flowingmixtures of air and pulverizedcoal. Trans. Amer. Soc. Mech.Engrs. 70, 65-79, 1948.

4. Farbar, L., Metering of PowderedSolids in Gas-Solid Mixtures. Ind.Eng Chem. 44, 2947, 1952.

5. Hariu, O. H., Molstad, M. C,Pressure Drop in Vertical Tubesin Transport of Solids by Gases,Ind. Eng. Chem. 41, 1148, 1949.

Continued on page 39

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Continued from page 9

APPENDIX C

Capital CostAuger bunk feeder $1,427.50Unloading wagon system 583.00Hand feed cart system .... 423.00

Operating Cost Per year

Auger bunk feeder 7.50Unloading wagon 24.00Feed Cart 0.00

I.) Pay-off period for auger vs.unloading wagon:

(1427.50 - 583.00)= 57.5 years

( 24.00 - 7.50)2.) Hand cart vs. onloading wagon

The hand cart system, having botha lower first cost and operating cost,is clearly indicated as the best selection.

APPENDIX D

Capital CostSelf feeder

Continued from page 16

economic advantage of such a program. If the economics are not favorable, then inducements must beadded, possibly through constructionor financial aid. This is another portion of the program that will requirea great number of years to accomplish.

Research activities must keepabreast or ahead of watershed devel

opment work. The development oftechniques and methods and the compilation and analysis of research information will need priority in anadequate program of water use andconservation.

Experience has shown that thereare two distinct phases to a waterdevelopment program, the engineering and the agronomic which must bebrought together. It is virtually impossible to deal with watershed development or with research in hydrology solely in terms of one or theother. Close co-operation is requiredto develop to the fullest the potentials of any watershed. Satisfactorilydesigned structures are but half thejob, compatable crops and croppingpractices are necessary to completethe picture. It is only through suchco-operation that the project economics can become favorable.

Modifications to 3 existingportable granaries $ 300.00

Feed cart system(Appendix C) 423.00

Net capital cost favoringself feeder 123.00

Yearly Feed CostsLimited feeding system

Grain (Appendix B) 3,865.00Roughage 112 Tons*

at $17.50/Ton .../.... 1,960.00

$5,825.00Self feeding system

Grain (Appendix B) 4,625.00Roughage 82 Tons (1)at S17.50/Ton 1,435.00

$6,060.00Net feed cost favoring

limited feeding 235.00Pay-off period 123

= i/2 year235

For additional comparisons:

Continued from page 14

6. Towle, W. L., Schweyer, H. E.,Moffatt, L. R., Viscosities ofliquid-solid systems. Ind. Eng.Chem. 29: 489-92. 1937.

7. Furukawa, J. et al, LiquidlikeProperities of Fluidized SystemsInd. Eng. Chem. 50, 821, 1958.

8. Segler, G., Pneumatic Grain Conveying, N.I.A.E., Silsoe Bedfordshire.

9. Lapple, C. E., and Shepherd, C.B., Calculation of particle trajectories Ind. Eng. Chem. 32, 605-617,1940.

10. Cramp, W., Priestly, A., Pneumatic Grain Elevators, The Engineer, 137, 34-35, 64-65, 89-90, 112-113. 1924.

11. Squires, L., Squires, W., The Sedimentation of thin Discs. Amer.Inst. Chem. Eng., 1937.

12. Flawksley, P. G. W., Brit. CoalUtil, Res. Ass. Bull. No. 4, ThePhysics of Particle Size Measurement: Part 1. The Fluid Dynamics and the Stokes Diameter.

13. Belden, D. H., Kassel, L. S., Pressure Drops Encountered in Conveying Particles of Large Diameter in Vertical Transfer LinesInd. Eng. Chem. 41, 1175, 1949.

14. Dalla Valle, J. M., Micromeritics,Pitman, New York, 1948.

15. Scheidegger, A. E., The Physicsof flow Through Porous Media,The University of TorontoPress, 1957.

39

1.) for unloading wagon

Pay-off period —(583 - 300)

= 1.35 years(235 - 24)

2.) for auger bunk feeder

Pay-off period =(1,427.50 - 300)

= 4.95 years( 235 - 7.50)

LIST OF REFERANCES

1. Cattle Finishing in Alberta -

University of Alberta, Edmonton1958.

2. Machinery and Allied ProductsInstitute, Chicago, Illinois.

3. Self Feeder for cattle, Alberta Department of Agriculture, Edmonton.

4. MTM Assosiation for Standard

and Research, 620 Penn. Ave.,Pittsburgh, Penn., 1950

16. McEwen, E., Simmonds, M. A.,Ward, G. T., Resistance to AirFlow of Beds of Agricultural Products. Trans. Inst. Chem. Eng.32, 130-140, 1954.

17. Babbitt, J. D., Observations onthe Absorbtion of Water Vaporby Wheat. Can. J. Res., 27F, 55-72, 1949.

18. Mehta, N. C, Pressure Drop inAir-Solid Flow Systems. Ind. Eng.Chem. 49, 986, 1957.

19. Crane, J. W., Carleton, W. M.,Predicting Pressure Drop in Pneumatic Conveying of Grains. Agric.Eng., 38, 168-171, 180, 1957.

20. Pinkus, O., Pressure Drop in thePneumatic Conveyance of Solids.Jour. Ajpp. Mech. 74, 425-31,1952.

21. Leffler, K. L., Thomas, W. A.,The Effect of Inclination Anglesin Capacity of Belt Conveyors,Flow, 12, 62, July 1957.

22. Agricultural Engineers Yearbook.23. Jenike, A. W., Flow of Solids in

Bulk Handling Systems. Bull.No. 64, Utah Eng. Exp. Sta., 1954.

24. Jenike, A. W., Better Design forBulk Handling, Chemical Engineering, 61, p. 175, 1954.

25. Leggett, R. F., Clogging of Bituminous Coal in Bunkers. Trans.A.S.M.E., 69, 525, 1947.

26. Wolf, E. F., von Hohenleiten, H.L., Experimental Study of Flowof Coal in Chutes at RiversideGenerating Station, Trans. A.S.-M.E., 67, 585, 1945.