2007 p. Kariuki Nyahoro

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PergamonSolar Energy Vol. 59, Nos. 1-3, pp. 11-17, 1997

0 1997 Elsevier Science LtdPII: SOO38-092X 96)00133-8 Printed in Great Britain. All rights reserved

0038-092X/97 $17.oO+O.C@

SIMUL TED PERFORM NCE OF THERM L STOR GE IN

SOL R COOKER

P. KARIUKI NYAHORO,* RICHARD R. JOHNSON** and JOHN EDWARDS***Appropriate Technology Centre, Kenyatta University, Box 43844, Kenya and **North Carolina State

University, Raleigh, NC 27695-7910, U.S.A.

(Communicated by Joachim Luther)

Abstract-An explicit finite-difference method is used to simulate the thermal performance of short-termthermal storage for a focusing, indoor, institutional, solar cooker. The cooker storage unit consists of acylindrical solid block. The block is enclosed in a uniform layer of insulation except where there arecavities on the top and bottom surfaces to allow heating of a pot from storage and heating of the storageby solar radiation. A paraboloidal concentrator focuses solar radiation through a secondary reflector ontoa central circular zone of the storage block through the cavity in the insulation. The storage is charged

for a set period of time and heat is subsequently discharged to a pot of water. In these simulations a potof cold water is placed on the hot storage block and the time then estimated until the water either boilsor the temperature of the water reaches a maximum value. Simulations are made for a given pot capacitywith the storage block made from either cast iron or granite (rock). The effects on cooker performanceare compared for a variety of height to diameter ratios of the storage block and size of the area of solarinput zone. 0 1997 Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

Cooking with fuel wood con sumes the bulk oftotal national energy demand in many develop-ing countries (Nandw andi, 1988). Due to the

increasing population in these countries,demand for both wood and agricultural landhas risen to such an extent that there is now anet depletion of wood resources with someserious present and potential consequences suchas soil erosion, food shortage, and fuel woodshortage. In addition to improving efficienciesof wood-based cooking devices and introducingother biomass-b ased fuels, the use of solar cook-ers is one way of reducing the deman d forfirewood. A dvantages of using solar cookers

include free availability of solar radiation, acces-sibility to this radiation even in remote placeswith little access to conventional energy sup-plies, and its higher abundanc e in hot, dry orarid areas with the greatest need for fuel woodsubstitutes.

Current designs of solar cookers normallyused are box cookers, concen trators, and flatplate collector cookers. Com mon problemsassociated w ith cooker designs (Kuhnke, 1987;Grupp, 1990) include limitation to daytime and

outdoor use during sunny w eather, small capac-ity, and slow cooking speed. In an attempt toovercome some of the problems, indoor institu-tional hy brid cookers of the type designed byScheffler (Oehler and Scheffler, 1994; Otienoand Scheffler, 1989) have been constructed and

installed (Fig. 1). The concentrator in thiscooker is parabolic-shaped, mild steel, grid sup-porting panels of aluminized plastic sheet. Bothpot and secondary reflector are mounted inside

a kitchen. If needed during non-sunny periodsthe pot is moved laterally over an adjacentcombustion chamber where burning wood orcharcoal may be used to sustain the cookingprocess. In the study reported here, a numericalmodel is used to predict the behavior of thistype of cooker when it is operated with short-term sensible heat storage. Since the addition ofa heat storage component makes the cookermore expensive and comp licated, it is necessaryto study the effects of thermal and geometric

parameters on the cooker performance in orderto provide guidance for the effective design ofsuch cookers.

potstorage tmct

mi

Iitchen -primary reflector

WOII enlcentrot

Fig. 1. Layout of components of an indoor focusing hybridcooker similar to the design by Scheffler (1994) but including

a thermal storage block.

11


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12 P. K. Nyahoso et d

A layout of components for the solar cookeris show n in Fig. 1. The layout of all but thestorage block is consistent with the basic Scheffler

talled at the Appropriate Technologynyatta University, Nairobi (Otieno

and Scheffler, 198 9). The parabolic concentratoris located outside the kitchen building. It is atruncated section of a parabola of revolutionsuch that the focal point is well away from theconcentrator itself. The rays from the sun aredirected through a hole in the kitchen wall onto an angled reflector, finally focusing on an areaon the bottom of the storage block.

Fig. 2. tieat storage elements of a solar cooker showing PO;.storage block, insulation, and glazing.

The concentrator is on a two axis mountingsuch that the dish can be manually reset to

follow the path of the sun. ecause the systemis not precisely focuse d, a slight adju stmentevery half hour works quite adequately. In fact,in cooking experience without storage, when thepot has already been t-ought to the boil, theconcentrator is often left untracked so as tolimit heating of the pot. The concentrator isconstructed from many flat, reflective panelsmounted in a shaped frame. The rays from theconcentrator strike a second ary reflector in thekitchen which redirects them on to the bottomof the storage block. Simulation studies arecarried out for only one concentrator size.

The storage block is a cylindrical shaped solidthat is enclosed in insulation At the bottomcenter of the block the insulation has beenremoved so that the solar rays can impinge onthe storage surface. T he solar input zone is setback in a cavity because of the thickness of theinsulation. The surface at the solar input zonemay be coated so as to be a selective absorber.The solar input zone cavity is g azed, just as insolar collectors, to reduce the heat lost from the

Rot surface. At the top surface of the storageblock there is another centrally located cavityin the insulation to allow for placement of thepot on the storage block. For simulationpurposes this area was assumed insulated duringsolar heating of storag e, and in intimate contact

insu‘lation is a so discretized in the same coor& -nate system, but with special nodal matchmgnear the interface between storag e and insula-.tion. The pot wall and bottom have a speciaset of one-layer nodes. The water in the pot isrepresented by one node at the bulk watertempe rature. The interfacial s urfaces betweenpot, storage and insulation were assumed tohave contact resistance. T he temperature T(r,z,t)is solved using an explicit finite differencescheme based on a heat balance at each node

and the heat conduction equation. The bound-ary conditions on the outside surface of theinsulation are specified a s free convection andradiation to the surroundings at 20°C. At theradiation input zone there Is a specified solarradiation input uniformly distributed. over the

heat loss by convection andradiation through the glazed cavity to the sur-roundings at 20°C. Heat transfer between theinterior pot wall and th e water in the pot waseither described ‘by equations for convect’lon orequations for boiling dep ending on the potsurface and water temperatures.

ot bottom when cooking. The layoutof pot, storage block, and insulation are shownin Fig. 2. The 0.05 m3 (50 1); steel pot is com-pletely fihed with water.

The heat transfer mechanism between theinterior pot surfaces and the water in the potmay either be natural convection or boihng.The descriptions of these heat transfer types,and the rest&ant heat transfer rates, are quitediEerenf, necessitating a criterion to select theright mod e, and protection against nu mericalinstability that may resuit from the step changein values.

3.1. Finite fSeerence model The coefficient fo r natural convection heatThe storag e block is discretized as a cylindri- transfer inside th e pot, h,, was evaluated from

cal coordinate system with axial symmetry. The the correlation suggested by Evans and Stefany


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Simulated performance of thermal storage in a solar cooker 13

(1966) and given as:

Nui = hi&- = 0.55Ra:‘4k,

(1)

Ra

IgH; C,,d, CL t) - T,l

hkl 2)

Here Nu, and Rai are the instantaneous Nusseltand R ayleigh nu mbers, T,(t) is the time-dependent, area-averaged, instantaneous tem-perature on the surface in contact with water,and T, is the initial tem perature of water in thepot. The value of C =0.55 in eqn (1) is the valuesuggested by Evans and Stefany (1966). Thetemperature-dependent properties of specificheat, conductivity, and viscosity for water were

taken from expressions given by Yaw s (1977)and the density from an expression given byPerry (Nyahoro, 1992) and evaluated at temper-ature T,.

Wh en the temperature of a hot surfaceimmersed in a liquid exceeds the saturationtemperature of the liquid by a few degrees, themechanism of heat transfer ch anges to one ofboiling. For water at atmospheric pressure thisoccurs w hen the surface reaches a thresholdtemperature of about 105°C (Incropera andDewitt, 1990). The transition occurs even if theliquid is sub-cooled relative to the saturationtemperature. For surface temperatures between105°C and 130°C there is likely to be poolnucleate boiling. During the boiling phase thevertical wall of the pot and the water are lumpedinto a compound node which comm unicates byconduction with the lid and side insulation, andby the boiling m echanism with the bottomsurface of the pot. The b oiling flux q,,“, used forsub-cooled nucleate boiling in water, from theheating at the bottom of the pot, is that byRohsenhow and given by Incropera and Dewitt(1990) as:

The excess temperature is calculated as AT,,=(K - 100) where T, is the inside surface temper-ature at the bottom of the pot. The empiricalconstant CSf is assigned a value of 0.132 for acombination of water and polished stainlesssteel (Vahon et a l . , 1968). The exponent n onthe Prandtl number has a value of 1.0 for waterand 1.7 for other liquids. Property terms withsubscripts 1 and v are, respectively, evaluatedfor liquid and vapor phases at saturation tem-perature. During boiling, each node at the

bottom of the pot is assigned a boiling heatremoval rate of:

where T,, is the temperature of the compoundnode (water tem perature) and A,(m ,n) is the areaof node (m,n) in contact with the w ater. T hisequation ensures that no node at the bottom ofthe pot is assigned a heat removal rate that isso high as to cause its temperature to dropbelow T,, at the end of a simulation time step.The criterion for transition to boiling, and theexpression for heat transfer during boiling, con-tain the area-weighted surface temperature T,,

that covers the heated zone of the pot bottom.The criterion for transitio n from free convectionto boiling was taken as T,= 105.5”C. Whenboiling was determined to occur at the heatedzone, it was assume d to occur throughout thewater in the pot.

3.3. M o d el f o r h ea t t r a n s f er a t t h e i n p u t z o n e

The zone where the input so lar radiation isincident on the storage block is a geometricallycomplicated region in which both radiation andconvection are important. There is a cavityformed in the insulation to allow access to thestorage block. The cavity may be left open or itmay be closed with a transparent glazing whichallows radiation to pass easily into the cavity,but restricts radiation and convection of heatout of the cavity. The open cavity is treated asa heat transfer zone with gray surfaces, viewfactors, and free convection coefficients. The raytracing technique is used to determine the effec-tive fraction of input radiation that is absorbed

at the elemental surfaces in the input cavity.The glazed cavity has the same features, butincludes the multiple reflections, refractions andabsorptions at the glazing and within the coverthat are analyzed in a way that is similar to thetechniques used in flat-plate so lar collectors asdescribed by Duffie and Beckman (1991).

3.4. C o n v e ct i o n a n d r a d i a t i o n r o m t h e ex p o s edout er surfaces

The dimensions and temperature range of thecooker system satisfy the condition proposedby Churchill and Chu to calculate natural con-vection heat transfer from exposed surfaces(Incropera and Dewitt, 1990) as given in theexpression for Nusselt number based on effective


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14 P. K yahoro et ui

height H of the vertical surface:

Nu,= y =O.6&+0.67Ra~‘4

419a

5)

Property terms for air in this correlation areevaluated at the arithmetic mean of the surfaceand room temperatures using expressionsapplied by Nyahoro (1992).

Radiation heat loss from the outside surfaceis dealt with by considering the surroundings to

be at 20°C. At the beginning of a new time step,the emissive power is calculatetemperature of the previous time step. Thematrix method (Incropera and Dewitt, 1990) isused to analyze ra iation exchange at thesurface.

he time step was initially set according toility condition based on the grid size

and boundary condition. The time step was

then dynamically reduced at the time of thetransition to boiling as describe d below . At thetime of transition from convection in the pot toa mechanism of boiling, there is quite a changein the heat balance on the elements that ma keup the pot wall The thermal resistance associ-ated with boiling is only a fraction of the thermalresistance assoc iated with natural con vection.

ecause eqns I) and (3 ) are different in struc-ture, there is not a logical mechanism for devel-oping a smooth transition from one to the other.If the time step is too large it can lead to asituation whic h is unstable. The description ofthe procedure employed will explain the prob-lem and the solution. After each time step inthe convection mode, the area-weighted averagetemperature of the pot bottom in the heatingzone is calculated and compared to 1055°C. Ifthe temperature has risen above 105.5” C, indi-cating the initiation of boiling, then the nexttime step is calculated according to the boilingequation, It is quite possible that the temper-ature at the pot bottom may decrease because

of the increased heat loss to the water. There isalso a time delay before there is a change inheat conducted to the pot from storage becauseof the explicit formulation of the numericalequations. If the temperature falls below 1 05.5” C

then it either means that th ere is a reversion toconvection or that the time step was too large.In the case that the tern~~rat~~~e goes from beingabove lO5.5”C to being elow 1055°C thecalculation is redone, but with a short er time

step. Should the time step be less than a presetrn~~~rn~rn limit and the ~~rn~erat~re is still lessthan IO§.YC then it is assumed that t~a~sit~~~did not take place and the calculation is replacedby a shorl Bime see ) cenvection c;alculation.CXherwise hbe calculation will prog ress with theassumption of boiling heat transfer. T he presetminimum time step for these ~ irn~lat~~~s is 6 s.

The procedure for transconvection is similar but

than 105 5°C for the present ~~~~~~~ timestep, then it is assumed Ihat the -mode willchange to convection on the subsequent tim~estep.

Shown in Fig. 2 is t

Insulation surrounds the solid to form an inf below the solid an

radiation input area OF zone.

4.9. Storage charging phase

Starting with the pot cavity filled with aof insulation (no pot), concentrated solartion at a constant rate of ar strikes the so

ough the open focal cavity fortime of charge. ~adiatio~ is

eat by a coating of solar absotance z f and thermal e~missiv~t~ Ed at theheating zone. At the end of the charging phase,the input cavity is sealed with a block of insula-tion. For simulations reported here the storagecharge time is fixed at 140 min.

4.2. Pot heating phase storage discharge)

After the storage char base is compl

ed from thereplaced with a metallic pot of co

water of diameter D, and he ight HP . An insu-r the pot. The pot isthe water either boils


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Table 1. Nominal dimensions and constant properties assumed for the storage materials and for the 50 1 pot

Cast ironGranitePotInsulation

Input zoneInput cavityPot cavityExternal surface

Specificheat

(J,$k)

420820480

loo0

Nominal NominalDensity Conductivity radius height Thickness

(kg?m’) (W&Q

Absorptance Emissivitylx E (Z (i) (5

1212 52 0.346 0.12640 3 0.346 0.18055 15.2 0.22 0.32 0.0025

160 0.1 0.060.95 0.100.10 0.05

0.95 0.2225 0.32250.57

or reaches a maxim um temperature below thatof boiling.

5. SIMULATED CASES AND RESULTS

5.1. Scope of simulations

Simulations are carried out for granite andcast iron storage blocks of equal size. For oneset of three simulations, the shape of the storageblock w as varied by changing the height todiameter ratio while maintaining the same mas sof storage. For another set of four simulations,the focal area on a nominally sized storageblock was varied while the total rate of heat

input remained the same. The properties andsizes of the components are presented in Table 1.The following parameters were fixed for allsimulations: 50 1 pot size, input radiation rateof 7 kW (estimated from two solar concentratorsaround midday in Nairobi), initial start-up tem-perature of 20°C storage charge time of 140 min,mesh size of 0.02 m, and tlrermal contact con-ductance of 3.600 W /(m”.C) (Incropera andDewitt, 1990).

5.2. ypical time history and energy inventory

Typical time histories for the temperaturewithin the nominally sized storage block, at anode located on the solid side of the solid-potinterface, are shown in Fig. 3. The d iameter ofthe focal zone is 0.46 m. At the 140 min mark,

Fig. 3. Time history of the pot-storage interface temper-ature for a cast iron (CI) storage unit and for a granite (Gr)storage unit. Charging occurs for the first 140 min, followed

by discharging for the remaining time.

the solar heating is cut off and the pot is placedon the storage block. The maximum temper-atures are 570°C for granite and 43 0°C for

cast iron.The temperatures from 140 min to the end ofthe simulation represent the time to bring thewater to boil. From Fig. 3 it can be seen that ittakes 56.0 min for granite and 5.5 min for castiron. The temperature values do not representthe average storage temperature. They representthe local temperature on the storage side of thestorage-pot interface. As can be seen fromthe times, the heating rate of the pot in the castiron case is an order of magnitude faster thanthat of the granite (heat added to the pot is thesame in each case). The higher conductivity ofthe cast iron allows free flow of heat fromthroughout the storage to the storage-pot inter-face. The difference of temperature valuesbetween the cast iron and granite at the stor-age-pot interface is enough to produce thedifference between natural convection (low rateof heat transfer for granite) and boiling heattransfer (high rate of heat transfer for cast iron)over a significant part of the water heatingphase.

Shown in Fig. 4 is a bar ch art which givessome perspective on what hap pens to the radia-tion incident on the storage block. For the140 min ch arge at 7.128 kW , the total incident

Cast IrOn GK M tf?

Fig. 4. Fractions of the 60 MJ energy input that are lost,left in storage, or used to heat water for two cases using

heat storage in either cast iron or granite.


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16 P. K. Nyahoro et al

energy is 60 MJ . f that, 16.8 NlJ goes intoheating the 5 0 1 of water in the pot from 20°Cto lW C. Part of the balance of 43.2 M J is lostto the surroundings, and part is left as storedenergy in the storage block. The figure shows

that for granite the proportion that is lost ismuch higher than that for cast iron because ofthe longer time to heat the water antemperature in the focal zone. The 29 .4 M J ofenergy remaining in the cast iron storage at theend of the simulation suggests that it is at anaverage temperature of 300 and that thereshould be an alternative corn ation of storagesize, pot size, and heating schedule to betterutilize the energy.

5.3. Ef lect o f s to ra ge b lock shape on per fo r m anceThe values in Table 2 reflect the results of

simulations for three height to diameter ratiosof the storage block while maintaining the sameblock volume. For these simulations there wasno selective coating in the focal zone and CQ=er= 0.95. The focal diameter is 0.46 m. The timeto heat the water from 20°C to a final temper-ature and the heat fraction lost both decreasewith mcreasing height to diameter ratio for thecast iron, sugges ting that H/D =0.2 is better

than a ratio less than 0.2. The granite datasuggests that at valuesof H/D ~0.2, it is doubtfulthat the water ever reaches the target 108°C forthe nonselective coating in the focal area.

5.4. E ec t s o ffoca l zone a rea on per fo r m ance

The s ize of the input zone area w ill determinethe concen tration ratio of the solar collector

and: therefore, the cost and complexity of theconcentrator. The values presented in Table 5

these simulations ere is a selective coating in

mum tern~e~at~~ ~ seen in the focal zone forgranite. W ith a small foca? area j&=% 15 m),the temperature is as high as l35O ” C. This value

nreasonably high for any selective coat@,produces high thermal stresses jn the gran-

ite. At a focal radius of 0.25 rn~ the temperatureis much reduced, but still very high at 78E”C.The cast iron show s much less variation in

gemperature with focal radius because the heatcan be electively ~onducte away from the focalzone. The cast iron shows little variation inabsorption efhciency with changes in focal size,suggesting that there is an even tradeoffbetween the additional area fm beat loss andthe reduced focal temperature. The focal area isimportant because it irectly relates to thesophistication (and therefore cost) of the solarconcentrator needed to deliver the solar radia-tion to the cooker. A larger focal area mea ns a

less sophisticated concentration system.

mulation of a focusing type solari thermal storage remonstrates d

characteristics for granite and cast iron as thestorage material. Cast iron is shown to have

Table 2. Simulation results for cooker with granite or cast iron for three storage shapes

Granite Granite GraniteCastII on

(thin)

Castiron

(medium)

Castiron

(thick)

Height, H, (m) 0.06Diameter, D, (m) 0.892Ratio, NJD, 0.07Time to heat water, t (min) ? 70Final temperature (“C) 77Neat fraction lost 0.70Heat fraction left 0.10Heat fraction in water 0.20

0.1 0.126 0.06 0.1 0.1260.692 ii.616 0.892 0.692 0.6160.15 0.20 0.07 0.15 0.20

295 253 15.0 7.2 6.893 100 .# IO@ 100

0.62 0.58 0.30 0.24 0.230.12 0.14 0.42 0.48 0.490.26 0.28 0.28 0.28 0.28

Table 3. Overall simulation results for cooker with granite or cast iron For four focal radii

Granite Granite Granite Granite

ast

Iron

Cast

II OX2

Cast

IiQn

Cast

h on

Focal radius, Rr (m) 0.15 0.19 0.23 0.25 O.i5 0.19 0.23 0.25Efficiency gab-s 0.55 0.63 0.68 0.70 0.87 0.86 0.86 0.86Time to heat water, t (min) 66.3 56.0 56.2 59. 5.0 5.2 5.5 5.6Focal temperature. Z f”C) 1350 1053 890 781 555 538 525 522


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shorter cooking times and less heat loss thangranite. W ith cast iron m uch of the heat rem ainsunused in storage by the end of the simulation,but is still available to sustain boiling thereafter.The results indicate that the height of the storageblock should be at least one-fifth of the diameterof the block.

For the same total heat input, the temperatureat the heating zone will decrease as the area ofthe focal zone increases. This result is importantbecause an increase in the focal zone areaimplies a reduction of the cost and complexityof the concentrator that results from a lesscritical concentration ratio. It is also an impor-tant design parameter that may be selected inorder to limit temperature, limit thermal stress,

or allow for the use of an absorber coating.Overall changes in performance, because of

different focal radii and shape, are all relativelysmall. Because the maxim um temperatures forthe granite are beyond the use of normalabsorber coatings, and the time taken to heatstorage and cook the meal are lengthy, thecooker with storage h as limited appeal in themode simulated here, but short-term storagecould serve in practice to store incoming energyin mom ents when the cooker is not in use, or

to dampe n temperature fluctuations due tochanging w eather conditions. W hereas the simu-lations done here reflect the ability to boil water,a cooker with solid storage m ay be better suitedto a baking situation, where the intention is toextend the baking period and not necessarily totime shift the cooking operation.

CPD

H

h

h f pk

N u

Pr

4I,

R4a

NOMENCLATURE

specific heat (J/kg.K)diameter (m)height (m)heat transfer coefficient W/m’.K)latent heat of vaporization (J/kg)thermal conductivity W/m.K)Nusselt numberPrandtl numberrate of heat transfer (W)rate of heat transfer per unit area (W/m’)Rayleigh number

temperature (“C)time (min)thermal absorptancethermal coefficient of expansion (l/K)thermal emissivitystorage efficiency: fraction of heat that approachesthe absorber that ends up in storagedynamic viscosity (N.s/m2)kinematic viscosity (m’/s)density ( kg/m3)surface tension at the liquid-vapor interface

Subscriptsa airb boilingf focal area (or input zone area)i insulation1 liquid (water)s surface

REFERENCES

Duffie J. A. and Beckman W. A. (1991) Solar Engin eeri ngof Thermal Processes, 2nd Edn. Wiley Interscience,New York.

Evans L. B. and Stefany N. E. (1966) An experimental studyof transient heat transfer to liquids in cylindrical enclo-sures. Chemical Engineering Progress, Symp . Seri es, 64,209-215.

Grupp M. (1990) Solar cooking: lessons from the past, hopefor the future. Proc. Fir st W orl d Renew able Energy Con-gress, Sayigh (Ed.). Pergamon Press, Oxford, Vol. 2,pp. 1325-1327.

Incropera F. P. and Dewitt D. P. (1990) In t roduc t ion toHeat Tr ansfer, 2nd Edn. John Wiley, New York.

Kuhnke K. (1987) Solar cooking: obstacles and opportuni-ties. Gat e: Questio ns, Answ ers, Infor mat ion, 1 i&14.

Nandwani S. S. (1988) Experimental and theoretical analysisof a simple solar oven in the climate of Costa Rica-I.Solar and Wi nd Technol ogy, 5 2), 159-170.

Nvahoro P. K. (1992) The effects of thermal, ootical and* geometric paiametkrs on the performance bf &I institu-

tional focusing solar cooker. M.Sc. Thesis, KenyattaUniversity.

Otieno H. and Scheffler W. (1989). Draw ings for a SolarCooker. Appropriate Technology Center, KenyattaUniversity.

Oehler U. and Scheffler W. (1994) The use of indigenousmaterials for solar conversion. Solar Energy M aterialsand Sol ar Cell s, 33, 379.

Vahon R. I., Nix G. H. and Tanger G. E. (1968) Evaluationof constants for the Rohsenhow pool boiling correlation.J. Heat Tra nsfer, 9ll C, 239-247.

Yaws C. L. (1977) Physi cal Propert ies: A Guide to he Physi-cal , Thermodynamic and Transport Property Dat a of ndu-

str i al l y Import ant Chemical Compounds. McGraw-Hill,New York.

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2007 p. Kariuki Nyahoro