Flow Distribution in Unglazed Transpired Plate Solar Air Heaters of Large Area.

7/29/2019 Flow Distribution in Unglazed Transpired Plate Solar Air Heaters of Large Area.

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Pergamon Solar Energy Vol. 58, No. 4-6, pp. 221-237, 1996PII: SOO38-092X96)00083-7 Copyright 0 1996 Elsevier Science LtdPrinted in Great Britain. All rights reserved0038-092X/96 $15.00+0.00

FLOW DISTRIBUTION IN UNGLA ZED TRANSPIRED PLATE SOLARAIR HEATERS OF LARGE AREA

L. H. GUNNE WIEK,? E. BRUNDR ETT and K. G. T. HOLLANDSDepartment of Mechanical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1,

(Communicated by Doug Hittle)Abstract-Unglazed transpired plate solar air heaters have proven to be effective in heating outside airon a once-through basis for ventilation and drying applications. Outside air is sucked through unglazedplates having uniformly distributed perforations. The air is drawn into a plenum behind the plate andthen supplied to the application by fans. Large collectors have been built that cover the sides of sizeablebuildings, and the problem of designing the system so that the air is sucked uniformly everywhere (ornearly so) has proven to be a challenging one. This article describes an analytical tool that has beendeveloped to predict the flow distribution over the collector. It is based on modelling the flow-field in theplenum by means of a commercial CFD (computational fluid mechanics) code, incorporating a specialset of boundary conditions to model the plate and the ambient air. This article presents the 2D versionof the code, and applies it to the problem of predicting the flow distribution in still air (no wind) conditions,a situation well treated by a 2D code. Results are presented for a wide range of conditions, and designimplications are discussed. An interesting finding of the study is that the heat transfer at the back of theplate can play an important role, and because of this heat transfer, the efficiency of a collector in non-uniform flow can can actually be greater than that of the same collector in uniform flow. Copyright 01996 Elsevier Science Ltd.

1. INTRODUCTIONDaytime heating of fresh air for industrial workspaces and for crop drying are two importantapplications where a non-storage solar systemcan be beneficially used to heat ambient airdirectly on a once-through basis. Since the late1980s a promising solar air heating system ha sbeen developed and commercialized for just thistype of application (Fig. 1). The systems collec-tor-called an unglazed transpired plate, orUTP, collector-consists of a perforated platebuilt onto a southward facing wall. Using tradi-tional building methods, the plates are fastenedto the wall with standoff hardware, therebyleaving a plenum behind the plates. The plenumis connected to a fan which provides suctionpressure and also positive pressure to deliverthe heated air to the load, e.g. to the factoryfloor, v ia a manifold system. Tests conductedon a prototype installation indicated that suchan unglazed collector would b e practical(Carpenter and Kokko, 1991). Since then, largescale systems totalling thousands of squaremetres have been installed and successfully oper-ated for fresh-air heating in Cana da, the UnitedStates, and Germany (Hollick, 1994).

The essence of the design is the use of con-TPresently at Hatch Associates Ltd., 2800 Speakman Drive,

Sheridan Science and Technology Park, Mississauga,Ontario, L5K 2R7, Canada.

struction-grade technologies to produce a lowcost, robust solar heating system. However, thecollector is also based on sound heat transferprinciples (Kutscher et al., 1993). It is efficientbecause the convective heat loss to the air infront of the plates is later recaptured when theboundary layer is sucked into the plenum, and

PERFORATEDAESO R BERPL AT E

AMBIENT AIR

PLENUM

Fig. 1. Drawing of unglazed transpired plate (UTP) collec-tor installed on a vertical wall.

22 7


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228 L. H. Gunnewiek et al.

the intimate heat transfer to this boundary layerkeeps the plate temperature low, thereby reduc-ing radiant heat loss to an acceptable level.

While models for the local heat transfer atthe plate have been developed in some detail(Kutscher, 1994; Cao et al., 1993; Golneshan,1995), modelling the overall performance hasreceived less attention. The main item determin-ing this overall performance is the uniformityof the flow across the whole collector face. It isknown that poor distribution can cause localhot spots, w here the radiative losses will domi-nate (Kokko and Marshal, 1992). Indeed, onsome parts of the collector, the flow may actu-ally be out of the plate, instead of in (this iscalled flow reversal), meaning that not only isthe solar energy absorbed on that part of theplate totally lost, but some of the energy alreadycollected in other parts of the plate is also lost.Dymond and Kutscher (1995) recently analyzedflow distribution using a pipe network model.Wh ile capable of fast simulation on a PC, sucha model is of uncertain accuracy, as it replacesthe actual plenum by a system of short lengthsof pipes. More accurate would be a model thatsolves the fundame ntal equations of conserva-tion of mas s, mom entum, and energy, througha computational fluid dynamics (C FD) code.Wh ile unrealistically slow for design work, suchan approach would provide fundamental in-sights and a valuable research tool that couldbe used, for example, for calibrating the pipenetwork model.

The present article describes the first step inthe development of a suitable C FD model. Thecomplete model is a 3D one, capable of model-ling the whole plenum flow, including the effectsof the wind (Gunnewiek, 1994). We present inthis article a description of only the 2D version,which can account for variations in the plenumflow only in the vertical direction. This versionis ideal for stud ying the effect of buoyancy,which is a major cause of flow maldistribution.

2. DEVELOPMENT OF THE CFD MODEL2.1. Ove rv i ew

The basic CFD code (TASCflow Version 2.2)used was provided by Advanced ScientificCom puting Ltd. TASCflow uses the finitevolume procedure and follows in broad termsthe methodology laid out by Patankar (1980)whereby the equations prescribing conservationof mass , mom entum, and energy are discretizedand solved in such as way as to be fully conser-

vative of these quantities. A 3D code, it can beeasily adapted to 2D problems, as in the presentinstance, and provides a full range of supportfeatures. The code provides for a number ofplatforms including the one used for this study,which was a Sun SPARC w orkstation computerwith 48 Mb of memory.

Mod elling a full-scale UTP collector mountedon a wall (Figs 1 and 2) required some sim plifi-cations in order to fit the model g rid within thelimits of the computer. The approach used wasto include only the p lenum region (i.e. the H byD by W region in Fig. 2) in the com putationaldomain of the code, and to model the plate andthe ambient surrounds by means of a pair ofspecial boundary conditions applied at thedomains front-face, i.e. at the boundary at x=t. The first of these front-face bound ary condi-tions was essentially an energy balance on theplate that included the absorbed solar radiation,the radiation to the surrounds, and a model forthe plate-to-air heat transfer rate. The secondwas essentially a mom entum balance that incor-porated a model for the hydraulic resistance ofthe plate and for the pressure distribution atthe exterior face of the plate. The two boundaryconditions were not standard TASCflow bound-ary conditions, but were specially coded forthe present task by Advanced ScientificComputing Ltd.

The transpiration of air through the platewas modelled as a continuous phenomenarather than a s a process occurring throughdiscrete holes. Without this simplification theproblem would no t have been solvable in anyreasonable time-the work of Cao et a l . (1993)having shown that modelling the flow around

plenum exitplenumextensionabsorberplatefrontfoceat plenum

back-well

plenum

Fig. 2. Dimensioned sketch of the UTPs plenum.


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Flow distribution in unglazed transpired plate solar air heaters of large area 229

even a single hole required several hours ofcomputer time and a highly refined grid. Thesame study showed that the velocity profiles aresmooth within a short distance of the holes sothe errors associated with this simplifyingassump tion should be modest.2.2. F ron t - f a ce bounda ry - cond i t i o ns

M omen tum ba l ance. Literature models for thelocal pressure drop across a perforated plate,AP,, as a function of the approach velocity, us,were examined in detail (Gunnewiek, 1994). Tomake the results of the present study ap plicableto the largest range of plate porosities, platethicknesses, and hole shapes and diameters pos-sible, it was decided to model only the paramet-ric form of this function, the values of the actualparameters being evaluated later for the particu-lar plate at hand. A suitable form wa s found tobe

AP, = pK,u; (1 )where n and K, are the parameters. This modelcaptures the well-known equations for AP, thatapply in the limiting cases of very high holeReynolds n umber (where n=2) and very lowhole Reynolds number (where n= 1). Fittingeqn (1) to the long-orifice plate correlation func-tion of Licharowicz e t a l . (1965), assumingtypical values of porosity, thickness, and holediameter, gave a value of n= 1.5 over a widerange of velocities of interest, so at least for thepurpose of the present study, this value of n of1.5 was used. For n = 1.5, the values of K, (calledthe plates hydraulic impedance) were found torange b etween 750 and 3750 (m/s)/. (In prac-tice, one can find K H for a particular plate byevaluating AP, from some suitable equation,such as that of Licharowicz e t a l . (1965) orKutscher (1994), using uave for us, and thensolving eqn ( 1) for KH . )

The front-face pressure w as obtained fromthe ambient hydrostatic pressure as follows:

P(O,z) = P,,,-P&Y (2)where the first term on the right hand side isthe ambient pressure at ground level. It is nowassume d that eqn (1) applies locally, so u, isnow the local face velocity at x = t: i.e. U,= u( t , z ) ,called the suction velocity, and also AP , is thelocal pressure difference P( 0 ,~) P( t , z ), whereP(O, z ) is given by eqn (2). Thus combining eqns(1) and (2) one obtains a relation between thex-velocity and the pressure at x= t, which isthe mom entum balance front-face boun dary-

condition. The non-slip boundary conditionu( t , z )=O is also assumed to apply at the front-face.

Energy balance. The plate boundary layeris assum ed to be in the asymptotic region(Kutscher, 1991) over the whole collector, solong-wave radiation is assum ed to be the onlyheat loss. An energy balance? on the plate gives

c tG =pC,u ( t , z ) (i ( t , z ) - T , )+hD p (z )-To )+ (q ,M ) (3 )

where qb is the heat flow (assumed uniform) atthe back of the plate, which can occur byconvection to the air in the plenum. The platetemperature 7(z) can be expressed in termsof the plate heat exchange effectiveness cH X(Kutscher, 1994; Cao, 1993), which is definedby

E x=(T(t,Z)--T,)/(Tp(Z)-T,) (4)and w hich is a measure of the intimacy ofheat transfer between the air and the plate.Com bining eqns (3) and (4) gives the energy-balance front-face boundary-condition:

T&z)= T,+(~G),l(PC,a(t,z)+ LJ,) (5)where U, is h r / eHX , and (aG), is the net of theabsorbed solar radiation and the back-of-the-plate heat transfer: (UC),= UC-q , / A . Both h ,and eHX are relatively insensitive to suctionvelocity (as well to other system variables, suchas solar radiation, ambient temperature, etc.),so we will treat U, as a constant for a givensimulation, and as a parameter of the totalstudy. Also, the qb could not be established apriori, so it was determined a posteriori, as willbe explained later. W e consider (NC), to be oneof the parameters of the problem.

(It should be noted that there is a slightinconsistency in the derivation of this energy-balance boundary-condition, in that the platetemperature has been made different from thetemperature of the air emerging from the holes.This approach will be strictly true only in thelimit of enx = unity, i.e. for continuous suction.)2.3. Fur t her aspec ts o f th e p lenum m ode l

The fan downstream of the plenum wasassume d to produce a prescribed total flowQ m3/s, to have a uniform velocity profile at itsinlet, and to be connected to the plenum bytEquation (3) assumes the surrounds temperature T. is equal

to ambient air temperature T,. If this is not the case,h,(T,- 7) should be added to (stG).


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means of a straight, slip-wall, adiabatic exten-sion of height h , as shown by hatched lines inFig. 2. The computational domain was extendedto include this extension. It was found that forh>O.llH, the flow in the plenum was indepen-dent of the actual value of h chosen, and allsubsequent simulations were made to satisfythis condition on h . Indeed, none of theseassump tions about the nature of the itemsdownstream of the plenum would be expectedto significantly effect the plate flow uniformity,which w as the quantity of interest.

In addition to the boundary conditionsdescribed so far, others must be prescribed: Theplane x = D + t represents the wall on which thecollector is mounted, and this is expected to bewell-insulated, so this face was modelled asadiabatic and, of course, as non-slip. The sameis true of the floor plane, z=O. This com pletesthe boundary-condition specification.

TASCflow requires the user to specify eitherlaminar or turbulent flow, and if turbulent, aturbulence model m ust be chosen. The plenumReynolds numbe r based upon the hydraulicdiameter and average velocity varied from zeroat the bottom of the plenum to a maxim um atthe top. An examination of expected Reynoldsnumbe rs indicated that the majority of theplenum flow would be turbulent for all cases,based upon classical smooth pipe observations.Further, the jetting action at the suction holesin the plate would probably induce early trans-ition. Thus a turbulence model w as indicated,and the TASCflow k - - E model was chosen.?2.4. Paramet r i c range s tud ied

A suitable range of parameters to be studiedwas determined by examining the current tech-nology and possible future extensions to it. Thecollector height rang e w as 3.0 IHI 6.0 m, andthe plenum aspect ratio range w as 10 5 H /D I 5 0.The range of U, was 1.0 2 U, 19.0 W/mK.(This represents the range from a non-selectivesurface plate with low heat exchanger effec-tiveness with high losses for U, =9.0, to anabsorber with a selective surface a nd a highheat exchange effectiveness for U, = 1.0.) Therange of Q (given in terms of the average suctionvelocity, u,,, = Q/A) was 0.005 I Q / A I 0.08 m/s.tit was pointed out during the review of this article that

certain studies [e.g. Kays and Crawford (1980) andKuroda and Nishioka (1989)] would indicate that thelaminar assumption may have been a better one, becausethe flow in the plenum is accelerating. However, noprevious study, to our knowledge, relates to this exactproblem. More work is required to clarify this point.

The plate hydraulic impedance range was7501K, I 3750 (m /s).. Finally, the netabsorbed solar irradiance range was 4001(crG), I900 W/m.2.5. Gr id genera t i on and ref i n ementThe first step in setting up the code is togenerate the grid planes on which the nodalpoints (the points at the centres of the controlvolumes) are located. For convenience, thegrid planes were made parallel to the threeco-ordinate planes, so there were x-grid planes(parallel to the yz-plane), y-grid planes (parallelto the xz-plane) and z-grid planes (parallel tothe xy-plane). The nodal points lie at the inter-sections of these three sets of planes. To makethe 3D TASCflow code simulate the 2D prob-lem, three y-grid planes were constructed: aty = 0, at y = W/2, and at y = W , and then symme-try boundary conditions are prescribed at they=O and the y= W planes.

Part of the art of placing the grid planesrelates to the turbulence model. Turbulencemodelling codes require all the nodal points tobe in the fully turbulent region. F or the pointsnear the wall(s) this requires that they be in thelogarithmic part of the wall boundary layer, acondition normally satisfied only if the distanced of the first node away from the wall is in therange 55v j k 2 Id s900v /k 1~2 , where k is theturbulent kinetic energy (solved for in the code)and v is the kinematic viscosity. The largest dsatisfying this condition (typically to be about0.15D) was found to occur at the bottom of theplenum, where k is smallest. Thus the distancefrom the plate to the first x-grid plane awayfrom the plate and the distance from the backwall to the first x-grid plane away from theback wall were fixed on the basis of turbulenceconsiderations, and grid generation then pro-ceeded for the remainder.

It was determined that, for speedier con-vergence, the spacing between the remaining x-grid planes should not be constant but shouldincrease with x. This was done in such a waythat the ratio of adjacent spacings w as heldconstant. This expansion is characterizedthrough a TASC flow parameter R , which is theratio that the spacing between the first andsecond x-grid planes out from the rear wallmakes to the spacing between the first andsecond x-grid planes out from the plate. Keeping5 I R I 10 was found to concentrate sufficientflux elements near the front-face, where the flowis turning sharply, w hile reducing unnecessary


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computational labour near the back wall, with-out making the grid too coarse there. So, sub-sequent simulations kept R in this range. Thez-grid planes were kept uniformly spaced.

The process of discretizing the governingequations introduces an error in the solutionthat decreases as the total n umber of nodalpoints N in the domain increases. A gridrefinement study is normally carried out, inwhich a value of N large enough to ensureacceptable accuracy is determined. In the pre-sent study, we carried out the grid refinementstudy by comparing solutions obtained fromthree successively finer grids. The first grid,denoted Nl, ad 20 flux elements (or grid-planespacings) in the x-direction, 3 flux elements inthe y-direction (as required by TASCflow for2D simulations) and 100 flux elements in the z-direction; we write this as Nl = 20 3 x 100. Theremaining grids were N2 = 30 x 3 x 150, andN3 = 40 x 3 x 200. Extrapolation methods werethen used to estimate the solution asymptote asN-+ co, which is assum ed to be the true solution,and the error of each of the three grids w asthereby estim ated. This grid refinement studywas carried out for the following parametricsettings: H = 4.5 m; HID = 30; Q/A 0.02 m/s;KH = 2250 (m/s)2, (NC), = 650 W/m; and U, =5 W/m K. The results sh owed that simulationsobtained using the Nl grid should give answersfor the local suction velocity that are withinabout 2% of the true value, for the mean gaugepressure at the collector top that are within 7%of the true value, and for the useful energycollected that are within 3% of the true value.Full details are given by Gunnew iek (1994).Since this accuracy was considered to be justcomm ensurate with the current needs, and inview of the long simulation times required forthe other grid s, all subsequen t 2D simulationsreported in this article were carried out with theNl grid.

3. RESULTSThe combined range of the six controlling

parameters was covered by simulating certaincombinations of the following values: H =3.0,4.5, 6.0 m; (aG), =400, 650, 900 W/m; U, = 1.0,5.0, 9.0 W/m K; Q/A=0.005,.01, 0.02, 0.04,0.08 m/s; H/D = 10, 20, 30, 40, 50; and K, = 750,1500,225 0,3000,37 50 (m/s)12. Not all combina-tions were simulated. Rather a typical configu-ration was selected as one with the following mid-line settings: H=4 .5 m, (aG),=650

W/m2, U, = 5.0 W/m2 K, u,,, = 0.02 m/s, H/D =30, , = 2250m/s).. Then each of the param-eters was allowed to vary over its range, withthe rest held constant at the mid-line values.Simulations were computed at each of the result-ing combinations.With a few exceptions, Figs 3 and 4 plot thesimulated velocity profiles obtained for thesevarious parametric settings. The quantity plot-ted is the normalized suction velocity, U*,defined by U* =u(~,z)/u,,,. The exceptions arethose parametric com binations having Q/A=0.01 and 0.005 m/s. All of the simulations havingQ/A= 0.005nd mo st of those hav ing Q/A=0.01 were found to experience flow reversal nearthe top of the plate: that is, the flow there wasout of the plate rather than into the plate.Reverse flow through any part of the plate ishighly undesirable because none of the solarenergy incident on this region is collected, andsome of the energy that has already been col-lected over the rest of the plate is actually lostto the environment. Clearly flow reversal shouldbe avoided in practice. That it can occur forthese low values of Q/A is one of the moreimportant findings of this study. The lowest Q/Athat would not cause reverse flow over at leastsome of the range of the other param eters wasfound to be 0.0125 m/s, and so this setting wasadded to the list of settings simulated (seeFig. 4).

To physically interpret this reverse-flow result,as well as the profiles p lotted in Figs 3 and 4,it is useful to think of the flow as the superpo sit-ion of two flows: a buoyant flow driven by thetemperature differences developed by theabsorbed solar energy, and a forced flow devel-oped by the fan. Acting on its own (i.e. withQ =0) the buoyant flow would be in at thebottom of the collector and ou t at the top. InFigs 3 and 4, this flow is best illustrated by theprofile shown in Fig. 4 for Q/A= 0.0125. Incontrast, the forced flow acting on its ownwould produce a profile having less flow at thebottom and m ore at the top. In Figs 3 and 4this flow is best illustrated by the profile sho wnin Fig. 4 for Q/A=0.08. his shape of flowprofile is well known in manifold design (Bajuraand Jones, 1976) and is brought about by theplenum pressures tendency to decrease in thevertical direction because the flow is acceleratingin the plenum and because of friction along theplenum walls. Both acceleration and frictiontend to increase as the plenum is made narrower,so the forced flow profile shape is also clearly


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taGI.= 650 w/m2taGI,~900 W/m*

H=3.0mH=4.5mH =6.0m

/y ,+/D=40

Fig. 3. Normalized suction velocity profiles for various parametric settings. Unless otherwise indicated,all parameters are at their mid-line values.

demonstrated at high H /D settings, e.g. by theH/ D = 5 0 profile in Fig. 3.

KH= 1500tm/s)05

RELATIVE HEIGHT UP COLLECTOR, Z/H

The fact that most of the flows illustrated inFigs 3 and 4 have more flow at the bottom andless at the top demonstrates that the buoyantflow is playing a very important role over mostof the parametric range investigated. Buoyanteffects would be expected to be most pro-nounced when the height H is greatest, and alsowhen the net absorbed radiation (LYG),s great-est, and this expectation is borne out in theFigures. The Figures also show that buoyancyeffects tends to be reduced when Q/ A is large(as would be expected). The buoyant effect isalso seen to be reduced when the plates hydrau-lic resistance Kn is large or whe n the aspectratio H/ D is large. This is because there is alimit to the equivalent pressure difference that

buoyancy can develop (to overcome friction,either at the plenum walls or in the plate),whereas the forced flow, having fixed Q, is notrestricted in this way, and hence it tends todominate in these instances. The heat lossparameter U, has only a very modest effect onthe velocity profile; like the absorbed solarradiation, it has an effect (in this case somewhatmodest) on the inlet temperature, and thereforeon the buoyancy.The profile of the temp erature of the airentering the plenum, T= T(t,z), is also of inter-est. Wh en the velocity profile is completelyuniform, this temperature, now denoted T,,, isalso un iform, and is given by eqn (5) withU(U) = n,,, . We can form a normalized inlettemperature 0 as

f3= (VU) - K )/Vi,, - TCO) (6)


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+ u,,, = 0.0125 m/s+ u,, = 0.02 m/s+ u,, = 0.04 m/s-c U,, = 0.08 m/s

Relative height up collector, z/HFig. 4. Normalized suction velocity and temperature profiles for various settings of the average velocity

u,,,, = Q/A. The other parametric settings are at their mid-line values.

and consider profiles of 8. The bottom part ofFig. 4 shows the &profile at various values ofQ/A, as determined from the simulations. Theshape of this profile is seen to be almost amirror image (in the x-axis) of the velocityprofile. This fact can be deduced from an energybalance on the plate and the definition of theplate effectiveness. Thus the normalized temper-ature profile is roughly the inverse of the velocityprofile. Gunnewiek (1994) has presented a fullset of graphs of 13 or the situations covered inFig (3).

(We can now examine the assumption thatu, = hrleux is roughly constant over the heightof the collector. The largest variation of U, willoccur in the case of u,,,=O.O 125 m/s in Fig. 4.In this case the suction velocity u(t,z) varies bya factor of two. Examining the data of Kutscher(1992), we find that a doubling of suction veloc-

ity typically causes a 17% change in eux; thismeans that ~ux will vary over the collector byonly +8.5% from its average value. Also, ele-mentary calculations show that the temperaturevariation exhibited in the bottom of the graphon Fig. 4 at u,,, =0.0125 would cause a +8.5%variation in h, . Moreover these variations arein phase, so h, and cnx both take on slightlylarger v alues at the top of the collector andslightly smaller values at the bottom, so theirratio U, will be even more co nstant. In view ofthis, and in view of the fact that U, h as only amodest effect in the flow distribution, it is con-cluded that the assump tion of constant U,should not introduce significant errors.)

The pumping pressure that the fan mustdevelop (at least the part associated with thecollector) isAP,,,(H)=P,-pgH--P,,,(H) (7)


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200--

175 -

150 -

H 4.5 m, = =G 650 W/m*, U, 5.0 W/m* K+ H/LI = 10, = 750 (m/~),~KH-.- H/LI = 10, K,, = 2250 (m/s). ,p+ H/D = 10, K, = 3750 (m/s). ,I-0- H/D = 30, K, = 750 (n-~/s).~ ,I

-o- H/LJ = 30, K, = 2250 (m/s).*I

-n- H/D = 30, K, = 3750 (m/~).~ /H/D = 50, K, = 750 (m/~).~H/D = 50, K, = 2250 (m/s).HID = 50, K, = 3750 (m/s).

25 -

0.02 0.04 0.06Average inlet suction velocity, uave [m/s]

0.08

Fig. 5. Pressure difference across the plate at the top, as a function of the flow rate, for various parametricsettings.

where P,,,(H) is the mas s averaged pressure a tthe plenum exit, z=H. Figure 5 plots BP,,, vsQjA for various combinations of HID an d Q/A,the main two param eters tending to fix APO ,,.Since AP,,, is also the driving force for thesuction flow at the top of the plate, negativevalues of AP,,, indicate flow reversal-a situa-tion that must be avoided. However, for AP,,,greater than but approximately equal to zero,there is no flow reversal and, at the same time,fan work is minimized.

The relevant measure of thermal performanceis the useful additional energy, 4, carried by theexiting air, and th is can be measured at twostations: at the plane x = t, where the air is justleaving the plate, and at the top exit plane z=H. We denote these energy rates by qin and q,,ut,respectively (see Nomen clature). (The one ofultimate interest is qout, but we will find it usefulto consider both.) Energy rates qin and qoutdifferby the amount of heat transfer qb taking placeat the back of the plate. Thus q,, can be foundfrom qb=qout-%n* It can also be found byintegrating - kdT/dx at x = t from z =O to z =H, the fact that the two m easures of qb gave

exactly the same answer acted as a check onthe integrity of the CFD code, which is pro-gramm ed to satisfy a global energy balanceregardless of the coarseness of the grid.

Figure 6 plots qb/&,, as a function of thequantity it was found to most uniquely depen-dent on: namely the ratio AP,,,JAP,,. Thenumerator in this latter ratio is the quantitydefined by eqn (7) and the denominator AP,,is the pressure drop across the plate when thesuction is uniform: i.e. APo= ~KH (~ave)1.5. It isclear from Fig. 6 that qb can represent a substan-tial part of the heat transferred to the air. It isalso to be noted that qb can be either positiveor negative; positive qb occurs when the plateheats the air as it rises up the plenum, andnegative qb occurs when the plate cools the air.Wh en the flow is completely uniform (whichwould correspond to AP,,,fAP,, being equalto unity, since AP, has to be uniform foruniform flow), qb is zero; this is because theplenum air is all at uniform temperature, andat the same temperature as the plate. However,when u(t,z) is varying, the air in the plenum atany h eight z will in general be different from


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0.6I

-0.6 1 I \I0.1 23 5 1.0 2 3 5 10.0APoutA PPo

Fig. 6. A plot showing the contribution of the heat transfer at the back of the plate, as given by qb/qin,as a correlated function of the pressure difference across the plate at the top divided by the nominal

pressure difference for uniform flow.

the plate temperature, and thus there is anopportunity for back heat transfer. When forced-flow dominates, there is greater suctionat the top of the collector than at the bottomso the air enters the bottom hotter than tha tentering at the top. This hot air tends to becooled by the cooler plate near the top. Thusqb is negative w hen forced-flow dominates. Thereverse is true when buoyant flow dominates,where qb is positive. The pressure rateAP,,,/AP,, is an excellent measure of whetherbuoyant flow or forced flow is dominan t (andby how much) and this explains the strongcorrelation shown in Fig. 6.There are two effects of non-uniform flow onthe thermal performance of a collector. The firstis its effect on the heat transfer on the back ofthe plate, as just discussed. The second is thatit may change the fraction of the net absorbedenergy (crG),,A hat is transferred to the air as itpasses through the plate. This effect is exempli-fied in Fig. 7, which plots rin/~uni where ?in isqr,a/(aG),A and qunr, the collector efficiencywhen the flow is uniform, is given by runi=a/( 1 + UJ~C,,U,,,). The quantity qin/uuni is seento be essentially unity for all the simulations.

The quantity on the horizontal axis in Fig. 7 isthe coefficient of variation c, of the suctionvelocity profile-the quantity that was found tomost closely correlate with Y]in/quni:

cv= 1(U*-l)dz*I (8)0Figure 7 show s that the non-uniform flow hasonly very small direct effect on the energycollected by the plate.

The main quantity of practical interest isYl/r/,i, where I], defined by g =q.,,J(GA )=aq,,,/(aGA), is the true collector efficiency-i.e.fraction of incident solar energy that is transfer-red to the fluid. The ratio ~/)l,ni characterizeshow this overall efficiency differs from what itwould be if the flow were uniform. The aG inthe denominator of the definition for 1 is theactual aG, and not (crG),, but one can get thisaG from the equation

aG = (aG), 1+ (qb/qin) E* 7 1 (9)mwhich is readily derivable from the variousdefinitions th at have been introduced up to


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mI,.-E

F 0.990\

0.985

0.980 G0.0 0.1 0.2 0.3 0.4 0.5 0.6COEFFICIENT OF VARIATION, cV

Fig. 7. A plot showing that the quantity ~~~~~~~~s essentially unity over all the simulations carried outwith u,,~ 20.0125 m/s.

now. Substituting eqn (9) for aG and qin+ qb forqout in the definition for Q+, ne ob tains, aftersome re-arrangement, the desired expressionsfor Vl yluni :

? B(l+M)-=Vuni l+ MB(VuniIa) (10)

where M = qb/qin (from Fig. 6), B = ~in/Y ],,i fromFig. 7) and I?,,i/C1= l/( 1 + Uo/(pcpu,,,)). AS isclear from Fig. 6, B is essentially unity over therange of parameters tested. A typical value foryI,,i/cI is 0 .75, SO typically Y//quni an range from0.8 to 1.09 as M goes from -0.5 to +OS, whichis the approximate range for M in Fig. 6.

This mean s that non-uniform flow can actu-ally increas e the collector efficiency, but toachieve this gain in practice, one must arrangefor the flow to be greater over the bottom partof the collector than over the top part; i.e. onemust ensure that the flow is buoyancy-dominated. Interestingly, this is also the condi-tion that minimizes the fan work so it has adouble benefit. On the other hand, this condi-tion is also close to the situation where reverseflow would occur (reverse flow occurs whenAP,,JAP ,, in Fig. 6 goes negative).

4. CONCLUSIONSThis article has described a commercial CFD

code that was specially adapted to examineunglazed transpired plate solar collectors oper-ating und er the influence of fan suction pressureand buoyancy. Special emp hasis h as been placedupon modelling the flow in the plenum locatedbehind the transpired plate, to determine theimportance of the six controlling parameters on

the suction velocity profile. The results show ahighly non-uniform profile for some of the set-tings of the parameters, and relatively uniformprofiles for other settings. The nature of theprofile depends on whether the flow is domi-nated by buoyancy or by the forced-flow pro-duced by the fan. Physical reasoning is used toexplain the various velocity profiles observedand the qualitative effect of each of the parame-ters. For Q/A less than 0.0125 m/s, reverse flowwas observed, so this or smaller flows shouldbe avoided in practice, when sizing the fan.Wh en the flow is non-uniform, the code pre-dicts that an important amount of heat transfercan take place from the back of the plate to theair rising up the plenum. This heat transfer caneither increase or decrease the collector effi-ciency, depending upon the polarity of theprofile-i.e. depending whether the suctionvelocity is higher at the bottom than at the top(in which case the efficiency is predicted to beincreased by the non-uniform flow), or viceversa (in which case the efficiency is predictedto decrease). The higher-at-the-bottom polar-ity is favoured in flows that are buoyancy-dominated. This means th at to get the increasedefficiency, the average suction velocity shouldbe small, the plenum relatively wide, and theplates hydraulic resistance relatively low. (SeeFigs 3 and 4 for a quantification of these param -eter settings.) Conditions under which the effi-ciency can be improved by non-uniform floware also those that keep the fan power at aminimu m, so there is a dual benefit. How ever,the conditions are also fairly close to thosecorresponding to reverse flow, so the design-er must take care in choosing the correctparameters.


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It should be noted that this analysis has beenbased on certain models for the problem (e.g.uniform qb/A, turbulent flow, constant proper-ties, etc.) and it should be checked againstexperimental measurements and other, improv-ed, formulations in the future.Acknowledgements-Financial support of this work wasprovided by the Natural Science and Engineering ResearchCouncil (Canada), through a Strategic Grant. We also thankAdvanced Scientific Computing enc., 554 Parkside Dr.,Waterloo, Ontario, Canada, N2L 3G1, for their assistance,and Professor Alfred Brunger for useful input.

AC,c,

qin40.1R

T(x,z)L

collector area, = WH, m2specific heat of air at constant pressure, J/kg Kcoefficient of variation of suction velocity profile(eqn (8))plenum depth (Fig. 2), mdistance of first node away from a wall, msolar irradiance on collector, W/m2gravitational acceleration, m/scollector height (Fig. 2) mheight of plenum extension (Fig. 2), mradiation heat loss coefficient from mate toambient, =e(aTi - p_)/(T,- T,), W/(mK)plates hydraulic impedance, (m/s)atmospheric pressure at ground (z = 0), Paaverage pressure at plenum outlet, PaPrer- pgH - P,,,, Palocal pressure drop across the plate, PaAP, when flow is uniformvolumetric flow rate drawn out of the top of theplenum by the fan, m3/si? C,&- L). WQ C&T,,,- T,). useful heat collector collected,Wgrid expansion factor in x-directionair temperature at location (x,z), Kaverage bulk temperature of air entering theplenum at x = t face, KT(Q) when u(t,z) = u,,,, Kaverage bulk temperature of air at plenum exit,K

Ir;,,TO tT,(z) local plate temperature, Ku:

thickness of plate, mh,lcnx, W/m K

rJ* normalized suction velocity, u(t,z)/u,,,u.w , = Q/.4 m/s

U(V) x-component of air velocity at (x,z), m/su, suction velocity, m/s

v(V) z-component of air velocity at (x,z), m/sW width of collector (Fig. 2), m

X,Y,Z coordinates (Fig. 2), mZ* zlH

Greek lett ersa plate solar absorptivity

(uG), uG-q,/A

NOMENCLATURE

c plate emissivitycuX plates heat exchange effectiveness (eqn (5))

r~ collector efficiency, = q&GA)rlin q,a/((aG),A)quni q for uniform flow, = a/( 1 + LTO/(pCpu,,,))

p density of air, kg/m30 normalized inlet temperature (eqn (6))

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Flow Distribution in Unglazed Transpired Plate Solar Air Heaters of Large Area.

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