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An approximate method for calculating the heat flux through a solar collector honeycomb
Solar Energy. Vol. 30, No. 2, pp. 127-131, 1983 Printed in Great Britain.
O038qO2X/83[020127-05503,00/O © 1983 Pergamon Press Ltd.
AN APPROXIMATE METHOD FOR CALCULATING THE HEAT FLUX THROUGH A SOLAR COLLECTOR HONEYCOMB STEVEN L. MARCUS Department of Atmospheric Sciences, University of California, Los Angeles, CA90024, U.S.A. (Received 21 October 1981; revision accepted 4 April 1982)
Abstract--An approximatecalculationof the radiative-conductivefluxof heat from the absorber to the cover plate of a fiat plate collector equipped with a convection-suppressinghoneycomb is developed, using an exponential kernel approximation to the passage transmittance function of the honeycomb cells. Fluxes predicted by this method are found to be in satisfactory agreement with the results of previously reported calculations and experiments. If a selective absorber plate is used in a collector of this type, heat transport by the air in the collector eliminates the radiation slipbetween the honeycomband the absorber plate, so that thermalemissionby the wallsof the honeycombin close proximityto the hot absorber may largelycancel out the advantagegainedby makingthe absorber plate selective. I. INTRODUCTION In a single-cover flat plate solar collector with a properly functioning honeycomb, convection is suppressed and heat flux from the absorber plate to the cover plate occurs mainly by radiation and conduction [l]. The standard method for computing this heat flux has been to calculate the fluxes separately for the radiative and conductive modes and then to add the results. McMurrin et al. [2], however, have noted that this method may give incorrect results when either the absorber or the cover plate has a low thermal emissivity. Marshall et al. [3], for example, used a vacuum apparatus to study the radiation emitted by a highly selective absorber plate in the absence of heat conduction by the air. They found that the thermal radiation reaching the cover plate was virtually unchanged when a honeycomb was placed above (but not in direct contact with) the absorber plate. Since the emissivity of the honeycomb material used was higher than that of the selective absorber plate, this result requires the honeycomb itself to be at a lower temperature than the absorber plate. In the presence of heat conduction by air, therefore, the result would no longer apply, since the temperature discontinuity (or "radiation slip") between the absorber plate and the honeycomb would be eliminated. Recently Hollands et a/.[4] have made a careful experimental and theoretical study of the radiative-conductive heat transport through a solar collector honeycomb. They showed that a separate mode analysis may seriously underestimate the heat flux when selective surfaces are used, and presented numerical and analytical methods for computing the combined mode heat flux. In this note a simplified analytical treatment for the combined mode flux is given, based on the passage transmittance function (PTF) introduced by Edwards and Tobin[5]. The mathematical basis of the method is given in Section 2, and the analytical expressions for the heat flux are given in the Appendix. The practical application of this method is limited, since it requires knowledge of the value of the PTF for the honeycomb configuration being used. Some values of the PTF for square tubes have been computed by Tobin[6] and are
reproduced in part here in Table 1. Calculations of the heat flux using this method, however, indicate that the total flux is not a very sensitive function of the PTF, so that for an approximate calculation it may be sufficient to use a value of the PTF interpolated from those given in Table 1. This point is discussed more fully in Section 3, where values of the heat flux computed by this method are compared with experimental data given in [4], and the method is further illustrated by consideration of the absorber-to-cover plate heat loss in a high-temperature flat plate collector of a type discussed by Symons and Gani[7]. 2. METHODOF CALCULATION The physical assumptions used in this study are essentially the same as those adopted in [4]. We consider initially an infinite honeycomb array between two parallel plates separated by a distance L, and choose one cell of the honeycomb to represent the entire array. The side walls of this representative cell are considered to be opaque and thermally insulated, with an effective reflectivity equal to the sum of the transmissivity and reflectivity of the actual honeycomb cell wall. All surfaces are considered gray, with side walls taken to be diffuse emitters and specular reflectors while the upper and lower bounding plates are taken as diffuse emitters and reflectors. For the case when the bounding plates are black, the net radiative flux at a distance x from the Table 1. The passage transmittance function for a square tube with specular, dielectric walls, from data given in [6]. Subscripts on the side wall reflectivity are explained in Section 3 of the text Aspect
Re f l e c t i v i t y
.O~n .O~p ,11~ ,111p .~On .~Op
127
Ratio
3
5
.03756 .05685 .04713 .06541 • 1015 .08961
,01438 .03474 .01865 .03961 .04628 .04521
10 .003699 ,01807 ,004889 ,02076 .01343 .02245
S. L. MARCUS
128 lower plate is given by [5] qr(x) = oT, %'v(x) - oT247p( L - X) --
fo~ o ' r ( x ' ) 4 drp(x ~ _ ~ 7 ~- x ' ) .OX,
+
fj
~ ~. ,.4 d r , , ( x ' - x)
dx',
(1)
problem to include non-black bounding plates by introducing the same generalization into the analogous gaseous problem. The radiation-conduction problem for a gas between two parallel plates of arbitrary emissivity has received considerable attention in the literature of heat transfer (e.g. [9]-[12]). The non-dimensional equations incorporating the exponential kernel approximation may be written as follows[12]:
where Tt and T2 are the respective temperatures of the lower and upper bounding plates and the passage transmittance function ~-p depends on the horizontal cross section of the cell and on the effective reflectivity of the side walls. The conductive heat flux is given by
j d20-9f - 3 0 4 = ] 6 y - a -~(s~
OT
where the effective combined thermal conductivity k of the honeycomb walls and the air contained in the cells is assumed to be independent of temperature. In this study we consider the temperature field to be one-dimensional, and following [7] we take the heat conducted through the honeycomb walls to be 10 per cent of that conducted through the air in the cells. The condition for thermal equilibrium of this system may be written as
(7)
X, = 1 - 1 - e , (~ +4fs,)
(8)
X2 : 024 + 1 - e2 (~b + 4fs2),
(9)
E2
where
(2)
(3)
and
(6)
a = 9f + 3 X, - 6fs, - ~ ~p
y = fix,
Yo = ilL,
0
T2 Oz= T~'
T = TZ'
while continuity of temperature at the boundaries requires T(0) = T,
+~(X, 3
q,. = - k ~ x ,
kd r dq dx 2 " dx = 0,
(5)
(d0],
and
k/3 T ( L ) = T>
(4)
We now consider an alternate physical system in which the honeycomb array between the two plates is replaced by a gray, non-scattering gas with thermal conductivity k and volumetric absorptivity /3. Thermal equilibrium for this system is also described by eqns (1)-(4), with the passage transmittance function of eqn (1) replaced by the gas transmittance function ~ ( x ) = 2E3(flx), where E3 is the third exponential integral. If we now
make the exponential kernel approximation for both of the transmittance functions rp and re, the two problems become formally equivalent. Although this approximation is less justified for T, than for its more common application to %, it has been used previously by Tien and Yuen[8] in a study of the radiative properties of honeycombs; Marshall et al.[3] reported good agreement between the results of [8] and their measurements of the radiative properties of honeycombs in vacuum. Following the procedure of [8], we generalize the honeycomb
f - 4o.T,3. E~ and E2 are the respective emissivities of the lower and upper plates, and 13 is determined by matching the two transmittance functions at the upper boundary, r,, (L) = 2E3(flL).
(10)
The total non-dimensional heat flux is then given by
-
q~ + q~ ~rTi---Z•
(11)
Since flat plate solar collectors are (unfortunately!) restricted to the regime 02 ~>0.8, eqn (5) may be linearized with little loss in accuracy. The resulting system of eqns (5)-(9) is readily solved analytically, once r,(L) is known (see Appendix). As a check on the accuracy of the analytical solution, numerical solutions to the full nonlinear system of eqns (5)-(9) were also obtained. Fluxes calculated using a least-squares fit to the non-linear term in eqn (5) were found to differ by less than 2 per cent from the corresponding numerical results for all cases considered in this study.
Heat flux through a solar collector honeycomb 3. R E S U L T S
Hollands et aL [4] have measured the heat flux through a honeycomb as a function of the aspect ratio LID, where L is the length of the honeycomb cell and D is the diameter of a circle with area equal to that of the cell. The honeycomb walls were made of a Mylar plastic film 0.1 mm thick, with normal reflectivity of 0.11 and transmissivity of 0.45. In one series of experiments, designated BB, the honeycomb was bounded above and below by "black" plates with a emissivity 0.88; a second series of runs, designated SS, used "shiny" bounding plates of emissivity 0.065, while the third series of runs (SB) used a "shiny" plate below a "black" plate above the honeycomb. An average temperature difference of 7.5°C was maintained across the honeycomb, at a mean temperature of 28.9°C. The data for hexagonal cells with effective diameter D = 0.95 cm, taken from [4], are plotted as triangles in Figs. 1-3• In order to evaluate the heat flux for these experiments by the method of Section 2, the value of rp(L) for the honeycomb being used is required. The passage transmittance function for a square tube with specular, dielectric walls is given in Table 1, as a function of aspect ratio and side wall reflectivity, from data provided by Tobin[6]. The subscript p on the reflectivity indicates that the effects of polarization and angle of incidence on reflection have been taken into account, while the subscript n indicates that these effects have been neglected. Since the effective reflectivity of the Mylar film used in [4] consisted mainly of its transmissivity, the subscript n Values were adopted in this case. For glass honeycombs the subscript p values would be appropriate, although the assumption made for the wall conductivity in this paper would not be accurate. To model a honeycomb cell of length L and effective diameter/9, we choose a square tube of length L and width D(~/2) °5 so that the respective areas will be equal. For purposes of comparison with the experimental data of [4], therefore, the aspect ratios given in Table 1 were scaled by a factor of (7r/2)°5= 0.886 before the corresponding analytical fluxes were plotted in Figs. 1-3. To evaluate the PTF for an effective reflectivity of 0.56, simple linear extrapolation from the values given in Table 1 was used. As a check on the likely errors introduced by this procedure, heat fluxes for the tabulated aspect ratios of 3, 5 and 10 were calculated for the BB, SB and SS cases (shown in Figs. 1-3, respectively). Reflectivities of 0.56n, 0.36n and 0.56p were assumed, and the corresponding fluxes are shown respectively by diamonds, crosses and plusses in the figures. The results show that the 0.56n values give the best overall fit, as expected, and also that the differences in predicted flux for the three assumed reflectivities are quite small. In view of this, any errors resulting from the assumptions made regarding the effects of polarization or the linearity of the transmittance with reflectivity are also likely to be small. For a given reflectivity, values of the PTF for arbitrary L were estimated by setting rp(L) = (ao + a,L + a2L 2) 1, where ao, a. and a2 were obtained by matching the tabulated PTF's at aspect ratios 3, 5 and 10. The fluxes SE Vol. 30, No. 2--D
129
calculated in this manner for a reflectivity of 0.56n are shown by the solid lines in Figs. 1-3. These fuxes are similar to the analytical results given in [4] (not shown here), in that the fluxes for the BB case are slightly too low while those for the SB and SS cases are too high, and all predicted fluxes are within 20 per cent of the corresponding experimental values. Possible reasons for the discrepancy between analytical and experimental results are discussed in [4]. For purposes of comparison, the fluxes predicted by a separate mode analysis are shown by the dashed lines in Figs. 1-3. Evidently this
5 oo
zX Experiment x Combined mode f l u x , R E F L = 0 3 6 N
4 O0
--~'~A
+ Combined mode flux, REFL=O 56 P -,-0,,-- Combined mode flux, REFL=O 56 N
~
-- --Seperate mode flux, REFL=O56 N
7 L~ 3 oo
~
~
_ •
n
"
~
2 O0
" t I00
oOO
200
I
L
400
600
I
I
8(30
I000
1200
L/D
Fig. 1. Heat fluxes for experiment BB, described in the text. Experimental data were taken from Hollands et al. [4]. For the calculated fluxes, aspect ratios given in Table 1 have been scaled by a factor of 0.886.
500
& x +
Experiment Combned mode flux, Combined mode flux, Comb~.~d mode flux, -- -- Seperate mode flux,
4OO
REFL=036 REFL=056 REFL=056 REFL=056
N P N
N
I bOO0
12oo
300
2OO
if_ IO0
OOO 200
I
I
I
400
6OO
800 L/D
Fig. 2. As in Fig. 1, for experiment SB.
500 [
4 oo _
~
3oo
A
Experiment
x Combined mode flux, R E F L = O Z 6 N + Cornl~ned mode flux, I~ZFL= 0 5 6 P - - ~ - - Combtned mode flux, REFL= 0 5 6 N - - - - S e p e r e t e mode flux, REFL=O.56 N
L
2 O0
ooo 200
400
600
800
lOGO
L/D
Fig. 3. As in Fig. 1, for experiment SS.
1200
130
S. L. MARCUS
approach is valid only when both of the bounding plates have high emissivity, a result noted previously by Cess[ll] for the gaseous radiation-conduction problem. To investigate the implications of the radiative-conductive mode interaction for the design of high-temperature fiat plate collectors, calculations of the flux were made for the configuration of Figs. 1-3 with a cover plate emissivity of 0.88 and the temperatures of the absorber and cover plates held fixed at 400 and 320°K respectively. The absorber-to-cover plate heat flux is shown by the solid contours in Fig. 4 as a function of the absorber plate emissivity and the honeycomb aspect ratio LID, where D = 0.95 cm. Evidently the heat flux may be decreased either by decreasing the absorber plate emissivity or by increasing the honeycomb aspect ratio (i.e. the separation between the plates). For a high honeycomb aspect ratio the 2~" steradian field of view "seen" by the cover plate is occupied mostly by honeycomb, so that reducing the absorber plate emissivity in this case gives a relatively small decrease in the total heat flux. Thus if it is desired to use a honeycomb for convection suppression, there may be little advantage in using a selective absorber plate unless the emissivity of the honeycomb material itself is very low. Further discussion of this point is given in [4]. For a highly selective absorber plate ( e - 0.1-0.2) used in the configuration of Fig. 4, calculations using a separate mode analysis (not shown here) were found to underestimate the heat flux by about a factor of 2. Thus, estimates of the efficiency of solar collectors of this configuration made using this type of analysis are likely to be too optimistic. Marshall et aL [3], for example, found that solar collectors using a honeycomb in combination with a highly selective absorber plate had measured efficiencies (under normal atmospheric pressure) 10--17 per cent lower than those expected on the basis of a separate mode analysis. The dashed contours in Fig. 4 show the percentage decrease in the flux when the effective reflectivity of the honeycomb walls is lowered from 0.56n to 0.36n. The smallness of the resulting change again illustrates that
I0 O0
I
REFERENCES
1. H. Buchberg and D. K. Edwards, Design considerations for solar collectors with cylincrical glass honeycombs. Solar Energy 18, 193-203 (1976). 2. J. C. McMurrin, A. A. Lewandowski, N. Djordjevic and H. Buchberg, Sensitivity of flat plate solar collector performance to design and operating variables, with emphasis on transparent honeycomb collectors. In Solar Diversification, Vol. I. American Section of the International Solar Energy Society, Inc., Newark, Delaware (1978). 3. K. N. Marshall, R. K. Wedal and R. E. Damann, Development of plastic honeycomb flat-plate solar collectors. N.T.I.S. Rept No. SAN/1081761 (1976). 4. K. G. T. Hollands, G. D. Raithby, F. B. Russell and R. G. Wilkinson, Methods for reducing heat losses from flat plate solar collectors: Phase III. Final report, 1 May 1977-31 Jan. 1979. Rep. No. COO-2597-5. University of Waterloo. 5. D. K. Edwards and R. D. Tobin, Effect of polarization on radiant heat transfer through long passages. J. Heat Transfer 89C, 132-138(1967). 6. R. D. Tobin, The effect of specular polarizing walls on radiation transfer in an enclosure. MS thesis, Department of Engineering,University of California, Los Angeles (1965). 7. J. G. Symons and R. Gani, Thermal performance predictions and sensitivity analysis for high temperature fiat-plate solar collectors. Solar Energy 24, 407-410 (1980). 8. C. L. Tien and W. W. Yuen, Radiation characteristics of honeycomb solar collectors. Int. J. Heat Mass Transfer 18, 1409-1413 (1975). 9. R. Viskanta and R. J. Grosh, Effect of surface emissivity on heat transfer by simultaneous conduction and radiation. Int. J. Heat Mass Transfer 5, 729-734 (1962). 10. W. Lick, Energy transfer by radiation and conduction. Proc. 1963 Heat Transfer and Fluid Mechanics Institute, pp. 14-26. Stanford University Press, Palo Alto, (1963). 11. R. D. Cess, The interaction of thermal radiation with conduction and convection heat transfer. In Advances in Heat Transfer, Vol. 1. Academic Press, New York (1964). 12. E. M. Sparrow and R. D. Cess, Radiation Heat Transfer. Hemisphere Publishing Corporation, Washington (1978).
I"
oo/_ V ,Z" t ,oo / / , 30
errors arising from the interpolation of the transmittance as a function of the reflectivity are likely to be small, at least for the range of parameters considered here. All of the plastic film honeycombs considered in [3], for example, have effective reflectivities in the range 0.41-0.61, and so should be treatable by this method. For honeycomb materials of higher effective reflectivity, the methods of [4] should be used.
APPENDIX
To find an analytical expression for the flux, eqn (5) is linearized by setting
z
I
04~b~+coO
(02-<0~< 1),
where, for example, a least squares fit is obtained from 4X3X 5 -- 5X2X 6
bp 20xlx3- 15x22' 010
020
o3o
o40
050
060
070
080
090
oo
and
Absorber e m i s s i v i t y
Fig. 4. Absorber-to-cover plate heat flux (Wm-~-°C-~) for a high temperature flat plate solar collector with a plastic film honeycomb, as described in the text. Solid lines: combined mode heat flux for effective honeycomb wall reflectivity 0.56n. Dashed lines: percentage decrease in the flux when the reflectivity is lowered to 0.36n.
lOx~x6-6x2x5 cp - 20xtx3 - 15xf' where x, =- 1 - Off.
Heat flux through a solar collector honeycomb The following definitions are made in addition to those of Section 2: 3 (2-el)
cl = ~ -
9p2 ao = 16[ct2 4[cl
(1 + atbOsl + (a2 + b2)sz = - (ao + bo + cl)
f(I - 02) + ~ (1 - 0z4) P2-
(c2al - bl)sl + (1 + cza~ - b2)s2 = c102 - czao + bo.
Pl
- e~
- e2
at - 4 p l c 1 2
a2 = 4 p l c l 2
e,p2- 9f + 3~% - 1) bo =
where s~ and s2 are determined from the simultaneous linear equations
c2 : 1 - Ctyo
Yo e~ + e2 Pt=~-+ 9
e,)
~ (l - ~ , bl = Cl \
The non-dimensional heat flux is then given by 16fcl 2 , . , @= T tao+ alsl ~- a2s2)
ez=~ (2-~2~
4f }
131
b2 =
e I e2 9plCl " -
O038qO2X/83[020127-05503,00/O © 1983 Pergamon Press Ltd.
AN APPROXIMATE METHOD FOR CALCULATING THE HEAT FLUX THROUGH A SOLAR COLLECTOR HONEYCOMB STEVEN L. MARCUS Department of Atmospheric Sciences, University of California, Los Angeles, CA90024, U.S.A. (Received 21 October 1981; revision accepted 4 April 1982)
Abstract--An approximatecalculationof the radiative-conductivefluxof heat from the absorber to the cover plate of a fiat plate collector equipped with a convection-suppressinghoneycomb is developed, using an exponential kernel approximation to the passage transmittance function of the honeycomb cells. Fluxes predicted by this method are found to be in satisfactory agreement with the results of previously reported calculations and experiments. If a selective absorber plate is used in a collector of this type, heat transport by the air in the collector eliminates the radiation slipbetween the honeycomband the absorber plate, so that thermalemissionby the wallsof the honeycombin close proximityto the hot absorber may largelycancel out the advantagegainedby makingthe absorber plate selective. I. INTRODUCTION In a single-cover flat plate solar collector with a properly functioning honeycomb, convection is suppressed and heat flux from the absorber plate to the cover plate occurs mainly by radiation and conduction [l]. The standard method for computing this heat flux has been to calculate the fluxes separately for the radiative and conductive modes and then to add the results. McMurrin et al. [2], however, have noted that this method may give incorrect results when either the absorber or the cover plate has a low thermal emissivity. Marshall et al. [3], for example, used a vacuum apparatus to study the radiation emitted by a highly selective absorber plate in the absence of heat conduction by the air. They found that the thermal radiation reaching the cover plate was virtually unchanged when a honeycomb was placed above (but not in direct contact with) the absorber plate. Since the emissivity of the honeycomb material used was higher than that of the selective absorber plate, this result requires the honeycomb itself to be at a lower temperature than the absorber plate. In the presence of heat conduction by air, therefore, the result would no longer apply, since the temperature discontinuity (or "radiation slip") between the absorber plate and the honeycomb would be eliminated. Recently Hollands et a/.[4] have made a careful experimental and theoretical study of the radiative-conductive heat transport through a solar collector honeycomb. They showed that a separate mode analysis may seriously underestimate the heat flux when selective surfaces are used, and presented numerical and analytical methods for computing the combined mode heat flux. In this note a simplified analytical treatment for the combined mode flux is given, based on the passage transmittance function (PTF) introduced by Edwards and Tobin[5]. The mathematical basis of the method is given in Section 2, and the analytical expressions for the heat flux are given in the Appendix. The practical application of this method is limited, since it requires knowledge of the value of the PTF for the honeycomb configuration being used. Some values of the PTF for square tubes have been computed by Tobin[6] and are
reproduced in part here in Table 1. Calculations of the heat flux using this method, however, indicate that the total flux is not a very sensitive function of the PTF, so that for an approximate calculation it may be sufficient to use a value of the PTF interpolated from those given in Table 1. This point is discussed more fully in Section 3, where values of the heat flux computed by this method are compared with experimental data given in [4], and the method is further illustrated by consideration of the absorber-to-cover plate heat loss in a high-temperature flat plate collector of a type discussed by Symons and Gani[7]. 2. METHODOF CALCULATION The physical assumptions used in this study are essentially the same as those adopted in [4]. We consider initially an infinite honeycomb array between two parallel plates separated by a distance L, and choose one cell of the honeycomb to represent the entire array. The side walls of this representative cell are considered to be opaque and thermally insulated, with an effective reflectivity equal to the sum of the transmissivity and reflectivity of the actual honeycomb cell wall. All surfaces are considered gray, with side walls taken to be diffuse emitters and specular reflectors while the upper and lower bounding plates are taken as diffuse emitters and reflectors. For the case when the bounding plates are black, the net radiative flux at a distance x from the Table 1. The passage transmittance function for a square tube with specular, dielectric walls, from data given in [6]. Subscripts on the side wall reflectivity are explained in Section 3 of the text Aspect
Re f l e c t i v i t y
.O~n .O~p ,11~ ,111p .~On .~Op
127
Ratio
3
5
.03756 .05685 .04713 .06541 • 1015 .08961
,01438 .03474 .01865 .03961 .04628 .04521
10 .003699 ,01807 ,004889 ,02076 .01343 .02245
S. L. MARCUS
128 lower plate is given by [5] qr(x) = oT, %'v(x) - oT247p( L - X) --
fo~ o ' r ( x ' ) 4 drp(x ~ _ ~ 7 ~- x ' ) .OX,
+
fj
~ ~. ,.4 d r , , ( x ' - x)
dx',
(1)
problem to include non-black bounding plates by introducing the same generalization into the analogous gaseous problem. The radiation-conduction problem for a gas between two parallel plates of arbitrary emissivity has received considerable attention in the literature of heat transfer (e.g. [9]-[12]). The non-dimensional equations incorporating the exponential kernel approximation may be written as follows[12]:
where Tt and T2 are the respective temperatures of the lower and upper bounding plates and the passage transmittance function ~-p depends on the horizontal cross section of the cell and on the effective reflectivity of the side walls. The conductive heat flux is given by
j d20-9f - 3 0 4 = ] 6 y - a -~(s~
OT
where the effective combined thermal conductivity k of the honeycomb walls and the air contained in the cells is assumed to be independent of temperature. In this study we consider the temperature field to be one-dimensional, and following [7] we take the heat conducted through the honeycomb walls to be 10 per cent of that conducted through the air in the cells. The condition for thermal equilibrium of this system may be written as
(7)
X, = 1 - 1 - e , (~ +4fs,)
(8)
X2 : 024 + 1 - e2 (~b + 4fs2),
(9)
E2
where
(2)
(3)
and
(6)
a = 9f + 3 X, - 6fs, - ~ ~p
y = fix,
Yo = ilL,
0
T2 Oz= T~'
T = TZ'
while continuity of temperature at the boundaries requires T(0) = T,
+~(X, 3
q,. = - k ~ x ,
kd r dq dx 2 " dx = 0,
(5)
(d0],
and
k/3 T ( L ) = T>
(4)
We now consider an alternate physical system in which the honeycomb array between the two plates is replaced by a gray, non-scattering gas with thermal conductivity k and volumetric absorptivity /3. Thermal equilibrium for this system is also described by eqns (1)-(4), with the passage transmittance function of eqn (1) replaced by the gas transmittance function ~ ( x ) = 2E3(flx), where E3 is the third exponential integral. If we now
make the exponential kernel approximation for both of the transmittance functions rp and re, the two problems become formally equivalent. Although this approximation is less justified for T, than for its more common application to %, it has been used previously by Tien and Yuen[8] in a study of the radiative properties of honeycombs; Marshall et al.[3] reported good agreement between the results of [8] and their measurements of the radiative properties of honeycombs in vacuum. Following the procedure of [8], we generalize the honeycomb
f - 4o.T,3. E~ and E2 are the respective emissivities of the lower and upper plates, and 13 is determined by matching the two transmittance functions at the upper boundary, r,, (L) = 2E3(flL).
(10)
The total non-dimensional heat flux is then given by
-
q~ + q~ ~rTi---Z•
(11)
Since flat plate solar collectors are (unfortunately!) restricted to the regime 02 ~>0.8, eqn (5) may be linearized with little loss in accuracy. The resulting system of eqns (5)-(9) is readily solved analytically, once r,(L) is known (see Appendix). As a check on the accuracy of the analytical solution, numerical solutions to the full nonlinear system of eqns (5)-(9) were also obtained. Fluxes calculated using a least-squares fit to the non-linear term in eqn (5) were found to differ by less than 2 per cent from the corresponding numerical results for all cases considered in this study.
Heat flux through a solar collector honeycomb 3. R E S U L T S
Hollands et aL [4] have measured the heat flux through a honeycomb as a function of the aspect ratio LID, where L is the length of the honeycomb cell and D is the diameter of a circle with area equal to that of the cell. The honeycomb walls were made of a Mylar plastic film 0.1 mm thick, with normal reflectivity of 0.11 and transmissivity of 0.45. In one series of experiments, designated BB, the honeycomb was bounded above and below by "black" plates with a emissivity 0.88; a second series of runs, designated SS, used "shiny" bounding plates of emissivity 0.065, while the third series of runs (SB) used a "shiny" plate below a "black" plate above the honeycomb. An average temperature difference of 7.5°C was maintained across the honeycomb, at a mean temperature of 28.9°C. The data for hexagonal cells with effective diameter D = 0.95 cm, taken from [4], are plotted as triangles in Figs. 1-3• In order to evaluate the heat flux for these experiments by the method of Section 2, the value of rp(L) for the honeycomb being used is required. The passage transmittance function for a square tube with specular, dielectric walls is given in Table 1, as a function of aspect ratio and side wall reflectivity, from data provided by Tobin[6]. The subscript p on the reflectivity indicates that the effects of polarization and angle of incidence on reflection have been taken into account, while the subscript n indicates that these effects have been neglected. Since the effective reflectivity of the Mylar film used in [4] consisted mainly of its transmissivity, the subscript n Values were adopted in this case. For glass honeycombs the subscript p values would be appropriate, although the assumption made for the wall conductivity in this paper would not be accurate. To model a honeycomb cell of length L and effective diameter/9, we choose a square tube of length L and width D(~/2) °5 so that the respective areas will be equal. For purposes of comparison with the experimental data of [4], therefore, the aspect ratios given in Table 1 were scaled by a factor of (7r/2)°5= 0.886 before the corresponding analytical fluxes were plotted in Figs. 1-3. To evaluate the PTF for an effective reflectivity of 0.56, simple linear extrapolation from the values given in Table 1 was used. As a check on the likely errors introduced by this procedure, heat fluxes for the tabulated aspect ratios of 3, 5 and 10 were calculated for the BB, SB and SS cases (shown in Figs. 1-3, respectively). Reflectivities of 0.56n, 0.36n and 0.56p were assumed, and the corresponding fluxes are shown respectively by diamonds, crosses and plusses in the figures. The results show that the 0.56n values give the best overall fit, as expected, and also that the differences in predicted flux for the three assumed reflectivities are quite small. In view of this, any errors resulting from the assumptions made regarding the effects of polarization or the linearity of the transmittance with reflectivity are also likely to be small. For a given reflectivity, values of the PTF for arbitrary L were estimated by setting rp(L) = (ao + a,L + a2L 2) 1, where ao, a. and a2 were obtained by matching the tabulated PTF's at aspect ratios 3, 5 and 10. The fluxes SE Vol. 30, No. 2--D
129
calculated in this manner for a reflectivity of 0.56n are shown by the solid lines in Figs. 1-3. These fuxes are similar to the analytical results given in [4] (not shown here), in that the fluxes for the BB case are slightly too low while those for the SB and SS cases are too high, and all predicted fluxes are within 20 per cent of the corresponding experimental values. Possible reasons for the discrepancy between analytical and experimental results are discussed in [4]. For purposes of comparison, the fluxes predicted by a separate mode analysis are shown by the dashed lines in Figs. 1-3. Evidently this
5 oo
zX Experiment x Combined mode f l u x , R E F L = 0 3 6 N
4 O0
--~'~A
+ Combined mode flux, REFL=O 56 P -,-0,,-- Combined mode flux, REFL=O 56 N
~
-- --Seperate mode flux, REFL=O56 N
7 L~ 3 oo
~
~
_ •
n
"
~
2 O0
" t I00
oOO
200
I
L
400
600
I
I
8(30
I000
1200
L/D
Fig. 1. Heat fluxes for experiment BB, described in the text. Experimental data were taken from Hollands et al. [4]. For the calculated fluxes, aspect ratios given in Table 1 have been scaled by a factor of 0.886.
500
& x +
Experiment Combned mode flux, Combined mode flux, Comb~.~d mode flux, -- -- Seperate mode flux,
4OO
REFL=036 REFL=056 REFL=056 REFL=056
N P N
N
I bOO0
12oo
300
2OO
if_ IO0
OOO 200
I
I
I
400
6OO
800 L/D
Fig. 2. As in Fig. 1, for experiment SB.
500 [
4 oo _
~
3oo
A
Experiment
x Combined mode flux, R E F L = O Z 6 N + Cornl~ned mode flux, I~ZFL= 0 5 6 P - - ~ - - Combtned mode flux, REFL= 0 5 6 N - - - - S e p e r e t e mode flux, REFL=O.56 N
L
2 O0
ooo 200
400
600
800
lOGO
L/D
Fig. 3. As in Fig. 1, for experiment SS.
1200
130
S. L. MARCUS
approach is valid only when both of the bounding plates have high emissivity, a result noted previously by Cess[ll] for the gaseous radiation-conduction problem. To investigate the implications of the radiative-conductive mode interaction for the design of high-temperature fiat plate collectors, calculations of the flux were made for the configuration of Figs. 1-3 with a cover plate emissivity of 0.88 and the temperatures of the absorber and cover plates held fixed at 400 and 320°K respectively. The absorber-to-cover plate heat flux is shown by the solid contours in Fig. 4 as a function of the absorber plate emissivity and the honeycomb aspect ratio LID, where D = 0.95 cm. Evidently the heat flux may be decreased either by decreasing the absorber plate emissivity or by increasing the honeycomb aspect ratio (i.e. the separation between the plates). For a high honeycomb aspect ratio the 2~" steradian field of view "seen" by the cover plate is occupied mostly by honeycomb, so that reducing the absorber plate emissivity in this case gives a relatively small decrease in the total heat flux. Thus if it is desired to use a honeycomb for convection suppression, there may be little advantage in using a selective absorber plate unless the emissivity of the honeycomb material itself is very low. Further discussion of this point is given in [4]. For a highly selective absorber plate ( e - 0.1-0.2) used in the configuration of Fig. 4, calculations using a separate mode analysis (not shown here) were found to underestimate the heat flux by about a factor of 2. Thus, estimates of the efficiency of solar collectors of this configuration made using this type of analysis are likely to be too optimistic. Marshall et aL [3], for example, found that solar collectors using a honeycomb in combination with a highly selective absorber plate had measured efficiencies (under normal atmospheric pressure) 10--17 per cent lower than those expected on the basis of a separate mode analysis. The dashed contours in Fig. 4 show the percentage decrease in the flux when the effective reflectivity of the honeycomb walls is lowered from 0.56n to 0.36n. The smallness of the resulting change again illustrates that
I0 O0
I
REFERENCES
1. H. Buchberg and D. K. Edwards, Design considerations for solar collectors with cylincrical glass honeycombs. Solar Energy 18, 193-203 (1976). 2. J. C. McMurrin, A. A. Lewandowski, N. Djordjevic and H. Buchberg, Sensitivity of flat plate solar collector performance to design and operating variables, with emphasis on transparent honeycomb collectors. In Solar Diversification, Vol. I. American Section of the International Solar Energy Society, Inc., Newark, Delaware (1978). 3. K. N. Marshall, R. K. Wedal and R. E. Damann, Development of plastic honeycomb flat-plate solar collectors. N.T.I.S. Rept No. SAN/1081761 (1976). 4. K. G. T. Hollands, G. D. Raithby, F. B. Russell and R. G. Wilkinson, Methods for reducing heat losses from flat plate solar collectors: Phase III. Final report, 1 May 1977-31 Jan. 1979. Rep. No. COO-2597-5. University of Waterloo. 5. D. K. Edwards and R. D. Tobin, Effect of polarization on radiant heat transfer through long passages. J. Heat Transfer 89C, 132-138(1967). 6. R. D. Tobin, The effect of specular polarizing walls on radiation transfer in an enclosure. MS thesis, Department of Engineering,University of California, Los Angeles (1965). 7. J. G. Symons and R. Gani, Thermal performance predictions and sensitivity analysis for high temperature fiat-plate solar collectors. Solar Energy 24, 407-410 (1980). 8. C. L. Tien and W. W. Yuen, Radiation characteristics of honeycomb solar collectors. Int. J. Heat Mass Transfer 18, 1409-1413 (1975). 9. R. Viskanta and R. J. Grosh, Effect of surface emissivity on heat transfer by simultaneous conduction and radiation. Int. J. Heat Mass Transfer 5, 729-734 (1962). 10. W. Lick, Energy transfer by radiation and conduction. Proc. 1963 Heat Transfer and Fluid Mechanics Institute, pp. 14-26. Stanford University Press, Palo Alto, (1963). 11. R. D. Cess, The interaction of thermal radiation with conduction and convection heat transfer. In Advances in Heat Transfer, Vol. 1. Academic Press, New York (1964). 12. E. M. Sparrow and R. D. Cess, Radiation Heat Transfer. Hemisphere Publishing Corporation, Washington (1978).
I"
oo/_ V ,Z" t ,oo / / , 30
errors arising from the interpolation of the transmittance as a function of the reflectivity are likely to be small, at least for the range of parameters considered here. All of the plastic film honeycombs considered in [3], for example, have effective reflectivities in the range 0.41-0.61, and so should be treatable by this method. For honeycomb materials of higher effective reflectivity, the methods of [4] should be used.
APPENDIX
To find an analytical expression for the flux, eqn (5) is linearized by setting
z
I
04~b~+coO
(02-<0~< 1),
where, for example, a least squares fit is obtained from 4X3X 5 -- 5X2X 6
bp 20xlx3- 15x22' 010
020
o3o
o40
050
060
070
080
090
oo
and
Absorber e m i s s i v i t y
Fig. 4. Absorber-to-cover plate heat flux (Wm-~-°C-~) for a high temperature flat plate solar collector with a plastic film honeycomb, as described in the text. Solid lines: combined mode heat flux for effective honeycomb wall reflectivity 0.56n. Dashed lines: percentage decrease in the flux when the reflectivity is lowered to 0.36n.
lOx~x6-6x2x5 cp - 20xtx3 - 15xf' where x, =- 1 - Off.
Heat flux through a solar collector honeycomb The following definitions are made in addition to those of Section 2: 3 (2-el)
cl = ~ -
9p2 ao = 16[ct2 4[cl
(1 + atbOsl + (a2 + b2)sz = - (ao + bo + cl)
f(I - 02) + ~ (1 - 0z4) P2-
(c2al - bl)sl + (1 + cza~ - b2)s2 = c102 - czao + bo.
Pl
- e~
- e2
at - 4 p l c 1 2
a2 = 4 p l c l 2
e,p2- 9f + 3~% - 1) bo =
where s~ and s2 are determined from the simultaneous linear equations
c2 : 1 - Ctyo
Yo e~ + e2 Pt=~-+ 9
e,)
~ (l - ~ , bl = Cl \
The non-dimensional heat flux is then given by 16fcl 2 , . , @= T tao+ alsl ~- a2s2)
ez=~ (2-~2~
4f }
131
b2 =
e I e2 9plCl " -