Solar Energy Vol. ... for the present work is that solar cells act as good heat collectors and are fairly good .... electric power per area that can be drawn from.
Pergamon
0038-092X( 95)00072-O
Solar Energy Vol. 55, No. 6, pp. 453-462, 1995 Copyright 0 1995 Elsevier Science Ltd Printed in the U.S.A. All rights reserved
0038-092X/95 $9.50 + 0.00
MODEL CALCULATIONS ON A FLAT-PLATE SOLAR HEAT COLLECTOR WITH INTEGRATED SOLAR CELLS TROND *Department
BERGENE*
of Physics,
University
and OLE MARTIN
LOVVIK*t
of Oslo, P.O. Box 1048, Blindern,
(Communicated
by VOLKER
N-0316 Oslo, Norway
WITTWER)
Abstract-A detailed physical model of a hybrid photovoltaic/thermal system is proposed, and algorithms for making quantitative predictions regarding the performance of the system are presented. The motivation for the present work is that solar cells act as good heat collectors and are fairly good selective absorbers. Additionally, most solar cells increase their efficiency when heat is drawn from the cells. The model is based on an analysis of energy transfers due to conduction, convection and radiation and predicts the amount of heat that can be drawn from the system as well as the (temperature-dependent) power output. Special emphasis is laid on the dependence of the fin width to tube diameter ratio. We attribute values to the model parameters, and show that hybrid devices are interesting concerning system efficiency as is also confirmed by previous experiments. Possible applications of such systems are also proposed.
1. INTRODUCTION
making quantitative predictions regarding the performance are presented. The approximations needed to obtain analytical expressions are explicitly stated. The present model is to some extent based on the models for flat-plate solar heat collectors presented by Duffie and Beckman ( 1991) and Sizmann (1991), giving attention to the radiation terms and to the necessary modifications due to the addition of solar cells. We discuss the validity of the model with numerical examples where realistic values are attributed to the model parameters. These examples are compared to some relevant experiments (Hayakashi et al., 1990; Lalovic, 1986-1987; Imre et al., 1993). We also propose possible applications of hybrid systems.
Flat-plate solar heat collectors have been widely investigated during the last decades (e.g. Duffie and Beckman, 1991 and references therein) and have even achieved commerical success in various applications (e.g. Sultanovic, 1993). This is also the case for solar cells. The efficiency of solar cells is generally temperature dependent, usually with decreasing efficiency as the temperature increases because of the temperature dependence on mobility, diffusion length and lifetime of minority charge carriers and on the saturation current (Fahrenbruch and Bube, 1983). For instance, crystalline silicon (Si) solar cells show a relative decrease in efficiency of about 15% when the temperature increases 30 K. Stabilizing the temperature at a low level is also highly desirable to achieve a stable current-voltage characteristic curve for the solar cells. This can be achieved by cooling the solar cells during operation, and the low-temperature heat that can be drawn from the system may be interesting with respect to various applications (see discussion). The aim of the present work is to give a general description of a hybrid photovoltaic/ thermal system. Although such hybrid systems have been studied both experimentally (Hayakashi et al., 1990; Lalovic, 1986-1987; Imre et al., 1993; Garg et al., 1990) and theoretically (Garg et al., 1990; Bergene and Bjerke, 1993) to some extent, no detailed analysis based on energy transfers between the different system components seems to have been given. The present paper provides this, and algorithms for tAuthor
to whom all correspondence
should
2. THE MODEL The system in this work consists of a cover “c”, a solar cell “S”, an absorber “A”, a fluid “F”, and the ambiance “a”. We consider an absorber with tubes filled with a fluid under the absorber sheet (see Fig. 1). The components have local temperatures Tc, T,, TA, T and T,, respectively. They are further characterized by the wavelength dependent material properties absorbance a(n), emissivity e(n), reflectivity p(A) and transmittance r(n). These generally obey p(A) + ~(2) + t(n) = 1 and a(1) = E(A) (Kirchoff’s law). There is a possibility of energy exchange due to radiation, conduction and convection beween the components. The conductive and convective terms are taken to be linear in the temperature difference, and are characterized by the generalized conductances U,, between the
be addressed. 453
454
T. Bergene
and 0. M. Lavvik
-D-
-DW
4
Lx Fig. 1. Geometric configuration as assumed in the model of the hybrid system. C is the cover, S the solar cell, A the absorber and F the fluid. L is the length of the whole system, 6 is the thickness of the absorbing fin, D is the diameter of one tube and W is the width of one unit.
components M and A? For example, Us, is the conductance between the solar cell and the absorber, while U,, is the sum of edge and bottom losses due to conductance and convection. Energy conservation in steady state for the area above the tubes gives, for the fluid,
Table 1. The terms in curly braces are due to multiple reflections and transmissions and must be represented by geometric series. The table lists the terms important for the model description.
{EAC(A}=E*
{E*%I = CA%
40-r
+ U,,(T-
T,) = U,,(T,
- T),
(1) {rcrs%I
- T) + U*,(C
= U,,(T,
- T,) + {c*KJaG
- G) + {cs~)Rs
+ {rcrs%IE
{wsl
+ Us,(T,
= {z,a,}E
+ qJ-‘,
- T,) + Cesols&
+ {EACIs}GT;,
(3)
and for the cover, U,,(T,
- T,) + {CKc~C!
= Us&
- G) + @cE + {cs~JRs
+ {e*zscrc}aT:.
-W?+-‘s2PAPc
>
= %%a.4
= TC%
PsPc)(l- PSPA) - &*Pc >
( (I-
( (1 -
1- PSPA + TSPA PsPdl - PSPJ - &WC
>
(2)
- TA) + Us& + {Eca,}aT~
1 - PSPC + rsnc ( (1 -PsPc)(l
+ {+rs~SaT:
for the solar cell, PK.)
PsPc)(l- PsP.4) - r:P,Pc >
(I-
1
for the absorber, U&T,
%fPs( 1 - PSPC) + 6Pc
l(
(4)
q is the heat per length in the fluid direction y, D is the diameter of one tube, o is the Stefan-Boltzmann’s constant and I’(&) is the electric power per area that can be drawn from the solar cell under the total irradiance E. qf is the heat per length that is brought to the tube from the fin, and is calculated below. R, is the radiation from the solar cell. The terms in curly braces represent geometric series due to multi-
ple reflections and transmissions between the different components. Table 1 lists the most important ones. We have also introduced the symbol YY,most conveniently described by an example: the term {~~cc~}R~is defined as all the radiation emitted by the solar cells that is not reabsorbed by the cells. The material properties are further wavelength dependent. For example, the terms {zcas}E and {E~~}cJT~ are shorthand notations for
s(
zcc(s ( 1-
1 - PSPA+ %PA r%6-k)(1 - PSPA)- &A/k
>
E, d/I (5)
and ccAh.(l ( 1 -
PSPC)(
X
-h&Z) 1 -
+ PSPA)
XL,&) dA
2,2PC) -
ziPAPC
(6)
455
Solar heat collector
respectively, where E, is the incident spectral radiation density and L, the spectral radiance from the absorber (see also Sizmann, 1991). Some approximations have already been made in the balance equations above. First, we have supposed that the heat transport normal to the collector plane is independent of the heat transport in the plane. All material properties are presumed to be independent of temperature and equal on both sides. The components are further thought to be thin enough to allow for neglecting temperature gradients through them. The ambient temperature is postulated to be equal on all sides of the collector. We also neglect all radiation in and out of the fluid. In addition to the above presumptions, many of the terms above are negligibly small, and for the further calculations it is necessary to do some more approximations. First, we suppose that the fluid is properly isolated from the ambiance, so that we can neglect the term U,,(T- T,). The absorbance of the cover (e.g. glass) is also usually negligible for wavelengths where the solar cell and the absorber radiate, so that the last two terms of eqn (4) should be neglected. Although the solar cell is at a potential I’ and radiates accordingly (De Vos, 1981; Haught, 1984), the Es-terms can also be
neglected because the solar cell only radiates in a window corresponding to the semiconductor band gap where ~1~and es are very small. An effective heat conductivity Us, is further introduced instead of USC, and the radiative cover effects are included in this term. This is equivalent to the top loss coefficient in Duffie and Beckman (1991). We also use that CCT, so that the approximation 4Pu*,)-’ T: = (T+ q(DUAF)-1)4 x T4 + 4qT3(DUAJ1
(7) is valid. We finally use an empirical relation for the power from a solar cell (Fahrenbruch and Bube, 1983): P(T,)=(rlo-c(T,-T,))E,
(8)
where q0 is the efficiency of the solar cell at the reference temperature T,, and the factor c governs the temperature dependence. The choice of z as the reference temperature simplifies the following calculations, but makes the efficiency y10dependent on T,. However, it should be noted that possible mismatch effects that arise when the solar cell temperature varies strongly over the system, for example with low inlet fluid temperatures and low flow rates, are not taken into account.
3. PHOTOVOLTAUJTHERMAL
PERFORMANCE
We here consider one unit, that is, one tube and two halves of a fin. x = 0 is defined as the midpoint of the fin. For simplicity, we invoke the assumption that all the heat transport in the fin is via the absorber. In order to find the transport of heat qf from the fin to the tube, we consider an infinitesimal segment with width Ax. The energy balance equation for this segment is U,,(T,,-
T,)Ax + {Q~}GT:~Ax=
U,,(T,-
TAp)Ax+ (rcz,a,}EAx (9)
where TAf is the (x-dependent) temperature of the absorber on the fin, k is the thermal conductivity and 6 is the thickness of the absorbing fin. The solar cell temperature may be found from the balance eqn (3). This yields the differential equation
d2%r
-=Uua,(TA,-T,)+F,oT:,-S,
(10)
ksdx2 where we have used the notation u,
_ u Aa -
FR=
S=
+ Aa
(
&~(&a
-cE)
UsA+ Ua-cE’
{eAG}
-
>
uSA iEAC(S UsA+ Us,-CE
{zCzsaA} + F(i:,l;,)+;)) SA
Sa
>’ E.
(11)
456
T. Bergene
and 0. M. Lsvvik
Ua, is the loss factor from the absorber when the loss through the solar cells is accounted for, FR is a radiation loss factor and S is the part of the insolation that is useful for the absorber. Due to the radiation term, the differential equation has no analytical solution, and we have to make an approximation. Around TAP= T, the right hand side is almost linear in (T’r - 7”) for any reasonable choice of values attributed to the model parameters, and a Taylor expansion is quite accurate: d2 qf
ks~dx2 The modified
z F,oT,4 - S + Uia(K)(TAf-
T,).
(12)
loss factor (13)
also accounts
for the radiation
losses. Together
dcf dx
= 0
with the boundary
and
TAr((W-
conditions
O)/2) = TA:
xEo
where W is the width of the unit, this gives
%(x)=T,+
S - F,aT,4
S - F,oT,4 UL,(T,) -
T,+
cosh(wx) - TA> cosh(o( W - 0)/2) ’
u;,(K)
(14)
where co2 = U;,(T,)(kG)-‘. The heat brought
(15)
to the tube from the two half fins is thus qf=
-2,fds’ dx
x=(W-0)/Z
=(W-D)F,(S-F,aT:-U;,(T,)(T,-T,)), where the fin factor Ff is defined
(16)
as F
=
f
tanh(dW - WV w(W-D)/2
(17)
’
It is a measure on how effectively the heat is transported from the fin to the tube via the absorber. Having found the heat from the two half fins, it is straightforward to solve the balance eqns (l)-(4) to find the following expression for the generated heat: q(T) = WF(T)[S The collector
efficiency
- FR(T)aT4 - U,(T-
T,)].
factor F(T) is given by
F(T) =
(19) TFfj
the effective radiation
(18)
+4F,nT3)/UAr’
loss factor as
F,(T)
=
>
FR
(20)
451
Solar heat collector
and the total conductive loss factor as
(21)
The efficiency factor should be as close to unity as possible whereas the total conductive loss factor and the effective radiation loss factor should be as low as possible to maximize the performance of the absorber. We note that the expression for q reduces to the one presented by Sizmann (1991) when we remove the fins and the solar cells by putting W = D, v], = c = 0, zs = 1 and let UsA go to infinity. Usa is then identified as the top loss coefficient (Duffie and Beckman, 1991), and the total conductive loss coefficient is U,_= Us, + UAa. In order to find the temperature variation in the direction of the fluid Y, a Taylor expansion of the generated heat is performed. We denote the approximated heat q(T): 4(T) = Q(T) = 4(K) + q’(T,)(Tq’ is short-hand
T,).
(22)
notation for the differentiated heat:
4’(7.)=~ _.
(23)
T-T,
In the y-direction, an infinitesimal segment of the fluid has thus the following approximate balance equation: kC,(T(y
+ dY)- T(Y)) = @V(Y))dY,
(24)
where rit and C, are the mass transport and heat capacity for the fluid, respectively, and dy is the length of the segment. This gives a simple differential equation where the solution is T(y)= T,---
4KJ dva)
(
(25)
K-T-$$)exp(T),
where T is the inlet fluid temperature. The rate of heat that is drawn from the system is then QT = tiC,(T,
- TL
(26)
where the outlet temperature TL is the fluid temperature at y = L. The thermal efficiency, most conveniently defined as the ratio of the generated heat to the incoming solar insolation, is given by
rizc,z-T--$$)(I--exp(e)).
QT vA=rW=
(27)
ELW
In a similar vein we find the total amount of power from the solar cells. We first examine the fin. The solar cell temperature is found from the fin balance equation being equivalent to eqn (3): T,,(&
Y)=(~,A+U,,--E)-'C({~,~,} +
where the y-dependence the equation
{EAQI~TA&,
Y)”+
-h+cT,))E+UsaT, USA&(X,
Y)I,
(28)
of TAp(x,y) in eqn (28) is due to the term TA which is y-dependent from
TA(Y)=T(Y)+~.
q(T(Y))
(29)
AF
The y-dependent solar cell temperature solar cell temperature is thus
above the tube is similarly found from eqn (3). The mean
(30)
458
T. Bergene
and 0. M. Lsvvik
and the total efficiency of the solar cells is rls=rlo-4%
r,).
(31)
The integrals above are in principle possible to solve exactly since they only consist of exponential functions, but it is more illuminating to study plots with various input parameters. Some typical plots are presented in the following section.
4.
DISCUSSION
To discuss the validity of the model it is essential to attribute realistic values to the model parameters, and our choice of values is listed in the caption of Fig. 2. These were found by a review of relevant literature (Sizmann, 1991; Lampert, 1979; Lampert, 1987; Bogaerts, 1983). Solar cells are selective absorbers and the values of {rcrs~(~}, {E~cL~}and {rc~} were estimated taking into account typical absorption properties of silicon (i.e. an indirect band gap of 1.1 eV). We do not vary these material properties as they influence the performance in a way qualitatively similar to their influence on the performance of pure thermal converters (Sizmann, 1991). The thermal conversion efficiency ye* is plot-
ted in Fig. 2. We have plotted the dependence of WD-’ (with W held constant) to show how the relative size of the fin influences the performance. Note that this gives different results than increasing W with constant D, since unchanged flow rate gives lower fluid velocity when D is increased. The thermal efficiency is approximately halved when the fin width to tube diameter ratio is increased from 1 to 10. This dependence as a function of WD-’ is clearly stronger than for pure thermal systems. The efficiency reduction is due to the conduction (and radiation) losses from the fin, because the fin is at a higher temperature than the tube. The absorber is more sensitive to the choice of WD-’ than a pure thermal absorber, probably because the solar cell absorbs solar energy essentially in
% 70
60
2
3
4
5
6
7
8
9
WD-’ Fig. 2. The thermal efficiency vA as a function of WD-‘, with W being held constant = 0.10 m and D varying from 0.01 to 0.10 m. The bold, solid and dashed curves correspond to IT;= 280, 300 and 320 K, respectively. The curves denoted a, b and c correspond to ti = 0.0003,0.001 and 0.015 kg s-l, respectively. The other parameters are chosen as follows: {seas} = 0.7, {E*G} = 0.1, {rcrsaA} = 0.15, {eAas) = 0.05, E = 800 Wm-2, q,, = 0.125, c = 5 x 10m4, T,=293K, VA,=1 Wm-‘K-l, U,,=200Wm-2K-1, Us, = 100 Wmm2 K-‘, Us, = 6 Wm-’ K-l, k = 385 Wm-’ K-’ (copper), C, = 4200 J kg-’ K-’ (water),
10
Solar heat collector
the same wavelength area as the absorber. Besides, we have in the model assumed that no heat is transported via the solar cells. This may give lower heat transport from the fin to the tube than what is real, and thus tend to lower the efficiency. However, the significance of the latter effect is somewhat uncertain, and should be further investigated. The fact that the speed of the cooling liquid increases when the tube diameter is decreased does not compensate for these losses from the fin. Also, the efficiency may be increased only with a factor 0.10 if the flow rate is increased from 0.001 to 0.075 kg s-l. Also, qA is difficult to interpret because the system is strongly coupled to its environment. In particular, for T < T, the system receives heat from the environment and the conductive loss factors are positive. Consequently, thermal conversion efficiencies greater than one can be obtained for certain values of T and ti. The solar contribution to ye+,can therefore not be interpreted unambiguously. The outlet temperature TL is shown in Fig. 3. It may be viewed as a measure of the quality of the generated heat and is an important quantity of the thermal part of the system. Increasing WD-’ from 1 to 10 affects the quality of the heat, i.e. TL decreases. The solar cell efficiency is, on the other hand,
459
not heavily affected by the fin size as is shown in Fig. 4. The most important parameters here are the flow rate and the inlet fluid temperature, since the cell temperature depends strongly on them. When the flow rate is around 0.001 kg s-l there is not very much to gain on increasing it further. At low flow rates the solar cell efficiency increases when WD -’ increases, whereas at high flow rates the opposite is the case. This can be understood by the following arguments: At a low flow rate an increase in WD-’ gives higher fluid velocity which leads to a strong decrease in T, with a corresponding increase in the efficiency of the solar cell. At a high flow rate an increase in WD-’ entails only a small decrease in ‘TL, but heating of the fins when increasing WD-‘, leads to a (small) decrease in the efficiency of the solar cell. A combination of these two effects gives rise to the interesting behaviour on solar cell efficiency as shown in Fig. 4. The solar cell efficiency is roughly in the range 10.4-12.7%. For solar cells that are not cooled, working temperatures of 60-80X are common, this corresponding to electrical efficiencies of 9.5-10.5%. The relative increase in solar cell efficiency as the result of cooling is therefore in the order of lo-30%. The efficiency diagram of Fig. 5 shows clearly the effects of varying T and fin size (WD - ‘). At
K 340
320 TL
310
b‘ 280
c
C
2
3
4
5
6
7
8
9
10
WD-’ Fig. 3. The outlet
temperature
TL as function of WD-‘, with W being held constant. curves are as in Fig. 2.
Parameters
and
460
T. Bergene
and 0. M. Lsvvik
C
ba
b a
10.5
2
3
5
4
a
-Mm-
----
_-_--
6
I
8
10
9
WD-l Fig. 4. The electrical
efficiency
7s as a function of WDm', with W being held constant. curves are as in Fig. 2.
Parameters
and
% 70
60
50 7l.A 40
30
20
10
(TL - z)/E
[10-2KmZ/W]
Fig. 5. The thermal efficiency as a function of (TL - T)/E. The bold, solid and dashed curves correspond to T = 280,300 and 320 K, respectively. The curves denoted a, b and c correspond to D = 0.01 m (WD-’ = IO), D = 0.05 m (WD-’ = 2) and D = 0.10 m (WD-' = l), respectively. Other parameters are as in Fig. 2.
extremely low flow rates, when TL approaches the saturation temperature, the validity of the model is limited. This is due to the Taylor expansion in eqn (22). All flow rates fi >0.0003 kg s-r
correspond to points on the curves in the almost linear region where the validity of the model is good. Our plots show that a realistic estimate of
Solar heat collector
the total photovoltaic/thermal efficiency should be between 50 and 80%. This is comparable to the efficiency of pure thermal converters which rarely achieve efficiencies above 80%. However, the total efficiency is largely dependent on the fin size, so this should be taken into account in a hybrid system. Since the solar cell efficiency is rather insensitive to variations in II’D-‘, there is not much point in increasing the cost of a system by decreasing WD-’ if the main point is to cool the cells. On the other hand, if the thermal efficiency is of any importance, its dependence of the relative tube diameter should be weighted against the cost of the tubes. This is, however, beyond the scope of this paper. One should also take into account that the hybrid system generates electricity that has a much higher quality than heat. The hybrid system and the pure thermal system can therefore not be directly compared as far as the qualities of the products are concerned. It should also be noted that increasing the high quality output (electricity) by increasing ti leads to an even lower quality of the output heat (lower TL), although the amount of heat itself increases Taking quality factors of the (VA increases). different products into account is very difficult, and it is perhaps impossible to quantify the above quality relationships. Nevertheless, such considerations are necessary to characterize the hybrid system further when concerning various applications. Works describing hybrid systems are sparse, but some experiments have been carried out to test the performance of hybrid systems. et al. (1990) found a long term Hayakashi PV + thermal efficiency of around 60%, but this is difficult to compare to our model, because they used a black (absorbing) liquid. Lalovic (1986-1987) also found a similar efficiency, although the photovoltaic part only had an efficiency of 4%. Imre et al. (1993) provided long term measurements, and found a total system efficiency of over 60%. For a reasonable choice of system parameters, we conclude that our model predicts efficiencies that are comparable to relevant experiments. However, direct comparisons are difficult as the papers describing the experiments do not explicitly state the parameters relevant to our model. We believe that hybrid systems will find their applications in the near future, especially when the price of solar cells decreases further. This will increase the importance of reducing costs associated with mounting and framing which is
461
important in all solar energy applications (Bloch and Melody, 1992). The most probable application will be as small scale (domestic) systems for combined production of electricity and low-temperature heat. The development of such systems will not require much new technology. A possible large scale application might be a system for combined production of fresh water (from sea water) and hydrogen (through electrolysis). There is often a high demand for fresh water in areas where the solar insolation is high (the demand being enlarged by an electrolysis farm) and the thermal energy may therefore show up to be useful for preheating of sea water. The storage of solar energy as a chemical fuel (hydrogen) is also interesting with respect to transport of energy to other areas. 5. CONCLUSION We have shown that a hybrid photovoltaic/ thermal system is interesting with respect to system efficiency, and algorithms that can be used in computer simulations are presented. The model predicts the performance of the system fairly well with system efficiencies, thermal + electrical, about 60-80%. However, direct comparisons with relevant experiments are difficult as the system parameters relevant to the proposed model are not explicitly stated in the description of the experiments. As possible applications we propose a domestic system for the combined production of electricity and low temperature heat (for example as a pre-heater in a hot water system) and a large-scale system that might be interesting regarding production of hydrogen and fresh water. However, the quality of the products is also important when various applications are considered. Acknowledgements-We thank the Norwegian Research Council for financial support and Professor John Rekstad for helpful discussions.
NOMENCLATURE c factor
governing the temperature dependent efficiency of the solar cell CPheat capacity for the fluid (J kg ’K ‘) D diameter of one tube (m) E irradiance (Wm-s) E* incident spectral radiation density (Wm-’ nm-‘) F collector efficiency factor Ff fin factor FR radiation loss factor k thermal conductivitv (Wm-’ K-r) L length of the absorber (m) LA spectral radiance from the absorber (Wm~2nm-‘) ti mass transport (kg s-r)
462
T. Bergene
and 0. M. Lsvvik
P electric power (Wm-‘) 4 heat per length in the fluid direction (Wm-i) qr heat per length brought to the tube from the fin (Wm-i) 4 approximated heat (Wm-‘) 4’ differentiated heat (Wm-r K-i) R, radiation from the solar cell (Wm-*) S the part of the insolation that is useful for the absorber T temperature of the fluid (K) T mean temperature (K) T,, temperature of the ambience (K) Ti temperature of the absorber (K) TAPtemperature of the absorber on the fin (K) Tc temperature of the cover (K) 7; inlet fluid temperature (K) TL outlet fluid temperature (K) Y& temperature of the solar cell (K) UMN generalized conductance between components M and N (Wm-’ K-i) Vi, loss factor from the absorber (Wm-’ K-l) II;, modified loss factor (Wm-’ K-i) U, total conductive loss factor (Wm-a K-‘) W width of one unit (m) Q, absorbance of component M G defined by {Ed&}; the fraction of radiation emitted by component M that is not re-absorbed by M 6 thickness of the absorbing fin (m) sDI emmisivity of component M qA thermal conversion efficiency qs total efficiency of the solar cell ‘lo efficiency of the solar cell at temperature T. M pnr reflectance of component o Stefan-Boltzmann’s constant (5.67 x lo-” Wm-* Km4) tM transmittance of component M
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