Solar Energy Vol. 26, pp. 375376, 1981
0038092X/81104037502502.0010 Pergamon Press Ltd.
Printed in Great Britain.
TECHNICAL
NOTE
The optimization of solar heating systemsf PAUL R. BARNES Solar and Special Studies Section, Energy Division, Oak Ridge National Laboratory, Oak Ridge, TN 37830, U.S.A.
(Received 21 July 1980; revision accepted 10 November 1980) l. INTRODUCTION The optimum collector area of a solar system is the area that minimizes cost and is a function of both economic factors and system parameters. A formulation for the optimization of solar beating systems has been developed by Chang and Minard [1]. An exponential relationship between auxiliary energy and collector area was first demonstrated and then an equation for the optimum area was derived. This approach simplifies the solar system design process; however, in practice it is often infeasible to use the optimum area due to building constraints and the collector panel areas available. In this paper an approach for optimizing solar heating systems with discrete collector areas is presented. 2. OPTIbflJM COLLECTORAREA The annual operating cost of a solar system can be written as[l]
C = (Co + Gb + Cy)AI + Qa.,,Q + M,
(2)
where i is the annual interest rate and n is the number of years considered in the analysis. C, accounts for other areadependent costs: pipes, controls, land, installation, etc. The federal income tax credit can be taken into account by multiplying the value of I by 0.6 for solar systems installed at a cost of $10,000 or less. The fuel cost can be modified to account for annual price increases by substituting C',,for C,, in eqn (1) where [2]
E + [l+e]= : ~o Li~J '
(6)
The optimum area is found by solving for A when dC a~ = 0.
(7)
The solution of eqn (7) gives the optimum area derived by Chang and Minardi:
Aop = 1 In 3,,
(8)
where Q'Ct~
Y=(Cc + bCr + C,)l"
(9)
Chang and Miuardi obtained good agreement between Aop (eqn 8) and fchart for 5.29<3'< 11.4. In the example problem presented later in this paper Aop is in good agreement with fchart for 3' = 3.46. Thus, good results can be expected over the range of about 3.4 < 3' < 11.4. 3. OPTIMIZINGwrrlt DISCRETECOLLECTION AREAS
In practice it is often impossible to achieve the optimum collector area due to the limited size options available for collector panels and/or other building limitations. The collector area must then be made either larger or smaller than Aop, i.e. Aop +AA. To examine the effects of nearoptimum collection areas on cost, we will first derive the optimum cost Cop by substituting eqns (8) and (5) into eqn (1):
(4)
where e is the annual fuel cost escalation rate. The auxiliary energy is approximately related to collector area by Q,~ = Q= exp ( ,~A),
~A =(C~ + C,b + C,)l AO, dfexp ( ~a).
(1)
where A is the collection area; Cc is the collector cost per unit area; Ct is the storage tank cost per unit volume; b is a proportiouality constant that relates storage volume to collector area; Cy includes other solar system costs; Q,,~ is the annual auxiliary energy required; Ct is the unit fuel cost; M is the cost relatively independent of collector areas such as annual maintenance cost, property taxes, insurance, etc.; and the Capital Recovery Factor, I, is given by t = i(1 + i)" (1 + i)"  1'
optimum area is, in general, within the accurate region. Higher accuracy over an extended range can be obtained by a secondorder exponential function[3]. The secondorder decay constant is about a factor of 103 smaller than ,L The substitution of eqn (5) into eqn (I) and differentiating with respect to A gives
Cop = Cs (Aop + ,~t) + 114,
(10)
where C, is the annualized solar system cost per unit area given by
(5)
C, =(Cc+Crb+Cy)l.
where a is the auxiliary energy consumption decay constant and Q,, is the auxiliary energy required for A = 0. Chang and Minardi demonstrated that eqn (5) is accurate to within a few per cent except for very large and very small collection areas and that the
(ll)
The cost with a solar system of area Aop +AA is
C(Aop +AA) = Cop + C, [+AA + ~5 (exp ( • ~,AA) 1)] (12)
fResearch sponsored by the U.S. Dept. of Energy under Contract W7405ENG26 with the Union Carbide Corp.
= Cop + ,~C,(AA)2/2, 375
(13)
376
Technical Note ORNLDWGOOq153t
16
I ANNUAL
LOAD~ 1
I
ORNLDWG 80H807 400
I
I
~" 3 0 0
g oo 
~
~8
/
~
~
ENERGYSUPPLIED BY THE SOLAR
I 200
SYSTEM

I
t' Aoot
1 TWO PANELS,I 4
az
_. . . . . . . . .
o/, O
tOO
o 2
4
6
8
10
COLLECTOR AREA (m2)
Fig. 1. Auxiliary and solar system supplied energies vs collector area.
where a Taylor series expansion of the exponential function in eqn (12) has been used to write eqn (13). For small values of AA such that A(AA)Z~Aop, i.e. Aop >10,~(AA)z, the cost function does not vary greatly from the optimum value. For large values of AA, should the collector area be larger or smaller than Aop? Consider the function F given by
F = C(Aop  AA)  C(Aop + AA) = C,(,it exp (,iAA) ,i~exp ( ,IAA) 2AA)
Notice that F is always positive, which indicates an advantage in selecting a collector area larger than A,~ rather than smaller, i.e. the cost associated with the larger area is less than that associated with smaller area. Equations (13) and (14) imply that for a collector panel with an effective area Ap, the optimum discrete collection area Aopt is that area consisting of m panels which is just greater than Aop, i.e.
mAp.

I I I 2
I
I~A j'' THREE PANELS
opI
I I
I 1
I
4 6 COLLECTOR AREA (m2)
1 8
Fig. 2. Annualized cost as a function of collector area.
For an economic analysis, the following values are used: Cc = $185/m z C, = $300/m 2 b =0.0~m Cy = $10/mz Ct = $11.11/GJ (4¢/kWhr) e = 9 per cent i = 12 per cent n = 20 yr,
(14)
= 2C,(a ~ sinh (aAA) AA).
Aop t =
I
(15)
where m is the minimum value such that mAp > Aop and Ap
A value of M = 0 is assumed for simplicity although normal values for M will not greatly affect the analysis. The annual cost as a function of collector area is shown in Fig. 2. The optimum area Aop is approx. 4 m 2. For collector panels with an effective aperture of 1.6 m2, the areas of two and three panels are shown in Fig. 2. Since the annual total costs are nearly equal, the larger area is selected as the optimum o r Aopt = 4.8 m 2 where m = 3 in eqn (15). Note that the single panel area of 1.6 m 2 is less than A,~ or 2.27 m2,thus the use of the discrete area collector panels does not result in a severe economic penalty. s. SUMMARY The optimization of a solar system with a limited number of discrete col'ection areas available has been discussed. This optimization is based on an exponential relationship between the auxiliary energy required and collector area. In most cases, the optimum area is that area formed by an array of collector panels which is closest to hut greater than Aor In many cases, an even larger collection area can be used without a severe economic penalty since the cost curve changes slowly as collector area is increased. Larger collector areas are also preferable from the standpoint of reducing our dependence on nonrenewable fuels. REFERENCES 1. K. K. Chang and A. Minardi, An optimizing formulation for solar heating systems. Solar Energy 24(1), 99103 (1980). 2. T. E. Copeland and J. F. Watson, Financial Theory and Corporate Policy, pp. 527. Addison Wesley, Reading, MA (1979). 3. G. F. Lameiro and P. Bendt, The GFL method for designing solar energy space heating and domestic hot water systems. Proc. 1978 Ann. Meeting, Am. Section of ISES, Denver, CO, Vol. 21, pp. 113119 (1978).