A ranking procedure for active solar heating systems

Solar Energy Vol. 35, No. 3, pp. 291-293, 1985 Printed in the U.S.A.

0038-092X/85 +3.00 + .00 © 1985PergamonPress Ltd.

TECHNICAL NOTE A ranking procedure for active solar heating systems MARK S. DREW Department of Computing Science, Simon Fraser University, Burnaby, B.C. Canada V5A 1S6 (Received 16 April 1984; revision received 30 January 1985; accepted 5 February 1985)

1. INTRODUCTION Over the past several years, a number of sizing and costing methods for active solar heating systems have been advanced, ranging from the complex[l-5] to the simple[6]. One of the most successful (cf. [7]) is the Relative Areas Method of Barley and Winn[8]. This success is based on the two features that (i) the method is as simple to use as earlier one-chart procedures, yet leads to annual performance estimates that correspond very well with the relatively more cumbersome f-Chart Method [9] from which it is derived; and (ii) an economic optimization based on annual figures is a built-in factor in the method. While the method has been extended to deal with a solar-load optimization involving tradeoffs between solar system costs and building energy conservation measures[10], there has been no extension in one crucial direction: to date, the solar system performance and cost estimates have been considered given, fixed quantities. Comparison between different collector systems, then, entails a complete recalculation of such indicators as solar fraction, optimum collector area, life-cycle cost, and return on investment. What is put forward here is a methodology which comes at the problem from just the opposite viewpoint: assuming a given heating load, what is the most cost-effective solar system for the consumer? Because this methodology is based on the Relative Areas Method, the numbers involved are site-specific. But within a given geographic locality, we lump into a single parameter X all those factors, both physical performance and cost, which are determined by the specific collector system chosen. The resulting life-cycle savings are given by a very simple curve in terms of X. This amounts, then, to a ranking tool for the evaluation of available solar systems. As is shown below, the graphs developed are in fact load-independent, allowing comparison of different solar systems to be performed in general in a given location, independent of the particular application in mind. In Section 2, the analysis is motivated by the observation that the life-cycle savings ( L C S ) per unit load f o r a system with collector area Ao, corresponding to a solar fraction of approximately 50%, can be expressed as a linear function of a lumped variable X. Here, X represents, essentially, the collector area-dependent cost, modified by a collector-climate efficiency factor derived from the collector characteristics "ra and UL and the Relative Areas Method site-specific efficiency factor Z. A detailed exposition of the Relative Areas Method will not be repeated here; instead, the reader is referred to [8] for a comprehensive tre~tment of this approach. For our purposes, suffice it to say that, based on local conditions, financial factors (viz. inflation rates, interest rates, discount rates, depreciation, economic lifetimes, downpayment fraction, salvage fraction, tax rate, tax credit, property tax, and insurance rate) are all taken into account via four present-value factors E~, E2, E3, E4 that bring costs and savings back to present value terms. The Relative Areas Method then presents a best-fit to collector-system performance based on two fit parameters cl, c2, as well 291

as an effective collector efficiency, called Z, and the Ao for a reference collector system, called a,. The parameters Z and a, summarize climate information about solar systems in a given geographical location. In Section 3, the analysis is extended from the case A = A0 to the case in which the collector area takes on the value Aopt, where this value optimizes the L C S (life cycle savings) over the economic lifetime of the solar system. Again, in terms of X the curve derived is simple, allowing a straightforward ranking of competing solar systems in a given location. The parameter X is a modified effective cost per square meter; the lower the value of X, the larger is the LCS. What is new about X is that, in a specific geographical location and in a given financial climate, it includes all performance and cost information needed to compare the economic worth of two systems at a glance.

2. COST-EFFECTIVENESS PARAMETER The economic analysis used in [8] is adopted here. In general, for given load, solar fraction, system costs, and financial environment, the life-cycle savings (LCS) is given by L C S = FLCyE4 - (Cb + AC,,)EI - ACoE2 - CmE3,

(1)

where collector area, m 2,

A

Ca=

collector area-dependent cost, $m-2,

Cb = solar heating system base cost, $,

Co= collector area-dependent operating cost, $m - 2yr- 1, C m

-m-

maintenance cost, $yr- 1,

© = fuel cost, $GJ-1 delivered by furnace, F=

annual solar load fraction,

L=

annual heat load, GJyr - I ,

and El = first cost present-value factor, E2, E3, E4 = cash flow present-value factors, yrs. For simplicity, in the following it will be assumed that C,, is combined with CO. Introducing the site-specific Relative Areas Method dimensionless factors Ch C2, as well as the site-specific parameters as, Z with dimensions m2CW - i, the annual solar load fraction can be written [8] F = c~ + c2 In (A/Ao),

(2)

292

Technical Note

where the area Ao is given by

Ao = a, UA/(Fkra - FkULZ),

(3)

with F k = collector/heat exchanger heat removal factor, -m = collector transmittance-absorbtance product, UL = collector heat loss coefficient, WC ~m 2,

UA = building heat loss coefficient, WC -~, cj, c2 = Relative Areas Method curve-fit parameters, a~, Z = Relative Areas Method parameters, m2CW 1. Substituting for F in (1), it is straightforward to determine LCS at the a r e a A = Ao when F ~ c~. It is convenient to assume, further, that for the purposes of identifying a lumped cost/efficiency factor, one can simply use L = (UA)(DD), where DD is the annual degree-days for the locality (expressed in C-sec × 10 9), This assumption is not really needed at this point, but a related transformation will be used in the next Section for the case when A is the optimized area. With the indicated substitution for the load, the expression (1) can be rearranged as

(LCS + CbEI) = (cICjE4DD - a,X) UA,

(4)

where all the factors involving variables that change when the collector system is changed are lumped into X,

X = (C,EI + CoE2)/(F~R"g~ - FkULZ).

(5)

The meaning o f (4) is as follows: in a given locality, if the collector system base cost is known, then life-cycle savings per unit o f UA vary linearly with X when A = Ao. The lumped parameter X ($m -2) is essentially an areadependent cost corrected for collector efficiency and the climate o f the given site. It includes both the relevant cost information as well as design and climate information. For a fixed financial environment (E~, E2, E3, E4) and given location (a, Z), any o f the cost and performance (system) parameters in (5) are free to take on those values determined by a particular manufacturer in order to arrive at the value of X for the specific solar system that is to be evaluated. The fact that the LCS is related so simply to X, for the special case A = Ao, is now extended to the more useful case of an area chosen to optimize the LCS.

Here, the presence of X in the denominator reflects the fact that as C~ increases, LCS decreases. Since the X - d e p e n d e n c e d i s p l a y e d in (7) is logarithmic, there is no optimum X (and hence no optimum collector system to choose) that maximizes LCS. Instead, systems may be compared on the basis o f their X-value, independent o f load. A typical distribution of LCS per unit L, versus X, is shown in Fig. 1. The comparison of systems derived here must be carried out separately for each loc a l i t y - - a type o f collector good in one site may not be best in another location. For Fig. I, the particular financial factors used were chosen to agree with the (outdated) values in [12] for purposes of validation of the results; the location used was Denver, with (cf. [8]) c~ = 0.538, c_, = 0.316, a,. = 0.03096, Z = 0.03476. From [8], it is straightforward to calculate the Et, E2, E3, E4 for any particular financial environment, and here we have Ej = 0.8157, E2 = E3 = 13.082, E4 = 18.182. With these values, it is easy to determine the range over which X varies, e.g., with Fkra = 0.78, FkUL = 3.40 one finds from (5) that if 6", = 270.00 and Co = 1.0, t h e n X = 352.5. The typical range for X is shown in Fig. 1; the range is found by taking standard ranges of F k r a and F]~UL (cf. [13]) with collector costs that reflect typical corresponding ranges for good- and poor-quality collectors. For Denver, using the values DD = 0.35767 and C / = 9.48, the curves generated in Fig. 1 are determined by eqn (7) for various values of the system base price, Cb, per unit load L, as shown labelling the curves. (The DD value has been increased by 24% to be able to easily compare with independent previous figures calculated directly from eqn (1) that included a D H W load.), e.g., for the value X = 352.5 determined above, the left hand side of eqn (7) equals 69.8, in units of dollars of LCS per GJ of load. Now, in order to rank systems in a given locality, from Fig. 1 the system with the smaller X value ranks higher, since it has the greater LCS, independent of the intended load. An important special case of eqn (7) and Fig. 1 is that for Cb = 0, corresponding to a supplier's bid that combines Cb and Co into a single area-dependent cost (cf. [7]), e.g. for a building with load = 100GJ in the city corresponding to Fig. 1 (in this example, Denver), for a collector system with X = 300 the optimum LCS is $7860, whereas for X = 400, it is $6290 in present-value dollars. For a load half as large, the figures for the LCS are also halved. Site-specific analyses on the order of Fig. 1 might

100 "-

10o

80-

80

60-

60

40-

40

20-

20

3. OPTIMUM-AREA SYSTEM RANKING Using the parameter X, it is simple to evaluate the LCS when the collector area is optimized. The optimum area Aop, that maximizes LCS for some given collector is given by (cf. [81). Aoot = czLCjE4/(CaEI + CoE2).

(6)

Substitution into (1) is complicated by the fact that eqns (6) and (3) combine to form an irreducible factor L/UA which stands in the way of a neat interpretation in terms of dollars per unit of energy. However, it is justifiable[l 1] and general practice[7, 12] to use just UA in (3) for combined space heating-DHW systems, instead of a modified UA reflecting the presence in L of the DHW load. Therefore, it is permissible to replace UA by L/DD in (3), yielding the result

(LCS + CbE1)/L =

{cj + c2[In(c2E4CjDD/a.~X)

J]}CyE4.

(7)

T -'9

(.9

o,

o

o 2O 3O

-20-

4(1

-20

50 -40

I

0

200

~

I

I

400 600 800 X , $ n q -2

-40 I 1000 1200

Fig. 1. Life-cycle savings per unit load at optimum collector area versus collector system cost-efficiency factor X, for a range of values o f base cost per unit load.

Technical Note most appropriately be carried out in the context of consumers' advocacy groups. These analyses would be based on the regional financial environment and on local climate. As well, specific costs pertaining to available collector system and backup fuels could he assembled so that specific brands of collectors could be shown marked on the curves. Finally, it should be noted that the restrictions[13] on allowable parameters for the f-Chart Method carry over here, since these restrictions are inherited in the Relative Areas Method.

6.

7.

8. REFERENCES

!. TRNSYS, A Transient Simulation Program. Report 38, Solar Energy Laboratory, The University of Wisconsin-Madison (March, 1976). 2. M. J. Brandemuehl and W. A. Beckman, Economic evaluation and optimization of solar heating systems. Solar Energy 23, 1-10 (1979). 3. E. Michelson, Multivariate optimization of a solar water heating system using the simplex method. Solar Energy 29, 89-99 (1982). 4. K. K. Chang, A. Minardi, and T. Clay, Parametric study of the overall performance of a solar hot water system. Solar Energy 29, 513-521 (1982). 5. A. E. McGarity, C. S. ReVelle, and J. L. Cohon, An-

9. 10. I I.

293 alytic simulation models for solar heating system design. Solar Energy 32, 85-97 (1984). A Practical Course in Solar Heating and Cooling of Residential Buildings, Training Manual. Solar Energy Applications Laboratory, Colorado State University (September, 1979). Practical Aspects of Solar Space Heating Systems and Domestic Water Heating Systems for Residential Buildings. Solar Energy Applications Laboratory, Colorado State University (December, 1978). C, D. Barley, and C. B. Winn, Optimal sizing of solar collectors by the method of relative areas. Solar Energy 21,279-289 (1978). W. A. Beckman, S. A. Klein, and J. A. Duffle, Solar heating Design by the f-Chart Method. Wiley, New York (1977). C. D. Barley, Load optimization in solar space heating systems. Solar Energy 23, 149-156 (1979). C. D. Barley, Relative Areas Analysis of Solar Heating System Performance. M.Sc. Thesis, Colorado State University (1977).

12. Solar Annual Performance and Life-Cycle Cost Programs for Handheld Calculators. Solar Environmental Engineering Co., Fort Collins, Colorado (1977). 13. FCHART User's Manual. EES Report 49-3, Solar Energy Laboratory, University of Wisconsin-Madison (June, 1978).