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Economic evaluation and optimization of solar heating systems
Solar Energy, Vol. 23, pp. 1-10 Pergamon Press Ltd.. 1979. Printed in Great Britain
ECONOMIC EVALUATION AND OPTIMIZATION OF SOLAR HEATING SYSTEMS M. J. BRANDEMUEHLand W. A. BECKMAN Solar Energy Laboratory, University of Wisconsin, Madison, WI 53706, U.S.A. (Received 28 March 1978; revision accepted 1 December 1978)
Almtraet--A procedure is developed for assessing the economic viability of a solar heating system in terms of the life cycle savings of a solar heating system over a conventional heating system. The life cycle savings is expressed in a generalized form by introducing two economic parameters, Pj and P2, which relate all life cycle cost considerations to the first year fuel cost or the initial solar system investment cost. Using the generalized life cycle savings equation, a method is developed for calculating the solar heating system design which maximizes the life cycle savings. A similar method is developed for determining the set of economic conditions at which the optimal solar heating system design is just competitive with the conventional heating system. The results of these optimization methods can be presented in tabular or graphical form. The sensitivity of the economic evaluation and optimization calculations to uncertainties in constituent thermal and economic variables is also investigated. 1. INTRODUCTION The technical feasibility of solar energy for domestic water heating and space heating has been well established. While economic feasibility is the major bartier restricting its widespread usage, solar heating may now be economically competitive with conventional heating fuels, depending on the individual ckcumstances of the user. This guarded position dictates the need for a simple procedure by which the individual engineer or architect can evaluate the applicability of solar heating for a particular situation. The methods of this paper provide such procedures. 2. THI~-I~HALANALYSIS Klein et al. have developed a method for estimating the long-term thermal performance of standard solar heating systems[l-3]. This procedure, referred to as the
f-chart method, uses monthly average meteorological data, monthly heating loads, and collector characteristics to calculate the monthly fraction of load supplied by solar energy. The annual fraction of load can be determined from these monthly values and the corresponding monthly heating loads. This study uses the f-chart method to investigate the combined heating system configurations shown in Figs. 1 and 2. The annual fraction of load supplied by solar energy is dependent upon meteorological conditions, building location, collector type, collector area, annual load, and the annual load distribution. This functional dependence can he greatly reduced, though, by noting that meteorological conditions, building location, and annual load distribution can be expressed as a general location dependence. Examination of the f-chart correlations then show that, for a given location and collector type, TO TAPS HOT WATER AUXILIARY
RELIEF VALVE
Air SU
.Y
Fig. I. Schematic diagram of combined solar space and domestic water heating system with liquid as the transfer medium.
M.J. BRANDEMUEHLand W. A. BECKMAN SUPPLY AIR TO HOUSE
AUXILIARY
I
FAN ~) DAMPER
HEAT EXCHANGER
DAMPER TO TAPS HOT WATER PREHI
AUXILIARY
PEBBLE BED
WATER SUPPLY
RETURN AIR FROM HOUSE
Fig. 2. Schematic diagram of combined solar space and domestic water heating system with air as the transfer medium. the annual fraction of load met by solar is a function of the ratio of collector area to annual load, A/L, rather than A and L separately. The assumption that the annual load distribution depends only on location, and not on the magnitude of the annual load, is built into many methods for calculating domestic water heating loads or space heating loads. Domestic water heating loads are usually calculated assuming a constant average daily hot water demand, and the difference between the delivered hot water temperature and the water main supply temperature (location dependent). Assuming constant delivered hot water temperature, the water heating load distribution depends only on location and the number of days per month. Studies have also shown that the average annual space heating load is proportional to the long-term average degree days[4]. The annual space heating load is then distributed according to the long term average monthly degree days. When considering combined space and domestic water heating systems, the combined load distribution is affected both by the space and water heating load distributions according to the relative size of the individual loads. For example, the load distribution when the space heating load is twice as large as the water heating load will differ from the distribution when the water heating load is twice as large as the space heating load. The assumption that the combined load distribution is a function only of location implies that, in a given location,
the ratio of the space heating load to the domestic water heating load is a constant. This assumption seems reasonable over a modest range of home sizes. Most new home construction will conform to building codes which specify minimum insulation requirements. The space heating load in a particular location will then depend mainly on the size of the house. Most larger residences will house more people with a corresponding larger hot water usage. In order to obtain representative load distributions for combined systems, space and domestic water heating loads have been estimated for a typical residence. Space heating loads were estimated for each location using the degree day method[4] for a residence insulated to ASHRAE 90-75 standards [5]. This typical residence is a single family, one story house with 1 5 0 m 2 floor area, 120m 2 wall area, and an infiltration rate of one air change per hour. This results, for example, in an overall house UA of 30.0 MJ°C-~ days -~ in Dallas, Texas. The water heating loads were calculated for aconstant daily average hot water demand of 300 !. per day heated from the water main temperature to 60°C. The annual average water main temperatures, which vary from city to city, were obtained from Ref.[6]. The information used to estimate the load distributions for 19 exemplary U.S. locations is given in Appendix A, Table 2. Strictly speaking, the annual fraction of load supplied by solar energy for a given load distribution will not be valid for a different distribution. Fortunately, the annual
Economicevaluation of solar heating systems solar fraction is not very sensitive to the time distribution for the load. If, for example, the ratio of water heating load to space heating load in Madison, Wisconsin is doubled, the difference in F vs A/L is always less than 0.032.
3
ment is already expressed in current dollars, P2 is unity for this example. The factors PI and P2 are then
P~ = PWF(NB, e, d)
(2)
P2 = 1
(3)
3. ECONOMIC ANALYS~S
The economic calculations for this study are based on comparisons of the life cycle costs[7] of conventional heating systems with those of solar heating systems. The life cycle savings of a solar heating system over a conventional heating system can be expressed as the difference between a reduction in fuel costs and an increase in expenses incurred as a result of the additional investment for the solar system.
LCS = P, CFLF- P2(CAA+ CE)
(1)
whereLCS is the cycle savings of a solar heating system over a conventional heating system, $; CF the unit cost of delivered conventional energy (including furnace efficiency) for the first year of analysis, $-GJ-1; L the average annual combined space and water heating load, GJ; F the annual fraction of load supplied by solar energy; CA the solar energy system investment costs which are directly proportional to collector area, $-m-2; A the collector area, m2; CE the solar energy system investment costs which are independent of collector area, $; P~ the factor relating life cycle fuel cost savings to first year fuel cost savings and P2 the factor relating life cycle expenditures incurred by additional capital investment to the initial investment. It is assumed that the costs of components which are common to both conventional and solar heating systems (e.g. the furnace, ductwork, blowers, thermostat), and the maintenance costs of this equipment, are identical. As a result, all references to solar heating system costs, or conventional system costs, refer to the cost increment above the common costs. The multiplying factors, P~ and P2, facilitate the use of life cycle cost methods in a compact form. Any cost which is proportional to either the first year fuel cost or the initial investment can be included. These factors allow for variation of annual expenses with time (e.g. inflation) and they reflect the time value of money by discounting future expenses to present dollar values. To illustrate the evaluation of P~ and P2, consider a very simple economic situation in which the only significant costs are fuel and system equipment costs. Assume that fuel costs escalate at a constant annual rate, and the owner pays cash for the system at the beginning of the analysis. Here, P~ accounts for fuel escalation and the discounting of future payments. The factor P2 accounts for investment related expenses which, in this case, consist only of the investment. Since the invest-
?Each paymentis assumed to be made at the end of the period and the present worth of the series of payments is as of the beginning of the first period.
where d is the annual market discount rate, e the annual market rate of fuel price escalation and N~ the years of economic analysis. The function PWF(NE, e, d) is defined as
PWF(NE, e,d)=d l~_e[l-(l + e~ N~]
\i-~)
-
J"
(4)
The function PWF(NE, e, d) is a present worth factor that accounts for inflating payments in discounted money. When multiplied by a first period cost (which is inflated at a rate, e, and discounted at a rate, d, over NE periods), the resulting value is the present worth life cycle cost?. When the inflation rate is zero, [PWF(Ns, 0, d)]-' is the capital recovery factor. A more complex analysis may be formulated to include a wide variety of expenses so that P, and P2 take the following forms. P, = (1 - Ct)PWF(NE, e, d)
(5)
- . -,PWF(N,,O,d) P2 = 0 + 0 - ~ ~ PWF(NL, O,t) •
+
PWF(N1, O,d)l + ~
j
1
(I-C':)MPWF(N~,g,d)
+ t(l - i) VPWF(NE, g,d)
_~DPWF(N2,0, d ) - ~ G
(6)
where C is the commercial or non-commercial flag (1 or 0, respectively); i the annual mortgage interest rate; g the general inflation rate; NL the term of loan; N, the years over which loan payments contribute to analysis; ND the depreciation lifetime; N2 the years over which depreciation deductions contribute to analysis; i" the effective income tax rate; t the property tax rate based on assessed value; D the ratio of down payment to initial investment; M the ratio of first year miscellaneous costs to initial investment; V the ratio of assessed value in first year to initial investment and G the ratio of salvage or resale value to initial investment. All other terms are as previously defined. In the expression for P2 of eqn (6), the first term represents the down payment; the second term represents the life cycle cost of the mortgage principal and interest; the third, income tax deductions of the interest; the fourth, miscellaneous costs (maintenance, parasitic power, insurance, etc.); the fifth, net property tax costs; the sixth, straight line depreciation tax deduc-
M. J. BRANDEMUEHLand W. A. BECKMAN
tion;f the seventh, salvage or resale value. These and other terms may be added to or deleted from an analysis, allowing a range of economic complexity. The contribution of loan payments to the solar heating system life cycle cost will depend on the relationship between the term of the loan, NL, and the duration of the analysis, NE. If NL-< NE, all NL loan payments will contribute. However, if NL > NE, only NE loan payments will be made during the analysis. The procedure used to account for the remaining loan payments depends on the rationale for choosing the value of NE. If N~ is chosen merely as a period over which to consider the discounted cash flow, with no concern for costs outside the period, only NE payments should be discounted and NI = N~. If NE is the expected operating life of the system, and loan payments will continue to be made as scheduled, all NL loan payments will be made and N~ = NL. If NE is chosen as the period of building ownership, with the building to be sold after N~ years and the remaining loan principal paid in full at that time, the life cycle loan cost will consist of NE loan payments plus the principle remaining in year NE+,. The remaining loan principle must then be included as a reduction in resale value. The contribution of depreciation deductions to the solar system life cycle cost will depend on the relationship between the depreciation lifetime, ND, and the duration of the analysis, NB. The value of N2 is determined by an argument similar to the above discussion of N,.
4. ECONO]~COPTIMIZATION For a given location, heating load, and economic situation, it is possible to optimize the system design variables to yield the maximum life cycle savings. While there are many variables influencing the solar system performance, the main solar heating system design variable is the collector area. The effect of collector area on the. life cycle savings is illustrated in Fig. 3 for four sets of economic conditions. Curve A corresponds to an economic scenario in which solar energy cannot compete. Clearly, the conventional heating system is the economic choice. Curve B exhibits a non-zero optimum area, but the conventional system is still the economic choice. Curve C corresponds to the "critical" condition, i.e. the optimum solar system design can just compete with the conventional system. Curve D corresponds to tStraight line depreciation is used in the example. It can also be shown that, for N2 = N~ double decliningbalance or sum-ofdigits may be used as follows: DB = ~
C[
2C[f
+ ~-~DI PWF(ND - 1, -2[ND, d)
_ PWF(ND - 1, -2IND, 0)] (1 + d) N~ J
2i
SOD
= N D ( N D + l ) [PWF(No,
400(~
200C ¢n ~9 z
D
to ._1 o>o
7'
w ~
C - ~'OOO
B
Fig. 3. Life cycle savings vs collector area for four sets of economic conditions. an economic scenario which is favorable to solar energy. In this case, the solar heating system is the economic choice. Each curve of l~ig. 3 begins with a negative savings for zero collector area. The magnitude of this loss is equal to P2CE, and reflects the presence of solar energy system fixed costs in the absence of any fuel savings. As collector area increases, all curves except curve A show increased savings until reaching a maximum at some optimum collector area. As the collector area is further increased, the fuel savings continue to increase, but the excessive system costs force the solar savings to decrease. The maximum life cycle savings, and hence the optimum collector area, is characterized by the point at which the derivative of the life cycle savings with respect to collector area is zero. (Since F is a monotonic function of A, this condition is both necessary and sufficient.) ~p O(LCS) - 0 = PICFLZ~, - P2CA. O,4
(7)
Rearranging, the maximum savings are realized when the relationship between collector area and the fraction of load supplied by solar satisfies the following. OF
P2CA
L~'~ = pt CF"
(8)
(This relationship is shown in Fig. 4.) Since the load is constant throughout the optimization, it can be incorporated into the derivative to give at the optimum: OF P~CA a(AIL) = P , CF"
(9)
O, D)
÷ ND - ! - PWF(ND - 1, O, d)] d J
By employing the assumption that the load distribution depends only on location, F can be expressed as a function of location, collector characteristics, and the
Economicevaluation of solar heating systems 1.0
, SLOPE
~>< a-o hi
p,~ •
P2C~P~Cp. If P2CA]P~CF, as calculated by the individual for a particular economic senario, is less than or equal to the critical value, then the optimization method cited previously can be used to determine the optimal system design for the particular economic scenario. For cases in which the value of P2CAJP~CF is greater than the critical value, solar energy is not yet competitive with the alternative. However, since P2CAJP~CF is a function of time, it is possible to predict when the critical condition will occur. In particular, if P~ and P2 are assumed to be constant, if the cost of fuel is assumed to escalate at an annual rate e, and the cost of solar system components (both CA and C~) is assumed to inflate at an annual rate g, then the value of P2CA/P~C~ at a time j years in the future can be calculated.
i A
|
~
/
o.s
~q 0 / Y l 0
, I
I
4000
,
I
] OPTIMUM 200 0 -I o
~_
5
o
C~ = CF(1 + e)j - 2000 0
I 25
I t
l
~
COLLECTOR
CA~= CA(I + g)J
IOO
AREA M 2
Fig. 4. Relationshipamong life cycle savings, collectorarea, and solar load fraction at optimum condition.
P2CAj P2CA(I + g)j P, Cm = P,C~ (1 + e)j'
ratio AlL. Equation (9) then implies that, for a given location and collector type, the area to load ratio at which maximum savings can be achieved is a unique function of one economic parameter, P2C~JPtCF. Equation (9) describes the optimum for any set of economic conditions. However, Fig. 3 shows that not all economic scenarios yield optimal designs with positive savings. There will be some set of economic conditions for which the life cycle cost of the optimal solar heating system design just equals that of the conventional system. This condition, called the critical or break-even condition, is defined by curve C of Fig. 3. Analytically, it is determined by simultaneous solution of eqns (7) and (10).
If e > g, there will be some point in the future when P2CA/PICn, will equal the critical value, Z*. Denoting this time as the year ./*,
LCS = 0 = P, CvLF - P2(CAA + CB).
/ ,PICF\ n~Z P - ~ a ) j* =
• [1 + g \
(14)
'nh- e )
This analysis indicates that the solar heating system will be just competitive with the conventional system j* •years in the future. If the year ]* is large, the assumption that PI and P~ are constant may not be valid. However, this value will serve as a first approximation of the time required for the critical value to occur.
(10) 5. OPTIMIZATIONTABLES
Rearranging,
P2CA F P, CF = AlL + C~tCAL"
(11)
Equating (9) and (11), the critical condition (denoted by an asterisk) is realized when the area to load ratiosatisfies the following.
O0~L)[
F* =Z*. ,~L,* = (ALL)* + ( C J C , L)
(12)
For a given location and collector type, eqn (12) shows the critical condition to be a function of one parameter, CEICAL. Equating (11) and (12),
( P2CA ~ * = Z* (location, collector, CFJCAL). (13) The critical economic condition serves as a basis for the a priori decision of whether solar heating is economically viable by specifying the maximum value of
The optimization methods discussed in the previous section can be presented in either tabular or graphic form. For a given location or collector type, the annual solar fraction of the load, F, and its derivative with respect to the area to load ratio, OFIO(AIL), can be presented as a function o f AlL. Similarly, the critical parameter, Z*, can be presented as a function of CdCAL. After a set of these tables or graphs has been generated, solar heating system performance, economic viability, and optimal system design can be determined with minimal calculations. For example, a solar heating system distributor, serving several locations with a particular type of collector, could generate a small set of tables or graphs for the given collector type in the various service locations (e.g. Appendix A, Tables 3 and 4). On the other hand, a local equipment dealer handling several types of collectors could produce tables or graphs for several collector types in the particular location (e.g. Fig. 5). In either case, all subsequent solar heating system calculations will be greatly reduced. The optimization tables of Appendix A serve to illus-
M. J. BRANDEMUEHLand W. A. BECKMAN 3.01
, , ~ , ~ ~ , , A'FI~(r=) = 0'5 F~ UL= 3.0 w/m2"C 8'PR~T~I)=0.6 PRUL=2.5 w,/m~C
*
t\.,/"B
a-~l~-
ti~\ \
, .
~----~-CA vs. A / L
v
7g
~
E
0
0.5
LO
AREA TO t.OAD RA'rlO A/L OR PARAMETER
[ M2/GJ ] CE/CAL
Fig. 5. Optimization graph for two air collectors in Madison, Wisconsin. trate the use of this method. Table 3 has been constructed for a particular air collector in the 19 United States locations given in Table 2. Similarly, Table 4 uses a particular liquid collector in the same 19 locations. Each table consists of two parts. The left half of each table gives the value of (P2C~P~Cp)*, i.e. Z*, satisfying eqn (12) as a function of C~JCAL.The right half of each table gives the values of F and OFlr~(A/L)as a function of A/L. By eqn (9), the optimum value of A/L, for a given value of P2CA/P~CF,can be determined from this half of the table. The optimization tables have been generated using the weather data from Ref.[3]. To use the tables, it is first necessary to evaluate PI and P2 from economic conditions. By consulting with a solar system equipment supplier, the installed system costs can be determined. Knowing the present cost of delivered fuel energy and the annual heating load, the values of P2CA/P~CF and CdCaL can be calculated. Entering the tables at the appropriate location and collector type, the critical value of (P2CA/P,CF)* = Z* can be determined corresponding to the value of CdCAL. If P2CA/P, CF -< Z*, the solar heating system is economically competitive with the conventional heating system. Using the right half of the table, the user can interpolate to determine the economically optimal area to load ratio and the corresponding fraction of load met by the optimized system for the particular value of P2CA/P~CF. If P2CA/P,CF > Z*, the solar heating system is not currently economical. However, eqn (14) can be used to determine the year in which it will become competitive with the conventional system. To illustrate this procedure, consider a homeowner in .Madison, Wisconsin, who is installing a solar air system with combined space and domestic water heating capabilities. The system has two cover non-selective. collectors (Fs'(ra), = 0.5, FR'UL= 3.0 Wm-2°C-I), with area dependent costs of $200m -2 and fixed costs of $1000. The water heating load is estimated to be 23.4GL The long term average annual degree days for Madison is
4294°C days. Using the typical house described in Section 2, the building UA from Table 2 is 25.4 MJ°C-a day-1 resulting in an annual space heating load of 109.1GJ. The total load is then 132.5GJ. The present cost of conventional energy is assumed to be $9.90 GJ-' (corresponding to electric resistance heating). The homeowner in this case has decided to include loan costs, miscellaneous costs, taxes, and salvage value. Assuming d = 0.08, e = 0.10, i = 0.09, g = 0.06, N~ = 20, NL = 20, t =0.30, t =0.02, D=0.10, M=0.01, V=0.7, G = 0.10, and C = 0 (residence), P1 is equal to 22.169 from eqn (5) and P2 is 1.172 from eqn (6). In this case, PeCalPjCv= 1.068. Using Table 3 with CdCAL=O.038, it is seen that Z*= 1.395. Since P2CAJP~CF< Z*, the combined solar heating system is economically viable. To determine the optimal ratio of A/L, the user must interpolate from the values in the table. For P2CA/P~CF= 1.068, the interpolation yields the values of ALL=0.252, and F=0.38. With L = 132.5 GJ, A = 33.4 m2, the life cycle savings from eqn (1) are $2051. The homeowner may also wish to compare the solar heating system with an oil furnace for which the delivered cost of energy (including furnace efficiency) is $5.40 GJ-l. In this case, ~he value of P2CalPICv is calculated to be
P2C~
- - =
PICF
1.958.
Comparing this value with the critical value of Z* = 1.395, the solar heating system is not competitive with the conventional oil furnace. However, the year at which the solar system will become competitive can be calculated from eqn (14) by assuming that the collector component costs will increase at the general inflation rate, g = 0.06. ., 1n(1.395/1.958) I = ~ = 9 . 2 .
That is, it will be 9.2 yr before any system design with these collector characteristics will be competitive with oil heating in Madison, Wisconsin. 6. SENsrrlVITYANALYSIS The economic evaluation and optimization methods presented here are based on a set of assumptions: load distribution assumptions, f-chart assumptions, cost modeling assumptions, etc. However, the more subtle, and probably most important assumption is that the user can assign realistic, reliable values to economic variables. There is an inherent uncertainty in predicting future expenses and benefits. This uncertainty is magnified by the instability of current international energy affairs. In light of this dilemma, the results of both the life cycle cost analysis and the optimization procedures mustbe accepted with discretion, as they may be strongly influenced by an unknown future. The effect of variable uncertainties on economic analysis results can be evaluated by expressing-the life
Economic evaluation of solar heating systems
cycle solar savings as a linearized Taylor series. For a given set of conditions, the change in life cycle savings, ALCS, resulting from a change in a particular variable, Axe, can be approximated by the following: ALCS = oLCS Ax. Oxj '"
(15)
The expression for aLCSlaxj can be obtained by direct differentiation of the life cycle savings equation. The life cycle cost model of eqns (1) and (4)--(6) will be used for the purpose of this discussion. The derivatives of P~ and P2 from eqns (5) and (6) are given in Appendix B. The expressions in Appendix B can be used to investigate either the overall sensitivity to small changes in the variables or the error propogated by uncertainties in particular variable values. To illustrate the use of these relationships, consider the example of Section 5. The variables describing this analysis and the corresponding sensitivity information are given in Table 1. The derivatives with respect to Pj, P2 and LCS have been calculated using the equations of Appendix B for the given set of variable values. The table also gives the change in solar system life cycle savings, ALCS, caused by a 10 per cent relative increase in each of the variables to facilitate better understanding of the sensitivity. The table shows, for example, that a 10 per cent increase in the discount rate from 8.0 to 8.8 per cent yields a decrease in the value of P~ of approx. 1.824 (i.e. -228 x 0.008), or a relative change of -8.2 per cent. The value of P2 decreases by approx. 0.068, or a relative change of -5.8 per cent. The value of LCS decreases by approx. $385, or a relative change of -19 per cent. It can also be noted that the 10 per cent increase in discount rate will cause a 2.6 per cent increase in the value of P2CA[PJCF, implying that the slope of the F vs A l L curve defining the economically optimal collector
7
area increases by 2.6 per cent. Consulting the optimization tables at P2CA]P~CF = 1.096, the optimal collector area would decrease to 31.8m 2 with the annual solar fraction of load of 37 per cent. The information of Table 1 can also be used to estimate the uncertainty in life cycle savings due to uncertainty in different variables. Consider, for example, uncertainties of the following magnitudes: ACE = _+$100, AM = _+0.005, Ad = _+0.005, Ae = -+0.01, Ag = _+0.005, A / = -+0.025, AL = -+10GJ, AF =-+0.02, AG = _+0.1. The maximum possible uncertainty in life cycle savings due to these uncertainties can be estimated by the following:
ALES.,× = ~, laLcs] Ax. ~=11 axj [ '" Using Table 1 and the above uncertainties, ALCSMAx = 117+599+240+ 1016+99+ 159+834+582 + 165 = $3811. Notice that the maximum uncertainty is greater than the life cycle savings for this example. A more reasonable savings uncertainty estimate caused by these variable uncertainties can be obtained by the following: ALCSprob =
= \--~xj
'1 J
A LCSr,rob= $1600. The explicit effect of any one variable on the life cycle savings is largely determined by the values of many other variables. Because of this complexity, "rule of thumb" sensitivity trends are not easily recognized, and a quantitative analysis is usually required.
Table I. Sensitivity analysis example Variables, xj I. Noncommercial or commercial flag (0 or 1) 2. Area dependent investment cost 3. Area independent investment cost 4. Ratio of downpayment to initial investment 5. Ratio first year's misc. costs to init. inv. 6. Ratio first year's assessed value to init. inv. 7. Ratio of salvage value to initial investment 8. Annual market discount rate 9. Annual market rate of fuel price increase 10. Annual interest rate on mortgage I 1. Annual rate of general inflation 12. Property tax rate 13. Effective income tax rate 14. Duration of economic analysis (years) 15. Term of mortgage (years) 16. Depreciation lifetime (years) 17. Cost of conventional fuel in first year (S/GJ) 18. Annual heating and hot water load (GJ) 19. Annual load fraction supplied by solar 20. Collector area tALCS for AxJxj = 0.10.
Nominal Values
aPt ~x.j
axl
OLCS Oxj
ALCSf
0.000 200.000
1000.000 0.100 0.010 0.700 0.100 0.080 0.100 0.090 0.060 0.020 0.300 20.000 20.000 20.000 9.900 132.500
0.380 33.400
0.00000
0.00000
0.00000 0.00000 0.00000 0.12842 0.00000 0.155% x 102 0.00000 0.21834 0 . 0 0 0 0 0 -0.21455 -0.22802x 103 -0.85389 x lip 0.20373x 103 0.00000 0.00000 0.45908x 10' 0.00000 0.25869x 10I 0.00000 0.76419x 10I 0 . 0 0 0 0 0 -0.83024 0.13242x 101 0.14384x 10I 0 . 0 0 0 0 0 -0.37943x 102 0.00000 0.00000 0.00000
0.00000 O. ~000 0.00000
0.00000
0.00000
-0.39137 x 102 -0.11718 x lip -0.98630 x 103 -0.11978 x 106 -0.16769 x 104 0.16477x 104 -0.48084 x liP 0.10155 x 10~ -0.35258 x liP -0.19867 x liP -0.58690 x liP 0.63763x 104 0.54961 x 103 0.29140× 102
-783 - 117
0.11162x 104 0.83398x 102 0.29080x 102 0.00000
1105 1105 1105 0
-
10
-
119
117 17 - 385 1016 - 317
-
-
119
117 191 1099 60
M. J. BRANDEMUEHLand W. A. BECKMAN
8
The economic optimization of a solar heating system is also affected by variable uncertainty. Moderate changes in the value of P2CA/P,C~ often produce substantial chances in the optimal collector area. Fortunately, the curve of life cycle savings vs collector area exhibits a broad optimum as seen in Fig. 4. Near the optimum, large uncertainty in collector area results in small uncertainty in the life cycle savings. In general, the effect of variable uncertainty on system optimization is overshadowed by the effect on the life cycle savings evaluation.
and the f-chart solar heating system design correlations to develop a tabular or graphic method for estimating the optimal collector area and evaluating the solar system economic effectiveness. Using this method, an engineer, architect, or solar equipment company can prepare a set of tables or graphs for location and collector types frequently used. Once these tables or graphs are constructed, computations for solar heating system thermal and economic evaluation are greatly reduced.
RE~ERENCES
7. CONCLUSION
The methods presented in this paper allow the individual architect or engineer to design an economically optimal solar heating system and evaluate the economic comparison with an alternative conventional heating system. Life cycle cost analysis with two parameters, P, and /'2, is used for economic evaluation. The use of these parameters requires one economic assumption: all costs which contribute to the life cycle costs of the solar heating system or conventional system are directly proportional to either the first year fuel cost or the initial solar system investment. The introduction of P, and/'2 presents the life cycle savings equation in a compact and manageable form. This form accomodates straightforward manipulation of the savings equation to determine the optimal system design. The optimization method presented here uses the parameterized savings equation
1. S. A. Klein, W. A. Beckman and J. A. Duffle, A design procedure for solar air heating systems. Solar Energy 19, 509 (1976). 2. S. A. Klein, W. A. Beckman and J. A. Duffle, A design procedure for solar air heating systems. Solar Energy, 19, 509 (1977). 3. W. A. Beckman, S. A. Klein and J. A. Duffle, Solar Heating Design. Wiley lnterscience. New York (1977). 4. ASHRAE Handbook and Product Directory, Systems, American Society of Heating, Refrigerating, and Air Conditioning Engineers, New York (1973). 5. ASHRAE Standard 90-75, Energy conservation in new building design. American Society of Heating, Refrigerating, and Air Conditioning Engineers, New York (1975). 6. W. D. Collins, Teml/erature of Water Available for Industrial Use in the United States. U. S. Geological Survey Water Supply Paper, No. 520F (1925). 7. R. T. Ruegg, Solar Heating and Cooling in Buildings: Methods of Economic Evaluation, U. S. Department of Commerce, NBSIR 75-712 (1975).
8. APPENDIXA
-
Table 2. Load calculation data for combined systems Water Annual Mains Degree Building HeatingLoads (UA) Water Space Location Temperature Days
Albuquerque, NM
13.
2384.
28.3
21.6
Atlanta, GA
18.
1719.
29.3
19.3
50.3
Bismark, ND Boston, MA
6. 9.
5024, 3130.
24.3 27.1
24.8 23.4
122.0 84.9
Boulder, CO Cleveland, OH Columbia, MO
67.6
9.
3078.
27.2
23.4
83.8
II.
3419.
13.
2803.
26.7 27.7
22.5 21.6
91.3 77.5
19.
1272.
30.0
18.8
38.1
7.
4306.
25.4
24.3
109.4
Lander, WY
6.
4372.
25.3
24.8
llO.7
Lincoln, NB
12.
3259.
27.0
22.9
87.8
Los Angeles, CA
19.
lOll.
30.4
18.8
9.
4294.
25.4
23.4
Dallas, TX Great Falls, r.ff
Madison, WI
30.7 lOg.1
Miami, FL
25.
liD.
31.8
16.3
3.8
Nashville, TN
16.
2053.
28.8
20~2
59.l
New York, NY
If.
2673.
27.9
22.5
74.6
Phoenix, AZ
17.
862.
Reno, NV
If.
3345.
Washington, DC
14.
2347.
30.6 26.9 28.3
Ig.7 22.5 21.1
26.4 89.9 66.5
TemDeratures in °C. Degree days in °C-days. UAs in Md-°c-l-day -1 Heating loads in GJ.
Economic evaluation of solar heating systems Table 3. Combined heating (liquid) optimization tables, Fk(ra),, = 0.60 F~UI = 4.00 Wm -2 °C-' TABULATED: f f ~
PtCF"
cat .00
TABULATED: ~
P I CF
.05
.80
.00
.854
.655
.710
.625
.501
.879
.694
.531
.652
.518
.422
.928
.735
.561
.934
.622
.493
.393
1.598 1.399 1.159
.767
.608
.484
TX
2.505 1.717 1.503 1.245
.738
.633
.505
GREAT FALLS
MT
2,567 1.802 1.582 1.310
.900
.709
.542
LANDER
iqY
3,346 2.296 2.010 1.665
.987
.870
.664
LINCOLN
NE
2.435 1,661
1.454 1.204
,811
.643
.513
LOS ANGELES
CA
2.899 2.I54
1.886 1.562 1.068
.840
.642
MADISON
WI
2.169 1.507 1.320 1.093
.748
.593
.453
MIA~ff
FL
2.677 2.026 1.774 1.469 1.041
.822
.628
NASHVILLE
TN
2.175 1.457 1.275 1.056
.626
.538
.428
NEW YORK
NY
1.684 1.195 1.046
.596
.472
.384 .641
3.707 .00 2.384 .00 2.609 .00 1.842 .00 2.693 .00 1.912 ,00 2.409 ,00 2.505 .00 2.567 .00 3.346 .00 2.435 .00 2.899 .OO 2.169 .00 2.677 .00 2.175 .00 1.684 .00 3.595 .00 3.406 .00 2.040 .00
3.707 2.331 2.041
.10
ALBUQUERQUE
N~I
ATLANTA
CA
2.384 1.653 1.447 1.199
~ ISFIARCK
ND
2.609 1+790 1.567 1.298
BOSTON
MA
1.842 1.309
.949
BOULDER
CO
2.493
1.832 1.604 1.328
CLEVELAND
OH
1.912 1.288 1.128
COI,UMRIA
MO 2.409
DALLAS
.30
1.690 1.002
1.146
.866
AZ
3.595 2.434 2.131
1.765 1.046
+846
RENO
NV
3.406 2.334 2.044 1.692 1.003
.866
.659
WASUINGTON
DE
2 . 0 4 0 1.418 1.242 1.028
.554
,444
PIIOENIX
ra 2 GJ- t
AND LOAD FRACTION AlL m 2 GJ I .50
LOCATION
.02
CRITICAL
rn2 GJ 1
.698
.10
.20
2.525 1.710 .31 .51 1.781 1.318 .21 .36 1.949 1.443 .23 .39 1.417 1.027 .16 .29 1.997 1.532 .22 .40 1,396 .938 .16 .28 1.731 1.148 .21 .35 1,847 1.344 .22 .37 1.986 1.494 .23 .39 2.421 1.798 .29 .50 1.806 1.288 .21 .36 2,347 1,819 .26 .47 1.643 1.191 .19 .33 2.211 1.791 .24 .44 1.561 1.115 .19 .32 1.291 .985 '.15 .26 2.629 1.777 .31 .53 2.551 1.814 .30 .51 1.526"1.132 .18 .31
.30
.40
1.290 .65 ,984 .47 1.126 ,52 .825 .38 1.210 .54 .769 .37 .951 .45 .917 .48 1.143 .53 1.441 .66 1.053 .48 1.389 .63 .955 .43 1.419 .60 .825 .41 .739 .35 1.102 .66 1.398 .67 .875 .41
1.028 .76 .755 .56 .848 .62 .730 .45 1.O21 .65 .635 .44 .778 .54 .761 .57 .959 .63 1.038 .78 .836 .57 1.022 .75 .828 .52 1.094 .73 .661 .48 .634 .41 .897 .76 1.OO1 .78 .754 .49
.60
.80
.513 .92 .595 .69 .597 .76 .529 .58 .614 ,81 .469 .54 .569 .67 .601 .70 .609 .78 .539 .93 .604 .71 .439 .90 .584 .66 .533 .89 .509 .59 .490 .53 .495 .89 .407 .92 .556 .62
.255 .99 .466 .80 .378 .86 .436 .68 .370 .91 .382 .63 .456 .77 .410 .81 .352 .87 .116 .99 ,475 .82 .161 .97 .429 .76 ,175 ,96 .419 .69
.403 .62 .278 ,96 .211 .97 .425 .71
Table 4. Combined heating (air) optimization tables, Fk(ra), = 0.50 FkUI = 3.0 Wm -"°C ' TABULATED: ~
PICF"
P,C , t TABULATED: ~ m- GJ
CRITICAL
P I C~-"
AND LOAD FRACTION A/L m" GJ- I
C ! L m'_ Gj t .00
LOCATION ALBUQUERQUE ATLANTA BISMARCK BOSTON BOULDER CLEVELAND COLLIMBIA DALLAS GREAT FALLS LA~DER LINCOLN LOS ANGELES >IADISON MIAMI NASHVILLE NEW YORK pHOENIX RFNO WARIIINGTON
.02
.05
.10
.30
.50
.80
.00
•865
.658
.?58
.642
.491
4.345 .00 2.763 .O0 3.029 .00 2~693
NM 4 . 3 4 5 2.472 2.165 1.793 1.062 GA
2.763
1.764 1.544 1.279
ND
3.029 1.889 1.653 1.369
.812
.696
.533
HA
2.123 1.383 1.211 1.003
.661
.523
.399
CO
2.895 1.952 1.709 1.415
.839
.734
.564
OH
2.202 1.348 1.180
.977
.579
.491
.377
~10
2,796 1.689 1.478 1.224
.726
.614
.471
TX
2.912
1.612 1.335
.791
.652
.498
MT
2.979 1.911. 1.673 1.385
.821
709
.543
~tY
3.909 2.431 2.128 1.762 1.045
.872
.663
NE
2.825 1.762 1.543 1.277
757
.649
.498
CA
3.373 2.316 2.027 1.679
.995
.848
.646
Wl
2.505 1.588 1.390 1.151
~754
.595
.455
FL
3.111 2.244 1.965 1.627
.964
.854
.650
TN
2.518 1.541
.662
551
.422
1.841
1.349 1.117 .918
"
"
"
NY
1.934 1.267 1 . 1 0 9
.602
.476
.379
AZ
4 . 2 2 2 2.616 2.290 1.897 l 124
.799
.657
NV
3.976 2.465 2.158
.869
.659
DC
2.357 1.505 1.318 1.091
,561
.429
'
1.187 I.O59 •
709
.10
2.515 .34 1.869 .23 1.952 .25 1.477 .18 2.895 2.101 .00 .25 2.202 1.423 .00 .18 2.796 1.784 .00 .23 2.912 1.952 .00 .24 2.979 2.053 .00 .25 3.909 2,455 .O0 .32 2.825 1.867 ,00 .23 3.373 2.495 .00 .29 2.505 1.689 ,00 .21 3.111 2.437 .00 .28 2.518 .O0 1.618 .2O
.20
.30
1.637 .54 1.268 .38 1.387 .41 1.011 .30 1.501 .43 .940 .29 1.138 .37 1.291 .40 1.444 .42 1•705 .53 1.242 .38 1.799 .51 1.169 .35 1.855 .49 1.021 .33
1,195 .68 .924 .49 1.O36 .53 .792 .39 1.133 .56 .719 .37 .916 .47 .882 .50 1.054 .54 1.289 .68 .986 .49 1.257 .66 .895 .45 1.365 .65 .780 .42
1,934 .948 ,00 + 1.349 .16 .28 4,222 2.791 1.572 ,00 .35 .57 3.976 2 . 0 3 2 1.705 .O0 .33 .54 2.357 1.064 .00 1.593 .20 .33
.40
.60
.886 .411 .78 .92 .728 .513 .58 .70 .817 .512 .62 .75 .667 .487 .46 .58 .895 .543 .66 .80 .597 .421 .44 .54 .712 .508 .55 .67 .725 .514 .59 .71 .834 .492 .63 .77 ,898 .415 .79 .92 .771 .536 .58 .71 .903 .455 .76 .90 .740 .503 .53 .65 .968 .443 .77 .90 .639 .445 .49 .60 .706 .598 .438 .36 .42 .53 1.O4t .803 .412 .69 .78 .89 1.247 .837 .444 .68 .78 .91 .840 .694 .490 .42 .50 .62
.B0 .189 .98 .373 .79 .333 .84 .364 .66
.337 .89 .316 .61 .363 .76 .374 .80 .322 .85 .185 .98 .377 .80 .197 .96 .357 .74 .083 .97 .347 .68 .335 .60 .217 .95 .150 .96 .356 .70
10
M.J. BRANDEMUEHLand W. A. BECKMAN 9
.
~
B
OP2 _ O--i--
The following equations can be derived from the expressions for P~ and P: of eqns (5) and (6). The present worth function PWF(a, b, c) is abbreviated fia, b, c).
OP,o_.=.d_(I - C[~odf(NE' e' d)'
OP' = ( l - C[~oef(N~' e'
ONeOP--£= (I-Cto~NJ(NE,
0P~=-Cf(N~,e,d) Ot
-
e,d),
(Nl,O,d)
-
.
OP2=~Dl - ( l - t ~ - t f ( N " " d ) [ i - f ( N Z O , OP2 = ~-~
.,,"OP2 ~ = rH
Ct)f(N~, g, d),
(! -
aP2 aO
aP,
--~ = ( D -
at
I ( i + d ) ~E'
I
oP~ 0
~
i)]
t)I(NB,g, d)
tf(Nt.,O,i)
'" t
1
+
t)
1
0
G
d)}
C{ o
+N~• (i + d)U~t - N--~-dli~,, O, d)
.
I
a
1
0
;)tV]~fiN.,g,d) .~.
I JL f ~ 2
~ )
tiN,, i, d) Io___ , ,
~
, [f( NL, O.i)]2JONL J " ' L '
o ~, i)
r)V]Zf(NE, g,d)+~ln(l+d) e ~ -J
T~?""
....
-.
'
I
D)t[, - ~ ] a - ~ / ( N , ,O
,,. d).
fiNL,O,i)
V]~df(N,,g,d , - ( l - D)[{/(N:0, i):d f ( N ' '
[ i- f( N: O,i)]-~f(N', i,
C/')M+(1-
-(1-
Note that since NI is usually equal to either NL or NE, the expression for the partial derivative of/'2 with respect to Nm will usually be added to either the expression for NL or NE. Most of the above equations involve derivatives of the present worth factor. Using eqn (4),
I-D 0 "N 0 f(NL'~,i) ~/( " ' d) + [(1 - Ci)M +t(i-
.
OP'-ffl aNe - "" - Ct)M + t(l ON,
]}
a -
= (D- l){ll - " tiN,, o, dl
aP2
-tVf(NE, g,d)-C[Mf(N,,g,d)+~----J(N2,0,d)] 0P2 ad
-
+f(Nb i, d)[ I
-~=(l-t)Vf(N~,g,d)
1)~f(NuO, d) +f(Nt i d)[i -
r, f(N1, O, d)
(D- IXl - tj[f(N~, O, i)l~ ~i/(NL' O, 0
0, d)
~af(a, b, C)
I [l+b\ °, [l+b\
a
1
0
1
a
a
i+b °
l+b °
]
~.~(a,b,c)= ~_ b[T-~--~c(T-~-~c ) - f(a,b,c)].
ECONOMIC EVALUATION AND OPTIMIZATION OF SOLAR HEATING SYSTEMS M. J. BRANDEMUEHLand W. A. BECKMAN Solar Energy Laboratory, University of Wisconsin, Madison, WI 53706, U.S.A. (Received 28 March 1978; revision accepted 1 December 1978)
Almtraet--A procedure is developed for assessing the economic viability of a solar heating system in terms of the life cycle savings of a solar heating system over a conventional heating system. The life cycle savings is expressed in a generalized form by introducing two economic parameters, Pj and P2, which relate all life cycle cost considerations to the first year fuel cost or the initial solar system investment cost. Using the generalized life cycle savings equation, a method is developed for calculating the solar heating system design which maximizes the life cycle savings. A similar method is developed for determining the set of economic conditions at which the optimal solar heating system design is just competitive with the conventional heating system. The results of these optimization methods can be presented in tabular or graphical form. The sensitivity of the economic evaluation and optimization calculations to uncertainties in constituent thermal and economic variables is also investigated. 1. INTRODUCTION The technical feasibility of solar energy for domestic water heating and space heating has been well established. While economic feasibility is the major bartier restricting its widespread usage, solar heating may now be economically competitive with conventional heating fuels, depending on the individual ckcumstances of the user. This guarded position dictates the need for a simple procedure by which the individual engineer or architect can evaluate the applicability of solar heating for a particular situation. The methods of this paper provide such procedures. 2. THI~-I~HALANALYSIS Klein et al. have developed a method for estimating the long-term thermal performance of standard solar heating systems[l-3]. This procedure, referred to as the
f-chart method, uses monthly average meteorological data, monthly heating loads, and collector characteristics to calculate the monthly fraction of load supplied by solar energy. The annual fraction of load can be determined from these monthly values and the corresponding monthly heating loads. This study uses the f-chart method to investigate the combined heating system configurations shown in Figs. 1 and 2. The annual fraction of load supplied by solar energy is dependent upon meteorological conditions, building location, collector type, collector area, annual load, and the annual load distribution. This functional dependence can he greatly reduced, though, by noting that meteorological conditions, building location, and annual load distribution can be expressed as a general location dependence. Examination of the f-chart correlations then show that, for a given location and collector type, TO TAPS HOT WATER AUXILIARY
RELIEF VALVE
Air SU
.Y
Fig. I. Schematic diagram of combined solar space and domestic water heating system with liquid as the transfer medium.
M.J. BRANDEMUEHLand W. A. BECKMAN SUPPLY AIR TO HOUSE
AUXILIARY
I
FAN ~) DAMPER
HEAT EXCHANGER
DAMPER TO TAPS HOT WATER PREHI
AUXILIARY
PEBBLE BED
WATER SUPPLY
RETURN AIR FROM HOUSE
Fig. 2. Schematic diagram of combined solar space and domestic water heating system with air as the transfer medium. the annual fraction of load met by solar is a function of the ratio of collector area to annual load, A/L, rather than A and L separately. The assumption that the annual load distribution depends only on location, and not on the magnitude of the annual load, is built into many methods for calculating domestic water heating loads or space heating loads. Domestic water heating loads are usually calculated assuming a constant average daily hot water demand, and the difference between the delivered hot water temperature and the water main supply temperature (location dependent). Assuming constant delivered hot water temperature, the water heating load distribution depends only on location and the number of days per month. Studies have also shown that the average annual space heating load is proportional to the long-term average degree days[4]. The annual space heating load is then distributed according to the long term average monthly degree days. When considering combined space and domestic water heating systems, the combined load distribution is affected both by the space and water heating load distributions according to the relative size of the individual loads. For example, the load distribution when the space heating load is twice as large as the water heating load will differ from the distribution when the water heating load is twice as large as the space heating load. The assumption that the combined load distribution is a function only of location implies that, in a given location,
the ratio of the space heating load to the domestic water heating load is a constant. This assumption seems reasonable over a modest range of home sizes. Most new home construction will conform to building codes which specify minimum insulation requirements. The space heating load in a particular location will then depend mainly on the size of the house. Most larger residences will house more people with a corresponding larger hot water usage. In order to obtain representative load distributions for combined systems, space and domestic water heating loads have been estimated for a typical residence. Space heating loads were estimated for each location using the degree day method[4] for a residence insulated to ASHRAE 90-75 standards [5]. This typical residence is a single family, one story house with 1 5 0 m 2 floor area, 120m 2 wall area, and an infiltration rate of one air change per hour. This results, for example, in an overall house UA of 30.0 MJ°C-~ days -~ in Dallas, Texas. The water heating loads were calculated for aconstant daily average hot water demand of 300 !. per day heated from the water main temperature to 60°C. The annual average water main temperatures, which vary from city to city, were obtained from Ref.[6]. The information used to estimate the load distributions for 19 exemplary U.S. locations is given in Appendix A, Table 2. Strictly speaking, the annual fraction of load supplied by solar energy for a given load distribution will not be valid for a different distribution. Fortunately, the annual
Economicevaluation of solar heating systems solar fraction is not very sensitive to the time distribution for the load. If, for example, the ratio of water heating load to space heating load in Madison, Wisconsin is doubled, the difference in F vs A/L is always less than 0.032.
3
ment is already expressed in current dollars, P2 is unity for this example. The factors PI and P2 are then
P~ = PWF(NB, e, d)
(2)
P2 = 1
(3)
3. ECONOMIC ANALYS~S
The economic calculations for this study are based on comparisons of the life cycle costs[7] of conventional heating systems with those of solar heating systems. The life cycle savings of a solar heating system over a conventional heating system can be expressed as the difference between a reduction in fuel costs and an increase in expenses incurred as a result of the additional investment for the solar system.
LCS = P, CFLF- P2(CAA+ CE)
(1)
whereLCS is the cycle savings of a solar heating system over a conventional heating system, $; CF the unit cost of delivered conventional energy (including furnace efficiency) for the first year of analysis, $-GJ-1; L the average annual combined space and water heating load, GJ; F the annual fraction of load supplied by solar energy; CA the solar energy system investment costs which are directly proportional to collector area, $-m-2; A the collector area, m2; CE the solar energy system investment costs which are independent of collector area, $; P~ the factor relating life cycle fuel cost savings to first year fuel cost savings and P2 the factor relating life cycle expenditures incurred by additional capital investment to the initial investment. It is assumed that the costs of components which are common to both conventional and solar heating systems (e.g. the furnace, ductwork, blowers, thermostat), and the maintenance costs of this equipment, are identical. As a result, all references to solar heating system costs, or conventional system costs, refer to the cost increment above the common costs. The multiplying factors, P~ and P2, facilitate the use of life cycle cost methods in a compact form. Any cost which is proportional to either the first year fuel cost or the initial investment can be included. These factors allow for variation of annual expenses with time (e.g. inflation) and they reflect the time value of money by discounting future expenses to present dollar values. To illustrate the evaluation of P~ and P2, consider a very simple economic situation in which the only significant costs are fuel and system equipment costs. Assume that fuel costs escalate at a constant annual rate, and the owner pays cash for the system at the beginning of the analysis. Here, P~ accounts for fuel escalation and the discounting of future payments. The factor P2 accounts for investment related expenses which, in this case, consist only of the investment. Since the invest-
?Each paymentis assumed to be made at the end of the period and the present worth of the series of payments is as of the beginning of the first period.
where d is the annual market discount rate, e the annual market rate of fuel price escalation and N~ the years of economic analysis. The function PWF(NE, e, d) is defined as
PWF(NE, e,d)=d l~_e[l-(l + e~ N~]
\i-~)
-
J"
(4)
The function PWF(NE, e, d) is a present worth factor that accounts for inflating payments in discounted money. When multiplied by a first period cost (which is inflated at a rate, e, and discounted at a rate, d, over NE periods), the resulting value is the present worth life cycle cost?. When the inflation rate is zero, [PWF(Ns, 0, d)]-' is the capital recovery factor. A more complex analysis may be formulated to include a wide variety of expenses so that P, and P2 take the following forms. P, = (1 - Ct)PWF(NE, e, d)
(5)
- . -,PWF(N,,O,d) P2 = 0 + 0 - ~ ~ PWF(NL, O,t) •
+
PWF(N1, O,d)l + ~
j
1
(I-C':)MPWF(N~,g,d)
+ t(l - i) VPWF(NE, g,d)
_~DPWF(N2,0, d ) - ~ G
(6)
where C is the commercial or non-commercial flag (1 or 0, respectively); i the annual mortgage interest rate; g the general inflation rate; NL the term of loan; N, the years over which loan payments contribute to analysis; ND the depreciation lifetime; N2 the years over which depreciation deductions contribute to analysis; i" the effective income tax rate; t the property tax rate based on assessed value; D the ratio of down payment to initial investment; M the ratio of first year miscellaneous costs to initial investment; V the ratio of assessed value in first year to initial investment and G the ratio of salvage or resale value to initial investment. All other terms are as previously defined. In the expression for P2 of eqn (6), the first term represents the down payment; the second term represents the life cycle cost of the mortgage principal and interest; the third, income tax deductions of the interest; the fourth, miscellaneous costs (maintenance, parasitic power, insurance, etc.); the fifth, net property tax costs; the sixth, straight line depreciation tax deduc-
M. J. BRANDEMUEHLand W. A. BECKMAN
tion;f the seventh, salvage or resale value. These and other terms may be added to or deleted from an analysis, allowing a range of economic complexity. The contribution of loan payments to the solar heating system life cycle cost will depend on the relationship between the term of the loan, NL, and the duration of the analysis, NE. If NL-< NE, all NL loan payments will contribute. However, if NL > NE, only NE loan payments will be made during the analysis. The procedure used to account for the remaining loan payments depends on the rationale for choosing the value of NE. If N~ is chosen merely as a period over which to consider the discounted cash flow, with no concern for costs outside the period, only NE payments should be discounted and NI = N~. If NE is the expected operating life of the system, and loan payments will continue to be made as scheduled, all NL loan payments will be made and N~ = NL. If NE is chosen as the period of building ownership, with the building to be sold after N~ years and the remaining loan principal paid in full at that time, the life cycle loan cost will consist of NE loan payments plus the principle remaining in year NE+,. The remaining loan principle must then be included as a reduction in resale value. The contribution of depreciation deductions to the solar system life cycle cost will depend on the relationship between the depreciation lifetime, ND, and the duration of the analysis, NB. The value of N2 is determined by an argument similar to the above discussion of N,.
4. ECONO]~COPTIMIZATION For a given location, heating load, and economic situation, it is possible to optimize the system design variables to yield the maximum life cycle savings. While there are many variables influencing the solar system performance, the main solar heating system design variable is the collector area. The effect of collector area on the. life cycle savings is illustrated in Fig. 3 for four sets of economic conditions. Curve A corresponds to an economic scenario in which solar energy cannot compete. Clearly, the conventional heating system is the economic choice. Curve B exhibits a non-zero optimum area, but the conventional system is still the economic choice. Curve C corresponds to the "critical" condition, i.e. the optimum solar system design can just compete with the conventional system. Curve D corresponds to tStraight line depreciation is used in the example. It can also be shown that, for N2 = N~ double decliningbalance or sum-ofdigits may be used as follows: DB = ~
C[
2C[f
+ ~-~DI PWF(ND - 1, -2[ND, d)
_ PWF(ND - 1, -2IND, 0)] (1 + d) N~ J
2i
SOD
= N D ( N D + l ) [PWF(No,
400(~
200C ¢n ~9 z
D
to ._1 o>o
7'
w ~
C - ~'OOO
B
Fig. 3. Life cycle savings vs collector area for four sets of economic conditions. an economic scenario which is favorable to solar energy. In this case, the solar heating system is the economic choice. Each curve of l~ig. 3 begins with a negative savings for zero collector area. The magnitude of this loss is equal to P2CE, and reflects the presence of solar energy system fixed costs in the absence of any fuel savings. As collector area increases, all curves except curve A show increased savings until reaching a maximum at some optimum collector area. As the collector area is further increased, the fuel savings continue to increase, but the excessive system costs force the solar savings to decrease. The maximum life cycle savings, and hence the optimum collector area, is characterized by the point at which the derivative of the life cycle savings with respect to collector area is zero. (Since F is a monotonic function of A, this condition is both necessary and sufficient.) ~p O(LCS) - 0 = PICFLZ~, - P2CA. O,4
(7)
Rearranging, the maximum savings are realized when the relationship between collector area and the fraction of load supplied by solar satisfies the following. OF
P2CA
L~'~ = pt CF"
(8)
(This relationship is shown in Fig. 4.) Since the load is constant throughout the optimization, it can be incorporated into the derivative to give at the optimum: OF P~CA a(AIL) = P , CF"
(9)
O, D)
÷ ND - ! - PWF(ND - 1, O, d)] d J
By employing the assumption that the load distribution depends only on location, F can be expressed as a function of location, collector characteristics, and the
Economicevaluation of solar heating systems 1.0
, SLOPE
~>< a-o hi
p,~ •
P2C~P~Cp. If P2CA]P~CF, as calculated by the individual for a particular economic senario, is less than or equal to the critical value, then the optimization method cited previously can be used to determine the optimal system design for the particular economic scenario. For cases in which the value of P2CAJP~CF is greater than the critical value, solar energy is not yet competitive with the alternative. However, since P2CAJP~CF is a function of time, it is possible to predict when the critical condition will occur. In particular, if P~ and P2 are assumed to be constant, if the cost of fuel is assumed to escalate at an annual rate e, and the cost of solar system components (both CA and C~) is assumed to inflate at an annual rate g, then the value of P2CA/P~C~ at a time j years in the future can be calculated.
i A
|
~
/
o.s
~q 0 / Y l 0
, I
I
4000
,
I
] OPTIMUM 200 0 -I o
~_
5
o
C~ = CF(1 + e)j - 2000 0
I 25
I t
l
~
COLLECTOR
CA~= CA(I + g)J
IOO
AREA M 2
Fig. 4. Relationshipamong life cycle savings, collectorarea, and solar load fraction at optimum condition.
P2CAj P2CA(I + g)j P, Cm = P,C~ (1 + e)j'
ratio AlL. Equation (9) then implies that, for a given location and collector type, the area to load ratio at which maximum savings can be achieved is a unique function of one economic parameter, P2C~JPtCF. Equation (9) describes the optimum for any set of economic conditions. However, Fig. 3 shows that not all economic scenarios yield optimal designs with positive savings. There will be some set of economic conditions for which the life cycle cost of the optimal solar heating system design just equals that of the conventional system. This condition, called the critical or break-even condition, is defined by curve C of Fig. 3. Analytically, it is determined by simultaneous solution of eqns (7) and (10).
If e > g, there will be some point in the future when P2CA/PICn, will equal the critical value, Z*. Denoting this time as the year ./*,
LCS = 0 = P, CvLF - P2(CAA + CB).
/ ,PICF\ n~Z P - ~ a ) j* =
• [1 + g \
(14)
'nh- e )
This analysis indicates that the solar heating system will be just competitive with the conventional system j* •years in the future. If the year ]* is large, the assumption that PI and P~ are constant may not be valid. However, this value will serve as a first approximation of the time required for the critical value to occur.
(10) 5. OPTIMIZATIONTABLES
Rearranging,
P2CA F P, CF = AlL + C~tCAL"
(11)
Equating (9) and (11), the critical condition (denoted by an asterisk) is realized when the area to load ratiosatisfies the following.
O0~L)[
F* =Z*. ,~L,* = (ALL)* + ( C J C , L)
(12)
For a given location and collector type, eqn (12) shows the critical condition to be a function of one parameter, CEICAL. Equating (11) and (12),
( P2CA ~ * = Z* (location, collector, CFJCAL). (13) The critical economic condition serves as a basis for the a priori decision of whether solar heating is economically viable by specifying the maximum value of
The optimization methods discussed in the previous section can be presented in either tabular or graphic form. For a given location or collector type, the annual solar fraction of the load, F, and its derivative with respect to the area to load ratio, OFIO(AIL), can be presented as a function o f AlL. Similarly, the critical parameter, Z*, can be presented as a function of CdCAL. After a set of these tables or graphs has been generated, solar heating system performance, economic viability, and optimal system design can be determined with minimal calculations. For example, a solar heating system distributor, serving several locations with a particular type of collector, could generate a small set of tables or graphs for the given collector type in the various service locations (e.g. Appendix A, Tables 3 and 4). On the other hand, a local equipment dealer handling several types of collectors could produce tables or graphs for several collector types in the particular location (e.g. Fig. 5). In either case, all subsequent solar heating system calculations will be greatly reduced. The optimization tables of Appendix A serve to illus-
M. J. BRANDEMUEHLand W. A. BECKMAN 3.01
, , ~ , ~ ~ , , A'FI~(r=) = 0'5 F~ UL= 3.0 w/m2"C 8'PR~T~I)=0.6 PRUL=2.5 w,/m~C
*
t\.,/"B
a-~l~-
ti~\ \
, .
~----~-CA vs. A / L
v
7g
~
E
0
0.5
LO
AREA TO t.OAD RA'rlO A/L OR PARAMETER
[ M2/GJ ] CE/CAL
Fig. 5. Optimization graph for two air collectors in Madison, Wisconsin. trate the use of this method. Table 3 has been constructed for a particular air collector in the 19 United States locations given in Table 2. Similarly, Table 4 uses a particular liquid collector in the same 19 locations. Each table consists of two parts. The left half of each table gives the value of (P2C~P~Cp)*, i.e. Z*, satisfying eqn (12) as a function of C~JCAL.The right half of each table gives the values of F and OFlr~(A/L)as a function of A/L. By eqn (9), the optimum value of A/L, for a given value of P2CA/P~CF,can be determined from this half of the table. The optimization tables have been generated using the weather data from Ref.[3]. To use the tables, it is first necessary to evaluate PI and P2 from economic conditions. By consulting with a solar system equipment supplier, the installed system costs can be determined. Knowing the present cost of delivered fuel energy and the annual heating load, the values of P2CA/P~CF and CdCaL can be calculated. Entering the tables at the appropriate location and collector type, the critical value of (P2CA/P,CF)* = Z* can be determined corresponding to the value of CdCAL. If P2CA/P, CF -< Z*, the solar heating system is economically competitive with the conventional heating system. Using the right half of the table, the user can interpolate to determine the economically optimal area to load ratio and the corresponding fraction of load met by the optimized system for the particular value of P2CA/P~CF. If P2CA/P,CF > Z*, the solar heating system is not currently economical. However, eqn (14) can be used to determine the year in which it will become competitive with the conventional system. To illustrate this procedure, consider a homeowner in .Madison, Wisconsin, who is installing a solar air system with combined space and domestic water heating capabilities. The system has two cover non-selective. collectors (Fs'(ra), = 0.5, FR'UL= 3.0 Wm-2°C-I), with area dependent costs of $200m -2 and fixed costs of $1000. The water heating load is estimated to be 23.4GL The long term average annual degree days for Madison is
4294°C days. Using the typical house described in Section 2, the building UA from Table 2 is 25.4 MJ°C-a day-1 resulting in an annual space heating load of 109.1GJ. The total load is then 132.5GJ. The present cost of conventional energy is assumed to be $9.90 GJ-' (corresponding to electric resistance heating). The homeowner in this case has decided to include loan costs, miscellaneous costs, taxes, and salvage value. Assuming d = 0.08, e = 0.10, i = 0.09, g = 0.06, N~ = 20, NL = 20, t =0.30, t =0.02, D=0.10, M=0.01, V=0.7, G = 0.10, and C = 0 (residence), P1 is equal to 22.169 from eqn (5) and P2 is 1.172 from eqn (6). In this case, PeCalPjCv= 1.068. Using Table 3 with CdCAL=O.038, it is seen that Z*= 1.395. Since P2CAJP~CF< Z*, the combined solar heating system is economically viable. To determine the optimal ratio of A/L, the user must interpolate from the values in the table. For P2CA/P~CF= 1.068, the interpolation yields the values of ALL=0.252, and F=0.38. With L = 132.5 GJ, A = 33.4 m2, the life cycle savings from eqn (1) are $2051. The homeowner may also wish to compare the solar heating system with an oil furnace for which the delivered cost of energy (including furnace efficiency) is $5.40 GJ-l. In this case, ~he value of P2CalPICv is calculated to be
P2C~
- - =
PICF
1.958.
Comparing this value with the critical value of Z* = 1.395, the solar heating system is not competitive with the conventional oil furnace. However, the year at which the solar system will become competitive can be calculated from eqn (14) by assuming that the collector component costs will increase at the general inflation rate, g = 0.06. ., 1n(1.395/1.958) I = ~ = 9 . 2 .
That is, it will be 9.2 yr before any system design with these collector characteristics will be competitive with oil heating in Madison, Wisconsin. 6. SENsrrlVITYANALYSIS The economic evaluation and optimization methods presented here are based on a set of assumptions: load distribution assumptions, f-chart assumptions, cost modeling assumptions, etc. However, the more subtle, and probably most important assumption is that the user can assign realistic, reliable values to economic variables. There is an inherent uncertainty in predicting future expenses and benefits. This uncertainty is magnified by the instability of current international energy affairs. In light of this dilemma, the results of both the life cycle cost analysis and the optimization procedures mustbe accepted with discretion, as they may be strongly influenced by an unknown future. The effect of variable uncertainties on economic analysis results can be evaluated by expressing-the life
Economic evaluation of solar heating systems
cycle solar savings as a linearized Taylor series. For a given set of conditions, the change in life cycle savings, ALCS, resulting from a change in a particular variable, Axe, can be approximated by the following: ALCS = oLCS Ax. Oxj '"
(15)
The expression for aLCSlaxj can be obtained by direct differentiation of the life cycle savings equation. The life cycle cost model of eqns (1) and (4)--(6) will be used for the purpose of this discussion. The derivatives of P~ and P2 from eqns (5) and (6) are given in Appendix B. The expressions in Appendix B can be used to investigate either the overall sensitivity to small changes in the variables or the error propogated by uncertainties in particular variable values. To illustrate the use of these relationships, consider the example of Section 5. The variables describing this analysis and the corresponding sensitivity information are given in Table 1. The derivatives with respect to Pj, P2 and LCS have been calculated using the equations of Appendix B for the given set of variable values. The table also gives the change in solar system life cycle savings, ALCS, caused by a 10 per cent relative increase in each of the variables to facilitate better understanding of the sensitivity. The table shows, for example, that a 10 per cent increase in the discount rate from 8.0 to 8.8 per cent yields a decrease in the value of P~ of approx. 1.824 (i.e. -228 x 0.008), or a relative change of -8.2 per cent. The value of P2 decreases by approx. 0.068, or a relative change of -5.8 per cent. The value of LCS decreases by approx. $385, or a relative change of -19 per cent. It can also be noted that the 10 per cent increase in discount rate will cause a 2.6 per cent increase in the value of P2CA[PJCF, implying that the slope of the F vs A l L curve defining the economically optimal collector
7
area increases by 2.6 per cent. Consulting the optimization tables at P2CA]P~CF = 1.096, the optimal collector area would decrease to 31.8m 2 with the annual solar fraction of load of 37 per cent. The information of Table 1 can also be used to estimate the uncertainty in life cycle savings due to uncertainty in different variables. Consider, for example, uncertainties of the following magnitudes: ACE = _+$100, AM = _+0.005, Ad = _+0.005, Ae = -+0.01, Ag = _+0.005, A / = -+0.025, AL = -+10GJ, AF =-+0.02, AG = _+0.1. The maximum possible uncertainty in life cycle savings due to these uncertainties can be estimated by the following:
ALES.,× = ~, laLcs] Ax. ~=11 axj [ '" Using Table 1 and the above uncertainties, ALCSMAx = 117+599+240+ 1016+99+ 159+834+582 + 165 = $3811. Notice that the maximum uncertainty is greater than the life cycle savings for this example. A more reasonable savings uncertainty estimate caused by these variable uncertainties can be obtained by the following: ALCSprob =
= \--~xj
'1 J
A LCSr,rob= $1600. The explicit effect of any one variable on the life cycle savings is largely determined by the values of many other variables. Because of this complexity, "rule of thumb" sensitivity trends are not easily recognized, and a quantitative analysis is usually required.
Table I. Sensitivity analysis example Variables, xj I. Noncommercial or commercial flag (0 or 1) 2. Area dependent investment cost 3. Area independent investment cost 4. Ratio of downpayment to initial investment 5. Ratio first year's misc. costs to init. inv. 6. Ratio first year's assessed value to init. inv. 7. Ratio of salvage value to initial investment 8. Annual market discount rate 9. Annual market rate of fuel price increase 10. Annual interest rate on mortgage I 1. Annual rate of general inflation 12. Property tax rate 13. Effective income tax rate 14. Duration of economic analysis (years) 15. Term of mortgage (years) 16. Depreciation lifetime (years) 17. Cost of conventional fuel in first year (S/GJ) 18. Annual heating and hot water load (GJ) 19. Annual load fraction supplied by solar 20. Collector area tALCS for AxJxj = 0.10.
Nominal Values
aPt ~x.j
axl
OLCS Oxj
ALCSf
0.000 200.000
1000.000 0.100 0.010 0.700 0.100 0.080 0.100 0.090 0.060 0.020 0.300 20.000 20.000 20.000 9.900 132.500
0.380 33.400
0.00000
0.00000
0.00000 0.00000 0.00000 0.12842 0.00000 0.155% x 102 0.00000 0.21834 0 . 0 0 0 0 0 -0.21455 -0.22802x 103 -0.85389 x lip 0.20373x 103 0.00000 0.00000 0.45908x 10' 0.00000 0.25869x 10I 0.00000 0.76419x 10I 0 . 0 0 0 0 0 -0.83024 0.13242x 101 0.14384x 10I 0 . 0 0 0 0 0 -0.37943x 102 0.00000 0.00000 0.00000
0.00000 O. ~000 0.00000
0.00000
0.00000
-0.39137 x 102 -0.11718 x lip -0.98630 x 103 -0.11978 x 106 -0.16769 x 104 0.16477x 104 -0.48084 x liP 0.10155 x 10~ -0.35258 x liP -0.19867 x liP -0.58690 x liP 0.63763x 104 0.54961 x 103 0.29140× 102
-783 - 117
0.11162x 104 0.83398x 102 0.29080x 102 0.00000
1105 1105 1105 0
-
10
-
119
117 17 - 385 1016 - 317
-
-
119
117 191 1099 60
M. J. BRANDEMUEHLand W. A. BECKMAN
8
The economic optimization of a solar heating system is also affected by variable uncertainty. Moderate changes in the value of P2CA/P,C~ often produce substantial chances in the optimal collector area. Fortunately, the curve of life cycle savings vs collector area exhibits a broad optimum as seen in Fig. 4. Near the optimum, large uncertainty in collector area results in small uncertainty in the life cycle savings. In general, the effect of variable uncertainty on system optimization is overshadowed by the effect on the life cycle savings evaluation.
and the f-chart solar heating system design correlations to develop a tabular or graphic method for estimating the optimal collector area and evaluating the solar system economic effectiveness. Using this method, an engineer, architect, or solar equipment company can prepare a set of tables or graphs for location and collector types frequently used. Once these tables or graphs are constructed, computations for solar heating system thermal and economic evaluation are greatly reduced.
RE~ERENCES
7. CONCLUSION
The methods presented in this paper allow the individual architect or engineer to design an economically optimal solar heating system and evaluate the economic comparison with an alternative conventional heating system. Life cycle cost analysis with two parameters, P, and /'2, is used for economic evaluation. The use of these parameters requires one economic assumption: all costs which contribute to the life cycle costs of the solar heating system or conventional system are directly proportional to either the first year fuel cost or the initial solar system investment. The introduction of P, and/'2 presents the life cycle savings equation in a compact and manageable form. This form accomodates straightforward manipulation of the savings equation to determine the optimal system design. The optimization method presented here uses the parameterized savings equation
1. S. A. Klein, W. A. Beckman and J. A. Duffle, A design procedure for solar air heating systems. Solar Energy 19, 509 (1976). 2. S. A. Klein, W. A. Beckman and J. A. Duffle, A design procedure for solar air heating systems. Solar Energy, 19, 509 (1977). 3. W. A. Beckman, S. A. Klein and J. A. Duffle, Solar Heating Design. Wiley lnterscience. New York (1977). 4. ASHRAE Handbook and Product Directory, Systems, American Society of Heating, Refrigerating, and Air Conditioning Engineers, New York (1973). 5. ASHRAE Standard 90-75, Energy conservation in new building design. American Society of Heating, Refrigerating, and Air Conditioning Engineers, New York (1975). 6. W. D. Collins, Teml/erature of Water Available for Industrial Use in the United States. U. S. Geological Survey Water Supply Paper, No. 520F (1925). 7. R. T. Ruegg, Solar Heating and Cooling in Buildings: Methods of Economic Evaluation, U. S. Department of Commerce, NBSIR 75-712 (1975).
8. APPENDIXA
-
Table 2. Load calculation data for combined systems Water Annual Mains Degree Building HeatingLoads (UA) Water Space Location Temperature Days
Albuquerque, NM
13.
2384.
28.3
21.6
Atlanta, GA
18.
1719.
29.3
19.3
50.3
Bismark, ND Boston, MA
6. 9.
5024, 3130.
24.3 27.1
24.8 23.4
122.0 84.9
Boulder, CO Cleveland, OH Columbia, MO
67.6
9.
3078.
27.2
23.4
83.8
II.
3419.
13.
2803.
26.7 27.7
22.5 21.6
91.3 77.5
19.
1272.
30.0
18.8
38.1
7.
4306.
25.4
24.3
109.4
Lander, WY
6.
4372.
25.3
24.8
llO.7
Lincoln, NB
12.
3259.
27.0
22.9
87.8
Los Angeles, CA
19.
lOll.
30.4
18.8
9.
4294.
25.4
23.4
Dallas, TX Great Falls, r.ff
Madison, WI
30.7 lOg.1
Miami, FL
25.
liD.
31.8
16.3
3.8
Nashville, TN
16.
2053.
28.8
20~2
59.l
New York, NY
If.
2673.
27.9
22.5
74.6
Phoenix, AZ
17.
862.
Reno, NV
If.
3345.
Washington, DC
14.
2347.
30.6 26.9 28.3
Ig.7 22.5 21.1
26.4 89.9 66.5
TemDeratures in °C. Degree days in °C-days. UAs in Md-°c-l-day -1 Heating loads in GJ.
Economic evaluation of solar heating systems Table 3. Combined heating (liquid) optimization tables, Fk(ra),, = 0.60 F~UI = 4.00 Wm -2 °C-' TABULATED: f f ~
PtCF"
cat .00
TABULATED: ~
P I CF
.05
.80
.00
.854
.655
.710
.625
.501
.879
.694
.531
.652
.518
.422
.928
.735
.561
.934
.622
.493
.393
1.598 1.399 1.159
.767
.608
.484
TX
2.505 1.717 1.503 1.245
.738
.633
.505
GREAT FALLS
MT
2,567 1.802 1.582 1.310
.900
.709
.542
LANDER
iqY
3,346 2.296 2.010 1.665
.987
.870
.664
LINCOLN
NE
2.435 1,661
1.454 1.204
,811
.643
.513
LOS ANGELES
CA
2.899 2.I54
1.886 1.562 1.068
.840
.642
MADISON
WI
2.169 1.507 1.320 1.093
.748
.593
.453
MIA~ff
FL
2.677 2.026 1.774 1.469 1.041
.822
.628
NASHVILLE
TN
2.175 1.457 1.275 1.056
.626
.538
.428
NEW YORK
NY
1.684 1.195 1.046
.596
.472
.384 .641
3.707 .00 2.384 .00 2.609 .00 1.842 .00 2.693 .00 1.912 ,00 2.409 ,00 2.505 .00 2.567 .00 3.346 .00 2.435 .00 2.899 .OO 2.169 .00 2.677 .00 2.175 .00 1.684 .00 3.595 .00 3.406 .00 2.040 .00
3.707 2.331 2.041
.10
ALBUQUERQUE
N~I
ATLANTA
CA
2.384 1.653 1.447 1.199
~ ISFIARCK
ND
2.609 1+790 1.567 1.298
BOSTON
MA
1.842 1.309
.949
BOULDER
CO
2.493
1.832 1.604 1.328
CLEVELAND
OH
1.912 1.288 1.128
COI,UMRIA
MO 2.409
DALLAS
.30
1.690 1.002
1.146
.866
AZ
3.595 2.434 2.131
1.765 1.046
+846
RENO
NV
3.406 2.334 2.044 1.692 1.003
.866
.659
WASUINGTON
DE
2 . 0 4 0 1.418 1.242 1.028
.554
,444
PIIOENIX
ra 2 GJ- t
AND LOAD FRACTION AlL m 2 GJ I .50
LOCATION
.02
CRITICAL
rn2 GJ 1
.698
.10
.20
2.525 1.710 .31 .51 1.781 1.318 .21 .36 1.949 1.443 .23 .39 1.417 1.027 .16 .29 1.997 1.532 .22 .40 1,396 .938 .16 .28 1.731 1.148 .21 .35 1,847 1.344 .22 .37 1.986 1.494 .23 .39 2.421 1.798 .29 .50 1.806 1.288 .21 .36 2,347 1,819 .26 .47 1.643 1.191 .19 .33 2.211 1.791 .24 .44 1.561 1.115 .19 .32 1.291 .985 '.15 .26 2.629 1.777 .31 .53 2.551 1.814 .30 .51 1.526"1.132 .18 .31
.30
.40
1.290 .65 ,984 .47 1.126 ,52 .825 .38 1.210 .54 .769 .37 .951 .45 .917 .48 1.143 .53 1.441 .66 1.053 .48 1.389 .63 .955 .43 1.419 .60 .825 .41 .739 .35 1.102 .66 1.398 .67 .875 .41
1.028 .76 .755 .56 .848 .62 .730 .45 1.O21 .65 .635 .44 .778 .54 .761 .57 .959 .63 1.038 .78 .836 .57 1.022 .75 .828 .52 1.094 .73 .661 .48 .634 .41 .897 .76 1.OO1 .78 .754 .49
.60
.80
.513 .92 .595 .69 .597 .76 .529 .58 .614 ,81 .469 .54 .569 .67 .601 .70 .609 .78 .539 .93 .604 .71 .439 .90 .584 .66 .533 .89 .509 .59 .490 .53 .495 .89 .407 .92 .556 .62
.255 .99 .466 .80 .378 .86 .436 .68 .370 .91 .382 .63 .456 .77 .410 .81 .352 .87 .116 .99 ,475 .82 .161 .97 .429 .76 ,175 ,96 .419 .69
.403 .62 .278 ,96 .211 .97 .425 .71
Table 4. Combined heating (air) optimization tables, Fk(ra), = 0.50 FkUI = 3.0 Wm -"°C ' TABULATED: ~
PICF"
P,C , t TABULATED: ~ m- GJ
CRITICAL
P I C~-"
AND LOAD FRACTION A/L m" GJ- I
C ! L m'_ Gj t .00
LOCATION ALBUQUERQUE ATLANTA BISMARCK BOSTON BOULDER CLEVELAND COLLIMBIA DALLAS GREAT FALLS LA~DER LINCOLN LOS ANGELES >IADISON MIAMI NASHVILLE NEW YORK pHOENIX RFNO WARIIINGTON
.02
.05
.10
.30
.50
.80
.00
•865
.658
.?58
.642
.491
4.345 .00 2.763 .O0 3.029 .00 2~693
NM 4 . 3 4 5 2.472 2.165 1.793 1.062 GA
2.763
1.764 1.544 1.279
ND
3.029 1.889 1.653 1.369
.812
.696
.533
HA
2.123 1.383 1.211 1.003
.661
.523
.399
CO
2.895 1.952 1.709 1.415
.839
.734
.564
OH
2.202 1.348 1.180
.977
.579
.491
.377
~10
2,796 1.689 1.478 1.224
.726
.614
.471
TX
2.912
1.612 1.335
.791
.652
.498
MT
2.979 1.911. 1.673 1.385
.821
709
.543
~tY
3.909 2.431 2.128 1.762 1.045
.872
.663
NE
2.825 1.762 1.543 1.277
757
.649
.498
CA
3.373 2.316 2.027 1.679
.995
.848
.646
Wl
2.505 1.588 1.390 1.151
~754
.595
.455
FL
3.111 2.244 1.965 1.627
.964
.854
.650
TN
2.518 1.541
.662
551
.422
1.841
1.349 1.117 .918
"
"
"
NY
1.934 1.267 1 . 1 0 9
.602
.476
.379
AZ
4 . 2 2 2 2.616 2.290 1.897 l 124
.799
.657
NV
3.976 2.465 2.158
.869
.659
DC
2.357 1.505 1.318 1.091
,561
.429
'
1.187 I.O59 •
709
.10
2.515 .34 1.869 .23 1.952 .25 1.477 .18 2.895 2.101 .00 .25 2.202 1.423 .00 .18 2.796 1.784 .00 .23 2.912 1.952 .00 .24 2.979 2.053 .00 .25 3.909 2,455 .O0 .32 2.825 1.867 ,00 .23 3.373 2.495 .00 .29 2.505 1.689 ,00 .21 3.111 2.437 .00 .28 2.518 .O0 1.618 .2O
.20
.30
1.637 .54 1.268 .38 1.387 .41 1.011 .30 1.501 .43 .940 .29 1.138 .37 1.291 .40 1.444 .42 1•705 .53 1.242 .38 1.799 .51 1.169 .35 1.855 .49 1.021 .33
1,195 .68 .924 .49 1.O36 .53 .792 .39 1.133 .56 .719 .37 .916 .47 .882 .50 1.054 .54 1.289 .68 .986 .49 1.257 .66 .895 .45 1.365 .65 .780 .42
1,934 .948 ,00 + 1.349 .16 .28 4,222 2.791 1.572 ,00 .35 .57 3.976 2 . 0 3 2 1.705 .O0 .33 .54 2.357 1.064 .00 1.593 .20 .33
.40
.60
.886 .411 .78 .92 .728 .513 .58 .70 .817 .512 .62 .75 .667 .487 .46 .58 .895 .543 .66 .80 .597 .421 .44 .54 .712 .508 .55 .67 .725 .514 .59 .71 .834 .492 .63 .77 ,898 .415 .79 .92 .771 .536 .58 .71 .903 .455 .76 .90 .740 .503 .53 .65 .968 .443 .77 .90 .639 .445 .49 .60 .706 .598 .438 .36 .42 .53 1.O4t .803 .412 .69 .78 .89 1.247 .837 .444 .68 .78 .91 .840 .694 .490 .42 .50 .62
.B0 .189 .98 .373 .79 .333 .84 .364 .66
.337 .89 .316 .61 .363 .76 .374 .80 .322 .85 .185 .98 .377 .80 .197 .96 .357 .74 .083 .97 .347 .68 .335 .60 .217 .95 .150 .96 .356 .70
10
M.J. BRANDEMUEHLand W. A. BECKMAN 9
.
~
B
OP2 _ O--i--
The following equations can be derived from the expressions for P~ and P: of eqns (5) and (6). The present worth function PWF(a, b, c) is abbreviated fia, b, c).
OP,o_.=.d_(I - C[~odf(NE' e' d)'
OP' = ( l - C[~oef(N~' e'
ONeOP--£= (I-Cto~NJ(NE,
0P~=-Cf(N~,e,d) Ot
-
e,d),
(Nl,O,d)
-
.
OP2=~Dl - ( l - t ~ - t f ( N " " d ) [ i - f ( N Z O , OP2 = ~-~
.,,"OP2 ~ = rH
Ct)f(N~, g, d),
(! -
aP2 aO
aP,
--~ = ( D -
at
I ( i + d ) ~E'
I
oP~ 0
~
i)]
t)I(NB,g, d)
tf(Nt.,O,i)
'" t
1
+
t)
1
0
G
d)}
C{ o
+N~• (i + d)U~t - N--~-dli~,, O, d)
.
I
a
1
0
;)tV]~fiN.,g,d) .~.
I JL f ~ 2
~ )
tiN,, i, d) Io___ , ,
~
, [f( NL, O.i)]2JONL J " ' L '
o ~, i)
r)V]Zf(NE, g,d)+~ln(l+d) e ~ -J
T~?""
....
-.
'
I
D)t[, - ~ ] a - ~ / ( N , ,O
,,. d).
fiNL,O,i)
V]~df(N,,g,d , - ( l - D)[{/(N:0, i):d f ( N ' '
[ i- f( N: O,i)]-~f(N', i,
C/')M+(1-
-(1-
Note that since NI is usually equal to either NL or NE, the expression for the partial derivative of/'2 with respect to Nm will usually be added to either the expression for NL or NE. Most of the above equations involve derivatives of the present worth factor. Using eqn (4),
I-D 0 "N 0 f(NL'~,i) ~/( " ' d) + [(1 - Ci)M +t(i-
.
OP'-ffl aNe - "" - Ct)M + t(l ON,
]}
a -
= (D- l){ll - " tiN,, o, dl
aP2
-tVf(NE, g,d)-C[Mf(N,,g,d)+~----J(N2,0,d)] 0P2 ad
-
+f(Nb i, d)[ I
-~=(l-t)Vf(N~,g,d)
1)~f(NuO, d) +f(Nt i d)[i -
r, f(N1, O, d)
(D- IXl - tj[f(N~, O, i)l~ ~i/(NL' O, 0
0, d)
~af(a, b, C)
I [l+b\ °, [l+b\
a
1
0
1
a
a
i+b °
l+b °
]
~.~(a,b,c)= ~_ b[T-~--~c(T-~-~c ) - f(a,b,c)].