# Optimization Using P1t P2 Method

As we have already seen in solar energy system design, the collector area is considered as the primary parameter for a given load and system configuration. The collector area is also the optimization parameter, i. e., the designer seeks to find the collector area that gives the highest life cycle savings. A method for the economic optimization was given in Section 12.2.3, in which life cycle savings

 FiGURE 12.3 Optimum collector area determination from the slope of the Fversus Ac curve.

are plotted against the collector area, Ac, to find the area that maximizes sav­ings. The optimization procedure can be simplified if life cycle savings (LCS) can be expressed mathematically in terms of the collector area. Therefore, the optimum is obtained when

d(LCS) = 0

dAc

or, by using Eq. (12.31) for LCS and Eq. (12.2) for Cs,

dF

PCp1L ^ ~ P2Ca = 0

Rearranging, the maximum savings are obtained when the relationship between the collector area and solar fraction satisfies the following relation:

dF = p2Ca

dAc PCFL

According to Eq. (12.37), the optimum collector area occurs where the slope of the F versus Ac curve is P2CA/P1CF1L. This condition is shown in Figure 12.3.

Example 12.12

For a residential liquid-based solar space heating system, the following infor­mation is given:

Annual heating load = 161 GJ.

First-year fuel cost rate, CF1 = \$8.34/GJ.

Area-dependent cost = \$210/m2.

Area-independent cost = \$1,150.

Market discount rate, d = 8%.

Mortgage interest rate, dm = 6%.

General inflation rate, i = 5%.

Fuel inflation rate, iF = 9%.

Term of economic analysis = 20 years Term of mortgage load = 10 years Down payment = 20%.

Ratio of first year miscellaneous costs to the initial investment, = 0.01. Ratio of assessed value of the system in the first year to the initial invest­ment, V1 = 1.

Ratio of resale value = 0.3.

Property tax, tp = 2%.

Effective income tax, te = 35%.

In addition, the solar fraction to the collector area varies as shown in Table 12.9.

 Area (m2) Annual solar fraction (F) 0 0 20 0.29 50 0.53 80 0.68 100 0.72
 Table 12.9 Solar Fraction to Collector Area in

 Example 12.12

Determine the optimum collector area that maximizes the LCS and the LCS. Solution

As the system is residential C = 0. The present worth factors can be estimated from Eq. (12.18) or the tables of Appendix 8 as follows:

PWF(ne, iF, d) = PWF(20, 0.09, 0.08) = 20.242 PWF(nmin, 0, d) = PWF(10, 0, 0.08) = 6.7101 PWF(nL,0, dm) = PWF(10, 0, 0.06) = 7.3601 PWF(nmin, dm, d) = PWF(10, 0.06, 0.08) = 8.5246 PWF(ne, i, d) = PWF(20, 0.05, 0.08) = 14.358

From Eq. (12.32),

P = (1 – Cte) PWF(n, iF, d) = 20.242

The various terms of parameter P2 are as follows:

P21 = D = 0.2

P2,4 = (1 – Cte)MjPWF(ne, i, d) = 0.01 X 14.358 = 0.1436

P2 5 = tp(1 – te)V1PWF(ne, i, d) = 0.02(1 – 0.35) X 1 X 14.358 = 0.187

Ct

p = PWF(nmin,0, d) = 0

nd

Finally, from Eq. (12.34),

From Eq. (12.37),

The data of Table 12.9 can be plotted, and the optimum value of Ac can be found from the slope, which must be equal to 0.00874. This can be done graphically or using the trend line of the spreadsheet, as shown in Figure 12.4, and equating the derivative of the curve equation with the slope (0.00874). By doing so, the only unknown in the derivative of the trend line is the collector area, and either by solving the second-order equation or through trial and error, the area can be found that gives the required value of the slope (0.00874).

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As can be seen, the optimum solution occurs at about a collector area of 30 m2 and F = 0.39. For this size, the total cost of the solar energy system is obtained from Eq. (12.2) as

Cs = 210 X 30 + 1150 = \$7,450

The life cycle savings are obtained from Eq. (12. 31) as

LCS = pCF1FL – P2CS = 20.242 X 8.34 X 161 X 0.39 – 1.1316 X 7,450 = \$2,170

Updated: August 29, 2015 — 5:41 pm