Due to the nature of the economic analysis, i. e., predicting the way various costs will occur during the life of a solar energy system, a number of uncertainties are involved in the method. The person responsible for the economic analysis of a solar energy system must consider a number of economic parameters and how these will develop in the years to come. A usual technique is to find how these parameters were modified during the previous years and expect that the same behavior will be reflected in the future years. These two periods are usually equal to the expected life of the system. Additionally, the prediction of future energy costs is difficult because international oil prices change according to the quantity supplied by the oil-producing countries. Therefore, it is desirable to be able to determine the effect of uncertainties on the results obtained from an economic analysis.

For a given set of economic conditions, the change in LCS resulting from a change in a particular parameter, say, Ax;-, can be obtained from

Подпись: ALCS = Дх. OXj — {P1Cf1LF – P2(CaAc + Cj)] Ax. (12.38)

image1718 Подпись: (12.39)

When uncertainties exist in more than one parameter, the maximum possible uncertainty is given by

Подпись: ALCSp image1721 image1722 Подпись: 2 Подпись: (12.40)

Therefore, the most probable uncertainty in LCS can be written as

From Eq. (12.38),

Подпись: (12.41)dLCS = d(P1CF1LF) _ d[P2(CgAc + Ct)] dx. dxt dxt

The partial derivatives of the ratios P1 and P2 can be obtained using Eqs. (12.32) and (12.34) for the most crucial parameters, as follows.

Подпись: dP1 diF image1727 Подпись: (12.42)

For the fuel inflation rate,

image1729 Подпись: (12.43)

For the general inflation rate,

dPL = 1 – Cte dR (1 + d)ne

Подпись: (12.44)

Подпись: For the resale value, Подпись: dLCS dF Подпись: P1CF1L Подпись: (12.45)

The partial derivative of the LCS with respect to the solar fraction is

The partial derivatives of all parameters can be seen in (Brandemuehl and Beckman, 1979).

Finally, it is necessary to know the partial derivatives of the PWF values. Using Eqs. (12.17) and (12.18), the following equations can be obtained, as given by Duffie and Beckman (2006).

If i = d,

dPWF(n, i, d) = = _1_ (12 46)

dn 1 + i 1 + d

Подпись:Подпись:dPWF(n, i, d) _ n(n — 1)

di 2(1 + i)2

dPWF(n, i, d) _ _ n(n + 1) dd 2(1 + d)2

If i Ф d,


image1739 Подпись: (12.50)


Подпись:Подпись: dPWF(n, i, d) dd Подпись: PWF(n, i, d)(12.51)

It should be noted that, in order to estimate the uncertainties of more than one variable, the same procedure can be used to determine the appropriate terms in Eqs. (12.39) and (12.40). A much easier way to determine uncertainties is to use a spreadsheet for the calculations. In this case, the change of one or more parame­ters in appropriate cells (e. g., cells containing i, d, iF etc.) causes automatic recal­culation of the spreadsheet, and the new value of LCS is obtained immediately.

Example 12.13

If, in a domestic solar energy system economic analysis, the fuel inflation rate is taken as 8%, find the uncertainty in LCS if the fuel inflation rate is ±2%. The solar energy system replaces 65% of the annual load, the first year fuel cost is $950, the initial installation cost is $8,500, ne = 20 years, d = 6%, P1 = 21.137, and P2 = 1.076.


The LCS of the system without uncertainty is obtained from Eq. (12.31):

LCS = P1CF1FL – P2CS = 21.137 X 950 X 0.65 – 1.076 X 8,500 = $3,906

The fuel inflation rate affects only P1. Therefore, from Eq. (12.42), for C = 0,

3PL = 8PWF(ne, iF, d)

diF diF

Подпись: dPWF(n, i, d) di image1745

From Eq. (12.50),


The uncertainty in LCS can be obtained from Eqs. (12.38)-(12.41), all of which give the same result, as only one variable is considered. Therefore, from Eq. (12.38),



F1 diF F

= 950 X 0.65 X 212.4 X 0.02 = $2,623

Therefore, the uncertainty in LCS is almost equal to the two thirds of the originally estimated LCS, and this is for just 2% in uncertainty in fuel inflation rate.


As an assignment the student is required to construct a spreadsheet program that can be used to carry out almost all the exercises of this chapter.

[1] This is according to a renewable energy-intensive scenario that would satisfy energy demands associated with an eightfold increase in economic output for the world by the mid­dle of the 21st century. In the scenario considered, world energy demand continues to grow in spite of a rapid increase in energy efficiency.

[2] The term biomass refers to any plant matter used directly as fuel or converted into fluid fuel or electricity. Biomass can be produced from a wide variety of sources such as wastes of agricultural and forest product operations as well as wood, sugarcane, and other plants grown specifically as energy crops.

[3] TOE = Tons of oil equivalent = 41.868 GJ (giga, G = 109).

[4] Meteorological data for various locations are shown in Appendix 7.

[5] + (2 X 2) + 0.723 – 1

0. 1 + 0.05 X 2(1 – 0.1) 0.88

= 2.306 W/m2-K

The difference between this value and the one obtained in Example 3.2 is only 4.6%, but the latter was obtained with much less effort.


The area lost at an incidence angle of 60° is:

Area lost = A tan(60) = 3.125 X tan(60) = 5.41m2

The geometric factor Af is obtained from Eq. (3.110):

[8] 2 52

= 2.5 X 0.625 + 0.625 X 2.5 1 +

[9] dT

q = Ur (T – To) + mc —

[10] Direct or open loop systems, in which potable water is heated directly in the collector.

• Indirect or closed loop systems, in which potable water is heated indi­rectly by a heat transfer fluid that is heated in the collector and passes through a heat exchanger to transfer its heat to the domestic or service water.

Systems differ also with respect to the way the heat transfer fluid is transported:

• Natural (or passive) systems.

• Forced circulation (or active) systems.

[11] 1 1

– +

UrAc UcAc UrAr (6.36)

which gives

UR = ———- L1———- (6.37)

—- + ————–

Uc Ur (Ar /Ac)

In some countries minimum U values for the various building components are specified by law to prohibit building poorly insulated buildings, which require a lot of energy for their heating and cooling needs.

Another situation usually encountered in buildings is the pitched roof shown in Figure. 6.3.

Using the electrical analogy, the combined thermal resistance is obtained from

Rtotal = Rceiling + Rroof


[13] Charge carrier. The charge carrier is the ion that passes through the electrolyte. The charge carrier differs among different types of fuel cells. For most types of fuel cells, however, the charge carrier is a hydrogen ion, H+, which has a single proton.

• Contamination. Fuel cells can be contaminated by different types of molecules. Such a contamination can lead to severe degradation in their performance. Because of the difference in electrolyte, catalyst, operating temperature, and other factors, different molecules can behave differ­ently in various fuel cells. The major contamination agent for all types of fuel cells is sulfur-containing compounds, such as hydrogen sulfide (H2S) and carbonyl sulfide (COS).

[14] The plants offer the lowest-cost solar-generated electricity for many years of operation.

• Daytime peaking power coverage and with hybridization could provide firm power, even during cloudy periods and night.

• Environmental protection is enhanced because no emissions occur dur­ing solar operation.

• Positively impacts local economy because systems are labor intensive during both construction and operation.

The negative impacts include:

• Heat transfer fluids could spill and leak, which, can create problems in the soil.

• Water availability can be a significant issue in the arid regions that are best suited for trough plants. The majority of this water is required for the cooling towers.

[15] 38

= 0.416 or 41.6%



Updated: August 29, 2015 — 5:53 pm