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## Q. 7.11

Design of a Solar-Power Generator

A solar power system capable of heating water into steam to power a turbine/ generator needs to be designed. The problem definition includes the following information and system requirements:
• Each second, 0.9 kJ of solar energy strikes a square meter of ground.
• The target value for the real efficiency of this system is 26%.
• The system needs to produce an average of 1 MW of power during a 24-h period.
• The heat engine releases its waste heat into the surrounding air.

(a) Design a solar power system to fulfill these requirements. (b) Assuming the resulting steam powers an ideal Carnot heat engine that operates between $T _{l} =25^°C and T_{ h} = 400^°C,$ calculate the upper bound on efficiency for the engine.

Approach
Following the general design process introduced in Chapter 2, we will first develop any necessary additional system requirements. Then, we will develop conceptual alternatives and converge on the most effective option.
We will then determine the more detailed geometric specifications of the system. Lastly, we will use Equation (7.13)

$\eta _{c}=1-\frac{T_{1}}{T_{h}}$                  (7.13)

to calculate the efficiency of the proposed engine.

## Verified Solution

(a) System requirements: In addition to the requirements given in the problem, we will need to make other assumptions about the system performance.
• We recognize that the solar power plant can operate only during the peak daylight hours, which is roughly one-third of a day. Therefore, to have an average output of 1 MW over the course of a full day, the plant must be sized to produce 3 MW during the daylight hours.
• A method to store the excess power during the day and retrieve it at night has to be devised, but we will not consider that aspect of the power plant’s design in this example.

Conceptual design : At this stage, we would develop a number of conceptual alternatives for the solar power system. This could include a system of solar cell panels to generate electricity to heat the water, a passive solar system to store heat in a thermal mass to heat the water, or a solar thermal energy system using mirrors to directly heat the water. Ideally, we would generate many alternatives here, potentially even developing ideas for new technologies. Simulation models can then be used to help mechanical engineers select the most effective option. We will assume that the solar thermal energy using trough-shaped mirrors has been determined to be the most effective option through preliminary cost, efficiency, and manufacturing analysis. See Figure 7.22.

Detailed design : We determine that parabolic trough-shaped mirrors will be used to collect the sunlight to heat water into steam. We choose to use standard $12-m^2$ mirrors and then need to determine the number of mirrors needed to meet the 3-MW power requirement. By using the target real efficiency of  $\eta = 0.26$, we will calculate the heat input and energy output of the power plant during a 1-s time interval. The heat $Q_{h}$ supplied to the system by sunlight is related to the plant’s output by Equation (7.12).

$\eta = \frac{W}{Q_h}$              (7.12)

Using Equation (7.5),

$P_{avg}=\frac{W}{\Delta t}$              (7.5)

the plant produces

$W = (3 MW) (1 s) \longleftarrow P_{avg}=\frac{W}{\Delta t}$

$= 3 MW . s$

$W = 3\left(\frac{MJ}{\cancel {s}} \right) (\cancel {s})$

$= 3 MJ$

of electrical energy each second. Given the assumed level of plant efficiency, the quantity of heat that must be supplied each second is

$Q_{h}=\frac{3 MJ}{0.26} \longleftarrow \left[\eta =\frac{W}{Q_{h}} \right]$

$= 11.54 MJ$

Sunlight will therefore need to be collected over the area

$A=\frac{1.154\times 10^{4 }kJ}{0.9 kJ / m^2}$

$=1.282\times 10^{4} (\cancel {kJ})\left(\frac{m^2}{\cancel {kJ}} \right)$

$=1.282\times 10^{4} m^2$

This area is equivalent to a square parcel of land over 100 m on a side.
The power plant would therefore require

$N=\frac{1.282\times 10^4 m^2}{12 m^2}$

$= 1068\frac{\cancel {m^2}}{\cancel {m^2}}$

$=1068$

individual collector mirrors.

(b) The ideal Carnot efficiency of a heat engine operating between the given low temperature (surrounding air) and high temperature (heated by sunlight) is

$\eta _{c}=1 – \frac{(25+273.15)K}{(400 + 273.15)K} \longleftarrow \left[\eta _{c}=1-\frac{T_{1}}{T_{h}}\right]$

$= 1-0.4429\left(\frac{\cancel {K}}{\cancel {K}} \right)$

$=0.5571$

or roughly 56%. Here we have converted temperatures to the absolute
Kelvin scale by using Equation (7.14)

$K = ^\omicron C + 273.15$

$^\omicron R = ^\omicron F + 459.67$                 (7.14)

as required in Equation (7.13).

Discussion
In our design, we have neglected the factors of cloud cover, atmospheric
humidity, and dirt on the mirrors, each of which would decrease the amount of solar radiation available for power conversion. The power plant’s footprint would therefore need to be larger than our estimate suggests, but from a preliminary design standpoint, this calculation begins to firm up the plant’s size. The ideal engine efficiency of 56% is the maximum value that physical laws allow, but it can never be achieved in practice. The power plant’s real efficiency is about half of this theoretically possible value. In that sense, the efficiency of 26% in this application is not as low as you might think at first glance.

A parabolic trough-shaped mirror system with

$N = 1068$
$\eta _{C} = 0.5571$