**March 17 Homework Solutions**
**1. If incident solar radiation averages 700 W/m**^{2} for 8 hours, estimate the area, A, of 80% efficient heliostats needed to provide 10 MW of electrical power (as in Solar One). Assume that the efficiency of converting solar heat to electrical energy is 35%. Calculate the land area that would be required for a solar thermal power plant delivering 1000 MW of electrical power under these conditions, assuming that the land area for the mirrors is twice the area of the mirrors themselves.
Assume that the question asks for 10MWe over an 8hr period or 80 MWh(e). With an efficiency of 35%, we need a heat input (80 MWh)/0.35 = 228.6 MWh. If the heliostats are 80% efficient, we need a solar input of (228.6 MWh)/0.8 = 285.7 MWh. The input to a solar collector over eight hours is (700 W/m^{2})(8 h) = 5600 Wh/m^{2} = 0.0056 MWh/m^{2}. The required area is the ratio of the total solar input required divided by the solar input per unit area

The required land area is twice the area of the mirror or **102040 m**^{2}** = 10.2 ha = 0.102 km**^{2}**. (**One hectare (1 ha) = 10^{4} m^{2} is sometimes called the metric acre. It is the area of a square that has a side of 100 m.)

**2. A single-story retail store would like to supply all its lighting requirements with batteries charged by photovoltaic cells. The PV cells will be mounted on the horizontal rooftop. The time average lighting requirements for the store are 10 watts per square meter of ceiling area. The annual average solar irradiance is 150 W/m**^{2}, the cell efficiency is 15%, and the battery charging-discharging efficiency is 80%. Assuming the roof area is the same as the ceiling area, what fraction of the roof area would be required to supply all the lighting?
Let A_{roof} be the area of the roof which is the same as the area of the ceiling. The annual lighting energy required is (10 W/m^{2})A_{roof}(8760 h/yr). The annual lighting power produced by solar energy is (15%)(150 W/m^{2})A_{PV}(8760 h/yr), where A_{PV} is the area of the photovoltaic solar collectors. The annual energy available, considering the 80% efficiency of battery charging and recharging (assuming that this is an average that applies all year, even when the solar output is directly feeding the lights) is (80%)(15%)(150 W/m^{2})A_{PV}(8760 h/yr Setting the solar energy generated equal to the lighting energy required gives

(10 W/m^{2})A_{roof}(8760 h/yr) = (80%)(15%)(150 W/m^{2})A_{PV}(8760 h/yr)

Solving this equation for the ratio of photovoltaic area to roof area gives **A**_{PV}**/A**_{roof}** = 0.556**

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**Problem 13.8 in text: Let’s assume a world of 10 billion people with an average demand for electricity of 1 kWe per person (720 kWh per month) for providing lighting, running refrigerators, other appliances, TVs, computers, etc. What surface area of polycrystalline silicon photovoltaic collectors would be required to provide the electrical needs for 10 billion people? How much electrical storage capacity would you estimate would be needed? If lead-acid batteries were used to store the energy, what mass of batteries would be required? (Hint: see Chapter 16, Table 16.1)**

From slide

8 in the global warming lecture, the global annual average solar radiation that reaches the earth’s surface is 198 W/m

^{2} (this is the sum of energy absorbed plus energy reflected;

for this problem, For simplicity, assume an average value of

200 W/m

^{2}. Table

13-3 shows polycrystalline silicon cells with an efficiency of 8-9% as deployed and 18.2% for experimental models. Assume an efficiency of 10% for this calculation. With a 10% efficiency and a 200W/m

^{2} solar irradiation the electric energy available in one month (

720 hours) is (10%)(200 W/m

^{2})(

720 h) = 14,400 Wh/m

^{2}_{ }=

14.4 kWh/m

^{2}. The energy demand for

10 billion people in one month would be 720x10

^{10} kWh. The area required then would be (

720x10

^{10} kWh) / (

14.4 kWh/m

^{2}) =

**5x10**^{11}** m**^{2}** = 500,000 km**^{2}. This is 20% larger than the 410,000 km^{2} area of California.
How much energy should we store? This depends on the deployment of the solar cells. There are many factors to consider: the imbalance in the world’s population between the northern and southern hemispheres, storing energy overnight and during seasons with low solar irradiation, the possibility of having floating oceanic arrays that could be towed to regions of maximum solar irradiation, the need for transmission lines, etc. Since we already have the number that one month’s energy demand is 720x10^{10} kWh, let’s estimate the battery mass required to store that much energy.

Table 16.1 gives the energy density of lead-acid batteries as ranging from 60 to 180 kJ/kg. Picking the high end of this range, 180 kJ/kg = 180 kWs/kg = 0.05 kWh/kg means that we would need (720x10^{10} kWh)/( 0.05 kWh/kg) = **1.44x10**^{14}** kg of lead-acid batteries**. According to __http://www.ldaint.org/factbook/chapter4.pdf__, the world production of lead in 2000 was 6.532x10^{9} kg.