To calculate the requested components, we'll use the following definitions and relationships in thermodynamics and solar energy:
10.1 Solar Power Generated
The solar power generated can be calculated using the formula:
\[ P_{\text{solar}} = A \cdot I \cdot \eta_c \]
Where:
- \(P_{\text{solar}}\) is the solar power generated (W)
- \(A\) is the collector area (m²)
- \(I\) is the solar irradiance (W/m²)
- \(\eta_c\) is the efficiency of the collectors (decimal)
10.2 Time to Produce 93.75 MWh of Solar Energy
First, we need to convert the energy requirement from MWh to Wh:
\[ E = 93.75, \text{MWh} = 93.75 \times 10^6, \text{Wh} \]
The time required to generate this energy can be calculated with:
\[ t = \frac{E}{P_{\text{solar}}} \]
10.3 Irradiance of the Region
The solar irradiance can be derived from the power output of the collectors:
Rearranging the solar power equation gives:
\[ I = \frac{P_{\text{solar}}}{A \cdot \eta_c} \]
Now let’s perform the calculations step-by-step.
Step 1: Calculate Solar Power Output Given:
- \(\eta_c = 0.65\)
- \(A = 40,000, m^2\)
We can plug \(P_{\text{solar}}\) into the equation using the Rankine plant's output power and efficiency.
Using the overall efficiency of the solar to electrical is given by:
\[ \eta = \eta_c \cdot \eta_r \]
Where:
- \(\eta_r\) is the Rankine cycle efficiency (given as 0.32).
The overall efficiency becomes:
\[ \eta = 0.65 \cdot 0.32 = 0.208 \]
Now using the output power from the Rankine cycle:
\[ P_{\text{solar}} = \frac{P_{\text{out}}}{\eta} = \frac{5\text{ MW}}{0.208} = \frac{5 \times 10^6, W}{0.208} \approx 24,038,462, W \text{ or } 24.04, MW \]
Step 2: Calculate Solar Irradiance Now we can find the solar irradiance:
\[ I = \frac{P_{\text{solar}}}{A \cdot \eta_c} = \frac{24,038,462, W}{40,000, m^2 \cdot 0.65} \approx \frac{24,038,462, W}{26,000, m^2} \approx 924.56, W/m^2 \]
Step 3: Calculate Time to Produce 93.75 MWh of Solar Energy Substituting into the equation for \(t\):
First, find \(P_{\text{solar}}\) again to find time, but taking into account the Rankine output as previously calculated:
\[ t = \frac{E}{P_{\text{out}}} = \frac{93,750,000, Wh}{5,000,000, W} \approx 18.75, hours \]
Summary of Results
- 10.1 Solar power generated: \(24.04, MW\)
- 10.2 Time it will take to produce \(93.75, MWh\) of solar energy: \(18.75, hours\)
- 10.3 Irradiance of the region: \(924.56, W/m²\)