Question

Part 1: A Carnot engine has a power output of 179 kW. The engine operates between two reservoirs at 8◦C and 228◦C. How much thermal energy is absorbed each hour? Answer in units of J.

Part 2: How much thermal energy is lost per hour? Answer in units of J.

Answer 1

To solve both parts, we first need to understand how a Carnot engine works. The efficiency (\( \eta \)) of a Carnot engine is given by the formula:

\[
\eta = 1 - \frac{T_C}{T_H}
\]

Where:
- \( T_H \) is the absolute temperature of the hot reservoir (in Kelvin).
- \( T_C \) is the absolute temperature of the cold reservoir (in Kelvin).

**Part 1: Calculate Thermal Energy Absorbed Per Hour**

1. Convert the temperatures from Celsius to Kelvin:
\[
T_H = 228^\circ C + 273.15 = 501.15 \, K
\]
\[
T_C = 8^\circ C + 273.15 = 281.15 \, K
\]

2. Calculate the efficiency of the Carnot engine:
\[
\eta = 1 - \frac{T_C}{T_H} = 1 - \frac{281.15}{501.15}
\]
\[
\eta \approx 1 - 0.56064 \approx 0.43936
\]

3. The power output (\( P \)) of the engine is 179 kW or \( 179,000 \) W.

4. The thermal energy absorbed (\( Q_H \)) can be found using the relationship between power, efficiency, and thermal energy:
\[
P = \eta Q_H
\]
Rearranging gives us:
\[
Q_H = \frac{P}{\eta}
\]

5. Substitute the values to find \( Q_H \) (in Watts, but we'll convert to Joules per hour):
\[
Q_H = \frac{179,000}{0.43936} \approx 407,422.69 \, W
\]

To convert this to Joules per hour:
\[
Q_H \text{ per hour} = Q_H \times 3600
\]
\[
Q_H \text{ per hour} \approx 407,422.69 \, W \times 3600 \, s \approx 1,465,135,684 \, J
\]
Thus, the thermal energy absorbed each hour is approximately:
\[
Q_H \approx 1.465 \times 10^9 \, J \text{ or } 1,465,135,684 \, J
\]

**Part 2: Calculate Thermal Energy Lost Per Hour**

The thermal energy lost (\( Q_C \)) can be calculated using the relationship:
\[
Q_C = Q_H - W
\]
Where \( W \) is the work done by the engine in one hour, and can be obtained from \( P \):
\[
W = P \times 3600 = 179,000 \, W \times 3600 \, s = 644,400,000 \, J
\]

Then using the thermal energy absorbed and work done:
\[
Q_C = Q_H - W
\]
\[
Q_C \approx 1,465,135,684 \, J - 644,400,000 \, J \approx 820,735,684 \, J
\]

Thus, the thermal energy lost each hour is approximately:
\[
Q_C \approx 820,735,684 \, J
\]

In summary:
- **Thermal energy absorbed per hour:** Approximately \( 1,465,135,684 \, J \)
- **Thermal energy lost per hour:** Approximately \( 820,735,684 \, J \)