Calculate the maximum possible efficiency of a heat engine that uses surface lake water at 20.7°C as a source of heat and rejects waste heat to the water 0.100 km below the surface where the temperature is 3.70°C.

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**Title: Calculating the Maximum Possible Efficiency of a Heat Engine**

**Problem Statement:**
Calculate the maximum possible efficiency of a heat engine that uses surface lake water at 20.7°C as a source of heat and rejects waste heat to the water 0.100 km below the surface where the temperature is 3.7°C.

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**Response Box:**

- **Numeric Response:** 
  - 94

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**Explanation:**

To determine the maximum possible efficiency of the heat engine, we use the concept of the Carnot efficiency, which is determined by the temperatures of the hot and cold reservoirs. Here is the formula to calculate Carnot efficiency (η):

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

Where:
- \( T_H \) = Temperature of the hot reservoir (in Kelvin)
- \( T_C \) = Temperature of the cold reservoir (in Kelvin)

**Given Temperatures:**
- Surface water temperature \( T_H = 20.7°C \)
- Deep water temperature \( T_C = 3.7°C \)

**Conversion to Kelvin:**
- \( T_H = 20.7 + 273.15 = 293.85 \, K \)
- \( T_C = 3.7 + 273.15 = 276.85 \, K \)

**Substitute values into the Carnot efficiency formula:**
\[ \eta = 1 - \frac{276.85}{293.85} \]

**Conceptual Analysis:**
This setup of using different temperatures at different depths is often considered when analyzing the efficiency limits of natural thermal gradients, such as those found in oceans or lakes, to assess potential energy extraction methods.
Transcribed Image Text:**Title: Calculating the Maximum Possible Efficiency of a Heat Engine** **Problem Statement:** Calculate the maximum possible efficiency of a heat engine that uses surface lake water at 20.7°C as a source of heat and rejects waste heat to the water 0.100 km below the surface where the temperature is 3.7°C. --- **Response Box:** - **Numeric Response:** - 94 --- **Explanation:** To determine the maximum possible efficiency of the heat engine, we use the concept of the Carnot efficiency, which is determined by the temperatures of the hot and cold reservoirs. Here is the formula to calculate Carnot efficiency (η): \[ \eta = 1 - \frac{T_C}{T_H} \] Where: - \( T_H \) = Temperature of the hot reservoir (in Kelvin) - \( T_C \) = Temperature of the cold reservoir (in Kelvin) **Given Temperatures:** - Surface water temperature \( T_H = 20.7°C \) - Deep water temperature \( T_C = 3.7°C \) **Conversion to Kelvin:** - \( T_H = 20.7 + 273.15 = 293.85 \, K \) - \( T_C = 3.7 + 273.15 = 276.85 \, K \) **Substitute values into the Carnot efficiency formula:** \[ \eta = 1 - \frac{276.85}{293.85} \] **Conceptual Analysis:** This setup of using different temperatures at different depths is often considered when analyzing the efficiency limits of natural thermal gradients, such as those found in oceans or lakes, to assess potential energy extraction methods.
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