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Chapter 15: Problem 37

An inventor proposes a heat engine to propel a ship, using the temperature difference between the water at the surface and the water \(10 \mathrm{m}\) below the surface as the two reservoirs. If these temperatures are \(15.0^{\circ} \mathrm{C}\) and \(10.0^{\circ} \mathrm{C},\) respectively, what is the maximum possible efficiency of the engine?

Short Answer

Expert verified
Answer: The maximum possible efficiency of the engine is approximately 1.74%.

Step by step solution

01

Convert temperatures to Kelvin

To calculate the maximum efficiency of the engine, we must first convert the temperatures from Celsius to Kelvin. To do this, we simply add 273.15 to each Celsius temperature. \(T_H = 15.0 + 273.15 = 288.15 \, \mathrm{K}\) \(T_C = 10.0 + 273.15 = 283.15 \, \mathrm{K}\) Where \(T_H\) is the temperature of the hot reservoir (water surface) and \(T_C\) is the temperature of the cold reservoir (10m below the surface).
02

Calculate the maximum efficiency

Now that we have the temperatures in Kelvin, we can use the formula for the maximum efficiency of a heat engine, given by the Carnot efficiency: \(\eta_{max} = 1 - \frac{T_C}{T_H}\) Plug the values of \(T_H\) and \(T_C\) into this formula: \(\eta_{max} = 1 - \frac{283.15}{288.15}\)
03

Compute the result

Perform the calculation: \(\eta_{max} = 1 - 0.9826 \approx 0.0174\)
04

Express the efficiency as a percentage

To express the maximum efficiency as a percentage, multiply the result by 100: \(\eta_{max} \% = 0.0174 \times 100 \approx 1.74 \%\) The maximum possible efficiency of the engine is approximately 1.74%.

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Most popular questions from this chapter

(a) How much heat does an engine with an efficiency of \(33.3 \%\) absorb in order to deliver \(1.00 \mathrm{kJ}\) of work? (b) How much heat is exhausted by the engine?
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