Question

An inventor claims to have devised a cyclical engine for use in space vehicles that operates with a nuclear-fuel-generated energy source whose temperature is \(920 \mathrm{R}\) and a sink at \(490 \mathrm{R}\) that radiates waste heat to deep space. He also claims that this engine produces 4.5 hp while rejecting heat at a rate of \(15,000 \mathrm{Btu} / \mathrm{h}\). Is this claim valid?

Short answer

Answer: Yes, the claim is valid as the claimed engine efficiency (0.4327) is less than the maximum possible efficiency (0.4674).

Step-by-step solution

Convert Given Power Output to BTU/h

To compare the given efficiency and maximum efficiency, we should have both values in the same unit. Convert the given power output (4.5 hp) to BTU/h using the conversion factor 1 hp = 2544.43 BTU/h. 4.5 hp = 4.5 * 2544.43 = 11449.935 \mathrm{Btu}/ \mathrm{h}

Calculate the Heat Input

To calculate the heat input, add the heat rejected (15000 BTU/h) to the power output (11449.935 BTU/h). Heat input = Power output + Heat rejected Heat input = 11449.935 \mathrm{Btu} / \mathrm{h} + 15000 \mathrm{Btu} / \mathrm{h} = 26449.935 \mathrm{Btu} / \mathrm{h}

Calculate the Claimed Engine Efficiency

Calculate the claimed engine efficiency using the formula: Claimed Engine Efficiency = Power output / Heat input Claimed Engine Efficiency = 11449.935 \mathrm{Btu}/\mathrm{h} / 26449.935 \mathrm{Btu}/\mathrm{h} = 0.4327

Calculate the Carnot Efficiency

Calculate the maximum possible efficiency using the Carnot efficiency formula: Carnot Efficiency = 1 - (Temperature of sink / Temperature of source) Carnot Efficiency = 1 - (490 \mathrm{R} / 920 \mathrm{R}) = 1 - 0.5326 = 0.4674

Compare Claimed Engine Efficiency with Carnot Efficiency

Compare the claimed efficiency with the maximum possible efficiency to determine if the claim is valid. If the claimed engine efficiency is less than or equal to the maximum possible efficiency, the claim is valid. Otherwise, it is not valid. In this case, the claimed engine efficiency (0.4327) is less than the maximum possible efficiency (0.4674). So, the claim is valid.