Question

A Carnot heat engine operates between a source at \(1000 \mathrm{K}\) and a sink at \(300 \mathrm{K}\). If the heat engine is supplied with heat at a rate of \(800 \mathrm{kJ} / \mathrm{min}\), determine \((a)\) the thermal efficiency and \((b)\) the power output of this heat engine.

Short answer

Based on the given information, we can conclude the following: (a) The thermal efficiency of the Carnot heat engine is 70%. (b) The power output of the heat engine is 9,333.33 W.

Step-by-step solution

Calculate the thermal efficiency of the Carnot heat engine.

To calculate the thermal efficiency of a Carnot heat engine, we use the formula: Thermal efficiency = \(1 - \dfrac{T_C}{T_H}\), where \(T_C\) is the temperature of the cold sink and \(T_H\) is the temperature of the hot source. In this case, \(T_C = 300\mathrm{K}\) and \(T_H = 1000\mathrm{K}\). Plugging these values into the formula: Thermal efficiency = \(1 - \dfrac{300}{1000} = 1 - 0.3 = 0.7\) The thermal efficiency of the Carnot heat engine is 0.7 or 70%.

Convert the heat input rate to watts.

The heat input rate is given in kJ/min, and we need to convert it to watts (W) to calculate the power output of the heat engine. We can do this by multiplying the given heat input rate by the conversion factor \(\dfrac{1000 \mathrm{J}}{1\mathrm{kJ}}\) and dividing by the conversion factor \(\dfrac{60\mathrm{s}}{1\mathrm{min}}\): Heat input rate = \(\dfrac{800\,\mathrm{kJ}\:/\: \mathrm{min} \times \dfrac{1000\,\mathrm{J}}{1\,\mathrm{kJ}}}{\dfrac{60\,\mathrm{s}}{1\,\mathrm{min}}} = \dfrac{800,000\,\mathrm{J/min}}{60\,\mathrm{s/min}} = 13,333.3\,\mathrm{W}\) The heat input rate is equal to 13,333.3 W.

Calculate the power output of the heat engine.

To find the power output of the heat engine, we'll use the following formula: Power output = Efficiency × Heat input rate We already found the thermal efficiency (0.7) and heat input rate (13,333.3 W), so we can plug them into the formula: Power output = 0.7 × 13,333.3 = 9,333.33 W The power output of the Carnot heat engine is 9,333.33 W. #Answer# (a) The thermal efficiency of the Carnot heat engine is 70%. (b) The power output of the heat engine is 9,333.33 W.