The power output of a car engine running at 2400 rpm is 500 kW. How much (a) work is done and (b) heat is exhausted per cycle if the engine's thermal efficiency is 20%? Give your answers in kJ.

Verified step by step guidance

1. Calculate the work done per cycle using the thermal efficiency. Thermal efficiency (\(\eta\)) is defined as the ratio of the work output (\(W_{out}\)) to the heat input (\(Q_{in}\)). The formula is \(\eta = \frac{W_{out}}{Q_{in}}\). Rearrange this formula to solve for \(W_{out}\) as \(W_{out} = \eta \times Q_{in}\).
2. Determine the heat input per cycle. Since the power output of the engine is given, and power is the rate of doing work, use the formula \(P = \frac{W_{out}}{t}\) where \(P\) is power and \(t\) is time. First, find the time for one cycle by using the engine speed in rpm (revolutions per minute) to convert it to time per cycle.
3. Convert the engine speed from rpm to cycles per second (Hz) by dividing the rpm value by 60. Then, calculate the time for one cycle as the reciprocal of this frequency.
4. Substitute the time per cycle into the power formula to find the total work done per cycle in joules, then convert this value to kilojoules by dividing by 1000.
5. Calculate the heat exhausted per cycle. Since \(Q_{in} = W_{out} + Q_{out}\) where \(Q_{out}\) is the heat exhausted, rearrange to find \(Q_{out} = Q_{in} - W_{out}\). Use the previously calculated values of \(Q_{in}\) and \(W_{out}\) to find \(Q_{out}\) in kilojoules.

Key Concepts

Power and Work

Power is the rate at which work is done or energy is transferred over time. In the context of engines, power output can be calculated using the formula P = W/t, where P is power, W is work, and t is time. Understanding this relationship is crucial for determining the total work done by the engine during its operation.

Thermal Efficiency

Thermal efficiency is a measure of how effectively an engine converts the energy from fuel into useful work. It is defined as the ratio of the work output to the heat input, expressed as a percentage. In this case, with a thermal efficiency of 20%, only 20% of the energy input is converted into work, while the rest is lost as heat.

Heat Exhaust

Heat exhaust refers to the thermal energy that is not converted into work and is expelled from the engine. It can be calculated by determining the total energy input and subtracting the useful work output. This concept is essential for understanding energy losses in engines and the overall performance of thermal systems.