The engine that powers a crane burns fuel at a flame temperature of 2000℃. It is cooled by 20℃ air. The crane lifts a 2000 kg steel girder 30 m upward. How much heat energy is transferred to the engine by burning fuel if the engine is 40% as efficient as a Carnot engine?

Verified step by step guidance

1. Convert the given temperatures from Celsius to Kelvin using the formula: \( T(K) = T(℃) + 273.15 \). For the flame temperature, \( T_{hot} = 2000 + 273.15 \), and for the cooling air, \( T_{cold} = 20 + 273.15 \).
2. Calculate the efficiency of a Carnot engine using the formula: \( \eta_{Carnot} = 1 - \frac{T_{cold}}{T_{hot}} \). Substitute the values of \( T_{cold} \) and \( T_{hot} \) in Kelvin to find \( \eta_{Carnot} \).
3. Determine the actual efficiency of the engine, which is 40% of the Carnot efficiency. Use the formula: \( \eta_{actual} = 0.4 \times \eta_{Carnot} \).
4. Calculate the work done by the crane to lift the girder using the formula: \( W = m \cdot g \cdot h \), where \( m = 2000 \ \mathrm{kg} \), \( g = 9.8 \ \mathrm{m/s^2} \), and \( h = 30 \ \mathrm{m} \).
5. Relate the work done to the heat energy transferred to the engine using the efficiency formula: \( \eta_{actual} = \frac{W}{Q_{in}} \). Rearrange to solve for \( Q_{in} \): \( Q_{in} = \frac{W}{\eta_{actual}} \). Substitute the values of \( W \) and \( \eta_{actual} \) to find the heat energy transferred.

Key Concepts

Carnot Efficiency

Carnot efficiency is the maximum possible efficiency of a heat engine operating between two temperature reservoirs. It is defined by the formula η = 1 - (T_c / T_h), where T_c is the absolute temperature of the cold reservoir and T_h is the absolute temperature of the hot reservoir. This concept sets an ideal benchmark for real engines, indicating that no engine can be more efficient than a Carnot engine operating between the same temperatures.

Heat Transfer

Heat transfer refers to the movement of thermal energy from one object or system to another due to a temperature difference. In the context of engines, heat is transferred from the combustion of fuel to the working fluid, which then performs work, such as lifting a load. Understanding how heat is transferred and utilized is crucial for calculating the energy output and efficiency of the engine.

Work Done by the Engine

The work done by an engine is the energy transferred to lift an object against gravity, calculated using the formula W = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height lifted. In this scenario, the crane lifts a 2000 kg girder 30 m, and calculating this work is essential for determining the total energy input required from the fuel burned, factoring in the engine's efficiency.