# Worksheet: Heat Engines

In this worksheet, we will practice calculating the work extracted by heat engines from the transfer of heat from high to low temperature reservoirs.

Q1:

A Carnot engine has an efficiency of 0.500. The temperature of the engine’s cold reservoir changes and its efficiency drops to 0.450. The initial temperature of the cold reservoir is . The temperature of the hot reservoir does not change.

What is the temperature of the hot reservoir?

What is the temperature of the cold reservoir after its temperature changes?

Q2:

It is found that an engine rejects 100 J while absorbing 125 J each cycle of operation.

What is the efficiency of the engine?

How much work does it perform per cycle?

Q3:

An engine operating between heat reservoirs at temperatures of and extracts 1,000 J per cycle from the engine’s hot reservoir. In modeling the heat flow in the engine, use three-significant-figure precision.

What is the maximum possible work that engine can do per cycle?

What is the minimum heating of the cold reservoir per cycle?

Q4:

A heat engine operates between two temperatures such that the working substance of the engine absorbs 5,000 J of heat from the high-temperature bath and rejects 3,000 J to the low-temperature bath. The rest of the energy is converted into mechanical energy of the turbine.

Find the amount of work produced by the engine.

Find the efficiency of the engine.

• A
• B
• C
• D
• E

Q5:

The Kelvin temperature of the hot reservoir of an engine is twice that of the cold reservoir, and work done by the engine per cycle is 50 J.

Calculate the efficiency of the engine.

Calculate the heat absorbed per cycle.

Calculate the heat rejected per cycle.

Q6:

Calculate the net work output of a heat engine that changes the state of a gas to correspond with the path , as shown in the diagram.

Q7:

A heat engine is found to have an efficiency of 0.620. The engine performs 300 J of work per cycle.

How much is the engine heated per cycle?

How much heat is rejected by the engine per cycle?

Q8:

A heat engine with an efficiency of 0.430 is heated by 450 J per cycle.

How much work does the engine perform per cycle?

How much heat does the engine reject per cycle?

Q9:

A heat engine has a cold reservoir that is at a temperature of 295 K. The engine’s efficiency is 0.370 and it is heated by 562 J per cycle.

How much work does the engine perform per cycle?

How much heat does the engine reject per cycle?

Q10:

A heat engine outputs 5 MJ of electrical energy while operating between two thermal baths of different temperatures. The working substance of the engine rejects 7 MJ of heat to the low-temperature bath. What is the efficiency of the engine?

Q11:

A Carnot engine operates in a Carnot cycle between a heat source at a temperature of and a heat sink at a temperature of . What is the efficiency of the Carnot engine?

Q12:

Estimate the Rankine efficiency of a steam power cycle. Dry and saturated steam is supplied at a pressure of 1.5 MPa. The pressure of the condenser is 400 kPa.

Q13:

Consider a constant volume Otto cycle of air. The temperature of air at the beginning of the compression is and the maximum temperature is . The compression ratio is 7. What is the work done per kg of air? Use a value of 1.4 for the ratio of the molar heat capacities of air.

Q14:

A heat engine outputs 0.30 MW of power when it is heated at a rate of 0.50 MW. The engine’s hot and cold reservoir temperatures are and respectively. Calculate the second-law efficiency of the heat engine.

Q15:

What is the thermal efficiency of a Rankine cycle if the temperatures of its cold and hot reservoirs are and , respectively?

• A
• B
• C
• D

Q16:

What is the thermal efficiency of a Sterling engine which uses an energy source whose temperature is and an energy sink whose temperature is ?

Q17:

Consider an air-standard Otto cycle, which has a compression ratio of 6.0. Its temperature and pressure at the beginning of the compression process are and 97,900 Pa, respectively. The heat addition per unit mass of air is 633 kJ. Calculate the thermal efficiency of the cycle.

Q18:

Consider a Carnot vapor power cycle using water as the working fluid. Saturated liquid enters the boiler at a pressure of 8.0 MPa, and saturated vapor enters the turbine. The pressure of the condenser pressure is 8.0 kPa. What is the thermal efficiency of this power cycle?

• A
• B
• C
• D

Q19:

What is the maximum theoretical efficiency for a heat engine operating between 373 K and 773 K?

Q20:

What is the thermodynamic efficiency of a heat engine that rejects heat at a rate of 20 MW when it is heated at a rate of 80 MW?

Q21:

What is the thermal efficiency of a Carnot cycle where the temperatures of the cold and hot reservoirs are and , respectively?

Q22:

A Carnot engine is used to measure the temperature of a heat reservoir. The engine operates between the heat reservoir and a reservoir consisting of water at its triple-point.

If 400 J per cycle are removed from the heat reservoir while 200 J per cycle are deposited in the triple-point reservoir, what is the temperature of the heat reservoir?

If 400 J per cycle are removed from the triple-point reservoir while 200 J per cycle are deposited in the heat reservoir, what is the temperature of the heat reservoir?

Q23:

In performing 100.0 J of work, an engine rejects 50.0 J of heat. What is the efficiency of the engine?