Heat engines transform input heat into work in accordance with the laws of thermodynamics. For instance, as we learned in the previous Concept, increasing the temperature of a gas at constant volume will increase its pressure. This pressure can be transformed into a force that moves a piston.

The mechanics of various heat engines differ but their fundamentals are quite similar and involve the following steps:

- Heat is supplied to the engine from some source at a higher temperature
(Th) . - Some of this heat is transferred into mechanical energy through work done
(W) . - The rest of the input heat is transferred to some source at a lower temperature
(Tc) until the system is in its original state.

A single cycle of such an engine can be illustrated as follows:

In effect, such an engine allows us to 'siphon off' part of the heat flow between the heat source and the heat sink. The efficiency of such an engine is define as the ratio of net work performed to input heat; this is the fraction of heat energy converted to mechanical energy by the engine:

If the engine does not lose energy to its surroundings (of course, all real engines do), then this efficiency can be rewritten as

An ideal engine, the most efficient *theoretically* possible, is called a Carnot Engine. Its efficiency is given by the following formula, where the temperatures are, respectively, the temperature of the exhaust environment and the temperature of the heat input, in Kelvins. In a Carnot engine heat is input and exhausted in isothermal cycles, and the efficiency is

The Stirling engine is a real life heat engine that has a cycle similar to the theoretical Carnot cycle. The Stirling engine is very efficient compared to a gasoline engine and could become an important player in today's world where green energy and efficiency will reign supreme.

where

**adiabatic expansion** is a process occurring without the exchange of heat with the environment

**isothermal expansion** is a process occurring without a change in temperature

**isobaric expansion** is a process occurring without a change in pressure

In a practical heat engine, the change in internal energy must be zero over a complete cycle. Therefore, over a complete cycle

When gas pressure-forces are used to move an object then work is done on the object by the expanding gas. Work can be done on the gas in order to compress it. If you plot pressure on the vertical axis and volume on the horizontal axis, the work done in any complete cycle is the area enclosed by the graph. For a partial process, work is the area underneath the curve, or

#### Example

A heat engine operates at a temperature of

a) What is the ideal efficiency of this engine?

We will plug the known values into the formula to get the ideal efficiency.

b) The engine drives a

To find the power of the engine, we will use the power equation and plug in the known values.

c) If the engine is operating at

First, we know that it is operating at

d) How much power is exhausted?

Since we know how much power is being put into the engine and how much energy is actually being used to lift the weight, we can determine how much energy is not actually being used to do work.

e) The fuel the engine uses is rated at

For this part of the problem, we need to convert the power being put into the engine into the amount of fuel being used(kW or kJ/s

### Interactive Simulation

### Review

Below is a graph of the pressure and volume of a gas in a container that has an adjustable volume. The lid of the container can be raised or lowered, and various manipulations of the container change the properties of the gas within. The points

- Consider the change the gas undergoes as it transitions from point
b to pointc . What type of process is this?- adiabatic
- isothermal
- isobaric
- isochoric
- entropic

- Consider the change the gas undergoes as it transitions from point
c to pointa . What type of process is this?- adiabatic
- isothermal
- isobaric
- isochoric
- none of the above

- Consider the change the gas undergoes as it transitions from point
a to pointb . Which of the following*best*describes the type of process shown?- isothermal
- isobaric
- isochoric

- Calculate the ideal efficiencies of the following sci-fi heat engines:
- A nuclear power plant on the moon. The ambient temperature on the moon is
15K . Heat input from radioactive decay heats the working steam to a temperature of975K . - A heat exchanger in a secret underground lake. The exchanger operates between the bottom of a lake, where the temperature is
4C , and the top, where the temperature is13C . - A refrigerator in your dorm room at Mars University. The interior temperature is
282K ; the back of the fridge heats up to320K .

- A nuclear power plant on the moon. The ambient temperature on the moon is
- A heat engine operates through
4 cycles according to thePV diagram sketched below. Starting at the top left vertex they are labeled clockwise as follows: a, b, c, and d.- From \begin{align*}a-b\end{align*} the work is \begin{align*}75 \;\mathrm{J}\end{align*} and the change in internal energy is \begin{align*}100 \;\mathrm{J}\end{align*}; find the net heat.
- From the a-c the change in internal energy is \begin{align*}-20 \;\mathrm{J}\end{align*}. Find the net heat from b-c.
- From c-d the work is \begin{align*}-40 \;\mathrm{J}\end{align*}. Find the net heat from c-d-a.
- Find the net work over the complete \begin{align*}4\end{align*} cycles.
- The change in internal energy from b-c-d is \begin{align*}-180 \;\mathrm{J}\end{align*}. Find:
- the net heat from c-d
- the change in internal energy from d-a
- the net heat from d-a

### Review (Answers)

- d
- c
- a
- a. \begin{align*}98\end{align*}% b. \begin{align*}4.0\end{align*}% c. \begin{align*}12\end{align*}%
- a. \begin{align*}1753 \;\mathrm{J}\end{align*} b. \begin{align*}-120 \;\mathrm{J}\end{align*} c. \begin{align*}80 \;\mathrm{J}\end{align*} d. \begin{align*}35 \;\mathrm{J}\end{align*} e. \begin{align*}-100 \;\mathrm{J}, 80 \;\mathrm{J}, 80 \;\mathrm{J}\end{align*}