## Heat engines

To convert heat into work, you need at least two places with different temperatures.  If you take in Qhigh at temperature Thigh you must dump at least Qlow at temperature Tlow.  The amount of work you get out of a heat engine is W = Qhigh - Qlow.  The maximum amount of work you can get out of a heat engine is the amount you get out of a reversible engine.

Wmax = (Qhigh - Qlow)reversible = Qhigh - QhighTlow/Thigh = Qhigh(1 - Tlow/Thigh).

W is positive if Thigh is greater than Tlow.

The efficiency of a heat engine is the ratio of the work obtained to the heat energy put in at the high temperature, e = W/Qhigh.  The maximum possible efficiency emax of such an engine is

emax =Wmax/Qhigh = (1 - Tlow /Thigh) = (Thigh - Tlow)/Thigh.

### Steam engines

A steam engine is a type of heat engine.  It takes heat from the hot steam, converts some of this heat into useful work and dumps the rest into the colder surrounding air.  The maximum fraction of heat that can be converted into work can be found using the laws of thermodynamics, and it increases with the temperature difference between the hot steam and the surrounding air.  The hotter the steam and the colder the air, the more efficient is the steam engine at converting heat into work.

In a typical steam engine a piston moves back and forth inside a cylinder.  Hot, high-pressure steam is produced in a boiler, and this steam enters the cylinder through a valve.  Once inside the cylinder, the steam pushes outward on every surface, including the piston.  The piston moves.  The steam does mechanical work on the piston and the piston does mechanical work on the machinery attached to it.  The expanding steam transfers some of its thermal energy to this machinery, so the steam becomes cooler as the machinery operates.

When the piston reaches the end of its range, the valve stops the flow of steam and opens the cylinder to the outside air.  The piston can then return easily.  In many cases, steam is allowed to enter the other end of the cylinder so that the steam pushes the piston back to its original position.  Once the piston is back at its starting point, the valve again admits high-pressure steam to the cylinder and the whole cycle repeats.  Overall, heat is flowing from the hot boiler to the cooler surrounding air and some of that heat is being converted into mechanical work by the moving piston.  The maximum efficiency of a steam engine is emax = (Tsteam - Tair)/Tsteam.  The actual efficiency is usually much lower.

#### Problem:

What is the maximum possible efficiency of a steam engine taking in heat at 100 oC and dumping it at room temperature of approximately 20 oC?

Solution:

• Reasoning:
The maximum efficiency of any heat engine is that of a  Carnot engine.  emax = (Thigh - Tlow)/Thigh.
• Details of the calculation:
100 oC = 373 K and  20 oC = 293 K.  The maximum possible efficiency is
(Thigh - Tlow)/Thigh =  (373 - 293)/373 = 0.21 = 21%.

### Internal combustion engines

An internal combustion engine burns a mixture of fuel and air.  The most common type is a four-stroke engine.  A piston slides in and out of a cylinder.  Two or more valves allow the fuel and the air to enter the cylinder and the gases that form when the fuel and air burn to leave the cylinder.  As the piston slides back and forth inside the cylinder, the volume that the gases can occupy changes drastically.

The process of converting heat into work begins when the piston is pulled out of the cylinder, expanding the enclosed space and allowing fuel and air to flow into that space through a valve.  This motion is called the intake stroke or induction stroke.  Next, the fuel and the air mixture are squeezed together by pushing the piston into the cylinder.  This is called the compression stroke.  At the end of the compression stroke, with the fuel and the air mixture squeezed as tightly as possible, the spark plug at the sealed end of the cylinder fires and ignites the mixture.  The hot burning fuel has an enormous pressure and it pushes the piston out of the cylinder.  This power stroke is what provides power to the engine and the attached machinery.  Finally, the burned gas is squeezed out of the cylinder through another valve in the exhaust stroke.  These four strokes repeat over and over again.  Most internal combustion engines have at least four cylinders and pistons.  There is always at least one cylinder going through the power stroke and it can carry the other cylinders through the non-power strokes.  The maximum efficiency of such an engine is emax = ( Tignition - Tair)/Tignition where Tignition is the temperature of the fuel-air mixture after ignition.  To maximize the fuel efficiency, you have to create the hottest possible fuel air mixture after ignition.  The highest efficiency that has been achieved is approximately 50% of emax.

#### Problem:

A heat engine absorbs 360 J of thermal energy and performs 25 J of work in each cycle.  Find
(a)  the efficiency of the engine and
(b)  the thermal energy expelled in each cycle.

Solution:

• Reasoning:
The amount of work you get out of a heat engine is W = Qhigh - Qlow.
The efficiency is e = W/Qhigh.
• Details of the calculation:
Qhigh = 360 J.  W = 25 J.  Qlow = Qhigh - W = 335 J.
(a)  The efficiency e = W/Qhigh = 6.9%.
(b)  The thermal energy expelled is Qlow = 335 J.