Heat Engine

Heat Engine

The generic definition of a heat engine is a system that converts heat (thermal energy) into work (mechanical energy).  The schematic below illustrates a heat engine.  The heat flows from the hot source into the engine system.  From there work is output or heat continues to flow to the cold sink.

heat engine diagram

Figure 1. Heat Engine Diagram


An example of a heat engine is a automobile engine.  The automobile engine operates in a cyclic manner, that is often illustrated on a pressure/volume diagram.


Figure 2. Pressure-Volume Diagram for a Heat Engine Cycle

As you can see in this diagram, the main steps of a cyclic heat engine are:

  1. Work is done on gas – compression of fuel
  2. Heat is added – fuel ignition
  3. Work is done by the gas – expansion of gas does work on piston
  4. Heat is extracted – exhaust valve opens

For the heat engine to be a complete cycle, the PV diagram needs to be a complete loop.  The area inside the loop represents the amount of work done during the entire cycle.





Second Law of Thermodynamics

Second Law:

It is impossible to extract an amount of heat QH from a hot reservoir and use it all to do work W . Some amount of heat QC must be exhausted to a cold reservoir. This precludes a perfect heat engine.

This version of the second law is often referred to as the Kelvin Statement.

In reference to entropy, the second law can be explained as: In any cyclic process the entropy will either increase or remain the same.

Second Law of Thermodynamics.



Heat Engine Efficiency & Engineering Details


The most efficient heat engine cycle is the Carnot cycle.  This process consists of two isothermal steps and two adiabatic steps.  The efficiency calculations are given for a single cycle.

heaeng2entcar1The above efficiency calculation relates the work and heat flow from the hot reservoir.  This equation can be rearranged to the second equation above referring to both the hot and cold heat flows.

For a Carnot cycle, the efficiency calculation can be further simplified to the below:

entcar2        * Note: Temperatures in Kelvin scale

If Q is heat added to the system, then the heat transferred from the system be negative.  So the Carnot cycle produces.

entcar4This can be further evaluated for reversible and irreversible cycles.


entcar5Clausius Theorem


entcar8Clausius Inequality

Engineering Details

When designing a heat engine, it is desirable to be as near to a Carnot cycle as possible.  This will maximize the efficiency of the engine.  The materials to be used for the engine must be capable to sustain the high temperatures and pressures that would occur during the cycle.  Also it is a concern that the engine materials not act a heat sink during the combustion process.  To maximize the efficiency of the engine, most of the heat produced in combustion should be able to be transferred to work, and not to the surroundings of the engine.



Environmental Impacts

When talking about environmental impacts of a heat engine, it is important to know what type of heat engine you’re talking about.  When thinking of a heat engine, most would think of an automobile.  However, there are other ways to think of a heat engine.

One example is actually Earth’s heat engine itself!  Earth’s atmosphere and hydrosphere are actually coupled processes that balance out solar heating through evaporation of water, convection, rainfall, winds, and ocean circulation.

The Hadley cell provides an example of a heat engine.  What that shows is with rising of warm and moist air near the equator, there is the descent of cold air in the subtropics.


Operating heat engines also has a negative impact on the quality of air, which is a topic that is mentioned quite often when talking about automobiles.  When dealing with the combustion of fuel, the main derivatives are carbon dioxide, water, and particulate matter.  An increase in particulate matter has been shown to cause asthma, lung cancer, heart problems, and premature death.  Engines can also produce nitrogen oxides, which is hazardous for plant and animal health, and leads to ground level ozone production.  This is not to be confused with the ozone layer, which protects the environment from UV rays.  Ground level ozone is very harmful to human health and the environment.  Obviously, automobiles are necessary for life today, but the pollution they cause contributes to the global greenhouse effect, which is a main concern in terms of global warming.

Is the Sun an Ideal Gas?

The schooling and in depth thermodynamic knowledge required to be able to do something like calculate the temperature or pressure at the suns core must be extensive right?  Well maybe not so much, what if you could estimate those properties using only the ideal gas law (PV=nRT)?  In fact solar physicist routinely do this.  If you are still skeptical I will attempt to convince you that this is a valid method to make those approximations.

Normally at low pressures the excluded volume of a gas (the space atoms physically take up) is very small compared to the total volume of the container, as a result the gas behaves similarly to an ideal gas (they act as point particles and the atoms them selves are not considered to occupy any volume).  However, as pressure become large the gas acts less and less ideally.  At higher pressures the excluded volume becomes much more significant, and the ideal gas law is no longer sufficient.  However the extreme pressure in the suns core allows for some interesting atomic events.

The sun core is made of mostly helium and hydrogen (64% helium and 36% hydrogen by mass).  In the core of the sun the temperatures are so extreme that the gas atoms are stripped of their electrons, and the matter undergoes a phase change from gas to plasma.  This is key because the atoms behave extremely different in the form of plasma than they do in the form of a gas.  In plasma the matter is composed of free particles, in this case hydrogen nuclei, helium nuclei, and free electrons act as individual particles.  The pressure at the core of the sun is so intense that the density at the core is about 158 g/cm3  (based on the standard model for the sun).  At the sun’s core the matter is assumed to be pure plasma, and by making that assumption it must also assumed that the matter is fully ionized.

Now the groundwork for this problem has been established and it is time to start describing the calculations required.  The goal is to prove that the exclusion volume is so small that the particles can be thought as point particles, at which point the ideal gas law can be applied.  In knowing the density, we can choose a volume of 1 cm3 and calculate the total mass, which unsurprisingly comes out to 158 g.  Once the mass is established, it is required to know how much mass each element attributes to the total.  We can accomplish this by multiplying the percent composition of each atom with the total mass (158 g) of our sample.  In 1 cm3 of plasma the mass attributed to hydrogen is 36% of 158g or 56.88g, and the mass attributed to helium is 64% of 158 g or 101.2 g.  In order to continue we must find the number of particles of each atom in their respective mass contribution to the sample.  This is again not an overly complicated step, which is accomplished by taking the mass contributed from one type of atom, and dividing it by the molecular mass for that atom.  From here, the resultant is multiplied by Avogadro’s number, and the total number of atoms in that mass can be determined.  The number of particles in 1 cm3 of plasma at the earth’s core is 3.397×1025 and 1.522×1025 for hydrogen and helium respectively.  The equation is displayed below.

Screen Shot 2014-12-10 at 10.27.18 AM

Screen Shot 2014-12-10 at 10.27.25 AM

In order to find the volume these particles occupy (or exclude from the total volume) the particles are assumed to be spheres with a radius of 5*10-3 pm (based on Ernest Rutherford’s gold foil experiment).  It is important to not that this is an estimation, which should be obvious because radius for a helium nucleus would be larger than the radius of a hydrogen nucleus due to a larger number of nuclear particles in helium.  The volume per sphere can be calculated using the standard geometric formula for a sphere which is (4/3)*π*r3.  The sum of these nuclei times the volume of a single nucleus will give the total exclusion volume for all particles present in the given volume.  The total calculated volume using this formula totaled 2.576*10-11 cm3.  This is a very small volume in fact it accounts for just 2.576*10-9 % of the entire volume!  The calculations are again displayed below.

Screen Shot 2014-12-10 at 10.26.43 AM

A more accurate radius can be used to try and further prove that the particles can be thought as point particles.  Using the same geometric equation for spherical volume the radius is replaced to 1.4*10-15 cm-3 multiplied by the cubed root of the mass number for that type of atoms.  The volume for a type of atom is then multiplied by the number of corresponding atoms in 1 cm-3.  This must be done for hydrogen and helium and then the summation of both atomic exclusion volumes is added to find the total excluded volume.  The total volume exclusion volume is calculated to be 1.094*10-12 cm3, or just 1.094*10-10 % of the total volume.  Again the calculations for this are below (click on them to enlarge equation).

Screen Shot 2014-12-10 at 10.26.05 AM

Screen Shot 2014-12-10 at 10.41.52 AM

So by using either method to find the exclusion volume, the overall outcome remains the same.  The excluded volume is extremely small in comparison to the total volume, and it is this fact that proves it is indeed possible to view the particles at the earth’s core as point particles.  In assuming this the ideal gas law can be used to estimate something like the temperature at the very core of the earth.

Personal Opinions

Heat engines depending on what kind you’re thinking of can be neutral to the environment, or bad for the environment.  If thinking about the effects of pollution from automobiles, sure it takes a toll on the environment, but it’s completely necessary for every day life.  The only rational solution is to find a way to have an efficient heat engine that doesn’t produce pollution and other toxins.

I think when people hear heat engine they automatically jump to combustion engine, but that is a very limited view for a heat engine.  It has to be remembered that a heat engine is a concept of physical properties that depict the harnessing of heat to do work on a system.  For instance in a combustion engine the heat comes from a combustion reaction, but couldn’t the heat source just as easily come from a renewable source such as solar thermal?  The amount of energy that solar thermal plants can produce is quite massive, but the concept is the same as a heat engine.  While yes, combustion engines probably the most common type of heat engine, I doubt they will be forever.  The push for renewable sources, and the technology being produced in the battery sectors are making it harder and harder to over look electric motor transportation.  The only hold up is stations to recharge your ride on a national scale.  Things are heading in the right direction though batteries can be charged faster than ever (I read the other day some batteries can be charged up to 70% in something like 2 min), and batter prices are beginning to decrease.  The only thing that would make the process ideal would be if the energy used to charge the battery cells was produced sustainably.

-Dan Larson



Chen Ph. D., F. (2014). Thermodynamics and Kinetics.

Engel, T. and Reid, P. (2010). Thermodynamics, statistical thermodynamics, & kinetics. New York: Prentice Hall.

Hyperphysics.phy-astr.gsu.edu, (2014). Carnot Cycle. [online] Available at: http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/carnot.html [Accessed 9 Dec. 2014].

Hyperphysics.phy-astr.gsu.edu, (2014). Carnot Cycle. [online] Available at: http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/carnot.html#c1 [Accessed 10 Dec. 2014].

The ideal gas law at the center of the sun, David B. Clark, Journal of Chemical Education, 1989, 66 (10), 826.

Hyperphysics.phy-astr.gsu.edu, (2014). Heat Engines. [online] Available at: http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/heaeng.html [Accessed 9 Dec. 2014].

Wikipedia, (2014). Heat engine. [online] Available at: http://en.wikipedia.org/wiki/Heat_engine [Accessed 7 Dec. 2014].