EGEE 102
Energy Conservation and Environmental Protection

The Carnot Efficiency

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A general expression for the efficiency of a heat engine can be written as:

Efficiency = Work HeatEnergy Hot MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8vrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aacaqGfbGaaeOzaiaabAgacaqGPbGaae4yaiaabMgacaqGLbGaaeOB aiaabogacaqG5bGaaeypamaalaaabaGaae4vaiaab+gacaqGYbGaae 4AaaqaaiaabIeacaqGLbGaaeyyaiaabshacaqGGaGaaeyraiaab6ga caqGLbGaaeOCaiaabEgacaqG5bWaaSbaaSqaaiaadIeacaWGVbGaam iDaaqabaaaaaaa@5325@

We know that all the energy that is put into the engine has to come out either as work or waste heat. So work is equal to Heat at High temperature minus Heat rejected at Low temperature. Therefore, this expression becomes:

Efficiency= Q Hot -Q Cold Q Hot MathTypeMTEF55+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8vrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aacaqGfbGaaeOzaiaabAgacaqGPbGaae4yaiaabMgacaqGLbGaaeOB aiaabogacaqG5bGaaeypamaalaaabaGaaeyuamaaBaaaleaacaqGib Gaae4BaiaabshaaeqaaOGaaeylaiaabccacaqGrbWaaSbaaSqaaiaa boeacaqGVbGaaeiBaiaabsgaaeqaaaGcbaGaaeyuamaaBaaaleaaca qGibGaae4Baiaabshaaeqaaaaaaaa5040

Where, QHot = Heat input at high temperature and QCold= Heat rejected at low temperature. The symbol is often (Greek letter eta) used for efficiency this expression can be rewritten as:

η ( % )=1 Q Cold Q Hot ×100 MathTypeMTEF55+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8vrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aacuaH3oaAgaqbamaabmaabaGaaiyjaaGaayjkaiaawMcaaiabg2da 9iaaigdacqGHsisldaWadaqaamaalaaabaGaamyuamaaBaaaleaaca WGdbGaam4BaiaadYgacaWGKbaabeaaaOqaaiaadgfadaWgaaWcbaGa amisaiaad+gacaWG0baabeaaaaaakiaawUfacaGLDbaacqGHxdaTca aIXaGaaGimaiaaicdaaaa4E48

The above equation is multiplied by 100 to express the efficiency as percent.

French Engineer Sadi Carnot showed that the ratio of QHighT to QLowT must be the same as the ratio of temperatures of high temperature heat and the rejected low temperature heat. So this equation, also called Carnot Efficiency, can be simplified as:

η ( % )=1 T Cold T Hot ×100% MathTypeMTEF55+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8vrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aacuaH3oaAgaqbamaabmaabaGaaiyjaaGaayjkaiaawMcaaiabg2da 9iaaigdacqGHsisldaWadaqaamaalaaabaGaamivamaaBaaaleaaca WGdbGaam4BaiaadYgacaWGKbaabeaaaOqaaiaadsfadaWgaaWcbaGa amisaiaad+gacaWG0baabeaaaaaakiaawUfacaGLDbaacqGHxdaTca aIXaGaaGimaiaaicdacaGGLaaaaa4EF7

Note: Unlike the earlier equations, the positions of Tcold and Thot are reversed.

The Carnot Efficiency is the theoretical maximum efficiency one can get when the heat engine is operating between two temperatures:

  • The temperature at which the high temperature reservoir operates ( THot ).
  • The temperature at which the low temperature reservoir operates ( TCold ).

In the case of an automobile, the two temperatures are:

  • The temperature of the combustion gases inside the engine ( THot ).
  • The temperature at which the gases are exhausted from the engine ( TCold ).

Please watch the following 4:40 presentation about how automobile engines work:

Click Here for Transcript of How an automobile engine works

From this diagram we are going to learn how, basically, an automobile engine works. This is a cylinder. This is basically one cylinder that we have and these are the walls of the cylinder and this is the piston that we have. And this piston is what moves up and down and we have two inlets. One inlet is for fuel and oxygen, fuel and air to come in and mixture. And this is a valve that opens and closes. And there is another outlet here, that takes out the exhaust gases and there is another valve that opens and closes at appropriate times. And when the fuel and air mixture is inside, this spark plug provides a spark to ignite the mixture. Now let's see in sequence what happens in this cylinder. This piston basically moves up and down. So the first step is to bring the fuel and oxidizer or air inside the mixture. So this valve opens up. This valve of course is closed. And this fuel and air mixture comes in as this piston is pushed down. This acts like a syringe, a medical syringe. When we push this down, the fuel and air mixture is basically sucked in and it fills up this entire wall unit as this piston comes down. That is the first step or stroke. This is a four stroke engine that we are talking about. First stroke is intake. And the second stroke now, the second step is we close both of these valves and push this mixture; this mixture is again, by pushing the piston up, this mixture that is in here is compressed. It is compressed in very high pressures. Roughly about 8 times to 9 times and this piston moves back up here and again we have to spend energy to move this piston up or compress this mixture here. When this mixture is compressed to the predetermined pressures, at that time, the spark plug ignites. This mixture and this flame front travels all the way and burns all this fuel and this fuel reaches very high temperatures and it pushes the piston all the way down. All the way down. So this is actually the power stroke. The first stroke is intake, the second stroke is compression - compression. And the third stroke is the power stroke. That is the stroke that gives us the power from this engine. In the first stroke and the second stroke, we have to supply the energy that is required to bring in the oxidizer and fuel and also to compress. After this mixture expands, or pushes the piston down, this valve, this exhaust valve will open and of course this valve is still closed and the piston again moves back up for which we have to supply the power and all these gasses go out through this exhaust. And once the exhaust gas surrounds this cylinder, this value closes again and this valve opens again and the entire cycle again repeats. So the fourth stroke is the exhaust stroke. So this is a four, basically, a four stroke engine that we are talking about. We need to know the four strokes and the important components of this engine cylinder. And there generally will be multiple cylinders. We are talking about one here and there could be four, there could be 6, there could be 8 or even 10 or even 12 sometimes. So as the engine becomes bigger and bigger you have more cylinders and that gives us more power.

Then, why should we operate the automobiles at low efficiencies?

It is not that we cannot achieve high temperatures, but we do not have the engine materials that can withstand the high temperature. As a matter of fact, we do not let the engine gases go the maximum that they can go even now and instead try to keep the engine cool by circulating the coolant.

So we are taking the heat out of the gases (thus lowering the Thot) and making the engine operate at cooler temperatures so that the engine is protected - but lowering the efficiency of an automobile.

It's like Taxes. The more money you earn (heat), the more money is taxed (cold), leaving you with less money to take home (efficiency). However, if you could earn more money (heat) and find a way to have less taxes taken out (better engine material), you would have more money to take home (efficiency).


Below are two temperature scales. The first scale, labeled "HOT," shows the range of temperatures for the combustion of gases in a car engine. The second scale, labeled "COLD," shows the range of temperatures at which gases are exhausted from the car engine.

Instructions: Select numbers from the range on the "HOT" scale and enter them (one at a time) into the text box labeled "Hot" below. At the same time, select numbers from the range on the "COLD" scale and enter them (one at a time) into the text box labeled "COLD" below. Try various combinations of hot and cold numbers and observe the graph to see the temperatures' effect on efficiency.

Example

For a coal-fired utility boiler, the temperature of high pressure steam (Thot)would be about 540°C and Tcold, the cooling tower water temperature, would be about 20°C. Calculate the Carnot efficiency of the power plant:

Solution:

Carnot efficiency depends on high temperature and low temperatures between which the heat engine operates. We are given both temperatures. However, the temperatures need to be converted to Kelvin:

T hot = 540 o C+273=813K T cold = 20 o C+273=293K η=[ 1 T cold T hot ]×100% η=[ 1 293K 813K ]×100% =64%

Practice

For a coal fired utility boiler, the temperature of high pressure steam would be about 540 degrees C and Tcold, the cooling tower water temperature, would be about 20 degrees C. Calculate the Carnot efficiency of the power plant.

Step 1
Convert the high and low temperatures from Celsius to Kelvin:

T hot = 540 o C+273 =813K

T cold = 20 o C+273 =293K

Step 2

Determine the efficiency using the Carnot efficiency formula:

η=[ 1 T cold T hot ]×100% η=[ 1 293K 813K ]×100% =64%
From the Carnot Efficiency formula, it can be inferred that a maximum of 64% of the fuel energy can go to generation. To make the Carnot efficiency as high as possible, either Thot should be increased or Tcold (temperature of heat rejection) should be decreased.

Practice Problem

Use the following link to generate a random practice problem.