Efficiency of a Stirling Engine

In summary, the efficiency of a Stirling engine working between two heat baths at two different temperatures is determined by the proportion of heat gained to heat lost.
  • #1
aurora14421
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0

Homework Statement



Derive an expression for the effiency of the cycle (of a Stirling Engine) working between two heat baths at temperatures T1 and T2 with volumes in the ratio V2/V1. Assume the working substance is a monatomic ideal gas.

Homework Equations



Work done=heat absorbed from going from V1 to V2 at constant temperature and is given by:

RT ln(V2/V1)

Heat absorbed in going from T1 to T2 is:

(3/2)R(T2-T1)

efficiency=1-Q2/Q1 where Q2 is the heat lost per cycle and Q1 is the heat gained per cycle.

The Attempt at a Solution



I have an expression but I have no idea if it was correct. I tried to google what the correct efficiency should be but I couldn't find it. My expression is:

[tex]\eta = \frac{(T_2-T_1) (ln(V_2/V_1)+3)}{T_2 ln(V_2/V_1) +1.5(T_2-T_1)}[/tex]

I don't think it's right, it looks a bit messy. If someone could tell me if it's correct or not that would be a great help. It would be an even greater help if someone could tell me where I have gone wrong or give me the correct expression so I can work out how to derive it.

Thanks.
 
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  • #2
I get a similar, but not identical, expression.

What did you get for Q1 and Q2?
 
  • #3
I had, Q1, the heat absorbed, is:

[tex] R T_2 ln(V_2/V_1) + 1.5R(T_2-T_1) [/tex]

And Q2, the heat lost, is:

[tex] R T_1 ln(V_2/V_1) + 1.5R(T_1-T_2) [/tex]
 
  • #4
I got, for the final term in Q2, 1.5R(T2 - T1)

But we should clarify something: are you, like me, using T2 for the hot temperature and T1 for the cold temperature?
 
  • #5
aurora14421 said:
I had, Q1, the heat absorbed, is:

[tex] R T_2 ln(V_2/V_1) + 1.5R(T_2-T_1) [/tex]

And Q2, the heat lost, is:

[tex] R T_1 ln(V_2/V_1) + 1.5R(T_1-T_2) [/tex]
The heat lost at constant temperature in the compression part of the cycle (V2 to V1) is equal to the work done on the gas (per mole) (dU = 0) :

[tex]\int_{V_2}^{V_1} PdV = RT_1\int_{V_2}^{V_1}dV/V = RT_1\ln{(V_1/V_2)}[/tex]

The heat gained at constant temperature in the expansion part of the cycle (V1 to V2) is equal to the work done by the gas (per mole) (dU = 0) :

[tex]\int_{V_1}^{V_2} PdV = RT_2\int_{V_1}^{V_2}dV/V = RT_2\ln{(V_2/V_1)} = - RT_2\ln{(V_1/V_2)}[/tex]

Therefore:

[tex]|Q_h| = C_v(T_2-T_1) + RT_2\ln{(V_2/V_1)}[/tex] and

[tex]|Q_c| = C_v(T_2-T_1) + RT_1\ln{(V_2/V_1)}[/tex]

So plug that into:

[tex]\eta = W/Q_h = (Q_h-Q_c)/Q_h[/tex]

I get:

[tex]\eta = \frac{R\ln{(V_2/V_1)}(T_2-T_1)}{C_v(T_2-T_1) + RT_2\ln{(V_2/V_1)}}[/tex]

For a monatomic gas:

[tex]\eta = \frac{2\ln{(V_2/V_1)}(T_2-T_1)}{3(T_2-T_1) + 2T_2\ln{(V_2/V_1)}}[/tex]

AM
 
Last edited:
  • #6
Thank you very much for all your help. Got it now.
 

1. What is a Stirling Engine?

A Stirling engine is a closed-cycle heat engine that operates by cyclic compression and expansion of air or other gas at different temperature levels. It was invented in 1816 by Robert Stirling and is known for its high efficiency and low emissions.

2. How does a Stirling Engine work?

A Stirling engine works by using a temperature difference between two separate chambers, one hot and one cold, to drive a piston back and forth. The heated air expands and pushes the piston, which then compresses the cooler air in the other chamber. This back-and-forth motion is converted into rotational motion, which can be used to power machinery.

3. What makes the Stirling Engine more efficient than other engines?

The Stirling engine is more efficient than other engines because it uses an external heat source, allowing for a high temperature difference between the hot and cold chambers. This results in a more efficient and continuous conversion of heat energy into mechanical energy, without the need for internal combustion or steam production.

4. What are the advantages of using a Stirling Engine?

Some advantages of using a Stirling engine include its high efficiency, low emissions, quiet operation, and ability to run on a variety of fuel sources such as solar, biomass, or waste heat. It also has fewer moving parts, making it more reliable and easier to maintain.

5. Are there any limitations to the efficiency of a Stirling Engine?

While Stirling engines have a high theoretical efficiency, the actual efficiency can be affected by factors such as friction, heat loss, and the quality of the materials used. The efficiency also decreases at higher power outputs, making it more suitable for smaller-scale applications rather than large-scale power generation.

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