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The Carnot Cycle
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The Carnot cycle is a generic cycle in the pressure-volume space that shows how in principle a thermodynamic machine can be built that converts heat flow into mechanical work.
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The Carnot Cycle
Description
The Carnot cycle is a generic cycle in the pressure-volume space that shows how in principle a thermodynamic machine can be built that converts heat flow into mechanical work.
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Equations
(ID 11135)
The efficiency ($\eta$) is a function of the heat supplied ($Q_H$) and the absorbed heat ($Q_C$), given by:
| $ \eta = 1-\displaystyle\frac{ Q_C }{ Q_H } $ |
We can express the heat supplied ($Q_H$) in terms of the low temperature ($T_C$), the low entropy ($S_C$), and the high entropy ($S_H$) as:
| $ Q_C = T_C ( S_H - S_C ) $ |
And using the high temperature ($T_H$) as:
| $ Q_H = T_H ( S_H - S_C ) $ |
If we substitute these expressions, we get:
| $ \eta = 1 - \displaystyle\frac{ T_C }{ T_H } $ |
(ID 11136)
Since the effective work ($W$) is equal to the integral along a closed path in the space of the absolute temperature ($T$) and the entropy ($S$), we have:
| $ W = \displaystyle\oint T dS$ |
Consulting the temperature-entropy graph, we can see that the absorbed heat the heat supplied ($Q_H$) is equal to the high temperature ($T_H$) due to the difference in entropy, i.e., the high entropy ($S_H$) and the low entropy ($S_C$):
| $ Q_H = T_H ( S_H - S_C ) $ |
(ID 11137)
As the effective work ($W$) is equal to the integral along a closed path in the the absolute temperature ($T$) and the entropy ($S$) space, we have:
| $ W = \displaystyle\oint T dS$ |
Consulting the temperature-entropy graph, we can see that the absorbed heat the absorbed heat ($Q_C$) is equal to the low temperature ($T_C$) due to the difference in entropy, i.e., the high entropy ($S_H$) and the low entropy ($S_C$):
| $ Q_C = T_C ( S_H - S_C ) $ |
(ID 11138)
Since the efficiency ($\eta$) with the effective work ($W$) and the heat supplied ($Q_H$) is
| $ \eta \equiv \displaystyle\frac{ W }{ Q_H } $ |
it can be replaced with the effective work ($W$) which, along with the heat supplied ($Q_H$) and the absorbed heat ($Q_C$), results in
| $ W = Q_H - Q_C $ |
yielding the following relationship:
| $ \eta = 1-\displaystyle\frac{ Q_C }{ Q_H } $ |
(ID 11155)
Examples
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