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Engineering › Thermodynamics ›

Thermodynamic cycles and the Carnot Cycle

A description of thermodynamic cycles and in particular the Carnot Cycle and the proof of it's thermal efficiency

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Contents

Thermodynamic Cycles
Carnot Cycle
To Improve Efficiency
Page Comments

Thermodynamic Cycles

A series of operations carried out on the Working Substance (WS) during which heat is supplied (Q) . There is a work output (W) after which the WS is returned to it's original state.

Expansion from

and a Work Ouput of

Compression from

and a Work input of

$\text{Net Work Output} = W_1 - W_2 = \text{Area of Cycle}$

(1)

The First Law Applied To A Cycle:

By Definition:

$P_1 \geq P_2\;\;\;\;\;\;\;\;E_1 = E_2$

(2)

$V_1 = V_2\;\;\;\;\;\;\;\;H_1 = H_2$

(3)

$T_1 = T_2\;\;\;\;\;\;\;\;Q_1 = Q_2$

(4)

The Energy of the WS at the start of the cycle + Heat supplied = Energy of the WS at the end of the cycle + WD +Heat losses.

or Work output = Heat supplied - Heat lost and rejected.

Drawing the cycle on a T $\Phi$ diagram

Heat supplied, $\Phi$ , increasing

$\text{Heat supplied} = \text{area under 1,B,2} = Q_S$

(5)

Heat rejected, $\Phi$ decreasing

$\text{Heat rejected} = \text{area under 1,B,2} = Q_R$

(6)

Area of diagram =

= Work Done

NOTE: the area of the P V diagram gives the Work Done in ft lbs. The area of the T $\Phi$ diagram gives the work Done in BTUs

The Thermal Efficiency Of A Cycle:

$\eta =\frac{Work\; Done}{Heat\;Suppplied}=\frac{Q_S-Q_R}{Q_S}$

(7)

therefore

$eta = 1 -\frac{Q_R}{Q_S}$

(8)

Carnot Cycle

We can obtain the efficiency of the Carnot Cycle and hence the efficiency of any Reversible cycle operating between the temperatures of

and

. This then represents the Ultimate Thermal Efficiency. This is then used to compare the efficiencies of other cycles operating between the same two temperatures. The importance of the Carnot Cycle in this role can not be under estimated.