Carnot Engine

Sadi Carnot (1796-1832), a French physicist, devised an engine which working in a four-stroke cycle under given conditions could be said to be the best possible heat engine to secure maximum useful mechanical energy out of the heat energy supplied to the engine. His engine worked between two reservoirs of heat, one at temperature T1 which we may call the source and the other at T3 which we may call the sink, with T1 > T3. The working substance was 1 mol of a gas which was contained in a cylinder one end of which was closed with a conducting lid while through the other end a frictionless piston could move in and out. The cylinder could be placed in contact with the heat source and sink by turn at will and could also be insulated from them when not required. The four stages of the operations and the manner in which they were performed are shown by an indicator (p, v) diagram (Fig. 3.2). The stages are as follows:

Stage 1. Place the cylinder in contact with the heat source T1 and extract a quantity of heat Q1 from it by a slow outward movement of the piston so that there is no change of temperature inside the cyclinder while the gas is being heated and its volume changing from v1 to v2 with the inside pressure being only a little higher than the outside pressure.

Stage 2. The cylinder is then removed from the heat source and placed in contact with the insulator and the gas is allowed to expand adiabatically under its own pressure from volume v2 to v3. The expansion leads to a fall of pressure as well as temperature from T2 (= T1) to T3.

Stage 3. The cylinder is then removed from the insulator and placed in contact with heat sink T3. The piston is then pressed gently inward so that a quantity of heat Q3 is delivered to the heat sink at temperature T3, while the volume changes from v3 to a volume v4

Fig. 3.2 The Carnot cycle: Indicator diagram

Fig. 3.2 The Carnot cycle: Indicator diagram

Stage 4. During this stage, an adiabatic compression will bring the volume v4 back to the original volume vi.

The work done by the gas during the different stages is as follows: Stage 1 Qi = W12 = Hp dv = R Ti P dv/v = R Ti ln (v2/vi) (3.6.1)

Jvi Jvi

Stage 2 W23 = P pdv = (p2v2Y /^ dv/vY = {(p2v2Y)/(i - y)} bi-y - v2i-Y]

= {(RT2/(i - Y)}[(v3/v2) - i] = {R/(i - Y)} [T3 - T2] = {R/(y- i)}(Ti - T3) (3.6.2)

The cylinder is now placed in thermal contact with the heat sink T3 and the gas compressed isothermally from volume v3 to v4, so that a quantity of heat Q3 is delivered to the sink. The work done in this process is negative, since work is done on the gas and not by the gas. So, the work done in

Stage 3 Q3 = W34 = T4 p dv = R T3 dv/v = RT3 ln (v4/v3) (3.6.3)

Jv3 J v3

The cylinder is now removed from the heat sink and placed on the insulator stand and compressed adiabatically from volume v4 to vi. The work done in fvi fvi Y Stage 4 W4i = p dv = (p4 v4Y) dv/vY

v4 v4

The work done in stage 4 also is to be reckoned negative, since it is done on the gas.

Thus, the total work done by the gas, from (3.6.1-3.6.4), is

since, the expressions in (3.6.2) and (3.6.4) cancel each other, and (v2/v1) =

In summary, the working substance in the Carnot engine drew a quantity of heat energy Q1 from the source, performed the useful work W, and returned the balance of heat Q3 to the sink before being adiabatically compressed to the original state.

In other words, Q1-Q3 = W, and the efficiency n of the engine is given by n = (W/Q1) = (Q1 - Q3)/Q1 = (T1 - T3)/T1 = 1 - (T3/T1) (3.6.6)

Also, it follows from (3.6.6) that

The Carnot engine being reversible is the most efficient of heat engines and it is not possible to invent an engine working reversibly between the temperatures T1 and T3 which can be more efficient than the Carnot engine. For, if it were possible to invent a super-Carnot machine, it would be possible to get work out of it continuously without transferring any heat to a sink. Such a possibility simply does not exist in nature. However, according to (3.6.6), the efficiency of the Carnot engine increases as the sink temperature T3 is lowered, and attains unity when it is brought down to Absolute zero. Thus, the Absolute scale of temperature, which we introduced earlier (2.4.2), is a temperature scale which can be deduced from thermodynamical reasoning. For this reason, it is also sometimes called the thermodynamic scale of temperature.

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