Carnot heat engine

Axial cross section of Carnot’s heat engine. In this diagram, abcdis a cylindrical vessel, cd is a movable piston, and A and B areconstant–temperature bodies. The vessel may be placed in con-tact with either body or removed from both (as it is here).[1]

ACarnot heat engine[2] is an engine that operates on thereversible Carnot cycle. The basic model for this enginewas developed by Nicolas Léonard Sadi Carnot in 1824.The Carnot engine model was graphically expanded uponby Benoît Paul Émile Clapeyron in 1834 and mathemat-ically elaborated upon by Rudolf Clausius in 1857 from

which the concept of entropy emerged.Every thermodynamic system exists in a particular state.A thermodynamic cycle occurs when a system is takenthrough a series of different states, and finally returnedto its initial state. In the process of going through thiscycle, the system may perform work on its surroundings,thereby acting as a heat engine.A heat engine acts by transferring energy from a warm re-gion to a cool region of space and, in the process, convert-ing some of that energy to mechanical work. The cyclemay also be reversed. The system may be worked uponby an external force, and in the process, it can transferthermal energy from a cooler system to a warmer one,thereby acting as a refrigerator or heat pump rather thana heat engine.

1 Carnot’s diagram

In the adjacent diagram, from Carnot’s 1824 work,Reflections on the Motive Power of Fire,[3] there are “twobodies A and B, kept each at a constant temperature, thatof A being higher than that of B. These two bodies towhich we can give, or from which we can remove theheat without causing their temperatures to vary, exer-cise the functions of two unlimited reservoirs of caloric.We will call the first the furnace and the second therefrigerator.”[4] Carnot then explains how we can obtainmotive power, i.e., “work”, by carrying a certain quantityof heat from body A to body B.

2 Modern diagram

The previous image shows the original piston-and-cylinder diagram used by Carnot in discussing his idealengines. The figure at right shows a block diagram of ageneric heat engine, such as the Carnot engine. In the dia-gram, the “working body” (system), a term introduced byClausius in 1850, can be any fluid or vapor body throughwhich heatQ can be introduced or transmitted to producework. Carnot had postulated that the fluid body could beany substance capable of expansion, such as vapor of wa-ter, vapor of alcohol, vapor of mercury, a permanent gas,or air, etc. Although, in these early years, engines camein a number of configurations, typically QH was suppliedby a boiler, wherein water was boiled over a furnace; QCwas typically supplied by a stream of cold flowing water

1

2 4 CARNOT’S THEOREM

Carnot engine diagram (modern) - where an amount of heat QHflows from a high temperature TH furnace through the fluid ofthe “working body” (working substance) and the remaining heatQC flows into the cold sink TC, thus forcing the working sub-stance to do mechanical workW on the surroundings, via cyclesof contractions and expansions.

in the form of a condenser located on a separate part ofthe engine. The output workW here is the movement ofthe piston as it is used to turn a crank-arm, which wasthen typically used to turn a pulley so to lift water out offlooded salt mines. Carnot defined work as “weight liftedthrough a height”.

3 The Carnot engine

Main article: Carnot Cycle

The Carnot cycle when acting as a heat engine consistsof the following steps:

1. Reversible isothermal expansion of the gas at the“hot” temperature, TH (isothermal heat addi-tion or absorption). During this step (1 to 2 onFigure 1, A to B in Figure 2) the gas is allowed toexpand and it does work on the surroundings. Thetemperature of the gas does not change during theprocess, and thus the expansion is isothermic. Thegas expansion is propelled by absorption of heat en-ergy Q1 and of entropy ∆SH = QH/TH from thehigh temperature reservoir.

2. Isentropic (reversible adiabatic) expansion ofthe gas (isentropic work output). For this step (2to 3 on Figure 1, B to C in Figure 2) the piston andcylinder are assumed to be thermally insulated, thusthey neither gain nor lose heat. The gas continues toexpand, doing work on the surroundings, and losingan equivalent amount of internal energy. The gas ex-pansion causes it to cool to the “cold” temperature,TC. The entropy remains unchanged.

3. Reversible isothermal compression of the gas atthe “cold” temperature, TC. (isothermal heat re-jection) (3 to 4 on Figure 1, C to D on Figure 2)Now the surroundings do work on the gas, caus-ing an amount of heat energy Q2 and of entropy∆SC = QC/TC to flow out of the gas to the low

temperature reservoir. (This is the same amount ofentropy absorbed in step 1.)

4. Isentropic compression of the gas (isentropicwork input). (4 to 1 on Figure 1, D to A on Figure2) Once again the piston and cylinder are assumedto be thermally insulated.

During this step, the surroundings do work on the gas,increasing its internal energy and compressing it, causingthe temperature to rise to TH. The entropy remains un-changed. At this point the gas is in the same state as atthe start of step 1.

4 Carnot’s theorem

Main article: Carnot’s theorem (thermodynamics)Carnot’s theorem is a formal statement of this fact: No

T

Liquid Vapour Liquid & Vapour

S S

T

TH

TC

real ideal engines (left) compared to the Carnot cycle (right). Theentropy of a real material changes with temperature. This changeis indicated by the curve on a T-S diagram. For this figure, thecurve indicates a vapor-liquid equilibrium (See Rankine cycle).Irreversible systems and losses of heat (for example, due to fric-tion) prevent the ideal from taking place at every step.

engine operating between two heat reservoirs can be moreefficient than a Carnot engine operating between the samereservoirs.

This maximum efficiency ηI is defined to be:

ηI =W

QH= 1− TC

TH(1)

where

W is the work done by the system (energy ex-iting the system as work),QH is the heat put into the system (heat energyentering the system),TC is the absolute temperature of the coldreservoir, andTH is the absolute temperature of the hot reser-voir.

A corollary to Carnot’s theorem states that: All reversibleengines operating between the same heat reservoirs areequally efficient.

3

It is easily shown that the efficiency η is maximum whenthe entire cyclic process is a Reversible process. Thismeans the total entropy of the total system consisting ofthe three parts: i)entropy of the hot furnace, ii)entropy ofthe “working fluid” of the Heat engine, and iii)entropy ofthe cold sink, remains constant when the “working fluid”completes one cycle and returns to its original state. (Inthe general case, the total entropy of this combined sys-tem would increase in a general irreversible process).Since the “working fluid” comes back to the same stateafter one cycle, and entropy of the system is a state func-tion; the change in entropy of the “working fluid” systemis 0. Thus, it implies that the total entropy change of thefurnace and sink is zero, for the process to be reversibleand the efficiency of the engine to be maximum. Thisderivation is carried out in the next section.The Coefficient of Performance (COP) of the heat engineis the reciprocal of its efficiency.

5 Efficiency of real heat engines

For a real heat engine, the total thermodynamic process isgenerally irreversible. The working fluid is brought backto its initial state after one cycle, and thus the change ofentropy of the fluid system is 0, but the sum of the entropychanges in the hot and cold reservoir in this one cyclicalprocess is greater than 0.The internal energy of the fluid is also a state variable, soits total change in one cycle, is 0. So the total work doneby the systemW , is equal to the heat put into the systemQH minus the heat taken out QC .

W = QH −QC (2)

For real engines, sections 1 and 3 of the Carnot Cycle; inwhich heat is absorbed by the “working fluid” from thehot reservoir, and released by it to the cold reservoir, re-spectively; no longer remain ideally reversible, and thereis a temperature differential between the temperature ofthe reservoir and the temperature of the fluid while heatexchange takes place.During heat transfer from the hot reservoir at TH to thefluid, the fluid would have a slightly lower temperaturethan TH , and the process for the fluid may not necessarilyremain isothermal. Let∆SH be the total entropy changeof the fluid in the process of intake of heat.

∆SH =

∫Qin

dQH

T(3)

where the temperature of the fluid T is always slightlylesser than TH , in this process.So, we would get

QH

TH=

∫dQH

TH≤ ∆SH (4)

Similarly, at the time of heat transfer from the fluid tothe cold reservoir we would have, for the magnitude oftotal entropy change ∆SC of the fluid in the process ofexpelling heat:

∆SC =

∫Qout

dQC

T≤

∫dQC

TC=

QC

TC(5)

Where, during this process of transfer of heat to the coldreservoir, the temperature of the fluid T is always slightlygreater than TC .We have only considered the magnitude of the entropychange here. Since the total change of entropy of the fluidsystem for the cyclic process is 0, we must have

∆SH = ∆SC (6)

The previous three equations combine to give:

QC

TC≥ QH

TH(7)

Equations (2) and (7) combine to give

W

QH≤ (1− TC

TH) (8)

Hence,

η ≤ ηI (9)

where η = WQH

is the efficiency of the real engine, and ηIis the efficiency of the Carnot Engine working betweenthe same two reservoirs at the temperatures TH and TC

. For the Carnot engine, the entire process is 'reversible',and Equation (7) is an equality.Hence, the efficiency of the real engine is always less thanthe ideal Carnot Engine.Equation (7) signifies that the total entropy of the totalsystem(the two reservoirs + fluid) increases for the realengine, because the entropy gain of the cold reservoir asQC flows into it at the fixed temperature TC , is greaterthan the entropy loss of the hot reservoir as QH leaves itat it’s fixed temperature TH . The inequality in Equation(7) is essentially the statement of the Clausius theorem.According to the second theorem, “The efficiency of theCarnot engine is independent of the nature of the workingsubstance”.