Carnot and Rankine's approach to heat engine

Carnot cycle is an interplay of heat and temperature in a reversible process, dS = Q rev/T

Carnot's idea was to conceive an ideal reversible heat engine with maximum efficiency that could become a benchmark for heat engines.

He conceived a cyclic machine that had two isothermal energy reservoirs that did not generate entropy. One is an isothermal expansion hot reservoir [ typically a boiler] to supply energy isothermally and the other is a cold reservoir with isothermal compression that also acts reversibly to act as an energy sink.

Energy moves in a clockwise direction from the hot reservoir first to an adiabatic expansion station to perform work reversibly. Being an adiabatic process there is no entropy generation.

Next, is a cold reservoir where the machine deposits the residual energy after performing mechanical work.

The next station is an adiabatic compression station which fills up the energy consumed by the machine while doing mechanical work and restores the initial energy before feeding back the gas to a hot reservoir.

The entropy generation in the hot reservoir is negated in the cold reservoir as the volume of expansion of gas in the hot reservoir equals the volume of compression of gas in the cold reservoir.

This completes a Carnot cycle.

Further explanation

Please refer to the image below.

Image credit: Google

Point 1-2: Isothermal Expansion. Heat is transferred reversibly from the high-temperature reservoir at constant temperature T1 (isothermal heat addition or absorption). During this step the gas is allowed to expand, doing work on the surroundings.?Although the pressure drops from points 1 to 2 the temperature of the gas does not change during the process because it is in thermal contact with the hot reservoir at T1, and thus the expansion is isothermal. Heat energy Q1 is absorbed from the high-temperature reservoir resulting in an increase in the entropy of the gas by the amount

Point 2-3: Isentropic (reversible adiabatic) expansion of the gas (isentropic work output). For this step, the gas in the engine is thermally insulated from both the hot and cold reservoirs. Thus, they neither gain nor lose heat, an 'adiabatic' process. The gas continues to expand by reducing pressure, doing work on the surroundings, and losing an amount of internal energy equal to the work done. The gas expansion without heat input causes it to cool to the "cold" temperature, T3. The entropy remains unchanged.

Point 3-4: Isothermal Compression. Heat is transferred reversibly to the low-temperature reservoir at constant temperature T3. (Isothermal heat rejection)

Now the gas in the engine is in thermal contact with the cold reservoir at temperature T3. Cooling does compression work. This causes an amount of heat energy Q2 to leave the system in the cold reservoir. The entropy of the system decreases by the amount dS2 = Q2/T3. This is the same amount of entropy gain in step 1.

Point 4-1: Adiabatic reversible compression. Once again, the gas in the engine is thermally insulated from the hot and cold reservoirs, and the engine is assumed to be frictionless, hence reversible. During this step, the surroundings do work on the gas. compressing it, and causing its temperature to rise back to T1 due solely to the work added to the system, but the entropy remains unchanged. At this point, the gas is in the same state as at the start of step 1.

This completes a Carnot cycle.

The notable point is Carnot chose isothermal heat reservoirs. The question is why he chose isothermal heat reservoirs despite fully realizing that for any isothermal process, the process must be very slow to be a reversible process.

The answer is he had no options other than isothermal heat reservoirs.

He did not consider constant-pressure heat reservoirs because these reservoirs are not reversible in nature. A constant pressure process has external intervention while keeping the pressure constant.

That is the reason Carnot went with isothermal reservoirs.

It is only the cyclic movement of internal energy that takes place in a Carnot cycle with zero change in internal energy dE = 0 in a cycle while the 'work' happens between hot and cold reservoirs. .

Carnot eliminated the ' work ' component of energy from his cycle because ' work ' associates itself with entropy. Whenever heat converts to work there is entropy generation. There is no net generation of ' work' at any point in the cycle.

In a reversible system dS = Q rev/T

Carnot cycle is an interplay of heat and temperature in a reversible process

Rankine had a more practical approach

He conceived a real-life cycle with similar four arms but not constrained to deliver a non-entropy reversible process. Rankine's focus was more on a workable practical machine. Rankine's cycle represents the movement of enthalpy which is total energy unlike the movement of internal energy in a cycle in Carnot's heat engine.

Rankine conceived constant pressure energy reservoirs to maximize the enthalpy delivery. For a constant pressure process

dH = delta Q, that is the total thermal energy

For an isothermal process

delta Q = delta W, that is work between two points.

The Rankine cycle is a well-known heat engine for power plants.

The issue is its low efficiency. It is about 40%. The efficiency is limited by the cooling tower water temperature that carries the enthalpy of compression ( condensation) of vapor in the constant pressure condenser for heat rejection.

These are some thoughts to explain Carnot and Rankine's different approaches to a heat engine.