## July 07, 2007

### Carnot Cycle - An Insight

Carnot - Can OR Cannot???????

This is in continuation of my previous article on 'Thermodynamics of Thermal Cycles' & is related to practical aspects of it.

All standard heat engines (steam, gasoline, diesel) work by supplying heat to a gas, the gas then expands in a cylinder and pushes a piston to do its work. The catch is that the heat and/or the gas must somehow then be dumped out of the cylinder to get ready for the next cycle.

We consider the standard Carnot-cycle machine having a piston moving within a cylinder, and having the following characteristics:
• A perfect seal, so that no atom escape from the working fluid as the piston moves to expand or compress it..............Wear & tear cannot be eliminated.
• Perfect lubrication, so that there is no friction...........Impractical.
• An ideal-gas for the working fluid................Not true at working conditions of pressure & temperature.
• Perfect thermal connection at any time either to one or to none of two reservoirs, which are at two different temperatures, with perfect thermal insulation isolating it from all other heat transfers......................Cannot be achieved practically.
• The piston moves back and forth repeatedly, in a cycle of alternating "isothermal" and "adiabatic" expansions and compressions, according to the PV diagram shown below:
This is basically
1. A high temperature Isothermal Expansion from A to B. - Heat is Supplied to the engine.
2. An Adiabatic expansion cooling down from B to C.
3. A low temperature Isothermal compression from C to D - Heat Removal step.
4. Adiabatic compression from D to A - work performed on fluid.

Carnot cycle works as an engine if heat is absorbed from A to B and rejected at C to D doing some useful work. It works as a heat pump if it is motor driven.

Isothermal Expansion

So the first question is: how much work is done by an isothermally expanding gas? Taking the temperature of the heat reservoir to be Th (h for hot), the expanding gas follows the isothermal path PV = n R Th in the (P, V) plane.

Hence the work done in expanding isothermally from volume Va to Vb is the total area under the curve between those values,

W = n R Th Ln(V2/V1)

Since the gas is at constant temperature Th, there is no change in its internal energy during this expansion, so the total heat supplied must be, the same as the external work the gas has done.

In fact, this isothermal expansion is only the first step: the gas is at the temperature of the heat reservoir, hotter than its other surroundings, and will be able to continue expanding even if the heat supply is cut off. To ensure that this further expansion is also reversible, the gas must not be losing heat to the surroundings. That is, after the heat supply is cut off, there must be no further heat exchange with the surroundings, the expansion must be adiabatic.

The work done in an adiabatic expansion is like that done in allowing a compressed spring to expand against a force—equal to the work needed to compress the spring in the first place, for a perfect spring, and an adiabatically enclosed gas is essentially perfect in this respect. In other words, adiabatic expansion is reversible. To find the work the gas does in expanding adiabatically from Vb to Vc, say, the above analysis is repeated with the isotherm replaced by the adiabatic equation PV^^k = Constant where k is adiabatic coefficient.

W = Pb Vb^^k * (Vc^^1-k - Vb^^1-k) / (1-k)

OR

W = (Pc Vc - Pb Vb) / (1-k)

(I do not know how to use exponent in this blogger thing....if anybody can help me he is welcome).

We’ve looked in detail at the work a gas does in expanding as heat is supplied (Isotherm) and when there is no heat exchange (adiabatically). These are the two initial steps in a heat engine, but it is equally necessary for the engine to get back to where it began, for the next cycle. The general idea is that the piston drives a wheel,which continues to turn and pushes the gas back to the original volume.

Download an animated version of Carnot Engine here.

The PV diagram for the complete cycle is given below.

Efficiency of the Carnot Engine
In a complete cycle of Carnot’s heat engine, the gas traces the path abcd. The important question is: what fraction of the heat supplied from the hot reservoir (along the red top isotherm) is turned into mechanical work? This fraction is called the efficiency of the engine.

The work output along any curve in the (P, V) plane is just --the area under the curve, but it will be negative if the volume is decreasing! So the work done by the engine during the hot isothermal segment is the area abfh, then the adiabatic expansion adds the area bcef, but as the gas is compressed back, the wheel has to do work on the gas equal to the area cdge as heat is dumped into the cold reservoir, then dahg as the gas is recompressed to the starting point.

The bottom line is that the total work done by the gas is the area bounded by the four paths: the curved "parallelogram" in the picture above. We could compute this area by finding P dV for each segment, but that is unnecessary—on completing the cycle, the gas is back to its initial temperature, so has the same internal energy. Therefore, the work done by the engine must be just the difference between the heat supplied at Th and that dumped at Tc.

Now the heat supplies along the initial hot isothermal path ab, equal to the work done along that leg, is (from the paragraph above on isothermal expansion):

Qh = n R Th Ln(Vb/Va)

and the heat dumped into the cold reservoir along cd is

Qc = n R Tc Ln(Vc/Vd)

Finally after some simple math

Efficiency = 1 - Tc / Th

It can be easily proved by second law of thermodynamics that no engine can be more efficient than carnot engine. As I have already shown in my previous article that Ideal carnot cannot go beyond 80% efficiency given the above assumptions are true, while practical combined efficiency of all mechanical system is ~40%. Thus, overall efficinecy drops to ~32%.

Clearly the major contributor is mechanical losses not thermodynamic losses. Hence, as per my analysis this is true only for said four step cycle and for thermodynamic efficiency operating between Th & Tc. Now engines have been devised with different no of thermodynamic steps, therefore reducing the total mechanical losses and improving overall system efficiency.

So if we can device a system where each mechanical component can either be improved or can be eliminated we can significantly increase the efficiency of overall cycle. (Fornuately it has been done, Read it here.....Waste Heat driven Engine)

However the above technology is still doubtful about its suitability for large size machines where a true HEAT ENGINE would be replaced by it.

Therefore, til that time we have to consider different cycles for improving the efficiency as I would present it in my next papers on Thermodynamics of heat engine ( Probably with exergy if I get some more time).

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