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The second law of thermodynamics, part 2

Second law of thermodynamics heat engine and Carnot cycle


The second law of thermodynamics, part 2

David Mark / tpsdave

The second law of thermodynamics, part 2

Our discussion of the second law of thermodynamics continues with Carnot cyclic processes and heat engines, expanding on the statements discussed in part 1.

We discussed three different statements of the second law of thermodynamics in part one and even examined the equivalence between two of those statements. Picking up from there, we are going to take a look at two of the earliest ideas to accompany the second law: cyclic processes and heat engines.

The thermodynamic cycle

At the heart of every engine is a cyclic process. Fundamentally, this is simply a series of processes that take a system through any number of stages that finally return to the initial stage, leaving the system as it was. To gain a good understanding of the typical, ideal cyclic process, one needs to grasp the relationship between heat and work, which we discussed in part one, and the idea of temperature.

While there are several such cyclic processes in thermodynamics, including the the steam engine, the Otto cycle, the diesel engine etc., we will restrict our discussion to what is arguably the most important of the lot, the Carnot heat engine or the Carnot cycle. This is also motivated by the fact that the most important statements of the second law of thermodynamics are built on the Carnot cycle. As a result, this cycles is sometimes also known simply as the thermodynamic cycle.

Carnot cycle PV diagram

PV-diagram of a typical Carnot cycle

With reference to the figure above, consider our system to be at A initially, with its pressure and volume given by (P_1,\,V_1). Say it is at a temperature T_1. The cyclic process it undergoes will be as follows:

  1. Isothermal expansion of the system takes place with an input of heat Q_1 which is manifested as a change in the volume and pressure of the system, or, simply, some work done by the system on its surroundings, implying that the temperature, T_1, remains constant. Say the system achieves the state (P_2,\,V_2) at the end of this, arriving at B.
  2. Isentropic expansion of the system now takes it to (P_3,\,V_3) at C with further increase in pressure and volume as the system continues to do some work on its surroundings and, in turn, sees a fall in temperature to some T_2 since no external heat is being supplied. Such a process is also known as an adiabatic process and completes half of our cycle. Having expanded in volume, our system is now ready to begin the remaining half of its process with contractions.
  3. Isothermal compression of the system drops its volume and pressurises it to some (P_3,\,V_3) with a loss of heat and maintenance of the same temperature, T_2, as work is done on the system this time.
  4. Isentropic compression finally returns the system to (P_1,\,V_1) as work is continued to be done on the system resulting in a rise in temperature to T_1 since no heat is lost. This is also known as adiabatic compression and sees our system complete its cyclic process to return to its initial state from which the cycle can repeat starting with another bout of isothermal expansion.

In this way, the system can, theoretically, keep going perpetually. However, as stated earlier, this is an ideal process, particularly because of the isentropic stages. (Isentropic processes are ideally defined adiabatic process, i.e. processes that see no heat or mass exchange, which further see frictionless work.) It is also worth noting that, while the process as a whole is cyclic, so is each process: the move from A to B, B to C, C to D and D to A are all individually reversible processes.

Revisiting Carnot’s statement

The simplest engine one can think of is a gas in a piston-cylinder system. Work is done by the gas is when it pushes against the piston and on the gas when we push the piston in. The entire thermodynamic cycle can be imagined this way. The product of pressure and volume is known as the pressure-volume work, which means the area of the graph (roughly equated to a rectangle)  gives the total work done.

Let us take a look back at the first statement of the second law of thermodynamics that we discussed in part one:

A reversible Carnot engine operating between two given temperatures is the most efficient engine possible; further, the efficiency of such a heat engine depends only on the temperatures between which it operates.

The idea of an engine, or even something as fundamental as a gas-filled piston-cylinder system, can be combined with the thermodynamic cycle to arrive at a physical picture for Carnot’s statement. The operating temperatures, according to our graph, are T_1 and T_2, which correspond to the source and sink temperatures.

What Carnot says, then, is that if one were to build an engine that operates according to the graph above, it would achieve 100% efficiency. In part one we defined efficiency as

    \[\eta = 1 - { T_L \over T_H}\]

which tends to one as the difference between the source and sink temperatures increase. For instance, if you had an engine operating between T_L = 298\,K, or, approximately, the temperature of a pleasant room, and T_H = 373\,K, somewhere around the boiling point of water, it would have an efficiency of 20%, which is not all that impressive.

What if we operate it between the cold of a vacuum (2.7\,K) and the heat of the sun’s surface (5,70,K)? We have an incredible engine running at 99.95% efficiency.

The heat engine

Notice that even the Carnot cycle does not achieve 100% efficiency, since T_L can never go to absolute zero. But, closer to reality, we had to operate between the Sun’s surface and outer space to even reach 99% efficiency. The Carnot cycle is but a conceptual entity. It is the ideal engine that no engine can be like, but the purpose of the Carnot cycle is not to be a exemplary but instead to help establish the maximum possible efficiency that an engine can achieve given its operating temperatures.

Let us attribute this work to a substance which we shall call, rather uninspiringly, the “working substance” or “working fluid”. The job of this substance (and indeed its definition) is that, through changes in pressure, volume and temperature, a working substance, which is often a fluid, helps run a thermodynamic process. In our example system above, the gas in the piston-cylinder setup is the working substance.

Carnot cycle heat engine

A generic heat engine

A Carnot engine, or an ideal heat engine, works as depicted in the figure above. It follows the Carnot reversible cycle, or the thermodynamic cycle discussed above and is therefore also known as a Carnot reversible heat engine. (It is worth noting that this is different from a steam engine, a diesel engine etc. and is more efficient that all of them — and also more idealistic.)

It draws heat Q_H from a high temperature reservoir, simply called the source, and does some work W while expelling heat Q_L = Q_H - W to a low temperature reservoir or, simply, sink. It is the working substance that, in going from A to B to C to D and back to A in its thermodynamic cycle, returns to its original state after experiencing a temperature change between T_1 and T_2 and heat gain and loss. All of this is directly related to the entropy of the work substance which we will discuss in detail in part three of this series. We will, then, also see just why no other heat engine operating between the same T_1 and T_2 can achieve a greater efficiency.

Refrigerators and the co-efficient of performance

The fact that the thermodynamic cycle is reversible means our ideal heat engine working on that principle is also reversible. If we then simply reverse the directions of all the arrows in the figure above, we have an engine that takes heat from the sink while some work is being done on its working substance and it then supplies heat to the sink. In other words, so long as we keep doing work on the engine, it cools the sink. In other words, we have with us a refrigerator.

The fact that work needs to be done on the working fluid (or, more generally, on the engine) is in agreement with the Clausius statement that heat passing from any body to a hotter one is not a spontaneous process. It further clarifies an ambiguity with the first law. Whereas the first law requires that heat be accounted for, it does not prevent spontaneous energy transfer from, say, the floor to a chair, or from ice to steam. It is the second law that puts a restriction on this, stating that spontaneous heat flow can only occur from a hotter body to a colder one and not vice versa.

This might seem like little more than common sense, but further exploration of the idea and the introduction of a mathematical basis (both of which we will do in part three) will quickly give us a more solid foundation on which to make such a claim.

What about the refrigerator we now have before us? To understand how good or bad it is, we can define, somewhat analogously to the efficiency, \eta, a new ratio called the co-efficient of performance, which has no attractive greek letter associated with it and is simply defined as

    \[\textrm{co-efficient of performance} = - { Q_L \over W }\]

or, in terms of temperature,

    \[\textrm{co-efficient of performance} = {T_L \over T_H + T_L}\]

which is simply the reciprocal of \eta. Consequently, it can also be much greater than one. Using our realistic room temperature system from the heat engine example, a refrigerator operating between 298\,K and 373\,K has a co-efficient of performance of \approx 3.97. (The sun-vacuum system, by contrast, has a co-efficient of performance of the order of 10^{-4}.)

In case of an actual refrigerator (the one in your house) the low temperature reservoir is the food, the high temperature reservoir exists on the back of the refrigerator (and gets pretty hot on occasion) and the coolant acts as a working substance flowing in cycles between the reservoirs and working just like our ideal heat engine in reverse, only with a much lower efficiency.

To understand the reason why spontaneous heat flow has a direction, to understand why the Carnot reversible cycle is the most efficient and to understand how scientists in the 19th century expected the universe to end, we have to explore the wonderful idea of entropy. This will be the subject of our discussion in part three of this series.

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V.H. Belvadi is an Assistant Professor of Physics. He teaches postgraduate courses in advanced classical mechanics, astrophysics and general relativity. When he is free he makes photographs and short films, writes on his personal website, makes music, reads voraciously, or plays his violin. He currently serves as the Editor-in-Chief of Physics Capsule.

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