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Valentine Costin
Valentine Costin

Modern Thermodynamics. From Heat Engines To Dis...



What motivated Carnot to attempt to calculate steam energyefficiency in 1820? Well, it was the timeof the Industrial Revolution, and the efficiency of your power supplydetermined your profit margin. Big engineswere primarily used in mass production of cloth, in factories called mills. Upto the late 1700's these mills were located by fast flowing rivers, the powersource was a large waterwheel, it turned a long rotating rod that stretched thelength of the factory. Ropes took powerfrom pulleys on this rod to turn individual looms, which were operated bysemiskilled laborers, often children. The picture here is much later (1914),and steam-driven, but shows the poweringscheme.




Modern Thermodynamics. From heat engines to dis...



But how does that relate to the energy expended producingthe heat in the first place? Well, Carnotknew something else: there was an absolute zero of temperature. Therefore, he reasoned, if you cooled thefluid down to absolute zero, it would give up all its heat energy. So, the maximum possible amount of energy youcan extract by cooling it from T H to T C is, what fraction is that of cooling it toabsolute zero?


In this chapter we consider a more abstract approachto heat engine, refrigerator and heat pump cycles, in an attempt todetermine if they are feasible, and to obtain the limiting maximumperformance available for these cycles. The concept of mechanical andthermal reversibility is central to the analysis, leading to theideal Carnot cycles. (Refer to Wikipedia: SadiCarnot a French physicist, mathematicianand engineer who gave the first successful account of heat engines,the Carnot cycle, and laid the foundations of the second law ofthermodynamics). For more information on thissubject, refer to a paper: AMeeting between Robert Stirling and Sadi Carnot in 1824presented at the 2014ISEC.


In the case of a heat engine heat QHis extracted from the high temperature source TH,part of that heat is converted to work W done on the surroundings,and the rest is rejected to the low temperature sink TL.The opposite occurs for a heat pump, in which work W is done on thesystem in order to extract heat QL fromthe low temperature source TL and"pump" it to the high temperature sink TH.Notice that the thickness of the line represents the amount of heator work energy transferred.


It is remarkable that the two above statements of theSecond Law are in fact equivalent. In order to demonstrate theirequivalence consider the following diagram. On the left we see a heatpump which violates the Clausius statement by pumping heat QLfrom the low temperature reservoir to the hightemperature reservoir without any work input. On the right we see aheat engine rejecting heat QL to thelow temperature reservoir.


If we now connect the two devices as shown below suchthat the heat rejected by the heat engine QL issimply pumped back to the high temperature reservoir then there willbe no need for a low temperature reservoir, resulting in a heatengine which violates the Kelvin-Planck statement by extracting heatfrom a single heat source and converting it directly into work.


Notice that the statements on the Second Law arenegative statements in that they only describe what is impossible toachieve. In order to determine the maximum performance available froma heat engine or a heat pump we need to introduce the concept ofReversibilty,including both mechanical and thermal reversibility. We will attemptto clarify these concepts in terms of the following example of areversible piston cylinder device in thermal equilibrium with thesurroundings at temperature T0, andundergoing a cyclic compression/expansion process.


For mechanical reversibility we assume that theprocess is frictionless, however we also require that the process isa quasi-equilibrium one. In the diagram we notice that duringcompression the gas particles closest to the piston will be at ahigher pressure than those farther away, thus the piston will bedoing more compression work than it would do if we had waited forequilibrium conditions to occur after each incremental step.Similarly, thermal reversibility requires that all heat transfer isisothermal. Thus if there is an incremental rise in temperature dueto compression then we need to wait until thermal equilibrium isestablished. During expansion the incremental fall in temperaturewill result in heat being transferred from the surroundings tothe system until equilibrium is established.


The simplest way to prove this theorem is to considerthe scenario shown below, in which we have an irreversible engine aswell as a reversible engine operating between the reservoirs THand TL, however theirreversible heat engine has a higher efficiency than the reversibleone. They both draw the same amount of heat QH fromthe high temperature reservoir, however the irreversible engineproduces more work WI than that of thereversible engine WR.


Note that the reversible engine by its nature canoperate in reverse, ie if we use some of the work output (WR)from the irreversible engine in order to drive the reversible enginethen it will operate as a heat pump, transferring heat QHto the high temperature reservoir, as shown in thefollowing diagram:


Notice that the high temperature reservoir becomesredundent, and we end up drawing a net amount of heat (QLR- QLI) from the lowtemperature reservoir in order to produce a net amount of work (WI- WR) - a Kelvin-Planckviolator - thus proving Carnot's Theorem.


Statistical mechanics has come a long way from these humble beginnings, but thermodynamics is still an important fieldin its own right. In this chapter, I will discuss some of the most important results of classical thermodynamics asseen from a modern statistical viewpoint.


We now have a third option: a system whose properties we can directly control. They change when we choose to changethem in exactly the way we choose. We are no longer in any way dealing with an isolated system. We are explicitlymaking an external change, reaching in from outside to alter the system. On the other hand, we only do that during arestricted time period. Before we make the change, the system (possibly including a heat bath) is isolated. After weare done making the change it is again isolated. But while we make the change, it is not isolated.


Classical thermodynamics is based on four laws that are very sensibly numbered starting at zero. When viewed from amodern perspective, three of them are trivial. The remaining one is extremely profound, and even today is oftenmisunderstood.


Suppose an amount of heat \(dQ\) flows from subsystem A to subsystem B. The entropy of A decreases and the entropyof B increases. The total change in entropy of the whole system is the sum of the two:


If \(T_A


So the only possibility is that \(T_A > T_B\). Heat is flowing from a warmer body to a colder one, and the overallentropy of the system increases. This leads us to the following very important conclusion: whenever heat flows betweentwo bodies, the total entropy of the system increases. As we will see shortly, this has important consequences foranyone trying to build a steam engine.


Let us analyze what happens during this cycle. Let \(Q_H\) be the heat absorbed from the hot bath, \(Q_C\) theheat expelled to the cold bath, and \(W\) the net amount of work performed during the whole cycle (that is, thework performed by the working body during steps 1 and 2, minus the work performed on the working body in steps 3 and4). Conservation of energy requires that \(W=Q_H-Q_C\).


Notice how little we assumed in deriving this: merely that the heat engine absorbed heat from one bath, expelled heat toanother, and did work. I described the Carnot cycle as an illustration, but no details of the cycle were required forthe derivation. Therefore, equation (9) applies equally well to heat engines that use differentcycles. I assumed nothing about the nature of the working body. It could be a gas, a liquid, a solid, or evensomething exotic like a supercritical fluid. I assumed nothing about how the working body performed work. It couldinvolve moving a piston, applying an electric field, or shooting ping-pong balls at a target.


There is no lower limit, of course. You can make a heat engine as inefficient as you want. You could connect the twoheat baths and allow energy to flow directly from one to the other without doing any work at all, thus achieving aspectacular efficiency of zero. The key to making an efficient engine is to minimize all transfers of heat except theones that are absolutely required for the operation of the engine.


Another interesting fact about heat engines is that you can run them backward, which turns them into heat pumps. Justperform the same steps in reverse order and changing the direction of movement (so that expanding becomes compressing,for example). In this case, heat flows out of the cold bath and into the hot bath. Heat is flowing opposite to itsnormal direction! It does not do this spontaneously, of course. Instead of the engine producing work, we now have todo work on it. You are probably already very familiar with this fact: refrigerators and air conditioners have anunfortunate habit of needing to be connected to an external source of energy.


Consider an even simpler case: just a working body and a single heat bath. Assume the working body has energy \(E\)and entropy \(S\), and that the heat bath has temperature \(T\). We now want to answer a simple question: whatis the maximum amount of work we can extract from the body?


The researchers plan to incorporate the TPV cell into a grid-scale thermal battery. The system would absorb excess energy from renewable sources such as the sun and store that energy in heavily insulated banks of hot graphite. When the energy is needed, such as on overcast days, TPV cells would convert the heat into electricity, and dispatch the energy to a power grid. 041b061a72


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