The reversible transfer of heat

Tc- This may require Carnot engines or heat pumps internal to the system that provide for the reversible transfer of heat from the temperature of the flowing fluid to that of the surroundings. Since Carnot engines and heat pumps are cychc, they undergo uo net change of state. 

The transfer of heat between two bodies requires a difference in their temperatures. The reversible transfer of heat between two bodies would require their temperatures to be the same. The question then arises of how we may reversibly add a quantity of heat to or remove a quantity of heat from a system over a temperature range. In order to do so we assume that we have an infinitely large number of heat reservoirs whose temperatures differ infinitesimally. Then, by bringing the system into thermal contact with these reservoirs successively and allowing thermal equilibrium to be obtained in each step, we approach a reversible process. 

Consider any number of systems that may do work on each other and also transfer heat from one to another by reversible processes. The changes of state may be of any nature, and any type of work may be involved. This collection of systems is isolated from the surroundings by a rigid, adiabatic envelope. We assume first that the temperatures of all the systems between which heat is transferred are the same, because of the requirements for the reversible transfer of heat. For any infinitesimal change that takes place within the isolated system, the change in the value of the entropy function for the ith system is dQJT, where Qt is the heat absorbed by the ith system. The total entropy change is the sum of such quantities over all of the subsystems in the isolated system, so... 

In conclusion, entropy is the physical quantity that represents the capacity of distribution of energy over the energy levels of the individual constituent particles in the system. The extensive variable entropy S and the intensive variable the absolute temperature Tare conjugated variables, whose product TdS represents the heat reversibly transferred into or out of the system. In other words, the reversible transfer of heat into or out of the system is always accompanied by the transfer of entropy. 

This derivation makes clear the difference between W , the shaft work of the turbine, and Wideal. The ideal work includes not only the shaft work, but also all work obtainable by the operation of heat engines for the reversible transfer of heat to the surroundings at T0. 

We base our present development on the discussion of Section 1.16 where a distinction was made between the entropy change of a system dS - dQr/T, associated with the reversible transfer of heat across the boundaries, and the entropy change arising from irreversible processes occurring totally within the system. As shown in Section 1.15, the heat transfer is subject to the relation 3Q < TdS whenever the inequality sign applies, one introduces the deficit entropy function 9 so as to render... 

Beyond this point one must be aware of important differences between the two laws. The performance of work is directly linked to changes in energy of a system, so that the integrating factor q relevant to the First Law is unity. Furthermore, changes in S are tracked by the reversible transfer of heat across the boundaries of the system. Other changes in S are incurred when irreversible processes occur this subject was treated in detail in Sections 1.12, 1.13, and 1.20. By contrast, alterations in E are tracked by performance of work, whether reversibly or irreversibly, under adiabatic conditions. Different changes in E are incurred when these processes take place under non-adiabatic conditions, as discussed in Section 1.7. 

The present volume involves several alterations in the presentation of thermodynamic topics covered in the previous editions. Obviously, it is not a trivial exercise to present in a novel fashion any material that covers a period of more than 160 years. However, as best as I can determine the treatment of irreversible phenomena in Sections 1.13, 1.14, and 1.20 appears not to be widely known. Following much indecision, and with encouragement by the editors, I have dropped the various exercises requiring numerical evaluation of formulae developed in the text. After much thought I have also relegated the Caratheodory formulation of the Second Law of Thermodynamics (and a derivation of the Debye-Hiickel equation) as a separate chapter to the end of the book. This permitted me to concentrate on a simpler exposition that directly links entropy to the reversible transfer of heat. It also provides a neat parallelism with the First Law that directly connects energy to work performance in an adiabatic process. A more careful discussion of the basic mechanism that forces electrochemical phenomena has been provided. I have also added material on the effects of curved interfaces and self assembly, and presented a more systematic formulation of the basics of irreversible processes. A discussion of critical phenomena is now included as a separate chapter. Lastly, the treatment of binary solutions has been expanded to deal with asymmetric properties of such systems. 

According to Eqn (3.9) this quantity is equal to the isosteric enthalpy, which turns out to be equal to the heat per mole evolved in the reversible transfer of an infinitesimal amount of adsorbate from the adsorbed phase to the gas phase at constant temperature, pressure, and area. 

For example, if a hot body (the system) is in contact with the surroundings at an infinitesimally lower temperature there will be an isothermal and reversible transfer of heat at Tj > and the loss in entropy of the hot body will be and the gain in entropy of the surroundings will be... 

On the other hand, we might think of a so-called reversible transfer of heat as an idealized transfer for vanishingly small temperature differences. As a consequence, then also the flux of transferred internal energy will be vanishingly small such that the idealized reversible transfer is an infinitely slow process. Since the entropy production is given as the product of the difference of reciprocal temperatures and the heat flux, it is small of second order and may be neglected for idealized re-... 

Since dS must be positive or zero, T > and thus the heat flows from the hotter to the cooler body, in agreement with experience. The creation of entropy in the system continues for as long as exceeds and the state of equilibrium requires equality of these temperatures. This is in accordance with the mecming of temperature as discussed in 1 4. A reversible transfer of heat thus requires that there shall be only an infinitesimal temperature difference. Thus if 2s<-2i is an infinitesimal, the increase of entropy in the above equation becomes equal to dqdT/T and is of the second order of smallness. 

In the special case where the only transfer of heat takes place reversibly at the constant temperature T the remainder of the process being adiabatic and reversible, (2 9) reduces to... 

We met the concept of reversibhity in Section 1.3, where we saw that it refers to the ability of an infinitesimal chcuige in a control variable to change the direction of a process. Mechaniccd reversibhity refers to the equality of pressure acting on either side of a movable wall. Thermal reversibility, the type involved in eqn 2.1, refers to the equality of temperature on either side of a thermally conducting wall. Reversible transfer of heat is smooth, careful, restrained transfer between two bodies at the same temperature. By mcJdng the transfer reversible, we ensure that there are no hot spots generated in the object that later disperse spontaneously and hence add to the entropy. 

Transfer of heat through the walls of the evaporator and condenser requires a temperature difference. This is shown on the modified reversed Carnot cycle (Figure 2.5). For temperature differences of 5 K on hoth the evaporator and condenser, the fluid operating temperatures would he 263.15 K and 313.15 K, and the coefficient of performance falls to 5.26. 

With this aim, Carnot proceeded to introduce the novel conception of a reversible cycle of operations, and arrived at the exceedingly important result that the motive power capacity for doing wrork] of heat is independent of the agents employed to develop it its quantity is determined solely by the temperatures of the bodies between which, in the final result [i.e., after the completion of the cycle], the transfer of heat occurs. Let there be given a source, and a refrigerator, at temperatures Ti, T2 respectively, where Tx > T2. In order that finite quantities of heat may be added to or taken from these without change of their temperatures, we may suppose them to consist of... 

To derive an expression for the change in entropy when a system is heated, we first note that Eq. 1 applies only when the temperature remains constant as heat is supplied to a system. Except in special cases, that can be true only for infinitesimal transfers of heat so we have to break down the calculation into an infinite number of infinitesimal steps, with each step taking place at a constant but slightly different temperature, and then add together the infinitesimal entropy changes for all the steps. To do this is we use calculus. For an infinitesimal reversible transfer dgrev at the temperature T, the increase in entropy is also infinitesimal and, instead of Eq. 1, we write... 

At the temperature of a phase transition, the transfer of heat is reversible. 

The equality applies when heat is transferred reversibly and the inequality refers to irreversible or natural or spontaneous transfer of heat. Thus,... 

Reverse the reverse of, or opposite to, the design intention. This could mean reverse flow if the intention was to transfer material. For a reaction, it could mean the reverse reaction. In heat transfer, it could mean the transfer of heat in the opposite direction to what was intended. 

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