Reversible processes heat transfer

Consider the process depicted in Fig. 4.1(a) on the next page. The system is isolated, and consists of a cool body in thermal contact with a warm body. During the process, energy is transferred by means of heat from the cool to the warm body, causing the temperature of the cool body to decrease and that of the warm body to increase. Of course, this process is impossible we never observe heat flow from a cooler to a warmer body. (In contrast, the reverse process, heat transfer from the warmer to the cooler body, is spontaneous and irreversible.) Note that this impossible process does not violate the first law, because energy is conserved. 

A reversible isothermal heat-transfer process between the Carnot cycle and its surrounding thermal reservoirs is impossible to achieve... 

Benedict, Pigford, and Levi have carried out mathematical analysis of the GS process. An exhaustive treatment of the process, including calculations for flow rates, dependence of composition on number of stages, effect of solubility and humidity on process analysis, temperature profile in cold towers, simultaneous heat and mass transfer in heat transfer section, concentration reversal in heat transfer section, corrosion, materials of construction, feed purification, and safety, etc. have been reviewed by Dave, Sadhukhan and Novaro. ... 

The work, W, can range from 2ero, if the engine is completely ineffective, to the limiting negative value attained for reversible operation. If IT = 0, then the process degenerates to one of simple heat transfer, for which... 

Since heat transfer with respec t to the surroundings and with respect to the system are equal but of opposite sign, = —Q. Moreover, the second law requires for a reversible process that the entropy changes of system and surroundings be equalbut of opposite sign AS = —AS Equation (4-356) can therefore be written Q = TcAS In terms of rates this becomes... 

Adiabatic A process for which there is no heat transfer between a system and its surroundings. An adiabatic process that is reversible is isentropic. 

Fig. 2.1. Reversible process with heat transfer at temperature To (to the environment) (after Ref. 5 ).

For any reversible process, the increase in entropy of any participating system is equal to the heat absorbed by that system divided by the absolute temperature at which the transfer occurred. That is, for a system, i. 

Under hot BW conditions this reaction is reversible, leading to a serious risk of carbonate scale depositing on heat transfer surfaces. Consequently, many large water utilities and industries around the world continue to use the old established, but effective lime (calcium hydroxide) and soda ash (sodium carbonate) processes to soften water by precipitating out insoluble hardness salts. 

As we have seen before, exact differentials correspond to the total differential of a state function, while inexact differentials are associated with quantities that are not state functions, but are path-dependent. Caratheodory proved a purely mathematical theorem, with no reference to physical systems, that establishes the condition for the existence of an integrating denominator for differential expressions of the form of equation (2.44). Called the Caratheodory theorem, it asserts that an integrating denominator exists for Pfaffian differentials, Sq, when there exist final states specified by ( V, ... x )j that are inaccessible from some initial state (.vj,.... v )in by a path for which Sq = 0. Such paths are called solution curves of the differential expression The connection from the purely mathematical realm to thermodynamic systems is established by recognizing that we can express the differential expressions for heat transfer during a reversible thermodynamic process, 6qrey as Pfaffian differentials of the form given by equation (2.44). Then, solution curves (for which Sqrev = 0) correspond to reversible adiabatic processes in which no heat is absorbed or released. 

In general the conditions under which a change in state of a gas takes place are neither isothermal nor adiabatic and the relation between pressure and volume is approximately of the form Pvk = constant for a reversible process, where k is a numerical quantity whose value depends on the heat transfer between the gas and its surroundings, k usually lies between 1 and y though it may, under certain circumstances, lie outside these limits it will have the same value for a reversible compression as for a reversible expansion under similar conditions. Under these conditions therefore, equation 2.70 becomes ... 

If there is no heat transfer or energy dissipated in the gas when going from state 1 to state 2, the process is adiabatic and reversible, i.e., isentropic. For an ideal gas under these conditions,... 

During process 4-1, heat is transferred isothermally from the working substance to the low-temperature reservoir at Tl. This process is accomplished reversibly by bringing the system in contact with the low-temperature reservoir whose temperature is equal to or infinitesimally lower than that of the working substance. The amount of heat transfer during the process is 641= f TdS = Ti Si — S4), which can be represented by the area 1-4-5-6-1 Q41 is the amount of heat removed from the Carnot cycle to a low-temperature thermal reservoir. 

---