True differential heat flow

KAT, true differential heat-flow rate to establish equal temperature between the reference and sample. 

In this discussion, we will limit our writing of the Pfaffian differential expression bq, for the differential element of heat flow in thermodynamic systems, to reversible processes. It is not possible, generally, to write an expression for bq for an irreversible process in terms of state variables. The irreversible process may involve passage through conditions that are not true states" of the system. For example, in an irreversible expansion of a gas, the values of p. V, and T may not correspond to those dictated by the equation of state of the gas. 

In practice, the true heating rates (dT/dt)ca and (dT/dt)cb are assumed to be equal to the programmed scan rate j3, and the true heat flow rate difference (heat flow rate difference, Ao, which reflects the intrinsic thermal asymmetry of the differential measuring system ... 

Strictly speaking, most of the equations that are presented in the preceding part of this chapter apply only to incompressible fluids but practically, they may be used for all liquids and even for gases and vapors where the pressure differential is small relative to the total pressure. As in the case of incompressible fluids, equations may be derived for ideal frictionless flow and then a coefficient introduced to obtain a correct result. The ideal conditions that will be imposed for a compressible fluid are that it is frictionless and that there is to be no transfer of heat that is, the flow is adiabatic. This last is practically true for metering devices, as the time for the fluid to pass through is so short that very little heat transfer can take place. Because of the variation in density with both pressure and temperature, it is necessary to express rate of discharge in terms of weight rather than volume. Also, the continuity equation must now be... 

In a truly countercurrent system with no phase change and constant heat capacities in both fluids, the proper mean temperature differential becomes the logarithmic mean of the two terminal temperature differentials (LMTD). In Section 9.S.2.4 on caustic cooling, we rely on this approach. The literature contains standard corrections to the LMTD for configurations that do not allow true countercurrent flow, as for example in multipass shell-and-tube exchangers [9]. 

The mean temperature differential (MTD), given true cocurrent or countercurrent flow, is independent of the type of apparatus and is easily calculated. A straightforward calculation gives the amount of heat that must be removed to cool the gas to any given temperature inside the exchanger. This information is often displayed in a duty curve, such as that of Fig. 9.3. This is a companion to the example given below, and it shows the temperature of the gas as a function of the percentage of the total... 

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