Possible difficulties with the energy equation

It does not make sense to treat the mixture specific heat capacity,

$$c_p = c_{pB} + m_* (c_{pA}-c_{pB}),$$ (2.17)

as a constant in the bulk advection term if the interdiffusion term is included. Two different difficulties would thereby arise, depending on whether it were treated as constant before or after the divergence of the energy flux was taken.

In the first case, constant $c_p$ in (2.15), it is not possible to use the species equation (2.5) to simplify the energy equation and an additional term appears on the right hand side of (2.16):

$$\rho D(c_{pA}-c_{pB})(T_*-T_{*r})\nabla^2_* m_*.$$ (2.18)

This would mean that the reference temperature, $T_{*r}$, the datum for the enthalpies, would appear in the energy equation, so that the temperature field would depend on a completely arbitrary quantity, which is obviously unphysical and unacceptable.

In the second case, constant $c_p$ in (2.16), if one attempts to write the energy equation (2.16) in the form of a divergence (2.11), a spurious source term appears:

$$\mbox{\boldmath$\nabla$}_*\cdot\mathbf{e}_* = -\rho D(c_{pA}-c_{pB})T\nabla^2_* m_*$$

$$= (c_{pA}-c_{pB})T(\mbox{\boldmath$\nabla$}_*\cdot\mathbf{j}_*)$$ (2.19)

This would mean that energy was not conserved. While the degree to which the conservation of energy (2.11) was violated might be small and within the limits of accuracy of the Boussinesq approximation, it is inconvenient for a number of reasons. For example, a global energy balance is often a useful check on the accuracy of an analytic or numerical solution (e.g. Incropera & DeWitt 1990, p. 198; de Vahl Davis 1983). Further, it would be necessary to specify the method by which the overall energy transfer rate was calculated.

Most importantly, however, the order of approximation involved in a constant mixture specific heat is inconsistent with the inclusion of interdiffusion. It should be clear by now--if not from the fundamental expression for the energy flux (2.12) then at least from the two possible errors exposed here--that the bulk advection of thermal energy and the interdiffusion of enthalpy are closely related. Indeed, they are an often convenient but somewhat artificial way of repartitioning the thermal energy transported by each of the species present when they move.

Another error was committed by myself (McBain 1995, 1997b) and Weaver and Viskanta (1991a): allowing the mixture specific heat to vary, but excluding the interdiffusion term. In the case of Weaver and Viskanta (1991a; discussed in §3.3.12), this may have caused their prediction of a spatial minimum in the gas-vapour phase temperature; an impossibility (as proven in §2.6.1) strongly suggestive of a spurious sink in the energy equation.

In conclusion, the interdiffusion term should be included in the energy equation if, and only if, the mixture specific heat is treated as a function of the vapour mass fraction. The question of when these should be included is taken up once the equations are nondimensionalized and some examples examined (ch. 6).