The energy equation

Neglecting the gravitational potential energy (since the cavities of interest are of limited vertical extent, see Spiegel & Veronis 1960), the steady-state energy equation is (Bird et al. 1960, p. 561):

$$\mbox{\boldmath$\nabla$}_* \cdot \mathbf{e}_* = 0,$$ (2.11)

where $\mathbf{e}_*$ is the energy flux with respect to a fixed frame of reference.

If the gas-vapour mixture is diathermanous (so that the radiant energy flux can be handled separately), and the Dufour effect, viscous stresses and the advection of kinetic energy are negligible, the energy flux is (Bird et al. 1960, p. 566):

$$\mathbf{e}_* = -\lambda\mbox{\boldmath$\nabla$}_* T_* + (h_A-h_B)\mathbf{j}_* + \rho h\mathbf{u}_*$$ (2.12)

where $\lambda$ is the thermal conductivity and $h_A$ and $h_B$ are the partial specific enthalpies of the gas and vapour and $h=h_B+m_*(h_A-h_B)$ is the specific enthalpy of the mixture (Guggenheim 1959, pp. 213-4).

If the gas and vapour form a `perfect gaseous mixture', so that the partial specific heat capacities, defined by (Guggenheim 1959, p. 212)

$$c_{pi}=\left( \frac{\partial h_i}{\partial T_*} \right)_{p_*}\qquad i=A,B,$$ (2.13)

are equal to the specific heat capacities of the pure components and independent of pressure (Guggenheim 1959, pp. 225, 118) and if the gas and vapour are calorically perfect, so that their specific heat capacities are independent of temperature, then the partial specific enthalpies appearing in the energy flux (2.12) can be expressed in terms of the temperature:

$$h_i=h_{ir}+c_{pi}(T_*-T_{*r}),\;i=A,B.$$ (2.14)

Whence,

$$\mathbf{e}_* = -\lambda\mbox{\boldmath$\nabla$}_* T_*$$

$$+ \rho(T_*-T_{*r})[c_{pB}+m_*(c_{pA}-c_{pB})]\mathbf{u}_*$$

$$- \rho(T_*-T_{*r})(c_{pA}-c_{pB})D\mbox{\boldmath$\nabla$}_* m_*$$

$$+ \rho h_{Br}\mathbf{u}_* +(h_{Ar}-h_{Br})\mathbf{n}_*$$ (2.15)

where (2.3) and (2.2) have been used for the definitions of $\mathbf{n}_*$ and $\mathbf{j}_*$.

In (2.15):

• $-\lambda\mbox{\boldmath$\nabla$}_* T_*$ is the Fourier conduction flux;
• $\rho(T_*-T_{*r})[c_{pB}+m_*(c_{pA}-c_{pB})]\mathbf{u}_*$ is the advection of thermal energy by the mean flow; and
• $-\rho(T_*-T_{*r})(c_{pA}-c_{pB})D\mbox{\boldmath$\nabla$}_* m_*$ is the advection of thermal energy by the vapour diffusion flux (the `interdiffusion' term).

The last two terms in (2.15) are divergence-free, by (2.6) and (2.5), and so have no effect on the energy equation (2.11). They must be included, however, in the energy flux at the boundary. Both terms are also of arbitrary magnitude, since enthalpies are only defined by their changes, rather than absolutely (Guggenheim 1959, pp. 11, 32). The reference enthalpy of the noncondensable gas, $h_{Br}$, may be set to zero, since all the cavities considered in this project are impermeable to the gas. The reference enthalpy of the vapour on the other hand must include the heat of vaporization or sublimation, since it is the term $h_{Ar}\mathbf{n}_*$ that accounts for the latent heat.

Substituting (2.15) into (2.11), with $\lambda$ assumed constant, the primitive thermal energy equation is:

$$\rho[c_{pB}+m_*(c_{pA}-c_{pB})]\mathbf{u}_*\cdot\mbox{\boldm... ...{\boldmath$\nabla$}_*m_*)\cdot(\mbox{\boldmath$\nabla$}_*T_*).$$ (2.16)

Multiples of the equations of continuity of the mixture (2.6) and the vapour (2.5) were subtracted to eliminate the terms proportional to $(T_*-T_{*r})$, which is important, as pointed out in the following paragraphs.