To understand the various relationships between fluxes and sources of mass , momentum and internal energy, I suggest you first consider a single component system. One has then three balance laws: mass, linear momentum and internal energy. Or alternatively one can start with mass, linear momentum and the total energy ( internal plus KE and potential energy) and then subtract from the total energy equation the mechanical energy equation (derived from the linear momentum equation) to get the balance law for internal energy. You can find the manipulation steps in many textbooks e.g., Bird Stewart and Lightfoot: "Transport Phenomena", de Groot and Mazur:  "Non Equilibrium Thermodynamics", Jou Casas-Vazquz and Lebonn: Extended Irreversible Thermodynamics" . If one does not like to work with the internal energy then using thermodynamic relations one can derive  from the internal energy equation the thermal energy equation where now temperature T is the dependent variable in the energy equation. Now at a gas-liquid (or solid-liquid or solid-gas) interface, continuity of flux (mass, momentum and thermal energy) leads to what are referred to as the "Jump Conditions". If there is no phase change then the normal heat flux on either side of the interface must be equal ( ignoring any surface source terms). If there is a phase change, one has bulk flow across a phase boundary (as in sublimation), and then the interfacial balance is

rho1*H1(v1-w).n+q1.n=rho2*H1(v2-w).n+q2.n

where rho1 and H1 are the density an enthalpy in phase one, q1 is the heat flux vector in phase 1, w is the velocity of the interface and v1 is the fluid velocity in phase one at the interface, and n the unit normal,  etc. Note in some case kinetic energy terms of evaporation and sublimation may be important and need to be included in the jump condition.

Conservation of mass at the interface gives

rho1(v1-w).n=rho2(v2-w).n

Combining these two equations we get

(q1-q2).n=Lambda rho1(v1-w).n=Lambda rho2(v2-w).n 

where Lambda=H2-H1, which is the enthalpy of vaporization of sublimation depending on the physics.

The above lays out the mathematical framework for solving phase change problems for single component systems and this framework commonly used in the literature. Of course, depending on the physics of the problem it may be convenient to work with integrated forms of the governing equations that simplify the analysis ( if appropriate). In the simplest case the sensible heat Cp

Thank you very much for your answers.

Most of my papers deal with the concept published in the book of de Groot and Mazur (about exergy, entropy and enthalpy of moist-air considered as an open, multi-components system).

My new questions correspond to new ideas I have about the turbulent fluxes at the bottom of the atmosphere, usually considered as equal to the sum of sensible plus latent turbulent heat fluxes (SH+LH) acting at the surface.

I have recently discovered that some issues described in Montgomery (1948) and Businger (1982) may be solved by using the Third-law based reference enthalpies I have computed and published in my new 2015 paper (see the attached recent lecture).

I arrive at a strange result indicating that, as expected by Montgomery and Businger, the sum SH+LH should be replaced by the flux of "h+phi". Moreover, when "h"  is computed with my 2015 reference enthalpies of dry air and water vapour, it appears that the impact of the new term "Lh Fv" could be of the order of 6 W/m2 on a global stage (in fact about 6 % locally)... and there is no place for "L_v F_v" versus "L_s F_v", depending on the kind of surface bellow the column of moist air...

May be I am wrong, but I do not see where is my mistake?