The total heat flux vector is defined as (Ref. 5):

where E is the internal energy. It is the sum of convective heat flux, ρuE, conductive heat flux, q, and radiative heat flux, qr. Hence, the total heat flux accounts for all three kinds of heat transfer described in The Physical Mechanisms Underlying Heat Transfer. Recall that the internal energy is related to the enthalpy, H, via the following for a fluid (see Thermodynamic Description of Heat Transfer):

The total heat flux vector, qtot, is more suited to check the heat balance as described in the next section Heat and Energy Balance.

Again, convective heat flux, ρuE, conductive heat flux, q, and radiative heat flux, qr are accounted in the sum. The additional terms that complete the total energy flux are the convected kinetic energy, ρu(u ⋅ u ⁄2), force potential energy, ρuΨ, and stress, −σu. For a fluid, this expression becomes (Ref. 5, chapter 3.5):

Introducing the total internal energy, E0, and total enthalpy, H0:

The potential Ψ has a simple form in some special cases — for example, for gravitational effects (Chapter 1.4 in Ref. 40) — but it is in general rather difficult to derive. Potential energy is therefore often excluded and the total energy flux is approximated by

The total energy flux vector, etot, is more suited to check the energy balance as described in the next section Heat and Energy Balance.