Your intuition is correct that the temperature may not be equal. However, a Biot number approach is not likely to help. Both the phases in this case are fluids, and if they have temperature gradients, the temperature gradients will induce convection, both between the phases and within the phases. The convection makes the isothermal assumption more accurate in nearly all cases. However, very careful work would require resolving the convection.

Convection currents, and convection-based heat transfer, are notoriously difficult to accurately describe. The most common approach is using Taylor-Green vortex descriptions for incompressible flow. This can, in many cases, provide reasonable looking flow descriptions, but their accuracy is often disappointing, especially when driven by buoyancy or temperature-gradient effects.

Aside from transient effects, the application you describe will depend most heavily on heat losses. It is difficult to predict where these losses occur in practice, but usually they occur around regions where insulation properties change (seams or voids) and where inlet/outlet/sensor/structural components attach to the tank. Sometimes they are larger in liquid region of the tank than in the vapor region, primarily because the vapor poorly conducts heat. However, it is generally rare that an analysis more sophisticated than a global heat loss term with otherwise equilibrium conditions can be justified based on measurements.

Mr. Baxter,

First of all, thank you for your detailed answer.

When you write : " The convection makes the isothermal assumption more accurate in nearly all cases", do you mean that if convection+conduction is fast compared to the heat flux entering the tank, the lumped model holds ?

At the interface, condensation/evaporation is used for heat/matter exchanges, is it correct in your opinion ? What I'm looking for is really a threshold to affirm "this tank can be thermal lumped" or not.

Regards,

interested