Imagine you have a solid that you are rapidly heating inside a vacuum chamber with some kind of thermal heater or a laser such that the material begins to evaporate or sublimate. Obviously radiative losses scale as $T^4$ but then energy is also lost via evaporation or sublimation. Is there an equation that allows for the energy lost by evaporation to be accounted for, given a temperature $T$ and perhaps the enthalpy of vaporization $\Delta H$?

In addition, how is this loss accounted for in the heat equation, is there an extra term to include alongside conduction and radiation?

  • $\begingroup$ The outgoing heat flux is the enthalpy of vaporization at that particular temperature. $\endgroup$ Commented Nov 24, 2021 at 16:21

1 Answer 1


In thermodynamics, work is defined as "energy that crosses the boundary of a system, without mass transfer, due to an intensive property difference other than temperature between the system and its surroundings". Heat is defined as "energy that crosses the boundary of a system, without mass transfer, solely due to a difference in temperature between the system and its surroundings". So, mass transfer is not considered as part of heat transfer (or work). (But for engineering calculations, a heat transfer coefficient that includes effects of evaporation/condensation in a film of constant mass is sometimes used; see the text Heat Transmission by McAdams.)

A rigorous evaluation of mass transfer- through evaporation, condensation, or other mass flow in/out of a system- is addressed using the thermodynamic "open" system, one for which mass transfer occurs in/out across the boundaries of the system. The first law of thermodynamics (energy conservation) includes terms for the "flow" energy in the open system by considering the enthalpy, velocity and elevation of the entering and exiting mass. See a good thermodynamics text such as Thermodynamics by Obert; also, a good text on transport phenomena such as Transport Phenomena by Bird, Stewart, and Lightfoot provides pertinent information.

  • $\begingroup$ To supplement, I recommend Knuiman et al.’s “On the Relation between the Fundamental Equation of Thermodynamics and the Energy Balance Equation in the Context of Closed and Open Systems,J Chem Ed (2012). $\endgroup$ Commented Nov 24, 2021 at 16:19
  • $\begingroup$ I know about open systems, but I am trying to figure out if this energy loss can be accounted for easily, given that I want to put this into a energy balance equation ($dU/dt$) then convert this into a weak form. Currently I have a mass term, $dm = \Gamma \Delta H M$ where $\Gamma$ is the mass evaporation rate and M is the molar mass of the liquid. Would this account for it? $\endgroup$
    – tjsmert44
    Commented Nov 25, 2021 at 11:07
  • $\begingroup$ You could approximate the energy loss as you say (see my comment about heat transfer coefficient for evaporation/condensation in my answer). If the loss of mass is small you can assume the system mass does not change. This is basically the same as using the energy balance for an open system and neglecting terms that are small. $\endgroup$
    – John Darby
    Commented Nov 25, 2021 at 16:08

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