I hope there’s a thermodynamic superhero to the rescue…

The problem seems simple. There’s an open control volume (of fixed volume), filled with an incompressible fluid, that changes its density with temperature. Since temperature does change over time, the mass of fluid in the control volume changes.

Now, in textbooks and examples there’s always one of these two statements:

- “Since mass in the control volume is constant, …”
- “Since only differences in internal energy […], reference states can be chosen arbitrarily.”

The latter statement also implies that one can calculate the difference of the internal energy by integrating the specific heat between the two temperatures, so no absolute value for the internal energy is ever needed.

Now, in the scenario at hand, the internal energy in the control volume varies both because of changes of specific internal energy, *and* of mass in the control volume:

Here, the internal energy pops up explicitly, and its absolute value does have an impact on the dynamics of the system. (On the right-hand side, there’s heat flows and mass flows into / out of the control volume, and one would like to rearrange the equation to have an expression for the change of the temperature as an ordinary differential equation.)

Do I get something wrong? Already made all colleagues here at work scratching their heads…