# When can the thermal and mechanical energies of a fluid control volume be considered separately?

For a fixed control volume of a fluid, the conservation of energy equation is typically written as something like $$\dot{Q} - \dot{W} = \frac{\text{d}}{\text{d}t}\iiint_{CV} \left( \hat{u} + \frac12 V^2 + gz \right) \rho \,\text{d}V + \iint_{CS} \left( \hat{h} + \frac12 V^2 + gz \right) \rho \, \mathbf{V}\cdot\hat{\mathbf{n}} \, \text{d}A.$$

Is there any criteria for which we can reasonably decouple this into separate conservation equations for thermal and mechanical energy? That is, into the form $$\begin{cases} \dot{Q} = \frac{\text{d}}{\text{d}t}\iiint_{CV} \hat{u} \rho \,\text{d}V + \iint_{CS} \hat{h} \rho \, \mathbf{V}\cdot\hat{\mathbf{n}} \, \text{d}A, \\ -\dot{W} = \frac{\text{d}}{\text{d}t}\iiint_{CV} \left( \frac12 V^2 + gz \right) \rho \,\text{d}V + \iint_{CS} \left( \frac12 V^2 + gz \right) \rho \, \mathbf{V}\cdot\hat{\mathbf{n}} \, \text{d}A. \end{cases}$$

I know that for some common types of analysis the mechanical terms are negligible compared to the thermal and ignored entirely. Does this imply that a separate mechanical energy conservation equation can be posed?

I suspect this depends on the relative magnitude of the viscous dissipation function. Is there some dimensionless number that quantifies this?

## 2 Answers

The energy equation can be simplified to the two equations you've referenced, given that in the first case there is negligible work, in the second case there is negligible heat transfer, and in both cases there is negligible viscous dissipation. For example, consider a heat exchanger with negligible inlet and outlet velocities and no elevation difference between the ports. One may opt to use the first equation you referenced since the mechanical work terms may be neglected. In a second example, consider a turbine or a pump. In many analyses, the device is considered adiabatic (no heat transfer to/from the device through control volume boundaries), therefore, the second equation you formulated may be used. I write this assuming you meant to include the static enthalpy terms in the second equation. In some types of flows (such as Poiseuille, Couette, etc) the viscous dissipation is not negligible and contributes to a heat up of the fluid. Given the type of flow assumed, the viscous dissipation function can be simplified.

• I think OP was looking for a single scenario in which both equations linked by the brace are true, not two opposing limits in each of which one of those two equations is true. – Daniel Hatton Apr 1 at 15:06
• @DanielHatton i think you are correct, now that i reread the original post. Thanks for keeping me honest! – mechcad Apr 1 at 18:44

You can't do it with conservation of energy alone: you need another physical principle, namely conservation of momentum in the form of the Navier-Stokes momentum equation (or, if the shear stresses are negligible, the Euler momentum equation). That conservation of momentum equation will give you an expression for the surface traction density at any given point on your control surface. Taking the scalar product of that with the local velocity gives an expression for the surface work rate density, then integrating over the control surface gives an expression for the total rate of work done by stuff outside the control volume on stuff inside the control volume. That expression should look remarkably like the second of the two equations you've got joined by the brace (how much like it depends exactly which stresses have been treated as negligible and which have been included in the equations); it may be made to look even more like it by using Gauss' theorem to convert between surface integrals and volume integrals. Then substituting that expression for the work rate into the energy conservation equation you've got at the start of your post will leave you with an expression for the rate of heat transfer that looks remarkably like the first of the two equations you've got joined by the brace.