Is step in thermodynamic fluid mechanic derivation correct?

Question

I am deriving an equation for a thermodynamic/fluid mechanics system and need some help deciding if I am allowed to do a certain step.

I have an expression involving a total differential that simplifies to

\begin{equation} \frac{D H}{Dt} = U P \Delta T dx \end{equation} When expanding the total differential I get \begin{equation} \frac{\partial H}{\partial x}V_x = U P \Delta T dx \end{equation} Can this be simplified to \begin{equation} \frac{\partial \dot{H} }{\partial x} = UP \Delta T \end{equation} I imagine starting with $Vx = \frac{dx}{dt} = \frac{\partial x}{\partial t}$ is a good place to start but I am a little unsure of the mathematics. If you could help me decide if this is correct or not it would be apprecated. $UP\Delta T$ can be viewed as constants. I know this should not be difficult, but I am struggling.

Answer 1

Up to your very last equation, the development is correct. But you are not trying to determine the rate of change of H with respect to t at a specific point. You want to determine how H is changing as a function of x (at steady state). So that is just$$\frac{dH}{dx}=\frac{UP\Delta T}{V_x}$$Typically, $\Delta T$ will not be a constant, so you would usually write:$$\frac{dH}{dx}=\rho C_p\frac{dT}{dx}=\frac{UP(T_w-T)}{V_x}$$where $T_w$ is the wall temperature.