Fluid Conservation of Energy

This is an application of the Reynolds Transport Theorem where we have

$$b=e⟶B=me=E$$

where

$$e=\frac{\text{Total Energy},e}{\text{mass},m}=u+\frac{V^2}{2}+gz$$

We then have

$$\begin{array}{rl}{B}_{\text{sys}}& ={\int }_{\text{sys}}e\rho d\forall =E_{\text{sys}}\\ B_{\text{C.V.}}& ={\int }_{\text{C.V.}}e\rho d\forall =E_{\text{C.V.}}\end{array}$$

The Energy Equation (E.E) is then:

$$\begin{array}{rl}\frac{D{E}_{\text{sys}}}{Dt}& =\frac{\partial }{\partial t}{\int }_{\text{C.V.}}e\rho d\forall +{\int }_{\text{C.S.}}e\rho \mathbf{V}\cdot \mathbf{n}dA\\ & =({\dot{Q}}_{\text{net, in}}+{\dot{W}}_{\text{net, in}}{)}_{\text{CV}}\end{array}$$

• The $\frac{D{E}_{\text{sys}}}{Dt}$ term is the energy change in the system
• The $\frac{\partial }{\partial t}{\int }_{\text{C.V.}}e\rho d\forall$ term is the energy change in the C.V.
• The ${\int }_{\text{C.S.}}e\rho \mathbf{V}\cdot \mathbf{n}dA$ term is the net energy flow rate through the C.S.

Some things to recall:

The work transfer rate (power) can be written as the product of the normal stress, $\sigma =-p$, and the fluid particle surface area , $\mathbf{n}\delta A$:

$$\delta \dot{W}=\sigma \mathbf{V}\cdot \mathbf{n}\delta A=-p\mathbf{V}\cdot \mathbf{n}\delta A$$

For all fluid particles on the control surface of at the instant considered, power transfer due to fluid normal stress can then be found as

$${\dot{W}}_{\text{stress}}={\int }_{\text{CS}}-p\mathbf{V}\cdot \mathbf{n}dA$$

This then overall gives us

$$\frac{\partial }{\partial t}{\int }_{\text{C.V.}}e\rho d\forall +{\int }_{\text{C.S.}}\left(u+\frac{V^2}{2}+gz\right)\rho \mathbf{V}\cdot \mathbf{n}dA={\dot{Q}}_{\text{net in}}+{\dot{W}}_{\text{shaft}}-{\int }_{\text{CS}}p\mathbf{V}\cdot \mathbf{n}dA$$

If only $p$ or normal stress $\sigma$ on the control surface is known:

$$\frac{\partial }{\partial t}{\int }_{\text{C.V.}}e\rho d\forall +{\int }_{\text{C.S.}}\left(u+\frac{V^2}{2}+gz+\frac{p}{\rho }\right)\rho \mathbf{V}\cdot \mathbf{n}dA={\dot{Q}}_{\text{net in}}+{\dot{W}}_{\text{shaft}}$$