# Frist Law of Thermodynamics.

1. Dec 15, 2007

### splitringtail

1. The problem statement, all variables and given/known data

Derive the heat equation for

Conduction

Convection/Condution

2. Relevant equations

The First Law of Thermodynamics

3. The attempt at a solution

This is the form of the 1st Law that we have been working with.

$$\frac{\partial E_{cv}}{\partial t}= \dot{Q_{cv}}-\dot{W_{cv}}+\dot{m_{in}}\left(h_{in}+\frac{v^{2}_{in}}{2} + gz_{in}\right)+\dot{m_{out}}\left(h_{out}+\frac{v^{2}_{out}}{2} + gz_{out}\right) + PowerProduction$$

However, I know I need it in the integral form and use Gauss's Theorem, but I am not sure of what it looks like in integral form. It is in none of my texts... so here's a stab at it... I think I am just not sure about the left hand side of the equation

$$\frac{\partial}{\partial t}\int \rho\left[u + \frac{v^{2}}{2} + g z\right] dr =-\int \vec{J}_{conduction} \bullet d\vec{s} - \int \vec{J}_{convection}\bullet d\vec{s}+ PowerProduction$$

CV stands for control volume and I just used a bullet to represent a dot product.