Frist Law of Thermodynamics.

[splitringtail]

Homework Statement

Derive the heat equation for

Conduction

Convection/Condution

Homework Equations

The First Law of Thermodynamics

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.

 

[Jhero]

I can help you derive the heat equation for conduction and convection/conduction. Let's start with the basics: the First Law of Thermodynamics. This law states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system, plus any mass flow in or out of the system, plus any power production within the system. In equation form, this can be written as:

\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

Where $E_{cv}$ is the internal energy of the control volume, $\dot{Q_{cv}}$ is the heat added to the control volume, $\dot{W_{cv}}$ is the work done by the control volume, $\dot{m_{in}}$ and $\dot{m_{out}}$ are the mass flow rates in and out of the control volume, $h$ is the enthalpy, $v$ is the velocity, $g$ is the gravitational acceleration, and $PowerProduction$ is any power produced within the control volume.

Now, let's focus on conduction. Conduction is the transfer of heat within a material or between two materials in direct contact. The rate of heat conduction is given by Fourier's Law:

\vec{J}_{conduction} = -k\nabla T

Where $\vec{J}_{conduction}$ is the heat flux, $k$ is the thermal conductivity of the material, and $\nabla T$ is the temperature gradient. This equation tells us that heat will flow from regions of high temperature to regions of low temperature, with the rate of heat transfer depending on the thermal conductivity of the material.

To incorporate this into our First Law equation, we can rewrite it in integral form and use Gauss's Theorem:

\frac{\partial}{\partial t}\int \rho u dV =-\int \vec{J}_{conduction} \bullet d\vec{s} + PowerProduction

Where $\rho$ is the density,