Equilibrium temperature distribution for rod of 2 materials

[melizzar]

1. Determine the equilibrium temperature distribution for a one-dimensional rod composed of two different materials in perfect thermal contact at x=1. For 0<x<1, there is one material (c$$\rho$$=1, K=1) with a constant source (Q=0), whereas for the other 1<x<2 there are no sources (Q=0, c$$\rho$$=2, K=2) with u(0)=0 and u(2)=0



I'm not sure to approach this problem. I believe I have to integrate two equations in order to find my solution for both materials but I'm not sure which ones. For my other ETD problems of one material, I've been using the equation KU"(x)+Q(x)=0. What should I do?

 

[mighty2000]

your first step would be to clearly define the problem and gather all the necessary information. From the given information, we know that we have a one-dimensional rod composed of two different materials in perfect thermal contact at x=1. The first material has a constant source (Q=0) and the second material has no sources (Q=0). We also know the values for the specific heat (c\rho) and thermal conductivity (K) for both materials.

Next, we can use the heat equation, which describes the temperature distribution in a given material, to solve this problem. The heat equation is given by:

d/dx (K dT/dx) + Q = c\rho dT/dt

Where T is the temperature, K is the thermal conductivity, Q is the heat source, c\rho is the specific heat, and t is time.

Since we are dealing with a one-dimensional rod, we can assume that the temperature distribution is only dependent on the x-axis. Thus, we can simplify the heat equation to:

K d^2T/dx^2 + Q = c\rho dT/dt

Now, since there are two different materials, we will have two different equations for each material. For the first material (0<x<1), we can use the heat equation as it is, since there is a constant heat source (Q=0). Thus, our equation becomes:

K1 d^2T/dx^2 = c\rho1 dT/dt

For the second material (1<x<2), there is no heat source (Q=0), so our equation becomes:

K2 d^2T/dx^2 = c\rho2 dT/dt

Now, we also know that the temperature at the boundaries x=0 and x=2 is given (u(0)=0 and u(2)=0). This means that we have two boundary conditions to satisfy for our equations.

For the first material (0<x<1), using the boundary conditions, we can write:

u(0) = 0, which means that T(0) = 0

u(1) = 0, which means that T(1) = 0

Thus, our equation becomes:

K1 d^2T/dx^2 = c\rho1 dT/dt, with boundary conditions T(0)=0 and T