# Heat diffusion with arrhenius source

1. Oct 28, 2009

### squaremeplz

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

a material occupies -L < x < L and has uniform ambient temperature T_a. A chemical reaction begins within the body leading to the 1-d heat equation:

$$pc \frac {\partial{T}}{\partial{t}} = k \frac {\partial^2{T}}{\partial{x^2}} + pQAe^\frac{-E}{RT}$$

with BC and IC

$$T(+/- L, t) = T_a$$ and $$T(x,0) = T_a$$

2. Relevant equations

3. The attempt at a solution

The book gives the following substitutions without any justification

$$\theta = \frac{E}{RT_a^2}(T-T_a)$$ and $$x = L*b$$

and arrives at the following linear equation

$$L^2 \frac {pc}{k}{ \frac {\partial{\theta}}{\partial{t}} = \frac {\partial^2{\theta}}{\partial{b^2}} + z*e^\frac{\theta}{1+y\theta}$$

then it asks to give the equations for z and y.

However, before I begin to solve this problem.. I would love to understand where the substitutions

$$\theta = \frac{E}{RT_a^2}(T-T_a)$$ and $$x = L*b$$

came from. I presume that x = L*b is used to make x dimensionless.. but what about theta well??

Im pretty sure I know how to solve the problem (seperation of variables on the last eq.) , but I really don't understand the substitutions. It seems that whoever wrote the book solved the problem first and then realized the substitutions were a good fit. Any information (about the substitutions) is appreciated.