# Homework Help: Chemical kinetics/heat equation

1. Nov 23, 2009

### squaremeplz

1. The problem statement, all variables and given/known data
the math in here is a bit over my head.

the equation is

$$\frac {d^2 \theta}{d x'^2 } = -y *exp(\theta)$$ eq. 1

first off, this is a steady state model. meaning, we consider the pre-explosion temperature to be small in comparison with the absolute temperature of the walls:$$\frac {\Delta T}{T} << 1$$

2nd, the reaction rate only depends on the deperature in accordance with exp(-E/RT)

3rd we regad the thermal conductivity of the walls as being infinitely large.

x' = x/r is the nondimensionalization of x, r is the half length (i.e radius for cylinder), not the derivative, for -L < x < L we have -1 < x' < 1. x' drops unit (i.e m, cm, ..)

theta is the nondimensionalization of temperature $$\theta = \frac {E}{RT^2_a} *(T - T_a)$$

y (although i used a different variable) is known as the frank kamenetskii parameter

$$y = \frac {Q}{d}*\frac {E}{R*T^2_a}*r^2*z* exp(\frac {-E}{RT_a})$$

E: activation energy
T_a: ambient temperature
Q: heat released
z: frequency of particle collision
r: radius or half width (depending on geometry)
R: gas constant
d: thermal conductivity

all uniform except Q, i think..

the book solves the differential equation 1, analytically, for a function $$\theta = f(y,x')$$ in case of high activation energy E. RT<<E

the book gives the following result.

$$exp(\theta) = \frac {a}{cosh^2(b \frac{+}{-} \sqrt \frac{a*y}{2} * x')}$$

im just trying to figure out what steps I need to take in order to arrive at the last solution.

seperation of vars?

2. Relevant equations

3. The attempt at a solution

Last edited: Nov 23, 2009