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Chemical kinetics/heat equation

  1. Nov 23, 2009 #1
    1. The problem statement, all variables and given/known data
    the math in here is a bit over my head.

    the equation is

    [tex] \frac {d^2 \theta}{d x'^2 } = -y *exp(\theta) [/tex] 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:[tex] \frac {\Delta T}{T} << 1 [/tex]

    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 [tex] \theta = \frac {E}{RT^2_a} *(T - T_a) [/tex]

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

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

    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 [tex] \theta = f(y,x') [/tex] in case of high activation energy E. RT<<E

    the book gives the following result.

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

    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
  2. jcsd
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