I need to understand the following model for in-situ combustion (1). The second of these is a molar balance equation.
Well, I've tried to use the following general balance equation:
Where $r$ is the rate of reaction and the $N$ is in moles (as mass).
I think that in case of the work there is no flow of immobile fuel inlet and outlet, so I think I could clear the two initial terms.
If $ \rho_f $ is the molar concentration of the fuel, the reaction rate in relation to the fuel is given by
$$ \dfrac {\partial \rho_f} {\partial t} = - k \rho_f ^ {\mu_f} $$
(as it is an elementary reaction, the order of the reaction in relation to the fuel is the coefficient itself)
The constant is given by the Arrhenius law:
$$ k_p ~ exp (-E_r / (RT)) $$
Therefore, considering $ \mu_f = 1 $, we have equation
$$ \dfrac {\partial \rho_f} {\partial t} = - k_p ~ exp (-E_r / (RT)) \rho_f = -W_r $$
Although it matches with the article if $ \mu_f = 1 $, I do not understand why in the article $ \mu_f $ is multiplying instead of being in the exponent.
Many thanks for any help!
(1) G. Chapiro, A.E.R. Gutierrez, J. Herskovits, S.R. Mazorche, W.S. Pereira (2016): "Numerical Solution of a Class of Moving BoundaryProblems with a Nonlinear Complementarity Approach", J. Optim. Theory Appl. 168(2), 534-550. doi:10.1007/s10957-015-0816-7