I'm stuck on this PDE: $$\rho \cdot C \cdot \dfrac{\partial T}{\partial \tau} + u \cdot \rho \cdot C \cdot \dfrac{\partial T}{\partial r} = \dfrac{\lambda}{r^{2}} \cdot \dfrac{\partial}{\partial r}\left(r^{2} \cdot \dfrac{\partial T}{\partial r}\right) + K - \dfrac{\rho \cdot C \cdot T}{V} \cdot \dfrac{\partial V}{\partial \tau}$$ This should describe the variation of enthalpy during a flash pyrolysis. I know all the parameters ($ \rho, C, \lambda, K$) and I can obtain $u$ and $V$ from analytic expression.
Assuming I know the initial condition and two boundary conditions, I'd like to solve it numerically using matlab.
Can anybody help me out? Thanks in advance.
IC and BC: $$T_{\tau = 0} = T_{0} \hspace{2cm} 0 \leq r \leq R $$ $$\dfrac{\partial T}{\partial r}\bigg|_{r=0} = 0$$ $$\dfrac{\partial T}{\partial r}\bigg|_{r=R} = C1 \cdot (T_{m} - T_{R})$$
Yes, $\tau$ is a time variable and $r$ is the space coordinate. I'm expecting a profile of $T$ with respect to both variables.
I'm sorry if I can't provide so much useful information, but I was given a scientific paper to study where these equations (there is a lot of equations, but this one is the trickiest) are displayed but there is no mention of how they solved and got to the results.