I am attempting to solve a heat storage problem using Matlab and the PDE Toolbox and could use some assistance with the math portion.
The task is to model a concrete thermal battery with an array of embedded pipes that carry steam from a boiler. The heat from the steam warms the battery so that the heat can be used later when low temperature steam flows through the pipes (while the boiler is ramping up for example). In order to model the entire battery, I figure a representative element needs solving first. In this case, a representative element is one pipe and the concrete directly surrounding it up to the radius of Re.
I'm still learning Matlab, so I looked around online for some examples I might be able to build off of and stumbled upon this example (Nonlinear Heat Transfer in a Thin Plate).
I think this example is similar because the plate can be revolved to match the shape of the concrete element and the bottom edge fixed at a high temperature is similar to the heat from the steam at the center of the element. So the figure showing the temperature of the plate would be modified with the x-axis as the axial (z) coordinate and the y-axis as the radial (r) coordinate.
The part that is giving me a problem (at least for now) is setting up the equation to be in the correct form for the PDE Toolbox. I have to use the specifyCoefficients function to map out values for m, d, c, a, and f in the general PDE:
$$m\frac{∂^2u}{∂t^2} + d\frac{∂u}{∂t} - ∇·(c∇u) + au = f$$
The energy equation of the concrete element is:
$$ρc_p\frac{∂T}{∂t} = λ(\frac{∂^2T}{∂r^2} + \frac{1}{r}\frac{∂T}{∂r} + \frac{∂^2T}{∂z^2})$$
I think I have some of the coefficients correct, but I would appreciate input from others.
I believe m = 0
I believe d = concrete_density x concrete_specific_heat x element_thickness
I believe c = concrete_thermal_conductivity x element_thickness
I'm not sure about a or f.
Thanks for the help.
The two equations you're written are not equivalent and cannot be compared.
If $c$ is constant than the first equation is
$$ m \frac{\partial^2 u}{\partial t^2} + d\frac{\partial u}{\partial t} - c\left(\frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{\partial^2 u}{\partial z^2}\right) + au = f $$
The second equation is
$$ -\lambda\frac{\partial^2 T}{\partial t^2} + \rho c_p\frac{\partial T}{\partial t} -\lambda\left(\frac{1}{r}\frac{\partial T}{\partial r} + \frac{\partial^2 T}{\partial z^2}\right) = 0 $$
The closest you can get is
\begin{align} m &= -\lambda \\ d &= \rho c_p \\ c &= \lambda \\ a &= 0 \\ f &= 0 \end{align}
But there's still a $\frac{\partial^2 T}{\partial r^2}$ missing in the second equation
If you actually meant to write
$$ \rho c_p\frac{\partial u}{\partial t} - \lambda\left(\frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{\partial^2 u}{\partial z^2}\right) = 0 $$
Then $c=\lambda$, $d = \rho c_p$ and every other constant is $0$