Temperature Distribution in a Semi-Infinite Wall

Any introductory heat transfer class solves for temperature distributions in plates, solids, etc.. The temperature distribution is a solution to the heat diffusion partial differential equation. The general form of the heat diffusion equation with no internal heat generation in Cartesian coordinates is $$\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} = \frac{1}{\alpha}\frac{\partial T}{\partial t}$$ The first series of problems typically tackled in the intro classes are the steady-state solutions. These are the cases where $\partial T / \partial t = 0$. And the very first problems even restrict us to only one dimension. Slowly we add in dimensions and time dependence. However, one problem that I find particularly interesting is the semi-infinite wall. The problem involves a half-space that represents a wall. A very thick wall. One might call it "infinite". So thick that the back end of the wall never receives any heat in the time span that we are concerned with. Therefore, we approximate it as an infinitely thick wall. Additionally, we will assume the wall's area is very large (or that it's boundary is so far away from the area that we're concerned with) and therefore the face of the wall is infinite. And due to symmetry, we can restrict the problem to be in only 1 space dimension: perpendicular to the face of the wall. Our partial differential equations can be reduced to $$\frac{\partial^2 T}{\partial x^2}=\frac{1}{\alpha}\frac{\partial T}{\partial t}$$ This is second order in position and first order in time. Therefore we require two boundary conditions and an initial condition. The initial condition is simple: we will assume that the wall starts off uniform in temperature. $$T(x,0)=T_0$$ Next we have to come up with the boundary conditions. In heat transfer there are three primary surface conditions: constant temperature, constant flux, and convection. The choice of surface condition depends on the physical context that you want to model. However, for the purpose of this solution we will assume that the surface condition has a constant temperature. Meaning that after $t=0$ we instantly change the temperature of the face of the wall to $$T(0,t)=T_s$$ Assuming either of the other two surface conditions changes very little, but requires us to introduce new constants and more complex algebra. As for the last boundary condition, we have already hinted at its nature: the wall at infinity is the same as it is at time $0$. $$T(x\to\infty,t)=T_0$$ But now we have to solve the PDE. But what if we don't know how to solve PDEs? This is what I find fascinating about this problem; we can turn this PDE of two variables into an ODE of 1 variable with a similarity variable ^[1] $$\eta=\frac{x}{\sqrt{4\alpha t}}$$ We now try to get $\frac{\partial^2 T}{\partial x^2}$ and $\frac{\partial T}{\partial t}$ which are in terms of $T,x,t$ into terms of just $T$ and $\eta$. $$\begin{align*}\frac{\partial^2 T}{\partial x^2}&=\frac{\partial}{\partial x}\left(\frac{\partial T}{\partial x}\right)=\frac{\partial}{\partial x}\left(\frac{\partial T}{\partial \eta}\frac{\partial \eta}{\partial x}\right)=\frac{\partial}{\partial x}\left(\frac{\partial T}{\partial \eta}\frac{1}{\sqrt{4\alpha t}}\right)\\ &=\frac{1}{\sqrt{4\alpha t}}\frac{\partial}{\partial x}\left(\frac{\partial T}{\partial \eta}\right)=\frac{1}{\sqrt{4\alpha t}}\left(\frac{\partial^2 T}{\partial x\partial \eta}\right)=\frac{1}{\sqrt{4\alpha t}}\left(\frac{\partial^2 T}{\partial \eta\partial x}\right)\\ &=\frac{1}{\sqrt{4\alpha t}}\frac{\partial}{\partial \eta}\left(\frac{\partial T}{\partial x}\right)=\frac{1}{\sqrt{4\alpha t}}\frac{\partial}{\partial \eta}\left(\frac{\partial T}{\partial \eta}\frac{\partial \eta}{\partial x}\right)\\ &=\frac{1}{\sqrt{4\alpha t}}\frac{\partial}{\partial \eta}\left(\frac{\partial T}{\partial \eta}\frac{1}{\sqrt{4\alpha t}}\right)=\frac{1}{4\alpha t}\frac{\partial^2 T}{\partial \eta^2} \end{align*}$$ The above depends on the continuity of $T$ and the independence of $x$ and $t$. Both are very appropriate assumptions for our problem. Likewise we can show $$\frac{\partial T}{\partial t}=\frac{\partial T}{\partial \eta}\frac{\partial \eta}{\partial t}=-\frac{x}{2t\sqrt{4\alpha t}}\frac{\partial T}{\partial \eta}$$ Putting these back into our heat equation $$\frac{1}{4\alpha t}\frac{\partial^2 T}{\partial \eta^2}=-\frac{1}{\alpha}\frac{x}{2t\sqrt{4\alpha t}}\frac{\partial T}{\partial \eta}$$ which can simplify to $$\begin{align*} \frac{\partial^2 T}{\partial \eta^2}&=-\frac{x\sqrt{4\alpha t}}{2\alpha t}\frac{\partial T}{\partial \eta}=-\frac{x}{x}\frac{x\sqrt{4\alpha t}}{2\alpha t}\frac{\partial T}{\partial \eta}=-2\frac{\sqrt{4\alpha t}}{x}\frac{x^2}{4\alpha t}\frac{\partial T}{\partial \eta}\\ &=-2\frac{x}{\sqrt{4\alpha t}}\frac{\partial T}{\partial \eta}=-2\eta\frac{\partial T}{\partial \eta} \end{align*}$$ And since it's all in terms of $\eta$ we can turn the partials back into total derivatives $$\frac{\mathrm{d}^2 T}{\mathrm{d} \eta^2}=-2\eta\frac{\mathrm{d}T}{\mathrm{d}\eta}$$ Now we are left to convert our boundary conditions into $\eta$. Our surface boundary condition, the condition where $x=0$ clearly corresponds to the case where $\eta=0$ and therefore $T(0)=T_s$. And the two conditions, $t=0$ and $x\to\infty$ clearly corresponds to the case where $\eta\to\infty$. Therefore, $T(\eta\to\infty)=T_0$. Thus the three conditions on the PDE are now two conditions in the ODE. I'm not going to solve this ODE because it's a standard separation of variables solution. I was only interested in sharing the conversion from the PDE of two variables into the ODE of one variable. I will however, spoil the ending and share the solution: $$ \DeclareMathOperator\erf{erf} \frac{T(\eta)-T_s}{T_0-T_s}=\erf{\eta}$$ where $\erf{\eta}$ is the error function (Go figure!). However, we can always substitute back in $x$ and $t$ because $T(\eta)=T(x,t)$. Therefore, in conclusion $$\frac{T(x,t)-T_s}{T_0-T_s}=\erf{\left(\frac{x}{\sqrt{2\alpha t}}\right)}$$

[1] Incropera, Frank P. Fundamentals of Heat and Mass Transfer. Hoboken, NJ: Wiley, 2013. Print.