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A straight rod is made of two parts, $[0,x_1]$ (green in the figure) with thermal diffusivity $\kappa_1$ and $[x_1,x_2]$ (blue) with thermal diffusivity $\kappa_2$. The rod is perfectly insulated. Zero $y$ and $z$ temperature gradients are assumed.

At $x=0$ temperature is maintained at constant $T_0$. At $x=x_2$ the rod is embedded into a perfect insulator ($\kappa=0$). At $t=0$ the rod has a uniform temperature $T(x,0)=T_i$.

Composite rod

Question: what is the temperature evolution of the rod?

1. The simple case where $\kappa_1=\kappa_2=\kappa$:

Let $u(x,t)=T(x,t)-T_0$.

Then Fourier's equation tells us:

$$u_t=\kappa u_{xx}$$

Boundary conditions:

$$u(0,t)=0$$ $$u_x(x_2,t)=0$$

Initial condition:

$$u(x,0)=u_i=T_i-T_0$$

Using the Ansatz $u(x,t)=X(x)\Gamma(t)$, separation constant $-k^2$ and the boundary conditions above, this solves easily to:

$$\Large{u(x,t)=\displaystyle \sum_{n=1}^{+\infty}B_n\sin\Bigg(\frac{n\pi x}{2x_2}\Bigg)e^{-\kappa \Big(\frac{n\pi }{2x_2}\Big)^2t}}$$

(for $n=1,3,5,7,...$)

The $B_n$ coefficients can easily be obtained from the initial condition with the Fourier sine series:

$$B_n=\frac{4u_i}{n\pi}$$

Back-substituting we get:

$$T(x,t)=T_0+\frac{4(T_i-T_0)}{\pi}\displaystyle \sum_{n=1}^{+\infty}\frac{1}{n} \sin\Bigg(\frac{n\pi x}{2x_2}\Bigg)e^{-\kappa \Big(\frac{n\pi }{2x_2}\Big)^2t}$$

(for $n=1,3,5,7,...$)

A plot for the first three terms at $t=0.1$:

Homegeneous rod temperature

2. The case where $\kappa_1\neq\kappa_2$:

We define two functions $u_1(x,t)$ for $[0,x_1]$ and $u_2(x,t)$ for $[x_1,x_2]$. We use the same Ansatz as under $1.$ We'll assume both functions have their own eigenvalues.

Boundary conditions:

$$u_1(0,t)=0\implies X_1(0)=0\tag{1}$$ $$\frac{\partial u_2(x_2)}{\partial x}=0\implies X_2'(x_2)=0\tag{2}$$

In addition (continuity):

$$u_1(x_1,t)=u_2(x_1,t)\tag{3}$$

With Fourier, the heat flux is the same at $x=x_1$:

$$\alpha_1\frac{\partial u_1(x_1)}{\partial x}=\alpha_2\frac{\partial u_2(x_1)}{\partial x}\tag{4}$$

Where $\alpha_i$ are the thermal conductivities.

a. for $u_1(x,t)$:

$$X_1(x)=c_1\cos k_1x+c_2\sin k_1x$$ $$X_1(0)=0\implies c_1=0\implies X_1(x)=c_2\sin k_1x\tag{5}$$

b. for $u_2(x,t)$:

$$X_2(x)=c_3\cos k_2x+c_4\sin k_2x$$ $$X_2'(x_2)=0\tag{2}$$ $$\implies -c_3k_2\sin k_2x_2+c_4k_2\cos k_2x_2=0\tag{6}$$ Using the additional conditions $(3)$ and $(4)$:

$$c_2\sin k_1x_1=c_3\cos k_2x_1+c_4\sin k_2x_1\tag{7}$$ $$c_2\alpha_1k_1\cos k_1x_1=-c_3\alpha_2k_2\sin k_2x_1+c_4\alpha_2k_2\cos k_2x_1\tag{8}$$

Problem:

$(6)$, $(7)$ and $(8)$ form a system of three simultaneous equations but with five unknowns: $c_2$, $c_3$, $c_4$, $k_1$ and $k_2$.

I'm tempted to set $c_3=0$ as it would yield $k_2$ from $(6)$. I think this would yield also the remaining unknowns. But can I a priori assume $c_3=0$? Or is there another approach possible?

I'm also left wondering whether perhaps $k_1=k_2$. The eigenvalues do not depend on $\kappa$, so perhaps the eigenvalues $k$ are common to both functions. Due to $(4)$, $u_1$ and $u_2$ would then still be distinct.

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  • $\begingroup$ The time scale for both regions has to be the same. So, I think you are going to find that $$\frac{k_1}{k_2}=\sqrt{\frac{\kappa_2}{\kappa_1}}$$ I don't know whether this will help or not. This is as far as I've gotten into the problem so far. $\endgroup$ – Chet Miller Oct 13 '16 at 12:41
  • $\begingroup$ @ChesterMiller: interesting thought! $\endgroup$ – Gert Oct 13 '16 at 12:43
  • $\begingroup$ Also. It looks like you are assuming that the thermal conductivities of the two regions are the same (even though the thermal diffusivities are different). Did you really want to do that? $\endgroup$ – Chet Miller Oct 13 '16 at 12:45
  • $\begingroup$ No, I'm not assuming that: see $(4)$. $\endgroup$ – Gert Oct 13 '16 at 12:47
  • $\begingroup$ Oh. OK. I missed that. $\endgroup$ – Chet Miller Oct 13 '16 at 12:48
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I can see how to do this now. You have 3 BC equations, and 3 unknown c's plus the unknown time eigenvalue $\lambda^2$ (You know that $k_1=\lambda /\sqrt{\kappa_1}$ and $k_2=\lambda /\sqrt{\kappa_2}$). You can eliminate 2 of the c's. Your third equation will give you a relationship with the 3rd c out front and a zero on the other side of the equation. This will allow you to determine $\lambda$. The last c is then determined by making good on the initial condition, using an infinite series.

ADDENDUM

$$\frac{1}{T}\frac{dT}{dt}=\frac{\kappa}{X}\frac{d^2X}{dx^2}=-\lambda^2$$ So, $$\frac{d^2X}{dx^2}+\frac{\lambda^2}{\kappa}X=0$$ If the time scale for both regions is the same (as expected), then: $$k_1=\frac{\lambda}{\sqrt{\kappa_1}}$$and$$k_2=\frac{\lambda}{\sqrt{\kappa_2}}$$

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  • $\begingroup$ Ok. I've developed another approach, so now I need to compare the two. Will try and post tonight, if not tomorrow. Thanks for your help! $\endgroup$ – Gert Oct 13 '16 at 15:57
  • $\begingroup$ A quick question: what's the rational basis for $\frac{k_1}{k_2}=\sqrt{\frac{\kappa_2}{\kappa_1}}$? $\endgroup$ – Gert Oct 13 '16 at 16:09
  • $\begingroup$ This problem should also be analyzed in detail in Carslaw and Jaeger. $\endgroup$ – Chet Miller Oct 13 '16 at 16:09
  • $\begingroup$ "Carslaw and Jaeger" Huh? $\endgroup$ – Gert Oct 13 '16 at 16:26
  • $\begingroup$ @Gert See ADDENDUM above. Conduction of Heat in Solids by Carslaw and Jaeger $\endgroup$ – Chet Miller Oct 13 '16 at 16:58
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This uses parameter symbols that I'm more accustomed to: k is thermal conductivity, $\alpha$ is thermal diffusivity, $\rho$ is density, and $c_p$ is heat capacity.

I followed the procedure which I outlined in my previous answer and obtained the following:

The characteristic equation for the n'th eigenvalue is: $$\tan\left[\frac{\lambda_n(x_2-x_1)}{\sqrt{\alpha_2}}\right]\tan\left[\frac{\lambda_nx_1}{\sqrt{\alpha_1}}\right]=\sqrt{\frac{k_1\rho_1c_{p1}}{k_2\rho_2c_{p2}}}$$ The n'th eigenfunction is: $$X_n=\cos\left[\frac{\lambda_n(x_2-x_1)}{\sqrt{\alpha_2}}\right]\sin\left[\frac{\lambda_nx}{\sqrt{\alpha_1}}\right]\tag{0 < x <x1}$$ $$X_n=\cos\left[\frac{\lambda_n(x_2-x)}{\sqrt{\alpha_2}}\right]\sin\left[\frac{\lambda_nx_1}{\sqrt{\alpha_1}}\right]\tag{x1<x<x2}$$ I leave it up to you to do the integation to get the n'th coefficient in the infinite series.

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