Boundary condition at junction of heated rods

[musemonkey]

1. 1D heat conduction problem: Two rods, the first of length $$a$$, the second of length $$L-a$$ with respective cross sectional areas $$A_1$$,$$A_2$$ and heat conductivities $$k_1$$, $$k_2$$, are joined at one end. There are some boundary conditions on the other ends of the rods, but my question is only: how to write the boundary condition at the juncture?

Homework Equations

Heat conduction eqn:

$$\alpha^2 u_{xx} = u_t$$

where $$\alpha^2 = \frac{k}{s \rho}$$, in which $$\rho$$ is the density and $$s$$ is the specific heat with units: energy / (mass degrees).

More fundamentally, heat flux through a cross-section per unit time is

$$H = - k A u_x$$.

General form of boundary condition at non-insulated rod end is given in the text as

$$u_{x} = h u$$
where $$h$$ is some constant (presumably with units inverse length).

The Attempt at a Solution

The idea of the $$u_{x} = h u$$ is intuitive enough: the heat flux from a surface is proportional to its temperature, but I can't figure out what $$h$$ is in terms of thermal conductivity k, specific heat s, cross-sectional area A, and mass density $$\rho$$. In other words, I can't even figure out how to set-up the simpler problem of a non-insulated rod end not connected to anything, much less two rods joined together, but my common sense tells me that the joined-rods problem reduces to the one rod end case because the effective conductivity and area at the junction should be $$min(k_1, k_2)$$ and $$min(A_1, A_2)$$ because what ever flows out of one rod has to be absorbed into the other. Actually I doubt this is right just from the way the problem is worded but it's the only thing that makes sense to me... To figure out $$h$$ I've tried playing with units:

Letting E = energy, L = length, T = temp, t = time, M = mass, I tried to combine

thermal conductivity $$k = \frac{E}{L\cdot T\cdot t}$$,

specific heat $$s = \frac{E}{M\cdot T}$$,

mass density $$\rho = \frac{M}{L^3}$$,

and cross-sectional area $$A = L^2$$

into units of inverse length. No success. Would much appreciate any guidance on how to write and think about the boundary conditions are for the two cases: one rod end and two joined rod ends with different material properties.

Just to be clear, the one rod end case is the one for which the text gives the form of the boundary condition. I'm puzzled as to how to think about it: Whatever heat is emitted has to be absorbed by something else, and how well that something can absorb one would think must be taken into account. So when we talk about one non-insulated rod end, does it mean, for examples, implicitly in a vacuum, in a medium with the same material properties, or in some ideal medium with perfect absorption? Lots of questions. Thank you for reading!

 

[musemonkey]

Would still much appreciate any help with this question.
-Musemonkey