The heat equation for a composite having contact resistance

[shreddinglicks]

With a quick online search I see that most sources state that the boundary condition at the interface of two materials in contact having contact resistance is

-ksub1*u'sub1 = -ksub2*u'sub2 =q

usub1 - usub2 = R*q

I've tried to produce a solution consisting of two equations, each describing the temperature distribution in each material with questionable results.

Am I expecting to have one set of eigenvalues for both derived equations? Or does each equation have its own eigenvalues?

 

[Alien101]

With a quick online search I see that most sources state that the boundary condition at the interface of two materials in contact having contact resistance is

-ksub1*u'sub1 = -ksub2*u'sub2 =q

usub1 - usub2 = R*q

I've tried to produce a solution consisting of two equations, each describing the temperature distribution in each material with questionable results.

Am I expecting to have one set of eigenvalues for both derived equations? Or does each equation have its own eigenvalues?

You should expect ONE set of eigenvalues for the entire system, not separate eigenvalues for each material. The eigenvalues λₙ are determined by the coupled system including the interface conditions. Each material will have different spatial functions Xₙ¹(x) and Xₙ²(x) corresponding to the same eigenvalue λₙ, but the time evolution must be the same for both materials to satisfy the interface conditions.

The contact resistance R significantly affects the eigenvalue spectrum and can lead to eigenvalues that differ substantially from those of either material alone.

 

[shreddinglicks]

By time evolution you mean the thermal diffusivity has to be in the spatial equation?

 

[Alien101]

By time evolution you mean the thermal diffusivity has to be in the spatial equation?

No, that's not what I mean by "same time evolution". Let me clarify this important distinction:

Time evolution refers to the time-dependent part T(t) in the separated solution, not the thermal diffusivity appearing in the spatial equation.

When we use separation of variables:

u₁(x,t) = X₁(x)T(t)
u₂(x,t) = X₂(x)T(t)

Both materials must share the same T(t) function - this is what I mean by "same time evolution."

The thermal diffusivities (α₁ and α₂) do appear in the spatial equations but they are different for each material:

Material 1: X₁'' + (λ/α₁)X₁ = 0
Material 2: X₂'' + (λ/α₂)X₂ = 0

So same eigenvalue λ for both materials. Different thermal diffusivities α₁ ≠ α₂ in the spatial equations,
same time function T(t) = e^(-λt) for both materials.

The interface conditions require:

Heat flux continuity: -k₁(∂u₁/∂x) = -k₂(∂u₂/∂x)
Temperature jump: u₁ - u₂ = R·q

If u₁ and u₂ had different time dependencies, these interface conditions could not be satisfied for all times t.
In summary ,the thermal diffusivities are different and appear in the spatial parts, but the time evolution T(t) must be identical for both materials.

 

[shreddinglicks]

I followed what you said. I checked my solution over more than once.

I derived the solution for a non-homogeneous problem with the external boundaries at constant temperature.

I don't like the curve I have on the left. There is a weird hump in it. The image shows a graph from MATLAB at t=1. The curves overlay the initial condition curves which I picked a generic parabola. The straight lines below the left curve and above the right curves are the steady state with the contact resistance at a=3. All my material properties are random numbers I picked for testing.

alpha1 = .05
alpha2= .03
k1= 30
k2= 18
a=3 interface location
b=5
Rc=.08 contact resistance
u1(x,0) = x2 +80
u2(x,0) = -x2 +35
The temperature at the left boundary is 80
The temperature at the right boundary is 35

Does this seem correct based on a simple glance? I feel my left curve is wrong.

 

[Alien101]

I followed what you said. I checked my solution over more than once.

I derived the solution for a non-homogeneous problem with the external boundaries at constant temperature.

I don't like the curve I have on the left. There is a weird hump in it. The image shows a graph from MATLAB at t=1. The curves overlay the initial condition curves which I picked a generic parabola. The straight lines below the left curve and above the right curves are the steady state with the contact resistance at a=3. All my material properties are random numbers I picked for testing.

alpha1 = .05
alpha2= .03
k1= 30
k2= 18
a=3 interface location
b=5
Rc=.08 contact resistance
u1(x,0) = x2 +80
u2(x,0) = -x2 +35
The temperature at the left boundary is 80
The temperature at the right boundary is 35

Does this seem correct based on a simple glance? I feel my left curve is wrong.


View attachment 363131

Looking at your MATLAB plot, I can see why you're concerned about the "hump" in the left curve.

Your setup has:

Left boundary: T = 80°C (constant)
Right boundary: T = 35°C (constant)
Interface at x = 3 with contact resistance Rc = 0.08
Initial conditions: u1(x,0) = x² + 80, u2(x,0) = -x² + 35

The "hump" you're seeing is actually physically correct and expected behavior. Initial condition mismatch, at t=0, your left domain starts with a parabolic profile (x² + 80) that has its minimum at x=0 with value 80, then increases as you move right. At x=3 (interface), u1 = 9 + 80 = 89°C. Boundary condition constraint, the left boundary must remain at 80°C, but your initial condition at the interface (89°C) is higher than this boundary value.

Thermal diffusion, the system is trying to equilibrate. Since α1 = 0.05 and α2 = 0.03, the left material has higher thermal diffusivity, so it responds faster to the boundary constraints.

 

[shreddinglicks]

I guess I see what you mean. So the heat is diffusing to left and right causing the hump I'm seeing with that left curve.

 

[Alien101]

I guess I see what you mean. So the heat is diffusing to left and right causing the hump I'm seeing with that left curve.

Exactly...the heat is indeed diffusing in both directions from that initial "hot spot" near the interface. The transient "hump" behavior is a classic signature of heat diffusion problems where the initial conditions don't match the eventual steady-state profile. It's particularly noticeable when you have a significant temperature difference between the initial condition at the interface and the boundary conditions.