Hi, I was trying to bring a transient conduction system problem to the frequency domain in order to facilitate the solution and I began to wonder: Can the analogy between electrical circuits and thermal circuits be extended to transmission lines for transient heat conduction? Let me explain my reasoning:
If one rearranges the terms of the Fourier conduction equation, in 1D cartesian coordinates:
q=-k*A*dT/dx
q/A * 1/k = -dT/dx
Now making q/A=q" (heat flux) and 1/k=R' (thermal resistance per unit length), one can write it as:
q"*R'=-dT/dx (1)
Writing out energy conservation, and considering the source term equal zero:
d/dx(k*dT/dx)=rho*cp*dT/dt
Identifying that k*dT/dx=-q" (the heat flux in an element), we can rewrite it as:
-dq"/dx=rho*cp*dT/dt
Now, calling C'=rho*cp (Thermal capacitance per unit length), the energy conservation can be rewritten:
-dq"/dx=C' * dT/dt (2)
Equations (1) and (2) are of the same form as the Telegrapher's equations for a 1D transmission line, which for the general case are:
-dV/dx=L' * dI/dt + R' * I (3)
-dI/dx=C' * dV/dt + G' * V (4)
Where I and V are the current and voltage, L' is the inductance per unit length, R' is the resistance per unit length, C' is the capacitance per unit length and G' is the shunt resistance per unit length.
The interesting part here is, if one makes L'=0 (no thermal inductance) and G'=0 (no thermal shunts in a flat plate), eq. (1) matches eq. (3) (V~T, q"~I) and eq. (2) matches eq. (4).
The consequence is that all features of a transmission line apply (given L'=G'=0) to thermal conduction: Wave propagation, wave distortion, etc. It also becomes relatively easy to associate different materials by using two-port networks and other basic electrical engineering concepts.
What do you think? Is that possible? If not, why? I'm really curious!
I think that the lack of coherence between the thermal carriers and the lack of anything approaching an electromagetic effect will mean that L=C=0 for all conditions and hence anything other than an exp(-kx) function will simply not apply.
Hi, Mr. Tony, thanks for your response,
I'm not suggesting the analogy extends to such an extent where the electromagnetic effect applies. I'm just putting that this transient thermal problem could be solved by an infinite RC network (L=0, but C is not equal zero as you suggested), as already pointed out by Robertson and Gross, in 1958 (link attached).
They also mentioned that those systems (1D thermal conduction systems) are indeed transmission line analogues, with the L=0 constraint that fundamentally transforms them into diffusion systems by eliminating the time derivative in one of the equations. But I didn't see a discussion of the consequences of that in the frequency domain, which is the point I'm curious about.
For example, if one uses the transmission line analog to plot the input impedance transfer function Zin(j*omega), the transfer function the comes off is one of magnitude Zin ~ 1/sqrt(omega) for higher frequencies.
That has the consequence that the function T/q" = Zin is very small in higher frequencies. As a possible application, one could concentrate power generation of a device in the higher frequencies (fast pulsed power applications), and still not have much temperature increase, which might be useful if one is worried about temperature damage to the device. Of course, the low-frequency average power is still positive (non-zero) in Joule-effect heat generating devices, so one cannot escape that. But that still could explain why fast pulsed lasers, fast LEDs and other devices are possible.
I mean, that's quite straightforward, but I couldn't find someone speaking about this specifically. Maybe somebody could point out which reference I'm missing.
http://nvlpubs.nist.gov/nistpubs/jres/61/jresv61n2p105_A1b.pdf
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