# Heat conduction model starting from DC RC-circuit

Thread Starter

#### johanhs

Joined Nov 4, 2016
2
Hi,

I'm a chemistry student and I'm trying to make a simple model for heat conduction,
using the analogy between heat conduction and electric conduction.
I already managed to draw the heat-current as a DC-circuit.

Now I want to make a V(t) (voltage in function of time) formula for the second capacitor.
But I don't know how to start. I understand the derivation of the V(t)-curve of a single DC RC-circuit,
but I don't know how to apply these principles on my circuit.
Any suggestions?

Best regards

Johan

#### WBahn

Joined Mar 31, 2012
26,398
Does your background include differential equations and/or Laplace transforms?

Thread Starter

#### johanhs

Joined Nov 4, 2016
2
Yes I have some experience with solving differential equations and Laplace.
My knowledge of electrical circuits is pretty limited however.
I've searched for similar circuits online, but haven't found one yet.

#### RBR1317

Joined Nov 13, 2010
625
Here is how to write the node equations for that circuit. Note that Vs will be the Laplace Transform of the applied voltage. Just solve for Vg and take the inverse Laplace transform of Vg to get the time domain response which you seek.

#### MrAl

Joined Jun 17, 2014
8,157
Hi,

I'm a chemistry student and I'm trying to make a simple model for heat conduction,
using the analogy between heat conduction and electric conduction.
I already managed to draw the heat-current as a DC-circuit.
View attachment 114760

Now I want to make a V(t) (voltage in function of time) formula for the second capacitor.
But I don't know how to start. I understand the derivation of the V(t)-curve of a single DC RC-circuit,
but I don't know how to apply these principles on my circuit.
Any suggestions?

Best regards

Johan
Hi,

I think there is still ongoing discussion about whether or not this kind of model is still realistic when relativistic effects come into play, but i also think this is an acceptable model for many applications even today. The main point being that heat travels through a conductor more like a wave too, so that would turn those resistances into transmission lines. It's good enough though i would bet for most things.

I am a little curious about what this system is modeling in the real world. Looks like a system inside another system.

I have to second the nomination for using Laplace Transforms as that makes the problem very simple. You can either work it from left to right or from right to left and then back again. Capacitors become Zc=1/(s*C) and Resistors stay as Zr=R.

It also depends on what kind of source you expect to drive this with. You show a battery so that would be modeled as E/s, but if you intend to drive it with another source like a sine then that has to change significantly. The source E/s would model an abrupt change in temperature from 0 to some value T.

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