Writing Nodal Equations for a Thermal System

[tesla93]

So I have to write out nodal equations for a system that I was given, and I started off by drawing out the flow diagram to see where everything was. I'm having trouble starting the nodal equations though.

The attachment has 2 diagrams: figure 1 is the correct diagram that models the thermal system, and figure 2 is what I drew out to try and write the nodal equations.

I started off at Q1

Q1= (P5-P4)+(P4-P1)+(P1-P5)=0

is this on the right track for a nodal equation? If anyone can explain how to write them or tell me if I'm at least on the right track that would help me so much. Thanks!

 

[rude man]

I see no reason to treat the several nodes as anything but temperatures. That's what they are. So why you choose to draw your diagram with P's instead of T's is a mystery.

Write the nodal equations just as you would if they were voltages and currents instead of temperatures and heat flows, if you're familiar with that. Irrespective, I trust you know that dQ/dt = ΔT/R where R is thermal resistance, and dQ/dt = CdT/dt where C is heat capacity.

 

[tesla93]

Sorry I was going off of my professors notes for a different system and he used P.

Would this be correct if I was looking at specifically at the node Ti?

0=-1/R1(Ti-T1)-1/R5(Ti-To)-Ci(Q(t)-Ti)

What I understood from nodal analysis is that you start at one node and look at what's going in and out at that point, and for the equation above, I wrote it assuming everything leaving the node is negative and everything entering is positive. Am I at least on the right track with this?

 

[rude man]

Sorry I was going off of my professors notes for a different system and he used P.

Would this be correct if I was looking at specifically at the node Ti?

0=-1/R1(Ti-T1)-1/R5(Ti-To)-Ci(Q(t)-Ti)

What I understood from nodal analysis is that you start at one node and look at what's going in and out at that point, and for the equation above, I wrote it assuming everything leaving the node is negative and everything entering is positive. Am I at least on the right track with this?

You need to appreciate what the quantities are. For example, you can't write Ci(Q(t) - Ti) at all. Why? Because Q and T have different dimensions. T has dimension of temperature and Q has dimension of energy. So that's a very fundamental no-no.

You're writing equations about heat rate of flow, dQ/dt, as I said last post. I.e. net heat flow into a node = rate of change of heat at that node. In your case the only source of heat flow into Ti is dQ/dt. So rewrite your equations using the correct terms, with consistent dimensions, showing the heat flow terms due to the thermal resistances, plus the buildup-of-temperature term, plus any sources of heat flow (in your case just the one going into Ti). Note that node To has no C arrow into it so there will be no dT/dt term for that node. (In physics we say that node To has zero heat capacity).