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).

 

[rezeew]

I would say that you are on the right track for writing nodal equations for a thermal system. However, there are a few things that can be improved upon.

Firstly, it is important to define your variables consistently. In your equation, you have used P1, P4, and P5 to represent pressure differences between different nodes. It would be clearer to use a consistent notation, such as P1-2, P2-3, etc. to indicate the pressure differences between specific nodes.

Secondly, it is important to consider the direction of heat flow in your equations. In a thermal system, heat flows from a higher temperature to a lower temperature. Therefore, the equations for Q1 and Q2 should have a negative sign to indicate that heat is flowing out of those nodes. Likewise, the equations for Q3 and Q4 should have a positive sign to indicate that heat is flowing into those nodes.

Additionally, it is important to consider the units of your variables. In a thermal system, heat is typically measured in watts (W) or joules per second (J/s). Therefore, your equations should have units of W or J/s on both sides.

Lastly, it is important to ensure that your nodal equations are consistent with the flow diagram. Looking at figure 1, we can see that the heat flow from Q1 to Q2 is also equal to the heat flow from Q4 to Q5. Therefore, your nodal equation for Q1 should also include the heat flow from Q4 to Q5, and your nodal equation for Q4 should include the heat flow from Q1 to Q2.

In summary, your nodal equations should be:

Q1 = -(P1-2 + P4-5)
Q2 = P2-3 + P5-6
Q3 = P3-4
Q4 = (P4-5 + P1-2)
Q5 = -(P5-6 + P2-3)

I hope this helps to clarify the process of writing nodal equations for a thermal system. It is important to carefully consider the variables, units, and direction of heat flow in order to ensure accurate and consistent equations.