Conductance
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In a GUNNS network, a conductance G creates flow w between two nodes due to the delta-potential dp between them as:
w = G·dp
This is a generic analog of Ohm’s Law for electrical networks:
I = V/R
since resistance R is the inverse of conductance G, and in electrical networks, potential is voltage V and flow is current I.
We define dp as the difference between the potentials of the Nodes 0 and 1 across the conductor, as:
dp = p0 – p1
so the flow w is positive for flow going from Nodes 0 to 1, which occurs when p0 > p1. In general, conductors can flow in either direction, from the higher potential node to the lower.
Similar to other effects, we define positive flow sign as travelling from port 0 to 1:
w = w0 = -w1
Writing the flow equation for each node, noting that the net flux on any node must be zero, by KCL:
Node 0: w0 = dp·G = 0
Node 1: w1 = –w0 = –dp·G = 0
Node 0: w0 = (p0 – p1)·G = 0
Node 1: w1 = –w0 = –(p0 – p1)·G = 0
By inspection we can write these equations in matrix form. This is the generic footprint of all conductance effects in the network’s system of equations:
The conductance effect in most links connects between two nodes, either two normal nodes, or a normal node and the Ground node, etc. There are exceptions. Some links have a built-in conductance between one of its port nodes and Ground without having to actually connect a port to the Ground node. Whether or not Ground is built-in to the link in this way or connected to by a port, since Ground is outside the system of equations the footprint of such a conductance effect in the network’s system of equations reduces to one row/column:
G·p t = 0.
If no other effects are on the node, that node’s row in the system of equations reduces to p t = 0, which means the potential in that node will go to zero if there are no other affects on it.
A link can combine multiple effects, and some links have more than one conductance effect connecting 3 or more nodes in different ways.
To avoid problems with the numerical solution method in GUNNS, you should apply some ‘sanity limits’ to the range of conductance values you use in your networks. Try to work within these limitations as much as possible:
- Don’t use conductance magnitudes larger than 10 15 or smaller than 10 -15. Any conductance smaller than 10 -15 should just be set to zero instead.
- All the conductances in your network should be within 15 orders of magnitude of each other. In general, the wider the range of magnitudes in your network, the more rounding and truncation errors you can get in the network solution. This normally isn’t noticeable until you start combining magnitudes with a difference greater than 10 or so.
- Don’t connect very large conductances to very small capacitive nodes, because similar rounding & truncation errors can result as above. This can cause the network to appear to leak quantity, and not conserve energy or fluid mass.
- If you think about it, connecting a very large conductance to a very small capacitance isn’t realistic anyway. It doesn’t make sense to connect a fluid duct with a conductance of 1 m 2 to a volume of 0.000001 m 3 — there must be some significant volume associated with the location you’re connecting that duct to, because the duct itself represents a volume. In the real world, there is always a significant capacitance associated with a significant conductance, so it’s best to model that capacitance realistically than to ignore it.
- Non-capacitive nodes are a similar case. Even though non-capacitive nodes don’t really exist in the real-world, they are useful to us, especially in electrical networks. In order for GUNNS to be able to solve for non-capacitive nodes, it has to treat them as having a very small non-zero capacitance in its numerical solution. This is called ‘conditioning’ the admittance matrix and is described more here. This causes the network to leak a small amount of quantity at non-capacitive nodes, which is why isolated non-capacitive nodes go to zero potential. The amount of leak is usually very small, but attaching large conductors to a non-capacitive node makes it worse.
- In summary: don’t use non-capacitive fluid nodes when strict conservation of mass is required.
See Admittance.