This section reviews some physical principles and how they are analogous to each other.

The idea of the Hydraulic-Electric analogy has been around for a long time. Real scientists may look down on it, but it is very useful for our purposes.

There are many physical concepts and mathematical relationships that are analogous between fluid mechanics and electricity, and to a lesser extent, thermodynamics. We take advantage of this to re-use a lot of the same maths for all 3 aspects. This is what allows GUNNS to be used for all aspects.

The most important analogy to understand is that of the relationships between Potential, Conductance and Flow. In electrical terms, this is represented by Ohm’s Law:

I = G·ΔV

where I is the electrical current through an electrically conductive path, ΔV is the voltage across the conductor (the difference in electrical potential), and G is the conductance (inverse of resistance), a property of the physical material and its geometry.

This is analogous to the thermal form, which is Newton’s Law of Cooling:

Q = k·ΔT

where Q is heat flux through a thermal conductive path, ΔT is the difference in temperature across the conductor (the difference in thermal potential), and k is the heat transfer coefficient of the path, which is really its conductance.

This is analogous to the fluid form, which is a linearization of a desired fluid flow equation, such as Bernoulli’s Equation:

ṁ = A·Δp

where ṁ (“m dot”) is the mass flow rate through a fluid conductive path (i.e. a pipe), Δp is the difference in pressure across the conductor (the difference in fluid potential), and A is roughly the cross-sectional area of the pipe. The analogy is less than perfect in the fluid case: A is really a function of much more, like length, obstructions, wall roughness and fluid properties, and most flow regimes are best modelled with a non-linear function which we must linearize (more on that here). These make the fluid aspect fundamentally more complicated than electrical and thermal, but this still works well enough for us.

Note the similarities in concepts between all 3 aspects:

Analogous Concept	Fluid Aspect	Electrical Aspect	Thermal Aspect
“Potential”	Pressure	Voltage	Temperature
“Flow”	Mass Flow	Current	Heat Flux
“Conductance”	Area (roughly)	Inverse of resistance	Heat transfer coefficient

We generalize this as the “Conductance Effect”:

w = G·Δp

where w is “flux” (generic flow), G is generic conductance, and p is generic potential. More details on the conductance effect here.

In all 3 aspects, what is physically flowing through the conductor?

• Fluid: molecules. In a fluid state (gas, liquid, plasma, etc.)
• Electrical: charge. Strictly speaking for positive current, positive charge carriers going in the opposite direction of electrons
• Thermal: heat (energy).

In all 3 aspects, there is a similar relationship between the change of quantity in a volume of space or material and the change in that volume’s potential.

Electrical capacitance gives the change in electrical potential due to current as:

I = C · δv/δt

where I is the electrical current into the space, C is the capacitance, and δv/δt is the change in voltage (potential) per unit time.

Fluid capacitance derives from the fluid’s equation of state, which is the relationship between density and pressure of the fluid. GUNNS may sometimes use a linear equation of state (i.e. the ideal gas law) or a non-linear state for liquids and non-ideal gasses. Regardless, GUNNS expresses the change in density to the change in pressure with a similar linear relationship as above, and then resolves any errors this linearization causes between the new quantity and the equation of state later. More on fluid capacitance here.

ṁ = C · δp/δt

Thermal capacitance, or heat capacity, is a property of a lump mass of material that is assumed to have uniform temperature and specific heat. The relationship between the lump mass’s change in temperature to change in heat is:

Q = C · δT/δt

where Q is the heat flux (power) into the mass.

Note the similarities in concepts between all 3 aspects:

Analogous Concept	Fluid Aspect	Electrical Aspect	Thermal Aspect
“Quantity”	Molecules	Charge	Heat (energy)
“Capacitance”	Fluid capacitance (n·β)	Electrical capacitance	Heat capacity (m·C p )

We generalize this as the “Capacitance Effect”:

w = C · δp/δt

where w is “flux” (generic flow), C is generic capacitance, and p is generic potential. More details on the capacitance effect here.

There are other less perfect analogs between physical parameters & effects that are still somewhat useful. Some of these are implemented or partially implemented in GUNNS, others are not. We won’t go into details, but here are a few ideas:

Analogous Concept	Fluid Aspect	Electrical Aspect	Thermal Aspect
“Power”	P = Q·Δp	P = I·V	P = Q (heat flux is power)
“Inductance”	Inertia	Internal inductance	none?
“External Field”	Gravity or body acceleration	Magnetic field	none?
“Ground”	perfect vacuum	electrical ground	absolute zero

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