# Chemical engineering equivalancy to the spring-mass-damper/RLC circuit

I understand the equivalency between the MSD and RLC circuits. Mass ~ resistance, spring ~ inductor, damper ~ capacitor.

With the equations of the form MSD:

$$m\ddot{x} + c\dot{x}+ kx = F$$

RLC:

$$L\ddot{i} + R\dot{i} + \frac{1}{C}i = V$$

I was thinking a bit about what the chemical engineering equivalent would be for a "plant circuit". For a capacitor, I thought it could be equated to a tank with a inlet at the top and the outlet at the opposite end (also at the top). It would fill without outputting anything until it is full. However, if the inlet stops, the outflow stops and thus doesnt work as a capacitor.

For resistance/mass, i thought the tank size might be the best representation.

I have no idea for an inductor/damper.

After some more thinking, it became clear that a single tank cannot possible approximate such a system, and it has to be a dual tank system. However, this complicates the ODE to such a point where a equivalency is not intuitive at all!

Would be awesome if someone could shed some light on this for me. I might be missing something somewhere.

## 2 Answers

It's not necessarily chemical engineering, but fluid and thermal dynamics have similar but not 100% equivalent systems: https://en.wikipedia.org/wiki/System_equivalence

Such a plant system could certainly be modeled as differential equations, but a chemical engineering equivalent would be concerning reaction rates and heat effects, not fluid flows, which I consider to be mechanical engineering (obviously there is overlap). I am not aware of equivalent fluid system equivalents to all of these.

• Mass/inductance: resists changes in direction. I suppose that pulsing fluids would have this but I've never seen fluid system modeled this way. Fluids do have inertia but no one cars about forces on them, at least in plants.
• Spring/resistor: differing force according to position. This would be height portion of the head loss equation
• Damper/capacitor: resistance to movement according to velocity. This is head loss due to friction.

Most fluid process systems run at steady state. Velocities don't changes or if they do it tends to be limited in its effects. Speed up a pump, open a valve, or fill a tank.