Timescale to propagate a perturbation through a system?

[tonydoss]

Hi,

I'm a geologist, and I really suck in physics so I would need some help, please! This is not a homework question, I'm an academic! This is for my research...

Let's say I have a "box" (a river catchment) with a mass, M, of material (sediments) in this box, which resides in the box for a time, T. There's a flux in the box, Fin, balanced by a flux out of the box, Fout, and equal to M/T (see attached PDF file)

Let's say I change Fin to a new value, Fin_new. I have the gut feeling it would take the time T for this perturbation to propagate through the box and for Fout to reach a new value (Fout_new = Fin_new)... but I have no idea how to put that in equations (for example, express Fout as a function of T and the old and new values of Fin).
Any help, please?

Thanks in advance

 

[skeleton]

If the catchment had zero volume, then surely Fout=Fin regardless of how Fin might vary in the future (eg: Fin_new). However, the purpose of the catchment (which has a non trivial volume) is to allow the absorption of excess mass (Fin_new - Fin_old) over an initial transition time (T'). Once the catchment has filled up, then Fin_new = Fout_new.

So, Fout_new gradually increases above Fout_old until T' has elapsed, thereafter Tout_new stabilizes and equals Tin_new. During that transition phase, the catchment was acquiring the excess Mass (delta_M).

 

[Andy Resnick]

I have seen this exact problem treated somewhere (I think it in the context of maintaining a fluid level), but I can't recall where- it's a "system with lag", and I think it's nonlinear.

If the change in input is a simple step (from one constant to another), you can readily determine the new equilibrium, but maybe not the detailed approach to equilibrium. If the input is allowed to vary continuously, the system becomes much more difficult to analyze.

 

[tonydoss]

say the change in input is a step function, how do I write Fout_new as a function of Fin_new, t (the time as a variable) and T (the residence time in the box)?

 

[Andy Resnick]

Like I said, I don't recall exactly where I saw it. Here's a PPT set of slides that treats a similar problem:

hotohke.ou.edu/~astriolo/ProcessDynamics&Control/Class08.ppt

The first 10-12 slides are great but then it gets a little rough. Hopefully it's enough to get you started.