- #1
danago
Gold Member
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I have a problem to solve where i am required to linearise a nonlinear system. Basically, the system involves a mixing tank of height H(t) and cross section area A with two input streams with flow rates Q1(t) and Q2(t) with concentrations C1 and C2 of some species (concentrations are constant). There is one outlet stream with flow rate Q(t) and concentration C(t). We are assuming that the tank is always perfectly mixed and that the outlet flow rate is proportional to [tex]\sqrt{H(t)}[/tex].
So what i need to do is express the concentration of the outlet stream as some nominal steady-state value plus an additional term for deviations from this nominal value
i.e. C(t) = C0 + c(t)
Where c(t) is a linear approximation. My first thought is to write a material balance on the species in the fluid:
i.e. [tex]\frac{d}{{dt}}\left( {CAH(t)} \right) = {C_1}{Q_1} + {C_2}{Q_2} - CQ[/tex]
It wouldn't be too hard to solve this DE for C(t), but I am not sure where to go from there. I guess i could then differentiate C(t) with respect to each variable to linearise it, but this seems a little tedious so I am not sure if there is something else i should be doing?
Any input is greatly appreciated.
Thanks,
Dan.
So what i need to do is express the concentration of the outlet stream as some nominal steady-state value plus an additional term for deviations from this nominal value
i.e. C(t) = C0 + c(t)
Where c(t) is a linear approximation. My first thought is to write a material balance on the species in the fluid:
i.e. [tex]\frac{d}{{dt}}\left( {CAH(t)} \right) = {C_1}{Q_1} + {C_2}{Q_2} - CQ[/tex]
It wouldn't be too hard to solve this DE for C(t), but I am not sure where to go from there. I guess i could then differentiate C(t) with respect to each variable to linearise it, but this seems a little tedious so I am not sure if there is something else i should be doing?
Any input is greatly appreciated.
Thanks,
Dan.