# Transfer function for a neutralization process

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#### Jaycee

dc3/dt = F1c1/V + F2c2/V -(F1+F2)c3

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#### Chris Schene

dc3/dt = F1c1/V + F2c2/V -(F1+F2)c3

This is the standard form of a first order system in any control text book:

(Assuming that F1, F2, F2, C2, C1 con be considered constants and c3 is the PV)

If those are not constants, you have a much more difficult transfer function: I would consider State Space, piecewise linear or non-linear approach if those variables cannot be considered constant.

Tau*(dx/dt) + x(t) = Ku(t)

c3(t) = K(1- exp(-t/Tau))

Rearranging:

dc3/dt + (F1+F2)c3= F1*c1/V + F2*c2/V

(1/(F1+F2))*(dc3/dt) + c3 = (1/[(F1+F2)*V])*( F1*c1+F2*c2)

Tau = 1/(F1+F2), Ku (the forcing function) = (F1*c1+F2*c2)/(F1*V+F2*V))

So, the whole equation:

c3 = (F1*c1+F2*c2)/(F1*V+F2*V))*(1-exp(-t*(F1+F2))

c3/F2 = 1/F2*((F1*c1+F2*c2)/(F1*V+F2*V))*(1-exp(-t*(F1+F2)))

It is hard to write these in plain text.

Ogunnaike and Ray, page 123. ISBN: 0-19-509119-1

J

#### Jaycee

thanks for the help

A

#### Anonymous

I really appreciate your help in deriving the transfer function but one of your assumption is wrong as F2 is not constant. I think that the expression for the transfer function C3/F2 shouldn't contain either c3 or F2. Anyone who can help , please help. I'd appreciate it very much. Thanks in advance

C

#### Chris Schene

Oh! you want c3 = F(F2) or c3(t)/F2(t), in other words c3 as a function F2? As your problem is stated, you are expressing F2, F1, c2, c2, V all as constants. A more precise way to state it would be:

dc3(t)/dt = F1c1/V + F2(t)c2/V -(F1+F2(t))c3(t)

In this way it is clear that F2 is not constant. Do any of the other parameters change?

Your problem is somewhat more complex now. I would approach this first via a numerical solution am I correct in assuming that:

1) c2, c1, c3 are concentrations of some reagent? Do c1 and c2 vary or can they be considered constant?
2) F1 and F2(t) are flow rates? (input and output? Two input streams?)
3) V is volume? You seem to be missing this term in (F1+F2(t))c3(t) as without it, the units don’t seem to match. Are you sure it isn’t (F1+F2(t))c3(t)/V ? Does volume change? Is its function of flow rate F1 and F2?

Are all the other variables considered to be constant for this expression?

1) I would choose an operating point, or in mathematical terms initial values for c1, c2, F1, F2(t)
2) Since I don’t know you process, I just chose some arbitrary values

I solved this numerically and the shape of the function varies quite a bit based on the initial values and F(t). The function is always exponential (first order) for a step in F2(t), but the time constant is a function of F2 and F1. For F2(t) as a ramp, the transfer function seems to have a shape similar to a parabola.

What initial values are you using for c1, c2, c3, F2, F1, V? What kind of function are you using for F2(t)? Step? Ramp? Something else? If it is a step function, then you have individual closed form transfer functions (c3(t)/F2(t) for each value of F2(t).

You could set this function up in Simulink or even excel to see what the effects of different initial condition are.

This is all I can tell you without more specific information about your process, constrains etc. You should find analysis techniques in a good controls book…..a schaums outline might have some good examples as well. If you understand basic engineering numerical techniques, it should be no problem.

J

#### Jaycee

yeah you're right i forgot to include the /V in the equation and I also forgot to mention that F2 is a function as it is the flowrate coming out of the controller. Can I have your email address so I can send more information about our topic. Many thanks.

C

#### Chris Schene

> "[email protected]", mailto:[email protected]

For those of you who may be following this discussion and have interest: I have created a block diagram in power point format which can transcribed to Simulink model, or persumably other modeling languages such as Lab View. If you provide your email and I will send it to you.

It is a useful example, as models like it are seem often in the controls world and it is a good Simulink training example as well.

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208469582

#### Alexpaok44

> "[email protected]", mailto:[email protected]

For those of you who may be following this discussion and have interest: I have created a block diagram in power point format which can transcribed to Simulink model, or persumably other modeling languages such as Lab View. If you provide your email and I will send it to you.

It is a useful example, as models like it are seem often in the controls world and it is a good Simulink training example as well.
Please send this to [email protected] . It would be of great help! thanks in advance!