Derivation Constants/Rate of Change

[rosaliexi]

Hi, I'm working on a process control question about a tank system at steady state. The part I'm having problems with is where I have derived a second order differential equation to model the system and have replaced the concentrations with derivation constants in that :
Actual Concentration at output (Ca) - Ideal Concentration at output (Cas) = Difference in Concentration at Output (Ca*), etc.
What I am struggling with is the Laplace transform of the equation; I need to know the value of Ca* and dCa*/dt at time = zero. To me, both will be zero, as at steady state Ca = Cas since the system has been running for a period of time. No information is given about this in the question that I can figure out, but I have done the transform several times and I can't get it to work.
Any help would be brilliant.
Thanks,
Rosie

 

[Chestermiller]

Hi, I'm working on a process control question about a tank system at steady state. The part I'm having problems with is where I have derived a second order differential equation to model the system and have replaced the concentrations with derivation constants in that :
Actual Concentration at output (Ca) - Ideal Concentration at output (Cas) = Difference in Concentration at Output (Ca*), etc.
What I am struggling with is the Laplace transform of the equation; I need to know the value of Ca* and dCa*/dt at time = zero. To me, both will be zero, as at steady state Ca = Cas since the system has been running for a period of time. No information is given about this in the question that I can figure out, but I have done the transform several times and I can't get it to work.
Any help would be brilliant.
Thanks,
Rosie

Show us more details of what you have done. Even if the system is at steady state to begin with, disturbances will upset the steady state, and these forcings must be included in your analysis and in your Laplace Transform.

 

[rosaliexi]

Show us more details of what you have done. Even if the system is at steady state to begin with, disturbances will upset the steady state, and these forcings must be included in your analysis and in your Laplace Transform.

It's all written out on Word so I have attached the relevant work. I know the disturbance and have done a general transform but I can't finish it and am assuming I am going wrong with cancelling the g(0) and g'(0) values because I keep getting the same answers otherwise.

 

[Chestermiller]

Maybe this will help:

$$\frac{s+2a}{(s+a)^2+w^2}=\frac{(s+a)}{(s+a)^2+w^2}+\frac{a}{w}\frac{w}{(s+a)^2+w^2}$$

 

[rosaliexi]

Ohhhh of course! Thank you, that's solved all my problems :D