Multivariable Linearization

1
Multivariable Linearization

• Solution of the Dilution Problem with more
  variables

2
Problem Description

• Consider the following Dilution problem
• The inflow concentration cin kg/m3 of A is
  time varying
• The flowrate F m3/s is NOT constant
• The material A in the fluid is not reduced by
  reaction or other means in the tank
• The Tank is well mixed
• Find the outflow concentration cout kg/m3 of A

3
Equation for multi-variable linearization
A mass balance for component A can be written
Inflow rate of A - Outflow rate of A
Accumulation rate of A
Note that the product Fc is non-linear, so we
must linearize the terms.
For the multivariable case, Taylors Series
truncated to first order terms is
Consider the function f(F,c)Fc
4
Application of multi-variable linearization
The linearized differential equation becomes
Transforming, the equation becomes
5
Use of Superposition
The transfer function for cout /cin where F is
constant is
The transfer function for cout / F where cin is
constant is
6
Block Diagram for multi-variable case
Because the system is linear, the effects of Cin
and F on Cout can be summed to calculate the
total effect on Cout