Analysis of a dynamic system

1
Analysis of a dynamic system

• Solution of the Tank Level Problem

2
Problem Description

• Consider the following Tank level problem
• The inflow is time varying
• Find the outflow given the inflow

3
Basic Equations
A continuity equation can be written assuming
conservation of mass and constant density
Inflow rate - Outflow rate Accumulation rate
or,
fout must be found in terms of fin. V must
expressed in terms of fout and fin.
For a tank with vertical walls, and
These three equations in four variables, fout ,
fin , V, and h allow us to write fout in terms
of fin . The problem arises that the last
equation is not linear. To solve this
analytically, it must be linearized.
4
Linearization
A plot of fout vs. h is shown to the right, where

If the tank level is constrained within a range
(for example 5 - 15), a linear approximation of
fout vs. h could be used as shown in orange
5
Taylors Series
Using a Taylors series representation,
Notice that this is in the form of y mx b
when we ignore the higher order terms (HOT).
Where b 0 and y is (f(x) - f(x0)) and x is (x
- x0 ).
Notice also that this is in the form of
The slope of the function at x0
6
Application of Taylors Series
Applying a truncated Taylors Series to
approximate Fout we get
This equation expresses f and h as deviations
from the operating point h0 and can be written as
Where
(a deviation variable)
(a deviation variable)
(a constant)
7
Expressing in terms of deviation variables
We can also combine the first two equations we
developed and write them in terms of deviation
variables
We can select our operating point at a steady
state where the system would be in equilibrium.
At that point
Subtracting this from the unsteady state equation
above
8
Expressing in terms of deviation variables
Note that this is also equal to
because h0 is a constant. We can write this
equation in terms of deviation variables also
Combining this with the linearized orifice
equation
We get
A first order constant coefficient differential
equation relating inflow to outflow.
9
Laplace Transformation
This equation can easily be solved using Laplace
Transform Techniques. Writing the equation in
functional notation we have
The initial conditions can be assumed to be
steady state and for the deviation variables,
their value is zero. Transforming the equation
(see pages 15-17 in SC)
Given some inflow rate function, the outflow rate
can be determined. Let the inflow rate be
changed by 1.3 ft3/s.
10
Transforming the Forcing Function
Transforming (see table 2-1.1)
Substituting inflow into the transformed
differential equation
Separating this equation into parts to be
transformed back to the time domain
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Partial Fraction Expansion
Multiplying by s and letting s0
Multiplying by
And let
Substituting a and b into the transformed
equation
12
Plotting the Solution
Transforming this equation back to the time
domain (see table 2-1.1
Plotting the response, notice that 63.2 of the
response occurs after A/K seconds