Project 1

Due Jan. 19, 2005  5pm

(For the simulations, you may use the spreadsheet to implement Euler methods. A tutorial (click here) is available in the web. )
1. A liquid of constant density is fed at a constant volumetric rate Fointo a conical tank of height Hmax
    and maximal radius Rmax. The flow out of the tank is Kvh1/2 where h is the height of the liquid in the
    tank and Kv is the valve coefficient (see Figure 1)

Figure 1. Conical Tank System.

a. Obtain a dynamic model which describes the height of liquid in the tank.
b. By analyzing the mathematical model (you do not need to solve the differential equation),
    how is the steady state value of h affected by changing Fo ?  Kv ? Rmax?
    and Hmax? Explain.
c. What is the effect of changing Rmax on the rate of change of h ? Why do you expect this ?
d. Using Euler method, simulate the system in order to verify your predictions using the
    following nominal parameters:
 
Rmax 5 ft
Hmax 10 ft
Kv 1 ft2 ft1/2/ s
h(0) 2 ft
Fo 2 ft3/s
( I would suggest using a sample time interval, Dt = 0.5.  Also, simulate the process
at least from 0 to 200 )

To check the effects of Rmax on steady state and rate of response, vary the values
of Rmax to 3, 4, 5, 6 and 7 ft. Next, put Rmax back to 5 ft and check the effects of
changing valve coefficient Kv to 0.9, 1.0 and 1.1. After putting Kv back to 1.0,
investigate the effects of the feed rate Fo to 1.7, 2, and 2.3 ft3/s. Finally, set Fo back
to 2.0 ft3/sand investigate the effects of starting the simulation at different initial
conditions, i.e. h(0) at 3, 4 and 5 ft. Make sure that the time length of your simulation
should at least go far enough to observe steady state conditions.

 Summarize the trends observed in a table, for example: (where response time is the time
it takes to reach 95% of its path from initial condition to steady state.)
 
Changes in Variable
Effect on Steady State
Effect on Response Time
As x increases

:
Increases

:
Shorter

:
2. First order linear systems. Liquid is fed to a storage tank in which the tank level (hence
    also tank volume) is held constant by a control system (not shown) in Figure 2

Figure 2. Storage Tank at Constant Volume

      a. Show that the component balance for A is given by
       
         where  t=V/F  is called the "residence time" of the mixing tank, or the "time constant".
         V is the volume of liquid in the tank. F is the volumetric flow rate in and out of the tank.
         CA is the concentration of A in the tank while CAin is the conentration of A in the feed.

      b. Simulate the system using Euler method for the following parameters:
       

      t 1 sec
      CAin 0.2 lb/ft3
      CA(0) 0.1 lb/ft3
           Obtain another plot using t=2 sec and CA (0)=0.4. What do you observe?
          ( I suggest using a sample time interval of 0.01, and simulate from t=0 to 10. )
      c. The time constant t is a very useful indicator of the speed of response. It represents
          the amount of time the process variable reaches 63.2 % from initial point to steady state.
          Using the two plots obtained for t=1 and t=2 secs, verify that at these times, the
          variable has indeed proceeded 63.2 % from CA (0) towards steady state.
    3. Second Order Systems. Suppose you are given a system described by
where z is a parameter of the system with initial conditions x(0)=0, dx/dt(0)=0.

a. Obtain the analytical solution, x(t), by solving the differential equation for the
    following cases: z=1.5, 0.5 and -0.1.  ( click here for a short review on solving high
    order linear ODEs and click here for an example of solving a second order ) .

    Plot your solutions for x(t) of each case from t=0 to 25 and
    summarize your observations for the three plots. ( I suggest obtaining a plot
    that overlays the three curves in one figure for easier comparison. )

b.  Now use Euler's method to obtain a numerical solution from t=0 to 25.
    ( I suggest using a sample time interval of 0.01.)
   To extend the method for higher order differential equation, one needs to
    first rewrite the equation to a set of first order differential equations.  For
    instance, first introduce an auxiliary variable, y, as follows:
    This allows one to then convert the original problem to be a set of first
    order equations:
    Simulate the following case: z=1.5, 1.0, 0.5, 0.2, 0 and -0.1.
    ( I suggest obtaining a plot that overlays the six curves in one figure
    to allow for easier comparison. ) You just need to plot x(t).

    Compare your numerical solutions (using Euler method) with your analytical
    solutions for the cases where z=1.5, 0.5 and -0.1, to assess the accuracy
    of the Euler method.

    Note: the parameter z is known as the damping coefficient and it is used to
    indicate whether oscillations are present or not. Summarize your observations.
 

 4. (OPTIONAL! - this means this number will not be graded and need not be submitted.
    Nonetheless, we encourage you to try out some of these suggested tricks from time to
    time - just for fun.) Sometimes, when investigating the effects of a parameter, the use of scrollbars can
allow more efficient observations of how parameters affect the responses.

Perform one spreadsheet simulation of the process given in 2b using one of the cases,
say t = 1.0, all the way to where a plot of the response has been drawn.

Now try including a scrollbar (click here for a short tutorial) that would manipulate
the value of t from 0.5 to 5.  You should be able to slide the scrollbar using the mouse
and concentrate on how the response changes, without being distracted by
keyboard-inputs to change t.