CHE381 Assignment 2 Solved

1.      Consider a conical tank of height H and radius R (as shown in figure) in which the liquid height is h. The outflow F = kh. Develop a linearized dynamic material balance equation for given system and find the steady-state gain and the time-constant in terms of known variables. 

 

2.      Use method of undetermined coefficients to solve 5 6𝑦 𝑓 when the forcing function f(t) is

a)      Unit step

b)     Ramp ft = t

 

 

3.      Solve for zero initial conditions

a)                        5    4𝑦           1

b)                       4    4𝑦           1

c)                       2    4𝑦           1

d)                       6    11           6𝑦           1

 

4.      Develop a simple pandemic dynamics model using your knowledge of reaction kinetics. Assume elementary kinetics and make reasonable assumptions so that an analytical solution is obtained for the time variation in the number of infected individuals. A possible simple reaction kinetic scheme is  

               

                              Infected + Uninfected ⎯⎯ 2 Infected (rate constant ki)

                             

                            Infected ⎯⎯ Recover                          (rate constant kr)

                             

                            Infected ⎯⎯ Dead                              (rate constant kd)

 

                                                             

5.      A series reaction A ⎯⎯ R ⎯⎯ S occurs in a batch reactor, R being the desirable product. The reactor is initially loaded with pure A. Obtain the time trajectory of the component concentrations (cA, cR, cS). Find the time at which cR is maximum. 

 

 

6.      Consider the unstable first order processes whose eqn. is as follow dy

                            y           Ku dt

Determine whether the proportional controller can stabilize this process. If yes then find the condition on KC (controller gain) for stabilization.

 

7.      For the second order underdamped system ODE 

                                                                                  𝑑 𝑦              𝑑𝑦

 𝜏 2𝜉𝜏 𝑦 𝐾𝑢 𝑑𝑡          𝑑𝑡

where 0 < ξ < 1. The response is characterized by the rise time (tR), peak time (tP), peak overshoot ratio (POR) and the decay ratio (DR), and the f% response settling time, as illustrated in the Figure.  

Second Order Underdamped Unit Step Response

 

 

Show that 

a)      The unit step response is given by 

                       𝑦       𝐾 1        𝑡     𝜙      with 𝜙      tan         

 b)  

c)  

d)      𝑃𝑂𝑅 𝑒               

e)      𝐷𝑅   𝑒               

                                        /

f)         

8.      Develop a dynamic model for two identical tanks in series as shown in Figure, relating cA2 to a change in the inlet concentration cA0. Assume constant F and V. Use the integration factor approach to solve the ODE system to obtain an analytical expression for cA2(t).

 

 

 

9.      Distributed parameter systems such as tubular reactors and heat exchangers often can be modelled as a set of lumped parameter equations. In this case an alternative physical interpretation of the process is used to obtain an ODE model directly rather than by converting a PDE to ODE form by means of a lumping method such as finite differences. As an example, consider a single centric-tube exchanger with energy exchange between two liquid streams flowing in opposite directions as shown in fig. below. We might model this process as if it were two small, perfectly stirred tanks with heat exchange. If the mass flow rate w1 and w2 and the inlet temperature T1 & T2 are known functions of time, derive the governing system of ODEs that need to be solved simultaneously. Assume that all liquid properties (1, 2, Cp1,Cp2) are constant, that the area for the heat exchanger in each stage is A, that the overall heat transfer coefficient U is the same in each stage, and the wall between the two liquids has negligible thermal capacitance.

10.  Consider PI temperature control of the heated tank, that has been considered in class. Draw a rough sketch for the locus of the two roots of the closed loop characteristic equation as KC is increased from 0 to ∞ for the case when τI < τ (ie (R 1)


11.  Consider P control of a process with the governing ODE 

a)      𝜏 𝑦 𝐾𝑢  

b)       𝜏 𝜏    𝜏 𝜏  𝑦 𝐾𝑢 

Both y and u are deviation variables so that for a P controller, 𝑢 𝐾 𝑦 𝑦 

For the characteristic equation of the controlled system, as KC is changed, the roots will change. Sketch the locus of the characteristic equation roots as |KC| is increased from 0 to ∞.