Gazi University Faculty of Engineering, Department of Chemical EngineeringChE 473 Process Control (Gr.-I-01)2011-2012 Fall SemesterDoç. Dr. N. Alper TAPANAsst. Duygu UYSAL

October 2011

PROBLEM SET I

1) A completely enclosed stirred-tank heating process is used to heat an incoming stream whose flow rate varies. The heating rate from this coil and the volume are both constant.

a) Develop a mathematical model (differential and algebraic equations) that describes the exit temperature if heat losses to the ambient occur and if heat losses at the ambient occur and if the ambient temperature (Ta) and the incoming stream’s temperature (Ti) both can vary.

b) Discuss qualitatively what you expect to happen as Ti and w increase (or decrease). Justify by reference to your model.

Notes: and Cp are constants. U, the overall heat transfer coefficient, is constant. As the surface area for heat loss to ambient. Ti>Ta.

2) A process tank has two input streams, Stream 1 at mass flow rate w1 and Stream 2 at mass flow rate w2. The tank’s effluent stream, at flow rate w, discharges through a fixed valve to atmospheric pressure. Pressure drop across the valve is proportional to the flow rate squared. The cross-sectional area of the tank, A, is 5m 2

and the mass density of all streams is 940 kg/m3.

a) Draw s schematic diagram of the process and write an appropriate dynamic model for the tank level. What is the corresponding steady-state model?

b) At initial steady-state conditions, with w1=2.0 kg/s and w2 1.2 kg/s the tank level is 2.25 m. What is the value of the valve constant (give units)?

c) A process control engineer decides to use a feed-forward controller to hold the level approximately constant at the set-point value (hsp=2.25 m) by measuring w1 and manipulating w2. What is the mathematical relation that will be used in the controller? If the w1 measurement is not very accurate and always supplies a value that is 1.1 times the actual flow rate, what can you conclude about the resulting level control?

3) The liquid storage tank has two inlet streams with mass flow rates w1 and w2 and an exit stream w3. The cylindrical tank is 2.5 m tall and 2 m in diameter. The liquid has a density of 800 kg/m 3. Normal operating procedure is to fill the tank until the liquid level reaches a nominal value of 1.75 m using constant rates : w1:120 k/min, w2:100 kg/min and w3:200 kg/min. At that point, inlet flow rate is w1 is adjusted to that the level remains constant. However, on this particular day, corrosion of the tank has opened up a hole in the wall at a height of 1 m, producing leak whose volumetric flow rate q4 (m3/min) can be approximated by

Where h is height in meters.

a) If the tank was initially empty, how long did it take the liquid level to reach the corrosion point?

b) If mass flow rates w1, w2 and w3 are kept constant indefinitely, will the tank eventually overflow?

Gazi University Faculty of Engineering, Department of Chemical EngineeringChE 473 Process Control (Gr.-I-01)2011-2012 Fall SemesterDoç. Dr. N. Alper TAPANAsst. Duygu UYSAL

October 2011

4) Using partial fraction expansion where required, find x(t) for,

a)

b)

c)

d)

e)

5) Find the solution of

Where

And x(0)=0. Plot the solution for values of h=1,10,100 and the limiting solution ( ) from t=0 to t=2. Put all plots on the same graph.

6)

a) The differential equation

Has initial conditions x(0)=1, x’(0)=2. Find Y(s) and without finding y(t), determine what functions of time will appear in the solution.

b) If , find y(t).

7) Consider the following transfer function;

a) What is the steady-state gain?

b) What is the time constant?

c) If U(s)=2/s, what is the value of the output y(t) when t → ∞?

d) For the same U(s), what is the value of the output when t=10? What is the output when expressed as a fraction of the new steady-state value?

e) If U(s)=(1-e-s)/s, that is, the unit rectangular pulse, what is the output when t → ∞?

Gazi University Faculty of Engineering, Department of Chemical EngineeringChE 473 Process Control (Gr.-I-01)2011-2012 Fall SemesterDoç. Dr. N. Alper TAPANAsst. Duygu UYSAL

October 2011

f) If u(t)=δ(t), that is, the unit impulse at t=0, what is the output when t → ∞?

g) If u(t)=2sin3t, what is the value of the output when t → ∞?

8) A tank having a time constant of 1 min and a resistance of 1/9 ft/cfm is operating at steady state with an inlet flow of 10 ft3/min. At time t=0, the flow is suddenly increased to 100 ft3/min for 0.1 min by adding an additional 9 ft3 of water to the tanks uniformly over a period of 0.1 min. plot the response in tank level and compare with the impulse response.

9) Consider a stirred tank reactor with one inlet stream (with C i, F) and one outlet stream (with Co, F). the reaction occurring is A → B and it proceeds at a rate r=k.Co

Where r = moles A reacting/volume.timek = reaction velocity constantC0(t) = concentration of A in reactor, moles/volumeV = volume of mixture in reactor

Further let F = constant feed rate, volume/timeCi(t) = concentration of A in feed stream

Assuming constant density and constant V, derive the transfer function relating the concentration in the reactor to the feed stream concentration. Prepare a block diagram for the reactor. Sketch the response of the reactor to a unit-step change in Ci.

10) A tank having a cross sectional area of 2 ft2 and a linear resistance of R=1 ft/cfm is operating at steady state with a flow rate of 1 cfm. At time zero, the flow varies as shown in Figure below.

a) Determine Q(t) and Q(s) by combining simple functions. Note that Q is the deviation in flow rate.b) Obtain an expression for H(t) where H is the deviation in level.c) Determine H(t) at t=2 and t=∞.

1

2q, cfm

t, min1 2 3