Problemas

EQ1035/2019
LECTURE 4
Problem nº1.
A system consists of two tanks as described in the figure below.
Assume that the outflow of each tank is proportional to the hydrostatic pressure inside. The tank
1 section is A1 and the tank 2 section is A2 (in m2).
Develop the input-output model for the system. Also determine the degrees of freedom and how
many control objectives exist.
Problem nº2.
A system consists of two tanks as described in the figure below.
Assume that the outflow of each tank is proportional to the hydrostatic pressure inside. The tank
1 section is A1 and the tank 2 section is A2 (en m2).
Develop the input-output model for the system. Also determine the degrees of freedom and how
many control objectives exist.
Problem nº3.
Develop the input-output model for the system shown in the figure, as well as determine the
number of degrees of freedom and the number of independent control objectives that can be
specified.
EQ1035/2019
All flow rates are volumetric and the tank sections are A1, A2 and A3 respectively.
Qv3 volume flow is constant and does not depend on z3, while all other effluent volumetric flow
rates are proportional to the hydrostatic pressures that cause the flow.
Problem nº4.
Find the solution of the following system of differential equations.
dx1
= 2 x1 + 3 x2 + 1
dt
dx2
= 2 x1 + x2 + et
dt
Contour conditions: x1(0) = x2(0) = 0.
Problem nº5.
Linearize the following systems of equations, for which ,  and  are constants.
dy
=  y +  y 2 +  ln y
dt
dy 1 − y
= m + ym 2 + sen( ) m
dt y
Problem nº6.
Determine the transfer function that relates Qvo and To, with Qv and T for the tank described in
the figure. Also draw the corresponding block diagram.
Consider the equation of linearization of the valve is as follows.
(z − z e )
QV = QVe +
R
EQ1035/2019
Problem nº7.
Using Laplace transforms find the dynamic behaviour of an isothermal batch reactor where the
following reactions take place.
A→ B→C
Assume first-order kinetics for the two reactions. Also, plot the concentration of A, B and C
versus time.
Problem nº8.
The following sequence of chemical reactions takes place in an isothermal continuous stirred
tank reactor. The previously developed kinetic studies have found that the first reaction is
second order with respect to A while the second reaction is first order with respect to A and B.
A→ B r1 = k1c A2
A+ B →C r2 = k2c AcB
One can assume that the reactor has a constant volume.

Obtain the transfer functions that relate the concentrations at the exit stream (c A, cB and cC) with
the concentration of feed (cAo).
Problem nº9.
The electrical resistance shown in the figure transfers heat primarily by radiation mechanism.
If the heat flow from the resistance to environment is q, and the temperatures of the resistance
and the environment are T and Ta respectively, the equation which models the dynamic
behaviour of the resistance can be expressed by the following relationship, where M, Cp and k
are constants.
dT
Mcp = q − k(T 4 − Ta4 )
dt
EQ1035/2019
Problem nº10.
The dynamic behaviour of a stirred tank reactor can be represented by the following transfer
function, where cA is the outlet concentration and cAo the concentration in the feed
cA 0.3
G(s) = =
cA0 4s + 1
Obtain the response cA = f (t) for a perturbation of the form presented in the attached figure,
considering that cAe = 5 kmol·m-3.
14
12
cAo (kmol·m-3)
10
8
-2 -1 0 1 2 3 4 5 6 7
time (min)
Problem nº11.
A balance of forces and moments on a mercury manometer leads to the following equation,
Where x is the displacement of the mercury with respect to its equilibrium position and P (t) is
the pressure acting on the manometer.
EQ1035/2019
d 2x dx
4 2
+ 0,5 + x = P (t )
dt dt
a) Find the transfer function relating x to P, assuming that the system is in equilibrium initially.
b) Determine the response time (time constant) and the damping factor.
c) Calculate the manometer response to a change of pressure given by P(t) = 2·e-4t if it occurs at
t = 0.
Problem nº12.
The dynamic behaviour of a process can be represented by the following transfer function.
y(s) 18
= 2
x (s) s + 3s + 9
a) After a step change in the form of x (t) = 3·step(t), what is the new steady state of y?
b) It is necessary that y'  10 for physical reasons. What is the biggest change in x that can
tolerate the process without exceeding this limit?
Problem nº13.
A change of pressure from 1.5 to 3.1 bar results in a response as indicated in the figure.
12,7
11,2
R(t)
2,3
8
t
Assuming a second order dynamics, estimate the parameters of the transfer function, where R' is
the deviation in the output of the system and P' is the pressure deviation.
R(s) K
= 2 2 P
P(s)  s + 2s + 1
EQ1035/2019
Problem nº14.
To study an element that is known to be critically damped, it has provided a unit ramp as input.
The output obtained in function of time (in deviation variables), is collected in the following
table. Calculate the time constant of the system.
time (min) c` (t) time (min) c` (t)

0,0 0,000 5,5 1,663
0,5 0,003 6,0 1,998
1,0 0,022 6,5 2,354
1,5 0,067 7,0 2,729
2,0 0,145 8,0 3,530
2,5 0,259 9,0 4,383
3,0 0,410 10,0 5,275
3,5 0,569 11,0 6,196
4,0 0,817 12,0 7,140
4,5 1,070 20,0 15,008
5,0 1,353 50,0 45,000

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EQ1035/2019

Contour conditions: x1(0) = x2(0) = 0.

One can assume that the reactor has a constant volume.

time (min) c` (t) time (min) c` (t)

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