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# Starting from the mathematical model of the system, determine all transfer functions that describe this process in deviation state in the open loop.

transfer functions that describe this process in deviation state

For the purposes of the assignment, the simplification assumption can be made that, in the expected
operating temperature range, the reaction rate is a lot more affected by changes in the temperature,
𝑇𝑇(𝑡𝑡), in the tank in comparison to the concentration of A, 𝐶𝐶𝐴𝐴(𝑡𝑡), so that the reaction rate can be
expressed with a pseudo-zero order expression where:
𝑟𝑟𝐴𝐴(𝑡𝑡) = 𝑘𝑘�𝑇𝑇(𝑡𝑡)�𝐶𝐶𝐴𝐴(𝑡𝑡) ≅ 𝑘𝑘�𝑇𝑇(𝑡𝑡)�𝐶𝐶𝐴𝐴𝐴𝐴
Numerical values (assumed constant):
 Overall heat transfer coefficient between the coolant and the fluid in the tank: 𝑈𝑈 = 425 W/m2
-K
 Heat transfer area between the coolant and the fluid in the tank: 𝐴𝐴 = 3.5 m2
 Inlet/Outlet flow rate of the fluid in the tank: 𝐹𝐹 = 6.30 × 10−4 m3
/s
 Volume of fluid in the tank: 𝑉𝑉 = 0.375 m3
 Density of the fluid in the inlet feed stream and in the vessel: 𝜌𝜌 = 880 kg/m3
 Specific heat of the fluid in the inlet feed stream and in the vessel: 𝑐𝑐𝑝𝑝 = 3680 J/kg-K
 Reaction frequency factor: 𝑘𝑘𝑜𝑜 = 2.87 × 105 s-1
 Reaction activation energy: 𝐸𝐸𝐴𝐴 = 6.47 × 104 J/mol
 Heat of reaction: 𝛥𝛥𝛨𝛨𝑟𝑟 = −2.79 × 104 J/mol
 Density of the coolant: 𝜌𝜌𝑐𝑐 = 1650 kg/m3
 Specific heat of the coolant: 𝑐𝑐𝑝𝑝𝑝𝑝 = 4190 J/kg-K
 Volume of coolant in the jacket: 𝑉𝑉𝑐𝑐 = 0.045 m3
F, CAi(t), Ti(t)
F, CA(t), T(t)
M
AC
k A B →
Tci(t)
Tc(t)
CAm(t)
CA(t)
T(t)

 Inlet/Outlet flow rate of coolant in the jacket: 𝐹𝐹𝑐𝑐 = 4.15 × 10−4 m3
/s
 Steady state outlet concentration of A: 𝐶𝐶𝐴𝐴𝐴𝐴 = 3300 mol/m3
 Steady state outlet temperature: 𝑇𝑇𝑠𝑠 = 105 o
C
 Transfer function for the final control element: 𝐺𝐺𝑓𝑓(𝑠𝑠) = 𝐾𝐾𝑓𝑓, where 𝐾𝐾𝑓𝑓 = −10 o
C/atm

 Transfer function for the online analyser: 𝐺𝐺𝑚𝑚(𝑠𝑠) = 𝐾𝐾𝑚𝑚, where 𝐾𝐾𝑚𝑚 = 1 (mol/m3
)/(mol/m3
)

a) Starting from the mathematical model of the system, determine all transfer functions that
describe this process in deviation state in the open loop. Determine initially any first order
dependencies between variables and calculate their respective gains and time constants.

Furthermore, combine these first order transfer functions so that the transfer functions
describing the effect of 𝐶𝐶𝐴𝐴𝐴𝐴
′ (𝑡𝑡), 𝑇𝑇𝑖𝑖

(𝑡𝑡) and 𝑇𝑇𝑐𝑐𝑐𝑐
′ (𝑡𝑡) on 𝐶𝐶𝐴𝐴
′ (𝑡𝑡), 𝑇𝑇′
(𝑡𝑡) and 𝑇𝑇𝑐𝑐

(𝑡𝑡) are derived. Clearly
show all steps taken and clearly indicate the units of all involved variables.
[Marks: 10/22]

b) Based on the obtained transfer functions in a) and using a generic PI controller transfer function
block construct in Simulink the block diagram of the closed-loop process. There are two
alternative but equivalent methods that the open-loop process can be drawn in this block
diagram. Include in the assignment report a screenshot of the Simulink diagram used.
[Marks: 2/22]

c) Consider that initially the system is at steady state and the following cases.
 If your Student ID number is even: At time 𝑡𝑡 = 0 there is a step change in the inlet
concentration of component A, 𝐶𝐶𝐴𝐴𝐴𝐴(𝑡𝑡), equal to 200 mol/m3
.
 If your Student ID number is odd: At time 𝑡𝑡 = 0 there is a step change in the temperature, 𝑇𝑇𝑖𝑖(𝑡𝑡),
equal to -10 o
C.

i. Using a P controller, simulate in Simulink the process response, under the above
described step change, for the following 𝐾𝐾𝑐𝑐 values: 0.001, 0.01, 0.1 and 1 atm/(mol/m3
).
Consider an adequate simulation time to reach a new steady state. Plot the responses of
𝐶𝐶𝐴𝐴
′ (𝑡𝑡), 𝑇𝑇′
(𝑡𝑡) and 𝑇𝑇𝑐𝑐

(𝑡𝑡) with time (one plot per variable including the responses for all 𝐾𝐾𝑐𝑐
values). Discuss the following: Presence or not and trend, if any, of an offset, presence or
not and trend, if any, of an oscillatory behaviour, stability or not of the response.
[Marks: 3/22]

ii. Starting from the general equation describing the closed-loop response of the feedback
controlled system and using the Final Value Theorem, calculate the offset for the various
Kc values considered in part i. Compare the calculated values with those predicted by
Simulink and discuss trends in relation to theory.
[Marks: 2/22]

d) Consider again that the system is at steady state. There is no change in the disturbances this
time. At time 𝑡𝑡 = 0 there is a step change of 200 mol/m3 in the set-point of the process.
i. Using a PI controller, simulate in Simulink the process response, under the above
described step change, for a constant 𝐾𝐾𝑐𝑐 value of 0.015 atm/(mol/m3
) and for the
following 𝜏𝜏𝐼𝐼 values: 1000 and 170 s. Consider an adequate simulation time to reach a
new steady state. Plot the responses of 𝐶𝐶𝐴𝐴
′ (𝑡𝑡), 𝑇𝑇′
(𝑡𝑡) and 𝑇𝑇𝑐𝑐

(𝑡𝑡) with time (one plot per
variable including the responses for both 𝜏𝜏𝐼𝐼 values). Discuss the following: Presence or
not and trend, if any, of an offset, presence or not and trend, if any, of an oscillatory
behaviour, stability or not of the response.
[Marks: 3/22]

ii. Starting from the characteristic equation of this closed-loop system, investigate its
stability for the 𝜏𝜏𝐼𝐼 values considered in part i. Compare the findings with those predicted
by Simulink and discuss trends in relation to theory.
[Marks: 2/22]