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8/3/2019 DBPS Project
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Dynamic Behaviour of
Process SystemsIsothermal CSTR Modelling and Simulation
Alastair Wong and Iniobong Akpan
15/01/2012
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DBPS project Alastair Wong and Iniobong Akpan 15/01/2012
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1. Dynamic Model Development In this project, a perfectly mixed isothermal CSTR is used to carry out the following methanol
synthesis reaction:
This reaction takes place in the presence of a solid catalyst implemented within a cage integrated
with the stirring mechanism of the CSTR system, as shown in the figure below:
Figure 1: Schematic of the heterogeneous gas/solid catalytic reactor system.
For the reaction over this particular catalyst, the reaction at the surface is rate-limiting, as a result
giving rise to a Langmuir-Hinshelwood-type rate expression [r is in kmol/kg-cat.h]:
⁄ ( )
where the values of the various kinetic parameters are tabulated below:
Table 1: Kinetic parameter values.
Parameter Valuekr 1.434x10
-4
KCO 4.111x10-2
KH2 1.227x10-2
KCH3OH 4.367x10-2
Keq 1.958x10-2
Furthermore, the outlet flowrate is determined by:
√
where
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[ ]
1.1. Dynamic model
The first part of this project requires a model capable of predicting time variation of composition inthe batch reactor. For simplicity of the model, the species CO, H2, and CH3OH are denoted ,
respectively.
Various assumptions had been made when developing the model. For instance, perfect mixing
assumption had been employed, as it is usually a reasonable assumption for a CSTR model. On the
other hand, ideal gas assumption had been employed due to the conditions the gas is under in this
problem. Following employment of the ideal gas assumption, Dalton’s Law becomes valid and was
used as one of the simplifying assumptions. Finally, as the CSTR of interest is in the form of a
confined reactor, the available volume inside the reactor was assumed to be constant. Hence, the
relevant equations constituting the model, along with their respective variables and assumptionsinvolved, are tabulated in the table below:
Table 3: Dynamic model of the system of interest.
Since the number of equations equals the number of variables, there is no degrees of freedom, and
the model is complete.
1.2. Notation Table
The notations used in this model along with their respective units and quantities in which they
represent are tabulated in the table below.
Equations No. of
equations
Variables No. of
variables
Assumptions
3 8 Perfect mixing
√
1 2
∑
1 0
⁄ ( )
1 3
3 1
[] 1 0
3 0 Dalton’s Law 1 1 Constant volume
1 0 Ideal gas lawTotal 15 15
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Table 4: Notation table for the dynamic model for the system of interest.
Variables Units Quantities
h Time
kmol Molar hold up of specie i
kmol/h Inlet molar flowrate of specie i kmol/h Outlet molar flowrate of gas
- Mole fraction of specie i in
reactor
kg-cat Catalyst weight
Stoichiometric constant of
specie i
kmol/kg-cat.h Reaction rate
- Valve position
K0.5
kmol/kPa.h Valve constant
K Temperature
kPa Pressure kPa Atmospheric pressure
kmol/kg-cat.h.(kPa)3
Reaction rate constant
1/(kPa) CO kinetic constant
1/(kPa) H2 kinetic constant
1/(kPa) CH3OH kinetic constant
1/(kPa)2
Equilibrium constant
kPa Partial pressure of specie i
kmol/h Total molar hold up
kmol/m3
Molar density
m
3Total volume
J/mol.K Molar gas constant
2. Determination of the valve constant cv For this part of the project, the value of the valve constant which would result in a reactor
pressure of 1300 kPa under steady-state conditions for inlet flowrates of 1.0x10-4
kmol/h of CO and
0.25x10-4
kmol/h of H2 was determined, given that the valve was half-open (i.e. ). In
addition, the following design parameters were given:
Table 5: CSTR design parameters.
Reactor Property Value
Catalyst Weight (w) 0.01 kg
Total Volume (V) 0.005 m3
T 475.15 K
2.1. Method
First of all, several variable types were defined, namely Molar_fraction, Molar_flowrate, Moles,
Pressure, Reaction_rate, Molar_density, and NoType, along with their respective lower and upper
bounds. In terms of the model, the following parameters and variables were listed and were given
their parameter and variable types, respectively, as follows:
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Table 6: Parameters used in the model along with their respective gPROMS identifier and parameter types.
Symbol Description Units gPROMS identifier Parameter type
Number of components - no_components INTEGER
Atmospheric
pressure
kPa Patm REAL (AS REAL
DEFAULT 101.325) Molar gas
constant
J/mol.K ideal_gas_constant REAL (AS REAL
DEFAULT 8.314) Reaction rate
constant
kmol/kg-
cat.h.(kPa)3
Kr REAL (AS REAL
DEFAULT 1.434E-4) Equilibrium
constant
1/(kPa)2
Keq REAL (AS REAL
DEFAULT 1.958E-2)
CO kineticconstant 1/(kPa) KCO REAL
H2 kinetic
constant
1/(kPa) KH2 REAL
CH3OH kinetic
constant
1/(kPa) KCH3OH REAL
Catalyst
weight
kg catalyst_weight REAL
Total volume m3
total_volume REAL Stoichiometric
constant of specie i
- reaction_stoichiometry REAL (AS ARRAY
(no_components) OFREAL)
Table 7: Variables used in the model along with their respective gPROMS identifier and variable types.
Symbol Description Units gPROMS identifier Variable type Inlet molar
flowrate of
specie i
kmol/h in_molar_flowrate Molar_flowrate(AS
ARRAY(no_components)
OF Molar_flowrate)
Outlet molar
flowrate of gas
kmol/h out_molar_flowrate Molar_flowrate
Mole fraction of
specie i in
reactor
- molar_fraction Molar_fraction (AS
ARRAY(no_components)
OF Molar_fraction Molar hold up
of specie i
kmol holdUp_moles Moles (AS
ARRAY(no_components)
OF Moles) Total molar
hold up
kmol total_holdup_moles Moles
Reaction rate kmol/kg-
cat.h
molar_reaction_rate Reaction_rate
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Partial pressure
of specie i
kPa partial_pressure Pressure (AS
ARRAY(no_components)
OF Pressure) Pressure kPa total_pressure Pressure
Valve position - valve_position NoType
Temperature K temperature REAL Density mol/m3
total_density Molar_density Valve constant K0.5
kmol/k
Pa.h
valve_constant NoType
In addition to the above, the equations mentioned in part (1) of the project was implemented into
the model. Afterwards, a process simulation entity was created, and the parameters and variables
were set and assigned, respectively. Since this part required a steady state model, the initial
conditions were set as
. Subsequently, the process simulation was run, and the value of
was found to be 3.7317966x10-6
.
For details of how the model and process simulation were set up, the reader is referred to the
entities Models – gas_phase_CSTR_question2 and Processes – Simulate_CSTR_question2 in the file
“CSTR Model”.
3. Estimation of steady-state timeFor this part of the project, the system was started from steady state. The inlet flowrates of the two
reactants, however, had changed at several specified times during operation according to the
following table:
Table 8: Operation procedure.
Time [h]: 0.0 1.0 10.0 1.0x10-4
0.1x10-4
1.2x10-4
0.25x10-4
2.75x10-4
0.25x10-4
0.0 0.0 0.0
Therefore, the problem in this part of the project required estimating the time it took for the reactor
to reach its new steady-state.
3.1. Method
In order to set up the model for this problem, the model used for the previous part of this project
was used, with some modifications. First of all, as the valve constant had been computed in the
previous problem, the value had been assigned in the process simulation
entity for this problem in gPROMS. Second of all, the schedule function was used to assign the
sequence for the inlet flowrates of the reactants as described in the table above. The assigned
flowrates after 10h, as specified in the last column in the table above, had remained steady for time
t>10h. As the system had started from steady-state, the initial conditions had remained for i
= 1, 2, 3.
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The system was set so that it had operated until time t=110h. Hence, the following graph of
variations of hold up moles of the various components over time was obtained. For such system
without the interference of any fluctuations, steady-state is defined as the state in operation where
various variables remain steady values over time. It can be seen from the graph below that this state
was reached at t=100h. Hence, it can be concluded that it had taken 90h for the reactor to reach its
new steady-state.
Figure 2: Graph of molar hold up vs. time for steady state-time determination.
For details of how the model and process simulation were set up, the reader is referred to the
entities Models – gas_phase_CSTR_question3 and Processes – Simulate_CSTR_question3 in the file
“CSTR Model”.
4. Proportional-integral controller tuningStarting from the steady-state computed in part (2), a proportional-integral (PI)controller was
introduced to manipulate the outlet valve position to effect a step change in the reactor pressure
from 1300 kPa to 1200 kPa:
∫
where , with being the pressure set-point and the error of the actual pressure
from the pressure set-point. Furthermore, is the steady-state valve position.
In order to assess the controller performance, the integral square error (ISE) was determined:
∫
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The constants and are the controller parameters of the PI controller, and the determination
and tuning of these parameters to minimise the ISE constitutes the problem in this part of the
project.
4.1. Method
First of all, in order to set up the model for this problem, the above equations and variables are
added to the model which was used in part (2) of this project. In addition, the steady-state valve
position and the pressure set-point were added to the list of parameters. Furthermore, since the
constants and were to be determined, they were placed in the variables section in this
problem and were then determined later through optimization.
In terms of the process simulation entity, the value of the valve constant was once again set as
. In addition, the pressure set-point was set as 1200 and the steady-state
valve position as 0.5. Since the constants and were to be optimized later, initial guesses were
assigned as -0.01 and 0.1, respectively. Note that the value of
is negative, since a positive change
in the manipulated variable (the opening of the valve) causes a negative change in the controlled
variable (the reactor pressure). Furthermore, the initial conditions were as follows:
Table 9: Initial conditions used for the process simulation.
Variables Values
0.0014240555 kmol/h
4.179494x10-5
kmol/h
1300 kPa
ISE 0
∫
0
Notice that the initial values of the molar hold up of species 1 and 2 were obtained from results in
part (2).
After setting up the optimizer to minimize the ISE value, setting the time horizon to 30 and control
type as time-invariant and allowable values as continuous, optimum values of the constants and
were found to be -3.0586x10-2
and 4.3334x101, respectively.
For details of how the model, process simulation, and the PI controller were set up, the reader isreferred to the entities Models – gas_phase_CSTR_question4, Processes –
gas_phase_CSTR_question4, and PI_controller_question4.
5. Estimation of K rIn order to validate the kinetic model, an experiment was performed corresponding to the
conditions described in part (3) of the project. The partial pressures of carbon monoxide and
methanol were measured at intervals of approximately 1h over a period of about 50h. The
measurements obtained (not shown) were used in re-estimating the value of , in which the
current available value was found to be particularly unreliable and could, in fact, be wrong by up toan order of magnitude.
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5.1. Method
First of all, the model developed for part (3) of this project was modified for this parameter
estimation; the reaction kinetic was moved to the variable section. Subsequently, it was assigned
an initial guess of 1.434x10-4
in the process simulation entity. On the other hand, in the parameter
estimation entity, the control variables for the experiments were the inlet flow rates, which were
given piece-wise constant control. On top of that, the upper and lower bounds of were set at
1.434x10-3
and 1.434x10-5
, as the error may be of an order of magnitude. It was assumed that the
standard deviation of the experiment was 5% and so the variance model was set as constant
relative variance with a value of 0.05.
5.2. Results analysis
The table below shows the results of the model parameter estimation.
It must be noted that:
Probability of parameter lying between (Final Value -α% Confidence Interval) and (Final
Value +α% Confidence Interval) = α%
The t-value shows the percentage accuracy of the estimated parameters, with respect to the
95% confidence intervals.
Table 10: Results of the model parameter estimation of part (5) of project.
Model
Parameter
Final
Value
Initial
Guess
Lower
Bound
Upper
Bound
Confidence Interval 95%
t-
value
Standard
Deviation90% 95% 99%
Reactor.
Kr0.00037584 0.0001434 0 1
4.079×10-
005
4.88×10-
005
6.472×10-
0057.701
2.45×10-
005
Reference t-value (95%): 1.6646
Hence, the final value of
is 0.00037584. In comparison to the estimated parameter value, the
confidence intervals and standard deviation are smaller and the 95% t-value is larger. Hence, the
final value generated from this estimation seems to be more accurate than the formerly given value.
However, the difference between the 95% t-value and the 95% reference t-value is not large, so the
generated value is not sufficiently accurate enough.
5.3. Lack of fit test
The table below shows the results of a lack of fit test carried out on the experimental data and
results:
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Table 11: Results of the lack of fit test.
Weighted Residual X2-Value (95%) Comment
98.658 * 98.484 Model may be inadequate representation of physical system
From the table, it can be seen that the weighted residual is larger than the X2-value, indicating that
the model may be inadequate representation of physical system.
For details of how the model, process simulation, and parameter estimation were set up, the reader
is referred to the entities Models – gas_phase_CSTR_question5, Processes –
Simulate_CSTR_question5, and Parameter Estimations – Parameter_Estimations_question5 in the
file “CSTR Model”.
6. Estimation of K r, K CO, K H2, and K CH3OH
For this part of the project, it was believed that the value of was already known with sufficient
accuracy from independent thermodynamic measurements. However, the values of , , and
were less reliable, probably varying by up to 50% from those mentioned in part (1) of the
project. This in turn affects the value of . Hence, this part of the project involved re-estimating the
values of , , , and .
6.1. Method
First of all, using the previous experimental data, these values were hence estimated. The model
developed for part (3) of this project was modified for this parameter estimation; the reactionkinetic parameters , , , and were moved to the variable section and was assigned
initial guesses of 1.434x10-4
, 4.111x10-2
, 1.227x10-2
, and 4.367x10-2
, respectively, in the assign
section of the process simulation entity for this part.
Since the estimated values of the above kinetic parameters may differ from those previously used by
50%, the upper and lower bounds of the parameters were assumed to be 50% and 150% of the
initial guesses. These calculated values are shown in the table below:
Table 12: Initial guesses and lower and upper bounds of the parameters to be estimated.
Parameter to be
Estimated
Initial Guess Lower Bound Upper bound
Reactor.KCH3OH 0.04367 0.021835 0.065505
Reactor.KCO 0.04111 0.020555 0.061665
Reactor.KH2 0.01227 0.006135 0.018405
Reactor.Kr 1.434E-4 1.434E-5 0.001434
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6.2. Results analysis
Similar to that in part (5) of this project, the table below shows the results of the model parameter
estimation.
Table 13: Results of the model parameter estimation for part (6) of project.
Model
Paramete
r
Final
Value
Initial
Guess
Lower
Bound
Upper
Bound
Confidence Interval95%
t-value
Standard
Deviatio
n90% 95% 99%
Reactor.
KCH3OH 0.040793 0.04367 0.021835
0.06550
5 6.711 8.029 10.65
0.005081 *
* 4.029
Reactor.
KCO
0.022264 0.04111 0.020555 0.06166
5
3.67 4.39 5.825 0.005071 *
*
2.203
Reactor.
KH2 0.0069093 0.01227 0.006135
0.01840
5 1.173 1.403 1.862
0.004924 *
* 0.7042
Reactor.
Kr
0.0002810
6
0.000143
4
1.434×10-
5
0.00143
4
0.134
7
0.161
2
0.213
9
0.001743 *
* 0.0809
Reference t-value (95%): 1.6655
It can be seen from the table the results of estimation of , , , and , shown to be
0.00028106, 0.022264, 0.0069093, and 0.040793, respectively. However, the 95% t-values are all
smaller than the reference t-value as stated in the table. This was probably due to the fact that too
many parameters were under estimation for the small amount of data, as a result leading to
inaccurate parameter estimation.
6.3. Correlation matrix
The table below shows the correlation matrix which was calculated from the variance-covariance
matrix:
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Table 14: Correlation matrix.
Correlation Matrix
Parameter No. 1 2 3 4
Reactor.KCH3OH 1 1
Reactor.KCO 2 1* 1
Reactor.KH2 3 0.998* 0.999* 1
Reactor.Kr 4 1* 1* 0.998* 1
From the table, the values in the off-diagonal are close to 1, indicating strong correlations between
any of two of the parameters , , , and . However, in an ideal case these valuesshould be closed to 0.
6.4. Lack of fit test
The lack of fit test results and conclusion is shown in the table below:
Table 15: Results of the lack of fit test.
Weighted Residual X2-Value (95%) Comment
93.567 95.081 Good fit: weighted residual less than X2-
Value
Since the weighted residual is lower than the X2-value, there is good fit between the model and the
representation of the physical system.
For details of how the model, process simulation, and parameter estimation were set up, the reader
is referred to the entities Models – gas_phase_CSTR_question6, Processes –
Simulate_CSTR_question6, and Parameter Estimations – Parameter_Estimations_question6 in the
file “CSTR Model”.