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Dynamic Behaviour of

Process SystemsIsothermal CSTR Modelling and Simulation

Alastair Wong and Iniobong Akpan

15/01/2012



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


OF Molar_fraction Molar hold up

of specie i

kmol holdUp_moles Moles (AS


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


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”.

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