Reactor Design and Control for Methanol Synthesis (Group 22

Reactor Design and Control Project:
Methanol Synthesis
Group 22
Joseph Chimento
Chin Yew Sian
Sebastian Gonzato
Kushagra Kohli
I live for the projects I won’t remember, with the people I won’t forget

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Contents
Abstract...................................................................................................................................................1
Introduction ........................................................................................................................................2
Reactor Design: Theory and Methodology.............................................................................................3
Sample Calculations Initial Gradients .................................................................................................4
Reactor Design: Results and Analysis......................................................................................................5
Trends .................................................................................................................................................5
Mass of Catalyst against ξ1..............................................................................................................5
Temperature against ξ1 ...................................................................................................................6
Pressure against ξ1 ..........................................................................................................................6
ξ2 against ξ1......................................................................................................................................6
Variation in Reaction Rate with ξ1..................................................................................................7
Conversion of Carbon Monoxide and Hydrogen ............................................................................7
Reactor Design: Sensitivity Analysis........................................................................................................8
Reactor Design: Alternative Designs.....................................................................................................10
Recycling ...........................................................................................................................................10
Increase in inlet temperature and isothermal behaviour.............................................................11
Decrease in Catalyst Mass ............................................................................................................11
Recycling and Pre- Cooling................................................................................................................11
Cooling followed by recycle ..........................................................................................................12
Recycle followed by cooling..........................................................................................................13
Comparison of designs..................................................................................................................13
Additional designs.............................................................................................................................13
Controller Design: Theory and Methodology .......................................................................................14
A. Transfer model identification .......................................................................................................14
B. Loop Pairing ..................................................................................................................................14
C. Outlet Temperature Control Loop................................................................................................15
D. Outlet Composition Control Loop.................................................................................................15
E. Multi-Loop Control........................................................................................................................15
Model Predictive Control (MPC).......................................................................................................16
Feed forward (FF) Control.................................................................................................................16
Result and Analysis ...............................................................................................................................16
A. Transfer function identification....................................................................................................16
B. RGA ...............................................................................................................................................16

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C. Outlet Temperature Control Loop................................................................................................17
D. Outlet Composition Control Loop.................................................................................................17
Determining Ultimate Gain and Critical Frequency......................................................................17
Feedback Controller:.....................................................................................................................18
Bode Plot.......................................................................................................................................18
E. Multi-Loop Control........................................................................................................................18
Advanced control techniques ...........................................................................................................19
..........................................................................................................................................................19
Test results........................................................................................................................................20
Discussion..............................................................................................................................................20
Transfer function identification ........................................................................................................20
Outlet Temperature Control Loop ....................................................................................................21
Outlet composition control loop...................................................................................................21
Detuning and Decoupling..............................................................................................................22
Advanced control systems............................................................................................................23
Controller Test ..................................................................................................................................23
Conclusion.............................................................................................................................................25
Nomenclature .......................................................................................................................................26
Appendix: Reactor Design.....................................................................................................................27
Appendix A: Design Equations Derivation ........................................................................................27
I. Mass of Catalyst.........................................................................................................................27
II. Mass Balances on Reaction 1:...................................................................................................27
III Mass Balance on Reaction 2 .....................................................................................................28
IV. Partial Pressures: .....................................................................................................................29
V. Pressure Equation.....................................................................................................................30
VI. Energy Balance.........................................................................................................................32
Appendix B: Justification of Plots......................................................................................................34
I. Temperature against ξ1 ..............................................................................................................34
Appendix C: Alternative Designs.......................................................................................................35
II. Inter-stage Cooling .......................................................................................................................37
Appendix D: Sample Calculations .....................................................................................................38
I. Initial gradients ..........................................................................................................................38
II. PFR vs CSTR volume ..................................................................................................................41
III. Cooling duty .............................................................................................................................42
Appendix: Process Control....................................................................................................................43
Appendix E: Transfer Function Identification ...................................................................................43

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I. Transfer Function of HE1 Flowrate ............................................................................................43
II. Matlab Code..............................................................................................................................44
Appendix F: Loop Pairing ..................................................................................................................44
I. Relative Gain Array.....................................................................................................................44
Appendix G: Outlet Temperature Control Loop ...............................................................................45
I. Cohen Coon Tuning Rules (Thornhill, 2015)...............................................................................45
II. Controller Performance ............................................................................................................45
Appendix H: Outlet Composition Control Loop ....................................................................................46
I. Ziegler- Nichols Tuning Rules (Thornhill, 2015) .........................................................................46
II. Critical Frequency......................................................................................................................46
III. Result and Analysis...................................................................................................................47
IV. Controller Performance...........................................................................................................47
......................................................................................................................................................48
V. Bode Plot...................................................................................................................................48
Appendix J: Multi Loop Control ........................................................................................................49
I. Detuning and Decoupling..........................................................................................................49
II. Advanced Control......................................................................................................................50
Appendix K: Multi Loop Control........................................................................................................53
I. Controller Testing.......................................................................................................................53
References ............................................................................................................................................53

GROUP 22 1
Abstract
Methanol has important uses in industry both as a feedstock and a fuel. It is synthesized by steam
reforming and subsequent reaction with Carbon Monoxide (CO). However IMDEAD, synthesizes
methanol by reacting carbon dioxide (CO2) and hydrogen (H2). Mass and energy balances were
conducted to obtain a set of ordinary differential equations for both extents of reaction and
temperature. MATLAB was then used to solve these ODE’s. It was found that given the high
operating temperature conditions, a packed bed reactor (PBR) would not meet the target production
since the catalyst would denature before the target production was achieved.
To overcome this, different designs such as cold shot cooling and recycling were tested. It was found
that a combination of recycling and pre-cooling minimized reactor size and maximised catalyst
efficiency while maintaining the outlet temperature 8K below the temperature limit. This led to a
robust, cheaper design able to handle changes in demand. In the sensitivity analysis that was carried
out it was found that the kinetic parameters, especially the exponential terms, were the most
sensitive because they affected the rate directly and thereby temperature, leading to a large effect
on the rate of methanol production.
Due to a mishap in the commissioning of the reactor, the second part of the report investigated how
to control a different reactor setup different to that in the first part. The main objective here was to
maintain a steady outlet methanol mass fraction of approximately 17% whilst keeping the outlet
temperature around 630K. This was done by manipulating the cooling water flowrate of two inter
stage heat exchangers the process can be controlled within a suitable range. Several controllers and
control methods were investigated such as PID, MPC and feed forward control. To test the
performance of the controller, several disturbances and set point changes were introduced into the
system (modelled in Simulink) and the observed response was evaluated and compared. The control
scheme chosen for the final controller test underperformed because the range of set point changes
investigated was not wide enough. It was found that MPC performed better in this test since it was a
more robust control scheme. However, feed forward control in conjunction with PI feedback control
was more useful in supressing the effect of disturbances.
Keywords: Voidage, Reaction Engineering, Simulation, Process Control, Kinetics

GROUP 22 2
Introduction
Methanol is mainly used as a feedstock for more complex chemicals but is also increasingly used as a
fuel (Institute, 2016). The majority of methanol is produced from the catalytic conversion of syngas
although it is can also be synthesized from biomass, in which case it can be considered to be a
biofuel.
IMDEAD uses neither of these synthesis routes and instead uses CO2 and H2 as reactants. The
objective of the packed bed reactor that IMDEAD is designing is to produce 300 tonnes day-1
of
methanol, equivalent to 108.51 mol s-1
. The process uses 2 reactions:
𝐶𝑂2 + 3𝐻2 ↔ 𝐶𝐻3 𝑂𝐻 + 𝐻2 𝑂
𝐶𝑂2 + 𝐻2 ↔ 𝐶𝑂 + 𝐻2 𝑂
Further objectives aside from the production target for the reactor design are:
 Ensure that the reactor temperature does not exceed 563K since temperatures higher than
this cause the catalyst to denature
 Lower capital cost by making our reactor as small as possible. This will also lead to a lower
mass of catalyst
 Conduct a sensitivity analysis to see how our design responds to changes in key parameters
 Prevent the reverse water gas shift if possible, since this uses up the reactants of reaction 1
The objective in the process control part of the report was to develop an effective control system
which would allow the revised process to operate within a suitable temperature range. The new
reactor system which features two inter stage coolers and three catalytic fixed-bed reactors is
shown below.
The reactions that are occur in this process are different to those in the reactor design part of the
report and are listed below. It is specified that essential control criteria is a steady state CH3OH
outlet mass fraction of 17% and an outlet temperature of 630K.
CO + 2H2 ↔ CH3OH
CO + 3H2 ↔ CH4 + H2O
Fig. 1 Schematic representation of the methanol production process. Manipulated variables are
denoted by “MV”, the controlled variables are denoted by “CV” and the measured disturbances are
denoted by “MD”
Tin (MD)
ΧCO (MD)
Ṁin (MD)
Reaction 1
Reaction 2 – Reverse Water gas shift.
Reaction 3
Reaction 4

GROUP 22 3
Reactor Design: Theory and Methodology
The Packed Bed Reactor (PBR) was modelled analogously to a Plug Flow Reactor (PFR) despite the
presence of a catalyst. This means the mass and energy balances are almost identical to that of a
PFR, with the addition of the Ergun equation to model the pressure drop across the reactor (which
was significant due to the presence of a catalyst). The same assumptions of plug flow, negligible
temperature and concentration gradients in the radial direction as well as negligible potential,
kinetic and work terms were used.
A mass balance over an infinitesimally small volume element was performed to obtain a differential
equation relating the extent of reaction 1 (ξ1) to distance along the reactor (z) and the same was
done for the extent of reaction 2 (ξ2). The rate was given in units of mol kgcat-1
s-1
which meant it
was multiplied by the density of catalyst and cross sectional area of the reactor to eliminate the kg
unit in the denominator, yielding a pre-factor of 840.
An energy balance was obtained using the same technique to obtain a differential relationship with
respect to temperature. The balance was simplified since the reactor was to be operated
adiabatically and 𝑑𝐻 = 𝐶 𝑝 𝑑𝑇 was used to relate the enthalpy of each species to temperature. The
heats of reaction had a correction term to address the fact that the reaction would occur at a
temperature other than 298K, which is the temperature at which the heat of reaction was
measured. The resulting differential equations are outlined below. Full details can be found in
Appendix A.
𝑑𝜀1
𝑑𝑧
= 𝑚 𝐶𝑎𝑡 𝑟1,𝐶𝐻3𝑂𝐻 = 840 𝑟1,𝐶𝐻3𝑂𝐻
𝑑𝜀2
𝑑𝑧
= 840 𝑟2,𝐶𝑂
𝑑𝑇
𝑑𝑧
= −840
𝑟1(∆𝐻 𝑟1
𝑇𝑟𝑒𝑓
+ 𝑣𝑖 ∫ 𝐶𝑝𝑖
𝑑𝑇
𝑇
𝑇𝑟𝑒𝑓
) + 𝑟2(∆𝐻 𝑟2
𝑇𝑟𝑒𝑓
+ 𝑣𝑖 ∫ 𝐶𝑝𝑖
𝑑𝑇
𝑇
𝑇𝑟𝑒𝑓
))
∑ 𝑁𝑖 𝐶𝑝𝑖
𝑛
𝑖
𝑑𝑚 𝑐
𝑑𝑧
= 𝜌𝑐 × 𝐴 × (1 − 𝜖) = 840
It should be noted that the heat capacity polynomials are not all valid over the temperature range
used; water is liquid at 298 K and 305 bar (although all other components are not) which means the
enthalpy of vaporisation has not been taken into account in our energy balance.
The relationship for pressure is given in the design brief the form of the Ergun equation and is shown
below. The ideal gas equation was used to calculate the density despite the high pressures involved.
Fig. 2 General schematic of PBR used for balances
Eqn. 3
(Energy
balance)
Eqn. 2
Eqn. 1
Eqn. 4
z z + Δz dz
(𝑛𝑎) 𝑧
(𝑛𝑎) 𝑧+∆𝑧
Correction
Terms

GROUP 22 4
𝑑𝑃
𝑑𝑧
= −
1.75𝐺2
(1 − 𝜖)
𝐷 𝑝 𝜌𝜖3
Using a MATLAB ODE solver, the four differential equations were solved to see how the four
dependent variables of interest vary along the reactor. As can be seen from Eqn 4, the length and
mass of catalyst display a linear relationship and therefore can easily be converted between each
other.
A number of other designs were also considered as outlined below.
Recycle
In a recycle reactor, a fraction of the product stream, α, is diverted and mixed with the inlet stream.
Modelling this requires performing a mass and energy balance over the mixing point in order to find
the new feed conditions. The pressure after the mixing point can be determined by adding the
weighted molar contributions of each stream being mixed (this can be done since the ideal gas law is
used). In practice, this means that if you have 1 kmol at 1 bar and 1 kmol at 3 bar the resulting
pressure is 2 bar. The recycle ratio is what is controlled in the design.
Pre-Cooling
Pre-cooling involves cooling the feed before it enters the reactor, usually through a heat exchanger.
The cooling duty is what is controlled in the design.
Sample Calculations
Initial Gradients
The initial gradients for our feed conditions are as follows. Full calculations can be found in Appendix
D.I.
𝑑𝑇
𝑑𝑧
= 8.10 °𝐾 𝑚−1
𝑑𝑃
𝑑𝑧
= −1.23 𝑏𝑎𝑟 𝑚−1
𝑑𝜉1
𝑑𝑧
= 0.35 𝑚𝑜𝑙 𝑠−1
𝑚−1
𝑑𝜉2
𝑑𝑧
= −36.96 𝑚𝑜𝑙 𝑠−1
𝑚−1
Calculating G
G (superficial mass velocity) is defined as mass flow rate per unit area per unit time.
Fig. 3 Recycling Schematic
Eqn. 5
Fig. 4 Pre Cooling Schematic

GROUP 22 5
𝐺 =
𝑁𝑡 × 𝑀
𝐴𝑟𝑒𝑎
Where Mis the mean molecular weight of the gas stream at any point in the reactor. Since steady
state was assumed and the cross sectional area of the PBR is constant, G is also constant and the
inlet value of G holds true anywhere in the PBR.
The inlet total molar flowrate NT0 is known to be 6,200 mol s-1
so only M0 is required:
𝑀0 = (0.22 × 28) + (0.71 × 2) + (0.01 × 32) + (0.02 × 18) + (0.04 × 44) = 10.04
Substituting into Eq.5 gives:
𝐺 =
6200 × 10.04 × 10−3
1
=
62.2𝑘𝑔
𝑚2 𝑠
Reactor Design: Results and Analysis
The solution to the set of ODEs for the feed conditions set in the Design Brief for Group 22 are
shown below (all plots are against ξ1).
Trends
Mass of Catalyst against ξ1
The plot below shows how the mass of catalyst, temperature, pressure and ξ2 vary with ξ1 in the
reactor. Obviously as the methanol production increases, more catalyst is needed explaining the
initial increase. The steep increase at 316 mol s-1
represents the maximum production rate of
Fig. 5 Plots of mass of catalyst, temperature, pressure and ξ2 against ξ1
Reaction 1
Eqn. 6

GROUP 22 6
methanol for the specified feed conditions. Adding more catalyst after this does not produce more
methanol.
Temperature against ξ1
Plotting temperature against methanol production produces an approximately linear relationship. At
the target production 109 mol s-1
the temperature outlet is approximately 583K, which exceeds
catalyst denaturing temperature. This problem is shown in Fig 7. This leads to an unsuccessful design
and so alternative designs were explored (see Alternative Designs).
It can be shown that if some assumptions are made, such as the Cp terms are assumed to remain
relatively constant with temperature (see Fig. 6), the relationship between temperature and ξ1 is
linear and this is indeed what is observed in Fig 4. A more detailed explanation can be found in
Appendix B.I.
Pressure against ξ1
It is clear that pressure decreases as the methanol production increases. Since the methanol
production increases down the reactor, this translates to a pressure drop through the reactor, which
is necessary to ensure a flow down the reactor. The reason for the spike at the end is similar to that
explained for the mass of catalyst, eventually the maximum production rate of methanol for the
feed conditions is reached.
ξ2 against ξ1
As reaction 1 progresses, the extent of reaction 2 becomes more negative implying that the water
gas shift reaction is taking place. By dividing Eqn. 2 by Eqn. 1 is it clear that the gradient of the graph
is simply the ratio of the rates, which remains constant based on the straight line obtained.
Fig. 6 Heat capacities from 500K to 560K. As
we can see, the heat capacities are roughly
constant over this range
Fig. 7 Temperature against extent of reaction 1 for
given conditions, with limits displayed. The
temperature should pass through the bottom right
quadrant in order for specifications to be met

GROUP 22 7
Variation in Reaction Rate with ξ1
Fig 8. shows how the variation in
reaction rate with ξ1 resembles a
quadratic, with a maximum value at
approximately 137 mol s-1
. Relating
this to the plot against z (Fig. 10), it
can be seen that the maximum value
of 137 mol s-1
relates to the maximum
rate of increase of ξ1 (i.e. the gradient
is a maximum here). The reason the
rate of production is maximised here is
because this corresponds to the locus
of maximum rate which is the peak
observed in Fig 8.
The rate reaches a maximum in Fig. 8 because of two competing effects. The first effect is that the
rate always increases with temperature due to the Arrhenius expression (and on a more
fundamental level because more molecules have enough energy for a successful collision). The
second effect is that in an exothermic reaction, a higher temperature favours the backwards
reaction which lowers the rate constant and lowers the rate. If the temperature is increased even
further the backwards reaction starts to occur which we also see in Fig 8, due to the rate turning
negative. These two effects combine to produce a maximum rate at approximately 630K.
Conversion of Carbon Monoxide and Hydrogen
The conversion of CO is always negative as Fig. 11 shows. This is because the water gas shift is taking
place as opposed to the reverse water gas shift, meaning CO is actually being produced. The plateau
that occurs at around 5 m down the reactor shows that reaction 2 slows down to a halt around this
point.
The conversion of H2 is negative for a small period after which it becomes positive again. It drops
below 0 as initially reaction the reverse water gas shift occurs to a greater extent than reaction 1.
Eventually (around 1.5 m down the reactor) the rate of reaction 1 increases enough to overcome the
production of H2 due to reaction 2. Specifically, the rate of reaction 1 has to be at least a 1/3 of the
rate of reaction 2 since 3 moles of H2 are consumed in reaction 1 as opposed to 1 mole in reaction 2.
Fig. 8 Rate of methanol production against temperature
Reaction 1
Fig. 9 Reaction rate against ξ1 Fig. 10 Temperature against z

GROUP 22 8
Reactor Design: Sensitivity Analysis
It is important to know how the experimentally determined parameters (such as rate constants or
equivalent particle diameter) affect the reactor model. Changes to the inlet flow rate are also
important, since production capacity may have to handle changes in market demand.
In order to do this, the parameters investigated were multiplied by factor and the absolute
percentage difference in ξ1 for all points up until 5m along the reactor was calculated. This gave us
the weighted average difference.
ξ1 was chosen in order to compare sensitivity since ultimately the production target is the most
important controlled variable, while an investigation interval of 5m was chosen since in all the
reactor designs the reactor length was shorter than this. A short interval also has the benefit of not
over exaggerating the sensitivity of parameters which made ξ1 flat line (see Fig. 11).
Parameter Value Currently
being used
Percentage
Change
Implemented (%)
Percentage change
in Extent of
reaction 1 (%)
Sensitivity
Pre Exponential
Factors of Rate
Constants
𝐴=0.55 𝐵=3453.38
𝐶=0.499
𝐷=6.62×10−11
𝐸=0.61×1010
±20% + 32.6%, - 54.8% High
Exponent Factors
of Rate Constants
𝐴=𝑒36696/ 𝑅 𝐶=𝑒17197/ 𝑅
𝐷=𝑒124119/ 𝑅 𝐸=𝑒−94765/ 𝑅
±5% + 99.7%, - 573.2% Very High
Heat Capacities Polynomial Expressions
given in Brief
±20% + 27.3%, - 51.3% High
Bed Voidage ε = 0.4 ±20% + 15.5%, - 16.5% Moderate
Inlet flow rate 6200 mol s-1
±20% + 30.3% , - 57.7% High
Equivalent
Particle Diameter
7.04 mm ±20% + 0.0%, -0.1% Very Low
Table. 1 Sensitivity Analysis showing sensitivity of parameters.
Fig. 11 Conversion of CO and H2 with length of reactor

GROUP 22 9
The general trend from Table 1 is the parameters that affect temperature through Eqn. 3 cause a
large change in ξ1 while everything else less so. It should be noted that the flow rate indirectly
affects temperature since it appears in the denominator of the energy balance. This trend is due to
the feedback loop between ξ1 and the temperature; since we have an exothermic reaction, if one of
these increases so does the other.
The fact that the equivalent particle diameter, which appears only in the pressure drop equation,
changes ξ1 very little implies that pressure is not as important as temperature in our rate equations.
These findings indicate that temperature is by far the most important variable in our model since it
affects the rate to the greatest extent.
With reference to Table 1, we see that the exponential kinetic factors are the most sensitive
parameters, with only a 5% change causing large changes ξ1. This is unsurprising since the rate
constants appear in the energy balance (and are themselves affected by temperature) and we are
changing the exponential terms in the rate constants which has an even greater multiplying effect
than simply multiplying the pre exponential factor.
In the left hand plots are considered in Fig (11), we notice two things. Firstly, a reduction (denoted
by negative change in legend) in exponent values causes both ξ1 and the temperature to flat line.
This is because the change in temperature is entirely due to the 2 reactions taking place, so reducing
kinetic parameters causes a massive reduction in reaction rate and therefore temperature gradient.
Fig. 11 Sensitivity Changes shown down the reactor. Negative change
means the multiplying factor is decreased
Limit ξ1
value
Commented [GS1]: Could probably phrase this differently
Commented [KK2R1]: Affects the rate significantly?

GROUP 22 10
Secondly, changing the kinetic parameters does not change the limit ξ1 value since for identical feeds
this is determined by the equilibrium constant (in an infinitely long reactor the yellow line
representing the negative change would reach this value as well). The heat capacities and pre
exponential factors of the rate constants produce similar (if less drastic) effects.
With reference to the right hand graphs, two distinct phenomena can again be observed. Firstly, a
lower inlet flow rate results in a steeper temperature gradient, which can be explained due to the
𝑛𝑖 𝐶𝑝𝑖 term in the denominator of the energy balance. We also see that changing the inlet flow rate
changes the limit ξ1 value by an equivalent amount, although this is unsurprising (less reactants
necessarily means less products at equilibrium, but the ratio stays the same). More importantly, the
limiting value is reached faster due to the higher temperature gradient.
Reactor Design: Alternative Designs
Recycling
As Fig. 12 demonstrates, recycling involves cycling back a fraction (α) of the product to the inlet of
the reactor. The graph below shows how the reactor length varies with recycle ratio, α. The change
in inlet temperature and outlet temperature for different α was also investigated. The top plot is the
outlet temperature while the plot underneath is the variation in the inlet temperature. It is
important to note that the degrees of freedom were limited by specifying ξ1 to meet the production
target 109 mol s-1
, so the length plotted corresponds to the length required to meet this target.
As can be seen Fig.12, a higher α results in a higher inlet temperature. This temperature increases
also means that the reactor length and by extension catalyst mass decrease leading to more efficient
catalyst usage and a cheaper reactor.
Unfortunately recycling does not decrease the outlet temperature significantly meaning the design
still is not feasible. However the explanations for these trends provide insight into our process.
Fig. 12 Temperature and Catalyst Mass against Recycle Ratio

GROUP 22 11
Increase in inlet temperature and isothermal behaviour
As α increases, more of the hotter outlet stream is mixed with the feed, causing the inlet
temperature to increase. As α approaches 1, the reactor approaches isothermal (or CSTR) behaviour.
This can be related this to the schematic below. F4 is specified by our production target, so as α
approaches 1 F3 gets increasingly high. This causes F5 to increase as F5 ≈ F3 for a higher alpha. Since F5
is now much higher than F1 the temperature of stream F2 is almost exclusively determined by F5
causing the temperature of F2 ≈temperature ofF5 = temperature of F3.
Decrease in Catalyst Mass
In Fig. 14 it can be seen that as α is increased the inlet temperature leads to a higher initial rate. This
means that the reactor operates closer to the locus of maximum rate which reduces the reactor
length and amount of catalyst needed.
Essentially, it has been shown that the
ideal reactor design for this reaction
would be a CSTR. This is counterintuitive,
since usually a PFR results in a smaller
reactor volume (see Appendix D.II).
However this result assumes isothermal
behaviour which means the concentration
gradient is the driving force behind a
reaction. The reactor is not isothermal
however, and the temperature plays a
much more important role in our rate of
reaction than the partial pressures.
Recycling and Pre- Cooling
It is clear that the temperature of the outlet needs to decrease if the target production (without
denaturing the catalyst) is to be met, meaning our reactor design requires cooling of some sort. We
were also motivated to try and cut costs using a smaller reactor which would also lead to more
efficient catalyst usage. From previous analysis we knew the latter could be achieved by recycling.
Fig. 13 Recycle schematic
Outlet
Temperature
α = 0 Inlet
Temperature
α = 0.4 Inlet
Temperature
α = 0.7 Inlet
Temperature
Fig. 14 Variation in rate of
methanol production with α

GROUP 22 12
With this in mind, it was decided to model a pre-cooling and recycle combination in the design,
although it was not clear whether to cool the inlet after the recycle stream or before. A general
schematic of both apparatus is shown below.
When cooling after the recycle, there is a lower risk of exceeding the temperature limit, as the entire
feed to the reactor is being cooled, but could lead to a higher cooling duty as more reactants need to
be cooled. The opposite is true if we cool prior to recycling.
Optimising this process is more complicated than in the case of simple recycle since there is an extra
degree of freedom in the form of cooler duty (Q) (the length of reactor is set by our production
target). Therefore α is set constant to observe variations in the cooler exit temperature and
subsequently set the cooler exit temperature constant to observe variations in α. The results for
both configurations can be found below.
Cooling followed by recycle
As it can be seen in trial 1 and 2, a higher recycle ratio corresponds to a lower mass of catalyst being
used up but a higher outlet temperature. The reason for this was explained in the recycling section
of alternative designs. It is important to recognise that the temperature of the recycle stream (F6) is
higher than F2 and therefore as α increases, the temperature to the reactor will increase.
Comparing trial 2 and trial 3 we note that higher exit temperature from the cooler results in a
decreased mass of catalyst and cooling duty but a higher outlet temperature. This is because a
higher cooler exit temperature results in a higher rate decreasing the mass of catalyst needed, but
also means that the feed temperature to the reactor is increased, causing the reactor exit
temperature to be higher.
To settle on a final design, degrees of freedom were limited by setting a recycle ratio of 0.9 and a
maximum outlet temperature of 555K. These were chosen because higher recycle ratios led to
unrealistically high flow rates for F6 while higher cooler outlet temperatures meant the catalyst was
close to its denaturing temperature.
Trial Cooler Exit
Temperature (K)
Recycle
Ratio, α
PFR Length
(m)
Mass of Catalyst
(kg)
Outlet
Temperature (K)
Q
(MJ)
Trial 1 470 0.5 3.77 3167 546 5.82
Trial 2 470 0.8 2.19 1840 546 5.82
Trial 3 479 0.8 1.69 1420 551 4.13
Trial 4 479 0.9 1.56 1310 555 4.13
Table 2: Investigation on Recycle subsequent to cooling
Fig. 15 Pre cooling with recycle schematics.
Commented [GS3]: But surely if this would happen we
just cool our feed more? I would get rid of this
Commented [GS4]: Is it the opposite? Maybe ‘similar
reasoning reason implies that the opposite…’ although that
still uses opposite… Can’t think of a better word

GROUP 22 13
Recycle followed by cooling
Surprisingly, if the recycle ratio increases the length of the PFR increases, meaning more catalyst is
required as well as an increased outlet temperature and cooling duty. This can be seen if trials 1 and
2 are compared. Because the cooler is introduced after the recycle, the outlet of the cooler is the
same as the inlet to the reactor meaning the temperature inlet is governed completely by the
cooling duty. As α increases, the total inlet flowrate increases but the reactor inlet temperature
remains the same as this is set exclusively by the cooler, leading to a lower temperature gradient
(due to the 𝑛𝑖 𝐶𝑝𝑖 energy balance) and ultimately a lower rate of reaction.
As the cooler exit temperature increases (as in trial 2 and 3), the cooler duty decreases and outlet
temperature decreases while the PFR length increases. This is again due to the rate of reaction being
lower at a lower temperature and therefore the length increases. However, a higher inlet
temperature to the reactor will lead to a higher reactor outlet temperature.
Setting the same upper limits as before of α = 0.9 and an outlet temperature of 555K, the two
approaches yield almost exactly the same results (the difference in values is due to heat capacities
changing over the temperature range and rounding errors). While a higher cooling duty was
expected for the second design due to a higher flow rate for F6, this is counterbalanced by the fact
that F6 needs to be cooled less due to a smaller temperature difference. It was determined that the
cooling duty required 1.83 L s-1
of water which was deemed reasonable (see Appendix D.III).
Comparison of designs
Given that the location of the cooler does not change the design, and the similar results obtained, it
makes more sense to cool the feed before recycling, since the cooler will not have to be as large to
accommodate a larger flowrate. Relating this to the original design, the catalyst mass required to
reach our target production (if we ignore the temperature limit) was 2430kg and the current catalyst
mass is almost half of this, while meeting the temperature criteria. This therefore provides a more
robust design that meets the target production which is also cheaper.
Additional designs
Additional design simulations, such as cold shot or inter stage cooling, where also investigated.
However, these were omitted from the report because they provided less useful insights. The
additional designs are outlined in Appendix C.
Trial Cooler Exit
Temperature
Recycle
Ratio, α
PFR Length
(m)
Mass of Catalyst
(kg)
Outlet
Temperature
(K)
Q
(MJ)
Trial
1
520 0.5 2.28 1916 557 3.53
Trial
2
520 0.7 2.89 2428 543 6.46
Trial
3
532 0.7 1.89 1588 555 4.14
Trial
4
547 0.9 1.57 1319 555 4.12
Table 3: Investigation on Pre-Cooling subsequent to Recycle

GROUP 22 14
Controller Design: Theory and Methodology
The process was simulated using a Simulink model. Note that in reality the results would have been
obtained by constructing a pilot plant to run the experiments.
A. Transfer model identification
A series of experimental step changes are carried out whereby a MV is changed step-wise to then
allow for the study of the step response in the CV’s. All the step responses can be modelled
approximately by a first order plus time
delay model (FOPTD). Fig. 16 (left) shows
how the data obtained for ΔCV with a ΔMV
would have actually be used to derive the
essential parameters in the transfer
function. The tangent drawn on the
transitional stage is located at the point
with the steepest gradient; this point was
found by exporting the data from Simulink
to Microsoft Excel and plotting various
trend lines until one with sufficiently steep
gradient was found. Td is the time delay
after which the step is made and τ is the
time constant (time at which 63% of the steady state effect has been achieved). Both are found by
analysing the intersection points of the steepest gradient with the CVinitial and CVfinal respectively.
B. Loop Pairing
Relative Gain Array was used to determine the best control structures for the CV’s. It was known
that both ΔMV’s (change in cooling water flowrate of HE1 or HE2) would affect both CV’s. It is
important to find which MV will have the most significant impact on the particular CV in question
(i.e. which MV will have a “direct” effect rather than “indirect” effect on CV). The following matrix
was proposed and provides the best pairing:
Fig. 16 The graphical construction for FOPTD where:
Δ𝐶𝑉
Δ𝑀𝑉
≈ 𝐾 𝑃
𝑒−𝑠𝑇𝑑
𝑠𝜏 + 1
𝑤ℎ𝑒𝑟𝑒 𝐾 𝑝 =
Δ𝐶𝑉𝑠𝑠
Δ𝑀𝑉
ΔCVss = CVfinal – CVinitial
ΔCV
ΔHE2
K
11
K
12
K
21
K
22
ΔHE1
ΔTout(ΔCV2ss)ΔXCH3OH (ΔCV1ss)
[
𝐾11 𝐾12
𝐾21 𝐾22
] 𝑤ℎ𝑒𝑟𝑒 𝐾𝑖𝑗 =
Δ𝐶𝑉𝑖𝑠𝑠
Δ𝐻𝐸𝑗
Relative gain array is now used: 𝑅𝐺𝐴 = [
𝜆11 𝜆12
𝜆21 𝜆22
]
𝑤ℎ𝑒𝑟𝑒 𝜆11 =
𝑑𝑖𝑟𝑒𝑐𝑡 𝑒𝑓𝑓𝑒𝑐𝑡
𝑖𝑛𝑑𝑖𝑟𝑒𝑐𝑡 + 𝑑𝑖𝑟𝑒𝑐𝑡 𝑒𝑓𝑓𝑒𝑐𝑡𝑠
𝑠𝑡𝑒𝑎𝑑𝑦 𝑠𝑡𝑎𝑡𝑒 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛𝑠ℎ𝑖𝑝𝑠: Δ𝐶𝑉1𝑠𝑠 = 𝐾11 𝛥𝐻𝐸1 + 𝐾12 𝛥𝐻𝐸2
Δ𝐶𝑉2𝑠𝑠 = 𝐾21 𝛥𝐻𝐸1 + 𝐾22 𝛥𝐻𝐸2
𝜟𝑯𝑬𝟐 = 𝟎 , 𝒇𝒐𝒓 𝒕𝒉𝒆 𝒅𝒊𝒓𝒆𝒄𝒕 𝒆𝒇𝒇𝒆𝒄𝒕: Δ𝐶𝑉1𝑠𝑠 = 𝐾11 𝛥𝐻𝐸1 + 0
∴ (
Δ𝐶𝑉1𝑠𝑠
Δ𝐻𝐸1
) 𝛥𝐻𝐸2=0 = 𝐾11 = 𝑑𝑖𝑟𝑒𝑐𝑡 𝑒𝑓𝑓𝑒𝑐𝑡
𝜟𝑪𝑽𝟐𝒔𝒔 = 𝟎 , 𝒇𝒐𝒓 𝒕𝒉𝒆 𝒅𝒊𝒓𝒆𝒄𝒕 + 𝒊𝒏𝒅𝒊𝒓𝒆𝒄𝒕 𝒆𝒇𝒇𝒆𝒄𝒕𝒔:
Δ𝐶𝑉1𝑠𝑠 = 𝐾11 𝛥𝐻𝐸1 + 𝐾12 𝛥𝐻𝐸2
0 = 𝐾21 𝛥𝐻𝐸1 + 𝐾22 𝛥𝐻𝐸2
∴ (
Δ𝐶𝑉1𝑠𝑠
Δ𝐻𝐸1
) 𝛥𝐶𝑉2𝑠𝑠=0 = 𝐾11 −
𝐾12 𝐾21
𝐾22
= 𝑖𝑛𝑑𝑖𝑟𝑒𝑐𝑡 + 𝑑𝑖𝑟𝑒𝑐𝑡 𝑒𝑓𝑓𝑒𝑐𝑡𝑠
𝜆11 =
𝐾11
𝐾11 −
𝐾12 𝐾21
𝐾22
Fig. 17 Loop pairing matrix
Commented [GS5]: Need to decide how we’re going to
represent Simulink and MATLAB (ie whether they need
italics). Just need to choose one way right at the end.

GROUP 22 15
The properties of RGA are that all the rows and columns sum to 1. Therefore 𝜆11 provides enough
information to find all 𝜆𝑖𝑗. If the loop pairing is effective then 𝜆𝑖𝑗 ≥ 1 for the respective pairs
C. Outlet Temperature Control Loop
A feedback controller was needed for the outlet temperature. The controllers that were tested had
the general transfer function as follows:
𝑜𝑝(𝑠) = 𝐾𝑐 (1 +
1
𝜏𝑖 𝑠
+ 𝜏 𝑑 𝑠) 𝑒(𝑠)
Where e(s) = error = set point - process variable.
Generally, 1/τi = I and τd = D, in alternative nomenclature. “I” is the integral term i.e. Kc/τi (for
parallel form) or 1/τi (for ideal form). Both have been used in this project to develop competency
with the different notation; for the outlet temperature loop the parallel form has been used while
for the outlet mass fraction loop the ideal form was used.
Cohen-Coon tuning rules can be used as a first approximation (see Appendix G.I.), but the
parameters are then refined by trial and error (tuning). This is done by testing the controller with a
series of step changes in the disturbances and set point to observe when the response is optimised.
The process variables in this system are the process outlet temperature or mass fraction.
D. Outlet Composition Control Loop
The Zeigler-Nichols tuning method was used to determine the parameters of feedback controller for
composition control loop. The tuning rules, as shown in the Appendix H.I., require the ultimate gain,
Ku and the critical frequency ωcg of the control system. This is done by testing the P - controller with
different Kc values until the system just reaches instability (i.e. oscillates at a constant amplitude).
The critical period, Pu was estimated by averaging the interval between troughs (see Appendix H.II.)
The experimentally determined Ku is then compared with Ku obtained from Bode stability criterion in
order to evaluate the accuracy of the transfer functions. Then a similar method to part C is carried
out to determine the controller transfer function.
E. Multi-Loop Control
It is clear to see in the matrix from RGA that each MV has
an effect on both CV’s. This causes loop interaction which
could cause the control system to fail when attempting to
maintain specified set points. In order to minimise this
interaction detuning was used to modify the Kc. Since it
was found from the RGA that 𝜆11 and 𝜆22 >1 the following
detuning formulae was used:
𝐾𝑐(𝑛𝑒𝑤) = (𝜆 − √ 𝜆2 − 𝜆) 𝐾𝑐(𝑜𝑙𝑑)
∗
To completely remove the interaction decoupling is
needed, which involves cancelling the indirect effect of a
MV on a CV using an additional transfer function. Usually
the “steady state” gains are used as they are sufficient to
decouple the dynamic system.
𝐾𝑖1 𝐾11 𝛥𝐻𝐸2 + 𝐾12 𝛥𝐻𝐸2 = 0 ∴ 𝐾𝑖1 =
−𝐾12
𝐾11
ΔHE2
K
11
K
12
K
22
ΔTout (ΔCV2ss)ΔXCH3OH (ΔCV1ss)
Ki
1
Fig. 18 Multi-loop control matrix.
Eqn. 7

GROUP 22 16
Sensor
Model Predictive Control (MPC)
MPC is an alternative controller type that has the advantage of predicting the future dynamic state
of the process. This means it can induce necessary control action to counter deviation from a target
set point (a complete derivation of MPC is beyond the scope of this report). The steps needed to
implement MPC can be found in appendix
J.II.2.
Feed forward (FF) Control
FF control works by measuring the
disturbances before they arrive at the
controlled system. This allows the actuators
to respond faster to the disturbance,
resulting a smaller deviation from the set
point. In order to determine the FF transfer
function, gff, the disturbance transfer
function gd has to be estimated beforehand.
The approximation model used was FOPTD
which is mentioned in part A. The formula of
gff is as below:
𝑔 𝑓𝑓(𝑠) = −
𝑔 𝑑(𝑠)
𝑔(𝑠)𝑔 𝑣(𝑠)ℎ 𝑑(𝑠)
Result and Analysis
A. Transfer function identification
As seen in Fig. 17, each MV is related to each CV
through a specific continuous time transfer function.
However, there are also three measured
disturbances which should also be included into this
matrix. A diagram including the MD’s will be omitted
as it is rather complicated, although it should be clear
that these MD’s will also have transfer functions that
relate them to the CV’s. In Appendix E.II. a MATLAB
script is provided which generates the final calculated
transfer functions and orders them into a resultant
matrix (“R”) as shown in Fig. 20.
B. RGA
The final matrix obtained is as follows:
𝑅𝐺𝐴 = [
𝜆11 = 2.027 𝜆12 = −1.007
𝜆21 = −1.027 𝜆22 = 2.027
]
This means that the cooling water flowrate of HE1
has the most direct effect on the outlet methanol
mass fraction and flowrate of HE2 has the most direct
on the outlet temperature. The relevant MV will
therefore be used in the control loop for its
respective CV (for which it affects the most).
Fig 20. “R” generated from the MATLAB script
Fig. 19 FF controller schematic.
MD
+
Controlled
Variable
Indirect Effect
Direct Effect
Measured Disturbance
Controller
Output
MV
hd(s)
gff(s)
gv(s)
gd(s
)
g(s)
CV
Commented [KK6]: Have you defined what this is before?

GROUP 22 17
C. Outlet Temperature Control Loop
We can see that as Kc was increased the response became more unstable. It was decided that Kc =
0.5 was optimal while the purple line shows the results from using Cohen-Coon parameters. It was
found that the optimised Kc = 0.5 in conjunction with a well-tuned I of -30 provided the best
response.
D. Outlet Composition Control Loop
Determining Ultimate Gain and Critical Frequency
While Kc = 850 results in oscillations with a greater amplitude than smaller values of Kc , there was a
slight kink (as shown in the circles) in the graph when the mass fraction reached close to 0.175. This
is due to the physical limit of the system that results in no further increase in amplitude. Therefore
the ultimate gain, Ku was determined at the point when the gain caused maximum oscillation
without a kink, which is shown above in the case of Kc = 750. The critical frequency ωcg was
determined experimentally is 22.5 hr-1
(see Appendix H.V.)
Fig. 22 Here shows the maximum amplitude of the responses with different P-controllers.
Figure 21. Results for the responses using the different P and PI controllers that were tested.
Commented [KK7]: Which kink? It’d be much easier to
annotate the graph here showing where it is.

GROUP 22 18
Feedback Controller:
The parameters of most controllers were estimated by Ziegler Nichols tuning method but PI (yellow)
was developed with further tuning. Kc = -337.5, I = 26.99 gave the best response. Although PID
responded fastest to the step change, it caused valve saturation within a shorter period of time. This
has restricted the control loop from changing the flowrates and therefore hindering recovery to the
set points. Refer Appendix H.IV for the effect of the step changes on the temperature.
Bode Plot
The ultimate gain and critical frequency determined from Bode Plot is 5000 hr ton-1
and 600 rad hr-1
while the experiments yield 750 hr ton-1
and 141 rad hr-1
respectively. (see Appendix H.V.)
E. Multi-Loop Control
With both control loops closed, the loop interaction was studied by using simultaneous changes in
the set point and disturbances at t = 0.1 hr. (see Appendix J.I.2 for step changes and disturbances)
Fig. 24 Plot of the response when both control loops are closed.
Fig. 23 The responses of different controllers to the step change of mass fraction to 0.2
Frequency (rad/hr)
Fig. 23 Bode plot for transfer function relating HE1 flowrate to outlet mass fraction.

GROUP 22 19
The detuned and single-way decoupled system gave the most accurate response. The refined
parameters used for detuning (red) and decoupled systems (yellow & purple) can be found in
Appendix H.III.
Advanced control techniques
FF fairs better overall since it achieves the target set point more quickly (see Appendix J.II.) It should
be noted that FF does not keep the system at its set point; this is achieved by the PI controllers, but
the overall control scheme shall be called FF for convenience.
Fig. 25 The different responses of the system to a disturbance step of -7K in the inlet
temperature at 0.1 hour.
Fig. 26 The response to a step change in the mass fraction set point (+0.015) at 0.1 hours for MPC and
0.5hours for FF

GROUP 22 20
Test results
The chosen control scheme was the PI controllers with FF control.
Discussion
Transfer function identification
It should be understood that the transfer functions found are approximate. All the step responses
were modelled using FOPTD, when in reality the responses could be higher order systems. This
approximation led to inaccuracies, especially with the measured disturbances. For example, a +5 ton
hr-1
change in the inlet mass flowrate produces this response:
Evidently the temperature response is not a FOPTD, and so for this particular response only the gain
was included in the transfer function. The response of the methanol mass fraction may seem FOPTD
Fig. 27 Results from the test for the chosen controller (in red) and the rejected MPC controller (in blue). It was
found that the MPC performs exceptionally well whilst the FF fails to meet target set points (yellow lines).
Fig. 28. The response of the system to a step change in the outlet temperature of +5 ton/hr (no control
implemented) at 0.1 hour.
Commented [KK8]: Should we say the yellow line is the
step/disturbance whatever
Commented [KK9]: Why?

GROUP 22 21
but in fact the time delay is negligible and is omitted. The models constructed are adequate for use
in FF and MPC control schemes which suggests they are a workable approximation.
Outlet Temperature Control Loop
With reference to Fig. 21, when the Kc is made more negative for the P-only controllers the response
becomes more unstable. Although the set point is actually reached at some times, this kind of
instability (with large error margins) is highly undesirable. When Kc = - 0.98 (Cohen-Coon parameter)
the instability is supressed but after tuning it is found that -0.5 is the most satisfactory controller
gain to use since the steady state error is reduced whilst maintaining relative stability.
Generally P-only is not used because a steady state error in the response remains. Most process
systems use PI or PID controllers which include an integral term to subtract the steady state error.
The response on the outlet mass fraction when using PI control can be found (in the Appendix G.II.).
This response is similar to that of the temperature. It can be observed that the Cohen-Coon
parameters (purple line) in Fig. 21 make the controller too aggressive i.e. causes instability as the set
point is reached too quickly. After tuning, the optimal parameters were found Kc = - 0.5 and I = - 30
which produces a more robust controller.
PID was also considered, with the Cohen-Coon parameters giving the best response:
It was found that for this test the mean steady state temperature for both controllers was 640.0 (4
s.f). The standard deviation was 0.65 for PI and 0.64 for PID (2 s.f). This is a negligible difference and
so the deciding factor is that PID is generally more risky than PI as it can cause “valve overloading”
(Thornhil, 2015). The finalised PI controller was put through several tests with different step changes
in the set point and with the introduction of disturbances in order to further tune it (see Appendix
G.II.).
Outlet composition control loop
As observed in Fig. 22, a higher amplitude is obtained with higher Kc. The kink that appears around
0.7 hour is due to the system reaching a theoretical limit. Higher temperatures could not be
physically achieved despite the increase in Kc. Therefore Ku is determined at the maximum amplitude
without damping or kink. The Ku and ωcg determined experimentally differ significantly to the result
Fig. 30 System responses using the best PI (Kc = -0.5, I = -30) and PID (Kc = -0.5, I = -30, D =-0.0036) controllers.
Commented [KK10]: Why is this 4 dp
Commented [KK11]: Nah change to minutes, or 0.7 time
or idk but can be phrased better.

GROUP 22 22
of Bode stability criterion. This is consistent with the fact that the FOPTD model being an
approximation.
As seen in Fig. 22, P-only control is not used because of the steady state error that remains in the
response. PID causes valve saturation at an earlier time creating the risk of a runaway reaction if the
if the temperature cannot be decreased fast enough.
Before any changes were introduced into the system, the PI (yellow line) was the best controller
because it gave the smallest standard deviation from the set point of 0.17 (see Appendix H.III.).
However, it is apparent that the alternative PI (orange line) responds faster to the set point change
and maintains a smaller deviation from the new set point over the time.
Detuning and Decoupling
Fig. 25 shows that the interaction of the indirect effect from each control loop results in more
instability (blue line), indicating detuning was required to synchronise the Kc of both controllers.
Although this improved the stability of system, it resulted in a more sluggish composition control
loop. Since the objective was to find a balance between stability and faster settling time, further
refining and retuning was required to determine a new set of parameters (red line) that gave a faster
response.
To completely remove the indirect effects, controller decoupling was also required, for which both
one way and two way decoupling were studied. This was done by determining the transfer function
of a decoupling controller using transfer functions of the direct and indirect effects. As mentioned
before, the FOPTD model is inaccurate meaning further tuning was necessary. The results showed
that two way decoupling was less desirable since the control scheme became more sluggish (yellow
line). This is because the decoupling system reduces the magnitude of the controller gain and so
decreases the controller output. It was found that the one way decoupling of the HE2 indirect effect
on the outlet mass fraction gave the best response (purple line); there is less overshoot in the outlet
temperature and mass fraction response to the same series of step changes described for Fig. 25.
(see Appendix J.I.). This is due to the fact that the mass fraction is more difficult to control than the
temperature since it is harder to control the reaction kinetics than just the temperature. Therefore
removing the indirect effect on the mass fraction has a greater effect on improving the response
than removing the indirect effect on the temperature.
Fig. 31 Here demonstrates the performances of P, PI and PID controllers with the parameters from Ziegler Nichols tuning
method. PID controller is unstable in this case. Therefore, this has affirmed that PI (Kc= -337.5, I=26.99) is the most
appropriate controller for this system.

GROUP 22 23
Advanced control systems
In Fig. 26, all the graphs eventually settle to a new operating point when the disturbance is
introduced (given enough time). It was found that MPC was able to delay the effect of the
disturbance but eventually settled to the new operating point relatively quickly. This is customary of
MPC since it has an ability to predict the control actions needed to stabilise the system. With FF
control (as with MPC) the disturbance is measured before it has its effect on the system. For the FF
control scheme, the FF controllers were tuned to counter the effect to a greater extent which meant
the system took longer to stabilise to the new operating point. It was found that in most cases the FF
control scheme was able to supress the effect of disturbances better than MPC.
It was decided that FF control was needed to reduce the initial deviation of mass fraction and
temperature when a disturbance was introduced. The FF controllers were well-tuned so that the
settling time and any large variation in outlet values (that may have caused system failure) were
reduced.
In Fig. 27 we can see that FF control scheme provided a lower sum of the squared error of the
difference between the set point and the outlet value, meaning the FF control scheme was also
preferable for the use in changing set points. As a result, the FF control scheme was chosen for the
controller test.
Controller Test
The series of changes made to the system (see Appendix K.I) could not be handled by the FF control
scheme used. The initial change in composition set point is not reached and causes the system to
destabilise to a lower operating point. Additionally the disturbances which intuitively aid in the
maintenance of the step point are supressed by the very well-tuned FF control. MPC handles the
changes much better as seen in Fig. 28.
The sum of the squared error difference between the outlet values and set point for the test using
MPC is 30,864 and 0.054 for the mass fraction. This is significantly better than the performance of
the chosen control scheme (786,274 and 1.01). The reason for this result is due to the over-tuning of
the PI controllers. Kc is over-tuned to achieve a very aggressive (quick) response for maintaining and
changing set points (within a small range; only +/- 5.9% was tested). This caused the valves to
saturate in our test causing the system to eventually destabilise to a lower operating point.
From Eqn. 7, it is clear that when the Kc is too negative the controller output will have too great a
magnitude meaning that the cooling water flowrate will attempt to decrease (or increase) to an
unattainable value (causing saturation). This then means that the system is pushed into a lower (or
higher) steady state operating that increases the magnitude of the error (and increases/decreases
the controller output further). At this point the control system fails and cannot recover; this is
especially the case for the chosen control scheme since any disturbances (which may aid in reaching
the set point) are supressed.
For the MPC, initially an outlet mass fraction of 0.18 (0.005 below target) is achieved and
maintained. Then as the disturbance is introduced the target is reached. With reference to Fig. 32
below, clearly the valves are saturated relatively fast with the FF scheme as the gradient for the
transition in the MV is too high. With the MPC there is a slower transition which allows it to stabilise
to an operating point that achieves 0.18 in the outlet mass fraction. At 1 hour the inlet temperature
Commented [KK12]: Just making sure you know which
figure this relates to cause I don’t see one lol

GROUP 22 24
is increased by 5K which allows for the transition to a new operating point where the target set point
is achieved.
With reference to Fig. 33 below, it was also found that when the decoupling system was removed
from the original design the aggression was reduced, since the additional controller output was not
included at the summing point. However the best resolution came when the PI controller gains were
also made less negative.
The sum of the squared error for the detuned PI controller was 69,787 and 0.026. In comparison to
MPC, the mass fraction error is 51.9% smaller whilst the temperature is 126.1% larger. Overall this
means MPC is still more favourable than simple control schemes.
Fig. 32 Behaviour of the heat exchangers during the test for the FF (above) and MPC (directly above).
Fig 33. When the original design is “simplified” with feed forward and decoupling removed, the system seems to respond
slightly better. It is found that when the PI controllers are “detuned” to make them less aggressive the system responds
much better and generally achieves the target set points.
Original controller
Tuned PI & No decoupling/FF

GROUP 22 25
Conclusion
We can conclude that given the feed conditions a simple reactor design would not work. The target
production would not be met due to the denaturing of the catalyst at high reaction temperatures.
Furthermore it was found that the kinetic reaction parameters A-E were very sensitive. This is due to
the fact that the kinetic terms, especially the exponents, affect the rate of reaction and the mass and
energy balances. The heat capacity terms were also found to be sensitive which affected the
temperature directly in the energy balance.
Additional designs were also explored. Most notably, a recycle stream was introduced and it was
found that an increase in the recycle ratio led to a decrease in reactor length but an increase in the
outlet temperature. The design found to be most successful and the choice of design was
introducing the recycle stream and subsequently cooling the feed. This led to an outlet temperature
of 555K and a required mass of catalyst of 1302 kg at steady state, as opposed to the 2430kg which
would have been obtained even if the temperature was not exceeded.
There were two main limitations to the models, namely using a heat capacity of water which not
valid for the liquid phase or pressure range considered. Also the use of the ideal gas law in the
calculations. The first limitation could be overcome by partitioning the heat capacity expression
between the enthalpy of vaporisation and superheating. The second limitation could be overcome
by using more accurate equations of state such as Van der Waal’s; since they would predict phase
change (Gallindo, 2015)(better models would also take into account the change in density due to
mixing). Incorporating a recycle into the reactor induces further limitations in the model. For a high
recycle ratio of α = 0.9, the inlet to the reactor is an order of magnitude greater than it would be
without recycle. However the pressure stays roughly the same (due to the assumptions). This
phenomenon is unphysical and in order to improve this experiments should be conducted on a small
scale recycle model to determine a better pressure relationship.
In the process control part of the report, it was found that the over-tuning of the PI controllers
weakened the performance of the control scheme when trying to reach a wider range of set points.
There were several issues with the chosen control scheme and in hindsight it was wrong to make the
control scheme overly aggressive for a small range of set point changes. In the test, the set point was
increased by 8.8% rather than the maximum 5.9% tested; further work would involve tuning the
control scheme to be more robust by testing a wider range of set point changes. However the fact
that disturbances are introduced to aid the maintenance of the set point is not an accurate
simulation of reality, since disturbance behaviour is random, unexpected and controlled upstream.
The use of well-tuned feed forward controllers would not have been a major issue if the
disturbances in the test were more realistic.
Our MPC performed well in the test but seemed to be less desirable when changing set points in a
smaller range (since it was not as aggressive) and for supressing disturbances. For the MPC
controller used in this report the linearization was based on the crude FOPTD transfer function
approximations on how the inputs effected the output. Improving the flaws of MPC would require a
more rigorous analysis of the process model requiring more time to be spent linearizing the model.
To conclude, MPC is the most favoured control scheme overall. It also has the advantage of having a
user friendly “Designer App” which allows the designer to tune the MPC structure very easily. With
the linearization carried out the MPC saves huge amounts of time in the construction of the control
scheme (only one controller is used and much less tuning is needed).

GROUP 22 26
Nomenclature
Symbol Name Units
CP,i Heat Capacity of substance i J mol-1
K-1
CV Controlled Variable -
Dp Equivalent Particle Diameter m
FF Feed forward Control -
G Superficial Mass Velocity kg s-1
H Enthalpy J mol-1
Ki Gain of process i Effect/Cause
Kij Gain of MVj on CVi -
Ku Ultimate Gain hr ton-1
MD Measured Disturbance -
MPC Model Predictive Control -
MV Manipulated Variable -
Ni Molar flow rate of species i mol s-1
P Pressure bar
Pu Critical Period hr
Q Cooling Duty W
R Ideal Gas Constant J mol-1
K-1
r1 Rate of Reaction 1 mol (kgcats)-1
r2 Rate of Reaction 2 mol (kgcats)-1
T Temperature K
V Reactor Volume m3
z Downstream reactor coordinate m
α Recycle Ratio -
β Fraction of Feed Diverted -
ε Void Fraction -
λ11 Element of Relative Gain Array -
ξ1 Extent of Reaction 1 mol s-1
ξ2 Extent of Reaction 2 mol s-1
τd Differentiation Factor hr
τi Integral Factor hr
νi Stoichiometric coefficient -

GROUP 22 27
Appendix: Reactor Design
Appendix A: Design Equations Derivation
I. Mass of Catalyst
The mass of the catalyst can be considered down the reactor. Note that all the differential equations
will be with respect to z as this is our independent variable.
𝑀𝑎𝑠𝑠 𝑜𝑓 𝑐𝑎𝑡𝑎𝑙𝑦𝑠𝑡 = 𝑚 𝑐 = 𝜌𝑐 × 𝑉𝑐 = 𝜌𝑐 × 𝑉𝑟 × (1 − 𝜖) = 𝜌𝑐 × 𝐴𝑟𝑒𝑎 × 𝑧 × (1 − 𝜖)
Where 𝜌𝑐 is the catalyst density, Vc is the volume of the catalyst, 𝜖 is the void fraction, z is the
distance downstream in the reactor.
We are dealing with constants in the final expression therefore if we take the differential with
respect to z we obtain
𝑑𝑚 𝑐
𝑑𝑧
= 𝜌𝑐 × 𝐴 × (1 − 𝜖)
Substituting in 𝜌𝑐 = 1400kgm-3
A=1m2
and 𝜖 = 0.4 we obtain
𝑑𝑚 𝑐
𝑑𝑧
= 840 𝑜𝑟 𝑚 𝑐 = 840𝑧
II. Mass Balances on Reaction 1:
Reaction 1 is considered as
𝐶𝑂2 + 3𝐻2 ↔ 𝐶𝐻3 𝑂𝐻 + 𝐻2 𝑂
If we consider the rate of methanol synthesis:
We know that the material balance for the product of a reaction is as follows:
𝐼𝑛𝑝𝑢𝑡 − 𝑂𝑢𝑡𝑝𝑢𝑡 + 𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = 𝐴𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛
Since we are considering a steady state in an infinitesimally small volume (diagram above) the
Accumulation term goes to 0.
Following the material balance:
𝑀 𝐶𝐻3𝑜𝐻 𝑁 𝑉 − 𝑀 𝐶𝐻3𝑜𝐻 𝑁 𝑉+𝑑𝑉 + 𝑀 𝐶𝐻3𝑜𝐻 𝑟1,𝐶𝐻3𝑂𝐻(𝜌𝑐 × (1 − 𝜖))∆𝑉 = 0
We have multiplied the rate by the mass of the catalyst in a particular volume as the units of the rate
expression is given in mole/(kg of catalyst x s). Note that 𝑁 𝑉 corresponds to the molar flow rate of
methanol at that specific volume and 𝑀 𝐶𝐻3𝑜𝐻 refers to the molecular mass of methanol.
Fig 1: General schematic of PBR used for balances
z z + Δz dz
(𝑛𝑎) 𝑧

GROUP 22 28
If we cancel the 𝑀 𝐶𝐻3𝑜𝐻 terms and rearrange we obtain the following:
𝑟1,𝐶𝐻3𝑂𝐻(𝜌𝑐 × (1 − 𝜖)) =
𝑁 𝑉+𝑑𝑉 − 𝑁 𝑉
∆𝑉
The right hand side corresponds to a differential which we can simplify:
𝑟1,𝐶𝐻3𝑂𝐻(𝜌𝑐 × (1 − 𝜖)) =
𝑑𝑁
𝑑𝑉𝑟
⇒
𝑑𝑁
𝐴𝑟𝑒𝑎𝑑𝑧
= 𝑟1,𝐶𝐻3𝑂𝐻(𝜌𝑐 × (1 − 𝜖))
Substituting A=1m2
and the other values previously we obtain
𝑑𝑁
𝑑𝑧
= 840𝑟1,𝐶𝐻3𝑂𝐻
Noting that 𝑑𝜀 =
𝑑𝑁
𝑣𝑖
where 𝑣𝑖 is the stoichiometric coefficient which is one for methanol and 𝜀 is the
extent of the reaction, yielding:
𝑑𝜀1
𝑑𝑧
= 840𝑟1,𝐶𝐻3𝑂𝐻
Where r1,CH3OH =
𝐴𝑃 𝐶𝑂2
𝑃 𝐻2
(1−
1
𝐾1
𝑃 𝐶𝐻3𝑂𝐻 𝑃 𝐻2 𝑂
𝑃 𝐶𝑂2
𝑃 𝐻2
3 )
(1+
𝐵
𝑃 𝐻2
𝑃 𝐻2 𝑂+𝐶𝑃 𝐻2
0.5+𝐷𝑃 𝐻2 𝑂)3
Plugging in the expression for the rate we obtain
𝑑𝜀1
𝑑𝑧
=
840𝐴𝑃𝐶𝑂2
𝑃 𝐻2
(1 −
1
𝐾1
𝑃𝐶𝐻3𝑂𝐻 𝑃 𝐻2 𝑂
𝑃𝐶𝑂2
𝑃𝐻2
3 )
(1 +
𝐵
𝑃 𝐻2
𝑃 𝐻2 𝑂 + 𝐶𝑃𝐻2
0.5
+ 𝐷𝑃 𝐻2 𝑂)3
For the values of the different coefficients check the next page.
III Mass Balance on Reaction 2
The following is essentially a repeat of the above with the species being changed.
Reaction 2 is defined as
𝐶𝑂2 + 𝐻2 ↔ 𝐶𝑂 + 𝐻2 𝑂
If we consider the rate of CO synthesis:
We know that the material balance for the product of a reaction is as follows:
𝐼𝑛𝑝𝑢𝑡 − 𝑂𝑢𝑡𝑝𝑢𝑡 + 𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = 𝐴𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛
Since we are considering a steady state in an infinitesimally small volume (diagram above) the
Accumulation term goes to 0.
Following the material balance:
𝑀 𝐶𝑂 𝑁 𝑉 − 𝑀 𝐶𝑂 𝑁 𝑉+𝑑𝑉 + 𝑀 𝐶𝑂 𝑟2,𝐶𝑂(𝜌𝑐 × (1 − 𝜖))∆𝑉 = 0

GROUP 22 29
We have multiplied the rate by the mass of the catalyst in a particular volume as the units of the rate
expression is given in mole/(kg of catalyst x s). Note that 𝑁 𝑉 corresponds to the molar flow rate of
carbon monoxide at that specific volume and 𝑀 𝐶𝑂 refers to the molar mass.
If we cancel the 𝑀 𝐶𝑂 terms and rearrange we obtain the following:
𝑟2,𝐶𝑂(𝜌𝑐 × (1 − 𝜖)) =
𝑁 𝑉+𝑑𝑉 − 𝑁 𝑉
∆𝑉
The right hand side corresponds to a differential which we can simplify:
𝑟1,𝐶𝐻3𝑂𝐻(𝜌𝑐 × (1 − 𝜖)) =
𝑑𝑁
𝑑𝑉𝑟
⇒
𝑑𝑁
𝐴𝑟𝑒𝑎𝑑𝑧
= 𝑟1,𝐶𝐻3𝑂𝐻(𝜌𝑐 × (1 − 𝜖))
Substituting A=1m2
and the other values previously we obtain
𝑑𝑁
𝑑𝑧
= 840𝑟2,𝐶𝑂
Noting that 𝑑𝜀 =
𝑑𝑁
𝑣𝑖
where 𝑣𝑖 is the stoichiometric coefficient which is one for methanol and 𝜀 is the
extent of the reaction, yielding:
𝑑𝜀2
𝑑𝑧
= 840𝑟2,𝐶𝑂
Where r2,CO =
𝐸𝑃 𝐶𝑂2
(1−
𝐾2 𝑃 𝐶𝑂 𝑃 𝐻2 𝑂
𝑃 𝐶𝑂2
𝑃 𝐻2
)
1+
𝐵
𝑃 𝐻2
𝑃 𝐻2 𝑂+𝐶𝑃 𝐻2
0.5+𝐷𝑃 𝐻2 𝑂
Plugging in the expression for the rate we obtain
𝑑𝜀2
𝑑𝑧
=
840𝐸𝑃𝐶𝑂2
(1 −
𝐾2 𝑃𝐶𝑂 𝑃 𝐻2 𝑂
𝑃𝐶𝑂2
𝑃 𝐻2
)
1 +
𝐵
𝑃 𝐻2
0.5
+ 𝐷𝑃 𝐻2 𝑂
𝐴 = 0.55𝑒
36696
𝑅𝑇
𝐵 = 3454.38
𝐶 = 0.499𝑒
17197
𝑅𝑇
𝐷 = 6.62 × 10−11
𝑒
124119
𝑅𝑇
𝐸 = 0.61 × 1010
𝑒
−94765
𝑅𝑇
𝐾1 = 10
3066
𝑇
−10.592
in bar-2
𝐾2 = 10
2073
𝑇
−2.029
IV. Partial Pressures:
One will notice that partial pressures are heavily involved in our extent of reaction balances. In order
to deal with this we will use the relationship:
𝑃𝑎𝑟𝑡𝑖𝑎𝑙 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 (𝑃𝑖) = 𝑇𝑜𝑡𝑎𝑙 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 (𝑃) × 𝑚𝑜𝑙𝑒 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 (𝑦𝑖)
The total pressure differential equation is given and will be shown later.
As far as the mole fraction is concerned, we will define this as

GROUP 22 30
𝑦𝑖 =
𝑚𝑜𝑙𝑎𝑟 𝑓𝑙𝑜𝑤 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑖 (𝑁𝑖)
𝑡𝑜𝑡𝑎𝑙 𝑚𝑜𝑙𝑎𝑟 𝑓𝑙𝑜𝑤 𝑟𝑎𝑡𝑒 (𝑁𝑡)
The molar flow rate at any point can be given by
𝑁𝑖 = 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑚𝑜𝑙𝑎𝑟 𝑓𝑙𝑜𝑤 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑖 (𝑁𝑖0) + 𝑣𝑖1 𝜀1 + 𝑣𝑖2 𝜀2
Where 𝑣𝑖1is the stoichiometric coefficient for reaction 1, negative for a reactant and positive for a
product. A similar definition is used for 𝑣𝑖2
Listing the molar flow rates:
𝑁𝐶𝑂2
= 𝑁𝐶𝑂20
− 𝜀1 − 𝜀2
𝑁 𝐻2
= 𝑁 𝐻20
− 3𝜀1 − 𝜀2
𝑁𝐶𝐻3𝑂𝐻 = 𝑁𝐶𝐻3𝑂𝐻0
+ 𝜀1
𝑁 𝐻2𝑂 = 𝑁 𝐻2𝑂0
+ 𝜀1 + 𝜀2
𝑁𝐶𝑂 = 𝑁𝐶𝑂0
+ 𝜀2
Summing over all flow rates we obtain
𝑁𝑡 = 𝑁𝑡0 − 2𝜀1
Where 𝑁𝑡0 = 𝑁𝐶𝑂20
+ 𝑁 𝐻20
+ 𝑁𝐶𝐻3𝑂𝐻0
+ 𝑁 𝐻2𝑂0
+ 𝑁𝐶𝑂0
Therefore 𝑦 𝐶𝑂2
=
𝑁 𝐶𝑂20−𝜀1−𝜀2
𝑁𝑡0−2𝜀1
=
𝑁 𝐶𝑂2
𝑁𝑡
Similarly 𝑦 𝐻2
=
𝑁 𝐻20
−3𝜀1−𝜀2
𝑁𝑡0−2𝜀1
=
𝑁 𝐻2
𝑁𝑡
𝑦 𝐶𝐻3𝑂𝐻 =
𝑁 𝐶𝐻3𝑂𝐻0
+𝜀1
𝑁𝑡0−2𝜀1
=
𝑁 𝐶𝐻3𝑂𝐻
𝑁𝑡
𝑦 𝐻2 𝑂 =
𝑁 𝐻2𝑂0
+ 𝜀1 + 𝜀2
𝑁𝑡0 − 2𝜀1
=
𝑁 𝐻2 𝑂
𝑁𝑡
𝑦 𝐶𝑂 =
𝑁 𝐶𝑂0
+𝜀2
𝑁𝑡0−2𝜀1
=
𝑁 𝐶𝑂
𝑁𝑡
Defining the partial pressures:
𝑃𝐶𝑂2
=𝑦 𝐶𝑂2
× 𝑃
𝑃 𝐻2
=𝑦 𝐻2
× 𝑃
𝑃𝐶𝐻3𝑂𝐻=𝑦 𝐶𝐻3𝑂𝐻 × 𝑃
𝑃 𝐻2 𝑂=𝑦 𝐻2 𝑂 × 𝑃
𝑃𝐶𝑂=𝑦 𝐶𝑂 × 𝑃
V. Pressure Equation
As can be seen all the partial pressure are dependent on the Total Pressure. The Pressure equation
given in the brief is as follows:
𝑑𝑃
𝑑𝑧
=
−1.75𝐺2
(1 − 𝜖)
𝐷 𝑝 𝜌𝜖3

GROUP 22 31
Where G is the superficial mass velocity (mass/(time x area))
Dp = equivalent particle diameter = 7.04 x 10-3
m
ρ = gas density.
The gas density changes as we go downstream in the reactor, as the composition of each species
changes.
We can use the ideal gas equation to calculate the density of the gas stream:
𝜌 =
𝑃𝑀
𝑅𝑇
Where T is the temperature of the gas mixture
R is the ideal gas constant (8.314J/molK)
M is the molecular weight of the gas mixture.
The molecular weight of the gas mixture changes as we go downstream. So does the Pressure and
the Temperature but these are taken care of through differential equations. We need to express the
molecular weight change through extent of reactions. To do this we will say that the molecular
weight of the gas mixture is a weighted average of the molecular weight of the components in the
gas mixture as follows:
𝑀 = ∑ 𝑦𝑖 ×
𝑛
𝑖
𝑀𝑖
Where Mi = Molecular weight of component ‘i’.
The mole fractions are already defined using extents of reaction above. Therefore we can substitute
this into the expression with the molecular weight of each component.
𝑀 = (𝑦 𝐶𝑂2
× 44) + (𝑦 𝐻2
× 2) + (𝑦 𝐶𝐻3𝑂𝐻 × 32) + (𝑦 𝐻2 𝑂 × 18) + (𝑦 𝐶𝑂 × 28)
We can substitute this expression into the expression for the density (ideal gas equation) and then
substitute that gas density into the pressure drop equation. We do not need to do this directly as we
already have specified the terms necessary and MATLAB will make the necessary substitutions. We
also know the feed pressure is 305 bar in order to solve the equation.
Looking at the G value we can see that due to conservation of mass the mass flow rate will stay
constant at all times during the reactor, and the cross sectional area will not change either.
Therefore we only need to evaluate this value at the inlet and then it can be used throughout. We
will do so using the following equation
𝐺 =
𝑁𝑡0 × 𝑀 𝑜
𝐴𝑟𝑒𝑎
Where M0 is the molecular weight at the inlet. To obtain this we look at our feed conditions for Nto
which in our case is 6200mol/s. The molar compositions at the inlet are specified, so we can use
these for M0
𝑀 = (0.22 × 28) + (0.71 × 2) + (0.01 × 32) + (0.02 × 18) + (0.04 × 44) = 10.04

GROUP 22 32
This gives a G of:
𝐺 =
6200 × 10.04 × 10−3
1
= 62.2 𝑘𝑔 𝑚−2
𝑠−1
VI. Energy Balance
The energy balance is done considering an infinitesimally small volume element as shown in Fig 34
and considering the various energies entering and leaving that element. The equation used to model
this is:
𝐼𝑛𝑝𝑢𝑡 − 𝑂𝑢𝑡𝑝𝑢𝑡 + 𝐻𝑒𝑎𝑡 𝑇𝑟𝑎𝑛𝑠𝑓𝑒𝑟 = 𝐴𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛
Plugging in terms we have:
𝑁 𝑇 𝐻|𝑧|−𝑁 𝑇 𝐻|𝑧+𝑑𝑧| + 𝑑𝑄 = 0
Where H is the total enthalpy coming into the volume element and Q is the specific heat being
transferred to/from the reactor. We can consider the distance down the reactor instead of the
volume as the cross-sectional area is just one metre squared. Dividing by dz we obtain:
𝑁 𝑇
𝐻
𝑑𝑧|𝑧|
−𝑁 𝑇
𝐻
𝑑𝑧|𝑧+𝑑𝑧|
+
𝑑𝑄
𝑑𝑧
= 0
Rearranging this expression we obtain the heat balance in differential form. Note that since the
reactor is adiabatic the Q term is neglected.
−
𝑑
𝑑𝑧
(𝑁 𝑇 𝐻) = 0
The total enthalpy coming into the element multiplied by the total molar flow rate can be expressed
using the sum of the enthalpy of each component multiplied by the molar flow rate of that
component. This is expressed below. Note that the enthalpy of mixing and enthalpy change due to
pressure drop have been neglected (Ideal gas).
𝑑
𝑑𝑧
(∑ 𝑁𝑖 𝐻𝑓𝑖
𝑛
𝑖
) = 0
Where 𝐻𝑓 𝑖
is the formation enthalpy for the component.
Fig. 34 General schematic of PBR used for balances
z z + Δz dz
(𝑛𝑡) 𝑧
(𝐻𝑓) 𝑧
(𝐻𝑓) 𝑧

GROUP 22 33
𝐻𝑓𝑖
= 𝐻𝑓 𝑖
𝑇𝑟𝑒𝑓
+ ∫ 𝐶𝑝𝑖 𝑑𝑇
𝑇
𝑇𝑟𝑒𝑓
Where Tref is the reference temperature for the standard head of formation.
Using the product rule:
∑ 𝑁𝑖
𝑑𝐻𝑓𝑖
𝑑𝑧
𝑛
𝑖
+ ∑ 𝐻𝑓𝑖
𝑑𝑁𝑖
𝑑𝑧
𝑛
𝑖
= 0
The highlighted term corresponds to the rate. However we are going to substitute the negative of
the rate as we will assume all species are consumed, due to the sign convention. We can also state
that the formation enthalpy differential dHfi = Cpi dT. This therefore simplifies to:
𝑑𝑇
𝑑𝑧
𝑛
𝑖
− 840 ∑ 𝐻𝑓𝑖
𝑟𝑖
𝑛
𝑖
= 0
Recalling our reactions 1 and 2 we can obtain expressions for each of the formation enthalpies.
∑ 𝐻𝑓 𝑖
𝑟𝑖
𝑛
𝑖
= 𝐻𝑓 𝐶𝑂2
𝑟𝐶𝑂21+ 𝐻𝑓 𝐻2
𝑟 𝐻21+ 𝐻𝑓 𝐶𝐻3𝑂𝐻
𝑟𝐶𝐻3𝑂𝐻1+ 𝐻𝑓 𝐻2𝑂
𝑟 𝐻2𝑂1+ 𝐻𝑓 𝐶𝑂2
𝑟𝐶𝑂22+ 𝐻𝑓 𝐻2
𝑟 𝐻22+ 𝐻𝑓 𝐻2𝑂
𝑟 𝐻2𝑂2+ 𝐻𝑓 𝐶𝑂
𝑟𝐶𝑂2
Where the first subscript refers to the species and the second subscript refers to the reaction
number.
We can define the rates of a reaction and equate the rates of each species to this rate to obtain a
more simplified expression:
𝑟1 = 𝑟𝐶𝑂21=
1
3
𝑟 𝐻21=− 𝑟𝐶𝐻3𝑂𝐻1= − 𝑟 𝐻2𝑂1
𝑟2 = 𝑟𝐶𝑂22= 𝑟 𝐻22=− 𝑟 𝐻2𝑂2= − 𝑟𝐶𝑂2
Therefore the sum can be simplified. We will also substitute the 𝐶𝑝𝑑𝑇 terms into the equation.
∑ 𝐻𝑓 𝑖
𝑟𝑖
𝑛
𝑖
= 𝑟1(𝐻𝑓 𝐶𝑂2
𝑇𝑟𝑒𝑓
+ ∫ (𝐶𝑝 𝐶𝑂2 𝑑𝑇
𝑇
𝑇𝑟𝑒𝑓
) + 3𝐻𝑓 𝐻2
𝑇𝑟𝑒𝑓
+ 3 ∫ (𝐶𝑝 𝐻2 𝑑𝑇
𝑇
𝑇𝑟𝑒𝑓
) − 𝐻𝑓 𝐶𝐻3𝑂𝐻
𝑇𝑟𝑒𝑓
− ∫ (𝐶𝑝 𝐶𝐻3𝑂𝐻 𝑑𝑇
𝑇
𝑇𝑟𝑒𝑓
)
− 𝐻𝑓𝐻2𝑂
𝑇𝑟𝑒𝑓
− ∫ (𝐶𝑝 𝐻20 𝑑𝑇
𝑇
𝑇𝑟𝑒𝑓
)) + 𝑟2(𝐻𝑓 𝐶𝑂2
𝑇𝑟𝑒𝑓
+ ∫ (𝐶𝑝 𝐶𝑂2 𝑑𝑇
𝑇
𝑇𝑟𝑒𝑓
) + 𝐻𝑓 𝐻2
𝑇𝑟𝑒𝑓
+ ∫ (𝐶𝑝 𝐻2 𝑑𝑇
𝑇
𝑇𝑟𝑒𝑓
)
− 𝐻𝑓𝐻2𝑂
𝑇𝑟𝑒𝑓
− ∫ (𝐶𝑝 𝐻20 𝑑𝑇
𝑇
𝑇𝑟𝑒𝑓
) − 𝐻𝑓𝐶𝑂
𝑇𝑟𝑒𝑓
− ∫ (𝐶𝑝 𝐶𝑂 𝑑𝑇
𝑇
𝑇𝑟𝑒𝑓
))
Noting that the Enthalpy of reaction is the enthalpy of the products – enthalpy of reactants, we can
substitute this into the above equation as a negative enthalpy of reaction with the CpdT correction
based on the stoichiometry of the species being considered in that particular reaction. This can be
written as

GROUP 22 34
∑ 𝐻𝑓 𝑖
𝑟𝑖
𝑛
𝑖
= 𝑟1(−∆𝐻𝑟1
𝑇𝑟𝑒𝑓
+ 𝑣𝑖 ∫ (𝐶𝑝𝑖 𝑑𝑇
𝑇
𝑇𝑟𝑒𝑓
)) + 𝑟2(−∆𝐻𝑟2
𝑇𝑟𝑒𝑓
𝑇
𝑇𝑟𝑒𝑓
))
where vi is the stoichiometric coefficient of a species in that particular reactant, defined as being
positive for a reactant and negative for a product in this case. (eg: 3 for H2 in reaction 1).
Substituting this back into the original equation we obtain:
𝑑𝑇
𝑑𝑧
𝑛
𝑖
− 840( 𝑟1(−∆𝐻 𝑟1
𝑇𝑟𝑒𝑓
+ 𝑣𝑖 ∫(𝐶𝑝𝑖
𝑑𝑇
𝑇
𝑇𝑟𝑒𝑓
)) + 𝑟2(−∆𝐻 𝑟2
𝑇𝑟𝑒𝑓
+ 𝑣𝑖 ∫(𝐶𝑝𝑖
𝑑𝑇
𝑇
𝑇𝑟𝑒𝑓
))) = 0
Rearranging for dT/dz we obtain:
𝑑𝑇
𝑑𝑧
=
840( 𝑟1(−∆𝐻 𝑟1
𝑇𝑟𝑒𝑓
+ 𝑣𝑖 ∫ (𝐶𝑝𝑖
𝑑𝑇
𝑇
𝑇𝑟𝑒𝑓
)) + 𝑟2(−∆𝐻 𝑟2
𝑇𝑟𝑒𝑓
𝑑𝑇
𝑇
𝑇𝑟𝑒𝑓
)))
𝑛
𝑖
Plugging in the rate we obtain:
𝑑𝑇
𝑑𝑧
=
840(
𝐴𝑃𝐶𝑂2
𝑃𝐻2
(1 −
1
𝐾1
𝑃𝐶𝐻3𝑂𝐻 𝑃 𝐻2 𝑂
𝑃𝐶𝑂2
𝑃𝐻2
3 )
(1 +
𝐵
𝑃𝐻2
𝑃𝐻2 𝑂 + 𝐶𝑃𝐻2
0.5
+ 𝐷𝑃𝐻2 𝑂)3
(−∆𝐻𝑟1
𝑇𝑟𝑒𝑓
𝑇
𝑇𝑟𝑒𝑓
)) +
𝐸𝑃𝐶𝑂2
(1 −
𝐾2 𝑃𝐶𝑂 𝑃𝐻2 𝑂
𝑃𝐶𝑂2
𝑃𝐻2
)
1 +
𝐵
𝑃𝐻2
0.5
+ 𝐷𝑃 𝐻2 𝑂
(−∆𝐻𝑟2
𝑇𝑟𝑒𝑓
𝑇
𝑇𝑟𝑒𝑓
)))
𝑛
𝑖
We are given:
∆𝐻𝑟1
𝑇𝑟𝑒𝑓
= -48980J/mol at 298K =Tref
∆𝐻𝑟2
𝑇𝑟𝑒𝑓
= 41000J/mol at 298K = Tref
CpCH3OH = 19.037 +9.146 x10-2
T-1.217 x 10-5
T2
-8.033 x10-9
T3
CpCO = 28.142 +0.167 x 10-2
T -0.537 x 10-5
T2
-2.22 x10-9
T3
CpH2 = 29.087 -0.191 x10-2
T +0.4 x 10-5
T2
-0.87 x 10-9
T3
CpCO2 = 19.8 +7.344 x10-2
T -5.602 x 10-5
T2
+17.15 x 10-9
T3
CpH20 = 32.217 +0.192 x10-2
T +1.055 x 10-5
T2
-3.593 x 10-9
T3
The molar flow rates have already been defined before on page 3.
The rates have previously also been expressed as functions of extents of reactions.
In order to conduct the integral of CpdT we will use the integral function in MATLAB from the T ref of
298K to the temperature at a particular point in z. We know the feed temperature is 501K to solve
the ODE.
Appendix B: Justification of Plots
Here we will justify why the plots created have the shape they do analytically.
I. Temperature against ξ1
As can be seen in Fig. ___ this produces a line of approximately constant gradient. We can relate this
back to the energy balance. If we divide through our energy balance with respect to z by our mass
balance with respect to z we obtain:
𝑑𝑇
𝑑𝜀1
=
840( 𝑟1(−∆𝐻 𝑟1
𝑇𝑟𝑒𝑓
𝑑𝑇
𝑇
𝑇𝑟𝑒𝑓
)) + 𝑟2(−∆𝐻 𝑟2
𝑇𝑟𝑒𝑓
𝑑𝑇
𝑇
𝑇𝑟𝑒𝑓
)))
840 𝑟1 ∑ 𝑁𝑖 𝐶𝑝𝑖
𝑛
𝑖

GROUP 22 35
Simplifying we obtain:
𝑑𝑇
𝑑𝜀1
=
((−∆𝐻 𝑟1
𝑇𝑟𝑒𝑓
𝑑𝑇
𝑇
𝑇𝑟𝑒𝑓
))
𝑛
𝑖
+
𝑟2
𝑟1
(−∆𝐻 𝑟2
𝑇𝑟𝑒𝑓
𝑑𝑇
𝑇
𝑇𝑟𝑒𝑓
)))
𝑛
𝑖
If the line we obtain is approximately a straight line, we must show that the right hand side of this
equation is approximately constant. If we look at the correction terms we can find that they are
actually negligible compared to the heats of reaction. For example for the highest temperature
permitted of 563K, the correction term is approximately 4 J mol-1
while the heats of reaction are 4
orders of magnitude higher. We did not cross them out in our energy balance as this would strictly
be incorrect but if we want to prove the gradient is approximately constant we can see that the
correction terms will not have a significant influence. Therefore we can simplify the equation to:
𝑑𝑇
𝑑𝜀1
=
(−∆𝐻 𝑟1
𝑇𝑟𝑒𝑓
)
𝑛
𝑖
+
𝑟2
𝑟1
(−∆𝐻 𝑟2
𝑇𝑟𝑒𝑓
)
𝑛
𝑖
We already know the standard heats of reaction are constants. The ∑ 𝑁𝑖 𝐶𝑝𝑖
𝑛
𝑖 terms can be shown to
be approximately constant. To do this we say ∑ 𝑁𝑖 𝐶𝑝𝑖
𝑛
𝑖 = ∑ 𝑁𝑖 × ∑ 𝐶𝑝𝑖 =𝑛
𝑖
𝑛
𝑖 𝑁 𝑇 ∑ 𝐶𝑝𝑖
𝑛
𝑖 .The
individual Cp terms can be shown to remain relatively constant as the plot shows (fig 16). Meanwhile
the total flow rate follows the expression NT0 -2ξ1. This changes by a relatively small amount down
the reactor. For example at the target production of 109mol/s we obtain a total flow rate of
5982mol/s compared to the feed flow rate of 6200mol/s, a 3.5% change which is relatively low.
Therefore if the ratio of the rates remains constant, we have shown that the gradient is constant. It
can be seen that the ratio is constant as Fig(5) shows through the constant gradient. This is because
the gradient of the graph of ξ1 against ξ2 is the ratio of the rates which is evident through the division
of equation 1 and equation 2.
Appendix C: Alternative Designs
I. Cold shot cooling
Cold shot cooling involves diverting a fraction of the feed, β, so that it can be mixed with the outlet
stream of a first reactor and then fed into a second reactor (Kogelbauer, 2015). This has the effect of
cooling the stream, and be can done for n reactors with the feed split into β/n parts (although we
only did this for 2 reactors). Like before requires mass and energy balances around the mixing point
in order to find the inlet conditions to the second reactor.
We attempted to model cold shot cooling as described in Fig. 35. There are 3 parameters that we
can change in cold shot cooling, the fraction of feed diverted, the length at which we cut off the 1st
reactor and by how much the feed is cooled (the outlet temperature of the 2nd
reactor is determined
Fig 35: Cold shot Cooling Schematic
Commented [KK13]: Need to include plot here.
Commented [KK14]: Is this necessary, we never even did
this lol

GROUP 22 36
by our production target). It proved difficult finding an optimum combination of these so instead we
plotted temperature against moles produced as well as length of the reactor to identify any general
trends.
In the plot below we have plotted outlet temperature against moles produced. We did this using a
simple energy balance at the outlet of the first reactor, followed by an analogous analysis to the first
reactor to determine the temperature at that outlet of the second reactor. The plotted results are
shown below in Fig. 36.
The blue lines correspond to the design target. As can be seen the design again fails to meet the
target specification, as the temperature exceeds the temperature limit for the catalyst before we hit
target production. We exceed the temperature by approximately 20K which is the same as originally.
For this we have used β=0.5 and stopped the reactor at 555K as is visible due to the green line
stopping at 550K. A reason for the increased temperature gradient is that there is a lower inlet feed
flow rate, causing the denominator of the temperature differential equation to decrease causing a
higher gradient. The reason for the almost identical end result is eventually, after the feed meets the
outlet stream of the first reactor, we have the same material in the reactor, and only the partial
pressures are slightly different. This causes the cold shot trend to converge to the adiabatic reactor
fairly quickly with only a small deviation due to partial pressure changes.
Interestingly, we can see that as cold shot cooling is observed, the length of the reactor decreases,
portrayed by Fig. 37. The Fig. corresponds to cold shot cooling at β=0.5. If we look at 100mol/s for
example, with cold shot cooling we hit this production at 2.25m while without cooling, this is at
2.8m. This also therefore improves the efficiency of the catalyst as the length is minimised, but again
the temperature limit means the technique isn’t possible in our design. The reason as to why the
length of the reactor decreases with cold shot cooling is a direct consequence of above. The lower
initial flow rate means a higher temperature can be reached quicker and therefore the target
production is met sooner. This can be seen since if we compare the cold shot cooling to the adiabatic
system, the gradient of the graph is steeper for cold shot compared to the adiabatic reactor only in
the first reactor. In the second reactor, when the total feed in the reactor is the same the gradients
are the same (lines are parallel). This evidences our theory of the flow-rate in the reactor being the
primary factor.
Fig 36. Temperature against moles of Methanol produced for β=0.5

GROUP 22 37
II. Inter-stage Cooling
Inter stage cooling uses 2 or more reactors with heat exchangers in between each one. Doing so
requires only an energy balance if we assume that no reaction takes place other than in the reactor,
which is reasonable since the reaction requires a catalyst to occur. The two parameters we can
change in this set up is the length of the first reactor and the cooling duty supplied between the
reactors.
F1 F2 F2 F3
The following graph (fig 39) shows the variation in temperature output of the second reactor (blue
curve) and reactor length (red curve), given that the inlet temperature of the second reactor is 555K.
Note that at every point, the production target of 108mol/s is being met.
Most importantly from this plot, we can see that the design target production is now being met
without a concern of the temperature increasing to deactivate the catalyst, as long as the heat
removed is greater than 4.5 MJ. In order to give an indication of how much coolant this required, if
saturated condensing steam is used at 1 bar, we require a flowrate of 2L/s or greater. In general, as
the heat removed increases, the reactor length required increases. This makes sense as the more we
cool the outlet of the reactor the slower the rate of reaction and therefore the reactor needs to be
longer to meet that target output. The temperature of the outlet decreases as the reactor cooling
duty increases as we are effectively decreasing the feed temperature of the second reactor resulting
in a linear change. The linear trend also implies that the heat capacities must not change to a
significant extent with temperature as otherwise the trend would have a varying gradient. If the heat
capacities remain relatively constant however, we can say that Q=Cp∫dT=Cp∆T. Fig (…) confirms the
releatively constant heat capcaities.
Fig 38. Inter stage Cooling Schematic
Fig 37. Comparison of Cold Shot Cooling with Normal Design for Reactor
Length
Commented [K15]: This defo needs some work. I feel like
we’ve done some smart shizz here but the graphs are way
too complicated rn. Ultimately we want to relate
Temperature to moles of extent of reaction, show our lines
of limiting value, and show that our operating point is below
this. I’m not sure how to put zhi 1 on these graphs however.
This should be the highlight of our report as this is how we
actually solve the problem. Not quite sure how to
communicate this section properly but perhaps Seb can help
with this.
Commented [GS16]: Nah, it isn’t necessary, put it in
Appendix

GROUP 22 38
Appendix D: Sample Calculations
I. Initial gradients
I1. Initial conditions
Temperature 𝑇0 = 501 °𝐾
Pressure 𝑃0 = 305 𝑏𝑎𝑟
Total feed flowrate 𝑛 𝑇0 = 501 °𝐾 with mole fractions 𝑦 𝐶𝑂 = 0.22, 𝑦 𝐻2
= 0.71, 𝑦 𝐶𝐻3 𝑂𝐻 = 0.01,
𝑦 𝐶𝑂2
= 0.04, 𝑦 𝐻2 𝑂 = 0.02
I2. Constants
These can be found in the design brief
I3. ξ1 and ξ2 gradient
For this we only need the rate of reaction 1 and 2.
First, the partial pressures which are simply the total pressure multiplied by the mole fractions:
𝑃𝐶𝑂 = 67.1 𝑏𝑎𝑟
𝑃 𝐻2
= 216.55 𝑏𝑎𝑟
𝑃𝐶𝐻3 𝑂𝐻 = 3.05 𝑏𝑎𝑟
𝑃𝐶𝑂2
= 12.2 𝑏𝑎𝑟
𝑃 𝐻2 𝑂 = 6.1 𝑏𝑎𝑟
Now for the equilibrium constants at 501 °𝐾:
𝐾1 = 3.371 × 10−5
𝑏𝑎𝑟−2
𝐾2 = 1.284 × 102
Fig 39. Comparing Temperature Outlet of 2nd
reactor and Heat against Reactor Length for
second Inlet temperature =510K

GROUP 22 39
The rate constants (units not given in brief):
𝐴 = 3.865 × 103
𝐵 = 3.453 × 103
𝐶 = 3.865 × 103
𝐷 = 30.98
𝐸 = 0.8030
We’re now able to evaluate the rates of reaction as outlined in the brief, giving:
𝑟1 = 4.1934 × 10−4
𝑚𝑜𝑙 𝑘𝑔−1
𝑠−1
𝑟2 = −0.04401 𝑚𝑜𝑙 𝑘𝑔−1
𝑠−1
The kg term comes from the fact that the rate is defined in terms of kg of catalyst present.
Using the design equation that was defined previously, the gradients therefore are:
𝑑𝜉1
𝑑𝑧
= 0.3523 𝑚𝑜𝑙 𝑠−1
𝑚−1
𝑑𝜉2
𝑑𝑧
= −36.96 𝑚𝑜𝑙 𝑠−1
𝑚−1
I4. Pressure gradient
For the pressure gradient, we first need to calculate the density. To do this, we use the following
equation:
𝜌 =
∑ 𝑀𝑟𝑖 𝑃𝑖
𝑅𝑇
We need to know the molar masses of each component:
𝑀𝑟𝐶𝑂 = 0.028 𝑘𝑔 𝑚𝑜𝑙−1
𝑀𝑟 𝐻2
= 0.002 𝑘𝑔 𝑚𝑜𝑙−1
𝑀𝑟𝐶𝐻3 𝑂𝐻 = 0.032 𝑘𝑔 𝑚𝑜𝑙−1
𝑀𝑟𝐶𝑂2
= 0.018 𝑘𝑔 𝑚𝑜𝑙−1
𝑀𝑟 𝐻2 𝑂 = 0.044 𝑘𝑔 𝑚𝑜𝑙−1
We now calculate the density using the partial pressures (making sure to convert partial pressure to
Pa):
𝜌 = 69.855 𝑘𝑔 𝑚−3
Putting this value back into the pressure drop equation:
𝑑𝑃
𝑑𝑧
= −1.288 𝑏𝑎𝑟 𝑚−1

GROUP 22 40
I5. Temperature gradient
The reaction doesn’t occur at standard temperature, so to use the heats of reaction given we need
to also integrate the heat capacities of our reactants and products to the temperature at which our
reaction is taking place. This is shown in our energy equation.
The integrated heat capacities are:
∫ 𝐶𝑝 𝐶𝑂 𝑑𝑇
𝑇=501
𝑇=298
= 5.995 × 103
𝐽 𝑚𝑜𝑙−1
∫ 𝐶𝑝 𝐻2
𝑑𝑇
𝑇=501
𝑇=298
= 5.870 × 103
∫ 𝐶𝑝 𝐶𝐻3 𝑂𝐻 𝑑𝑇
𝑇=501
𝑇298
= 1.077 × 104
∫ 𝐶𝑝 𝐶𝑂2
𝑑𝑇
𝑇=501
𝑇298
= 8.358 × 103
∫ 𝐶𝑝 𝐻2 𝑂 𝑑𝑇
𝑇=501
𝑇298
= 6.995 × 103
The heats of reaction at temperature
∆𝐻𝑟1
𝑇=298
+ 𝜐𝑖,1 ∫ 𝐶𝑝𝑖 𝑑𝑇
𝑇=501
𝑇
= −5.718 × 104
∆𝐻𝑟2
𝑇=298
+ 𝜐𝑖,2 ∫ 𝐶𝑝𝑖 𝑑𝑇
𝑇=501
𝑇=298
= 3.976 × 104
Here 𝜐𝑖,𝑗 is the stoichiometric coefficient of component I in reaction j.
It should be noted that in the derivation of the energy balance, a difference in definition leas to a minus sign in
front of the heats of reaction. The above definition is used since this is how it was written in the MATLAB code,
however both are correct.
We also need to know the molar flow rate multiplied by the heat capacities, since this is the denominator of
our energy balance. First we calculate the heat capacities at 𝑇 = 501 °𝐾:
𝐶𝑝 𝐶𝑂(𝑇 = 501) = 30.04 𝐽 𝑚𝑜𝑙−1
𝐶𝑝 𝐻2
(𝑇 = 501) = 29.03 𝐽 𝑚𝑜𝑙−1
𝐶𝑝 𝐶𝐻3 𝑂𝐻(𝑇 = 501) = 60.79 𝐽 𝑚𝑜𝑙−1
𝐶𝑝 𝐶𝑂2
(𝑇 = 501) = 44.69 𝐽 𝑚𝑜𝑙−1
𝐶𝑝 𝐻2 𝑂(𝑇 = 501) = 35.38 𝐽 𝑚𝑜𝑙−1
Our flow rates are simply the initial mole fractions multiplied by our initial flow rate of 6.2 𝑘𝑚𝑜𝑙 𝑠−1
:
𝑛 𝐶𝑂 = 1364 𝑚𝑜𝑙 𝑠−1

Reactor Design and Control for Methanol Synthesis (Group 22

Reactor Design and Control for Methanol Synthesis (Group 22

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