Characterization and scale-down of flow reactors for applications at small-scale manufacturing Margarida Marques da Silva Dias Coutinho Thesis to obtain the Master of Science Degree in Chemical Engineering Supervisors: Doutora Susana Isabel Massena do Nascimento Professor Doutor Francisco Manuel da Silva Lemos Examination Committee Chairperson: Professor Doutor Carlos Manuel Faria de Barros Henriques Supervisor: Doutora Susana Isabel Massena do Nascimento Members of the Committee: Professor Doutor José Manuel Félix Madeira Lopes November 2017
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Characterization and scale-down of flow reactors for
applications at small-scale manufacturing
Margarida Marques da Silva Dias Coutinho
Thesis to obtain the Master of Science Degree in
Chemical Engineering
Supervisors:
Doutora Susana Isabel Massena do Nascimento
Professor Doutor Francisco Manuel da Silva Lemos
Examination Committee
Chairperson: Professor Doutor Carlos Manuel Faria de Barros Henriques
Supervisor: Doutora Susana Isabel Massena do Nascimento
Members of the Committee: Professor Doutor José Manuel Félix Madeira Lopes
November 2017
i
Acknowledgments
First of all I would like to thank Hovione FarmaCiência for the opportunity of this internship. This
experience was undoubtedly very rich not only in terms of technical knowledge, but also in terms of
personal experience. To meet many different people with different functions in the company made me
understand the importance of each one to ensure the good functioning and production of the company.
I want to express my gratitude to Ruth and Susana for the orientation during the internship, for the
time dedicated to me whenever I needed and for the knowledge shared. What I have learnt with them
will be a major benefit in the future.
I would like to thank professor Francisco Lemos, my academic supervisor for the precious help in
my thesis, for the patience and for the long hours dedicated to me. I also want to thank professor Amélia
Lemos, who although not being my supervisor, always received me so well and helped me when I
needed.
An infinite “thank you” to Mafalda for being my big company, support and friend during the
internship; to have shared with me all the moments during these months and to made this internship
much happier.
A word of appreciation to the chemists Rudi and Ana who helped me a lot during the internship and
always received me so well. In addition, to Rudi for the laboratory tests which data I used in the
development of this thesis.
Last but not least, I would like to thank my family for all the support and love. For always listening
to my problems and giving their opinion that I truly appreciate.
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iii
Resumo
A produção em contínuo tem vindo a ganhar destaque na indústria farmacêutica pela excelente
transferência de calor e massa que oferecem, que permite a possibilidade de intensificar os processos
usando novas gamas de operação que resultam num aumento da conversão e da qualidade do produto,
assegurando também a sua homogeneidade. O aumento da segurança e a redução de resíduos típicos
da indústria farmacêutica são também fatores fundamentais para a aplicação desta tecnologia.
O objetivo desta tese é caracterizar reatores contínuos e desenvolver uma metodologia de
scale-up/scale-down de processos químicos em contínuo. Para concretizar este objetivo, os reatores
foram caracterizados relativamente ao fluxo, com testes de distribuição de tempos de residência e à
transferência de calor. Para descrever as distribuições de tempos de residência obtidas foram usados
o modelo de uma bateria de reatores, o modelo de duas baterias em paralelo e o modelo da dispersão.
Em paralelo, a reação foi também caracterizada a partir dos dados da produção em batch e em reatores
contínuos de laboratório. Para o primeiro foi usado um modelo cinético baseado em balanços mássicos
e entálpicos e para os últimos, o modelo da segregação total e o modelo do pistão ideal.
A cinética da reação obtida com os dados em batch e em contínuo foi comparada, detetando-se
algumas diferenças que não eram expectáveis. Foram identificadas possíveis razões para estas
diferenças e foi proposta uma metodologia de scale-up e scale-down de reatores contínuos baseada
num abordagem de modelação.
Palavras-chave: reatores contínuos, scale-up/down, distribuição de tempos de residência,
transferência de calor, modelação, cinética química.
iv
v
Abstract
Continuous manufacturing is gaining increased attention in the pharmaceutical production due to
the excellent mixing and heat transfer offered that leads to the possibility of intensifying the process
using a range of new operating conditions that result in the increase of conversion and the quality of the
product, as well as ensuring a higher uniformity. Additionally, among the major advantages of continuous
manufacturing are the enhanced safety and reduction of waste which is crucial in the pharmaceutical
industry.
The scope of this thesis is to characterize flow reactors and develop a scale-up and scale-down
methodology for continuous reactors. In order to achieve this goal, a characterization of different
reactors and of the reaction itself was done. To study the dynamics of the reactors, residence time
distribution and heat transfer tests were performed in continuous laboratory reactors. In order to describe
the residence time distributions obtained, the model of tanks in series, the model of two batteries in
parallel and the dispersion model were applied. In parallel, the reaction kinetics was studied in the batch
reactor using a kinetic model based in the mass and energy balances and in continuous laboratory
reactors using the segregation model and the ideal PFR model.
The apparent kinetics of the same reaction performed in batch production mode and in the
continuous reactors was compared and some unexpected differences were found. Possible reasons for
this difference were identified and a scale-up/scale-down methodology procedures is purposed based
on a modelling approach.
Key-words: continuous reactors, scale-up/down, residence time distributions, heat transfer,
Resumo ................................................................................................................................................... iii
List of Contents ....................................................................................................................................... vii
List of Figures .......................................................................................................................................... ix
List of Tables ........................................................................................................................................... xi
List of Schemes ..................................................................................................................................... xiii
Table 20. Summary of the parameters for the Dispersion Model for reactor coil 1/8. .......................... 55
Table 21. Summary of the parameters for the Dispersion Model for reactor coil 1/16. ........................ 57
Table 22. Summary of the dimensionless numbers analyzed in this chapter. ...................................... 58
Table 23. Parameters used in the Heat Transfer Models ..................................................................... 58
Table 24. Parameters adjusted for optimization of the Heat Transfer Model considering only dynamics
of the sensor. ......................................................................................................................................... 62
Table 25. Parameters adjusted for optimization of the Heat Transfer Model considering dynamics of the
sensor and heat losses. ......................................................................................................................... 63
Table 26. Kinetic parameters obtained from the Segregation Model for reactor coil 1/16. ................... 66
Table 27. Kinetic parameters obtained from the Ideal PFR Model for reactor coil 1/16. ...................... 68
Table 28. Kinetic parameters obtained from the Ideal PFR Model for reactor MicR............................. 68
Table 29. Summary of the kinetic parameters obtained from batch and continuous mode. ................. 69
Table 30. Temperatures assumed as real and estimated by the Segregation Model and Ideal PFR Model
for reactor coil 1/16 and MicR. .............................................................................................................. 70
Table 31. Relationship between a chemical reaction and the reactor. ................................................. 71
xii
xiii
List of Schemes Scheme 1. Thesis work methodology ................................................................................................... 22
Scheme 2. Proposed methodology for Scale-Up of continuous reactors. ............................................ 73
xiv
1
1. Introduction
1.1. Objectives and Motivation
Pharmaceutical manufacturing operates typically in batch mode. The versatility and flexibility
offered by batch processes constitute significant advantages that justify the selection of this operation
mode. However, in the recent years, continuous manufacturing is increasingly addressed in the
pharmaceutical industry as a way to modify traditional batch processes, reflecting the endorsement
being made by Regulatory Agencies, mainly Food and Drug Administration (FDA), and the press
releases from big pharma and biotech companies announcing big investments in the field. If improved
mixing and heat transfer can be achieved, the process can be intensified and new operating conditions
can be used, increasing conversion and the quality of the product, as well as ensuring a higher uniformity
and throughput. Additionally, continuous flow processing might also allow accessing a range of reactions
conditions that would otherwise not be accessible, such as processes which are a combination of high
temperature, high pressure and short reaction times, for instance.
One of the major advantages of continuous manufacturing is the enhanced safety not only because
of the reduced inventory of raw materials and solvents, but also because the hold times between
operations can be eliminated, leading to no unstable or hazardous intermediate accumulation. The real
time control and higher automation make sure all the parameters are within the range and the product
is being manufactured as intended, thus improving the quality of the product [1], [2].
Hovione is building capabilities to use continuous manufacturing for API synthesis and drug product
to use it as a differentiator element. Moving a process from batch to a continuous processing mode
requires a profound knowledge on the reactor performance, matching its characteristics to the reaction
kinetics.
The scope of this thesis is to characterize two main types of flow reactors for small-scale
manufacturing of Active Pharmaceutical Ingredients (APIs) in order to purpose a scale-up/scale-down
methodology to make scale-up and scale-down operations easier and more reliable. Every time a
specific product is to be manufactured in the production scale, preliminary studies have to be carried
out in the laboratory to allow the correct design of the manufacturing procedure and to select the most
suitable reactors for the necessary reactions. However, in some cases, the conditions that work well in
a laboratory scale do not work so well in the production scale. Laboratory equipment has a very small
size, which leads to an almost perfect mixing and an excellent heat and mass transfer, due to the larger
area/volume ratio. On the other hand, when the reaction is carried out in a production scale reactor, due
to its larger size, heat and mass transfer become more heterogeneous and it is more difficult to ensure
that all the reaction mixture is well mixed. For this reason, the conditions used in the production scale
have to be adapted.
In order to achieve the goal of making a correct scale-up and/or scale-down of continuous reactors,
this work focused on a contribution for the characterization of different reactors and of the reaction itself.
To study the dynamics of the reactors, flow characterization tests were performed in two different
continuous laboratory reactors, and heat transfer tests in one them. For the flow characterization, RTD
2
tests were carried out and the data was fitted applying the model of tanks in series, the model of two
batteries in parallel and the dispersion model. The reaction kinetics in the batch reactor and in two
different continuous laboratory reactors was also studied, and the segregation model and the ideal PFR
models were applied.
The apparent kinetics of the same reaction, performed in batch manufacturing mode in the
production scale was compared to the results obtained for the continuous laboratory reactors and
possible reasons for the differences observed were addressed.
A set of recommendations was established to provide a methodology for scale-up/scale-down
procedures.
1.2. Thesis Layout
A brief explanation about the structure of this thesis will be given in this section.
Chapter 2 provides an overview of the available literature on the most relevant concepts, including
the issues involved in the transition of the pharmaceutical production from batch to continuous
manufacture, the characteristics of commercial continuous reactors applied to pharmaceutical industries
and methodologies for scale-up/scale-down continuous processes. In chapter 3, an explanation of the
experimental part is given, including a description of the reaction and materials used as a case study,
as well as the experimental tests performed. Chapter 4 includes the results and discussion of the
experimental results obtained. Chapter 5 presents the results and explains the models applied to the
data presented in previous chapter. In Chapter 6 a methodology for scaling-up/scaling-down continuous
processes is presented. An overall discussion of the results and final remarks describing the impact of
the developed work and addressing suggestions for future work is presented in Chapter 7.
3
2. Literature Review
2.1. Batch vs. continuous
Batch manufacturing is a long process which uses large-scale equipment and is the preferred mode
of operation for the production of Active Pharmaceutical Ingredients (APIs). However, in the recent
years, continuous manufacturing has been encouraged to be used as a more efficient process, to create
a more robust and flexible method capable of manufacturing high-quality APIs [3], [4].
Concerning the advantages of batch manufacturing, flexibility and a readily reconfigurable set of
multipurpose unit operations are often thought of. A large number of products can be manufactured in
a single plant with multiple stirred tank reactors [1], [5]. High conversions can be achieved by keeping
the reactants for a long time in the reactor. Very slow transformations that cannot be accelerated by
increased heating and cooling are often best performed in batch reactors [6].
Nevertheless, the production in batch mode takes a longer time due to the existence of several
reaction steps as well as isolation and purification work-up operations between each reaction step [7].
Because of the hold times between steps, unstable species can be accumulated, making it a less safe
process. Another disadvantage of the batch manufacturing is the high technical challenges of scaling-up
batch operations from the laboratory to pilot scale and to manufacturing scale due to the difficulty in
maintaining mixing and heat transfer conditions along the production scales. Hence, the reaction
conditions can vary with location in the reactor leading to undesirable side products [1], [5].
The expectation is that more companies will invest in continuous manufacturing technologies with
the aim of gaining a significant competitive advantage. As the pharmaceutical manufacturing industry
progresses, higher-quality drugs will be produced faster and more cost effectively, benefiting patients
around the world [8]. Current estimates suggest a general increase in industrial continuous
manufacturing applications from 5% to 30% over the next few years [9].
In continuous manufacturing, due to the efficient mixing and excellent heat exchange of the reactors,
extreme conditions might be used in a safe way, leading to a reduction of raw materials and solvents
usage and, consequently, less costs. New reaction conditions can be applied, and processes that are
simply not viable in traditional batch mode operations will be able to be exploited [10]. These new
patterns include a new range of substrates and operational conditions, such as higher concentrations,
temperatures and pressures that will hopefully increase conversion and the quality of the product. This
is called process intensification, which is a chemical engineering development that leads to a smaller,
cleaner, and more efficient process [11].
Due to the small size of the flow equipment, a small amount of hazardous intermediates is formed
at any instant, what leads to an improvement of safety issues. Also, the better control of exothermic
reactions improve safety what constitutes a major advantage of continuous manufacturing [9], [12]. The
small amount of hazardous intermediates also reduces the environmental footprint and helps to
minimize issues of waste and energy usage typical of a pharmaceutical industry [12].
4
A key advantage of flow reactor technology is the ability to accurately control and monitor reaction
parameters. It is of upmost importance to consistently manufacture a product that has a uniform
character and quality attributes within specific limits [2], [5], [13]. This is possible thanks to the high
automation and in-line control implicit in continuous technology.
The reaction temperature, pressure, concentration, flow rate and residence time are very important
operating parameters for maintaining reaction under control and ensure selective product formation and
preserve intermediates degradation. Furthermore, the total flow of the streams ensures proper
residence time and the ratio between the flows of each stream ensures proper stoichiometry of the
reagents. The control of the flow rate and therefore of the residence time is crucial for handling highly
reactive intermediates [1], [10], [14].
In a continuous reactor, mixing is rapid and heat can be readily added or removed from the reactor,
which results in a higher yield [9] [10]. The high surface area also allows for an excellent control of
exothermic reactions. As well as increasing the rate of mixing, decreasing the reactor channel diameter
results in a high surface to volume ratio, what leads to a rapid dissipation of the heat generated during
the reaction [13].
Despite the innumerous advantages associated with continuous manufacturing mode enumerated
above, it needs to be noted that it is far from certain that flow technology will solve the current problems
in the pharmaceutical sector [14]. The current inventory of available batch manufacturing facilities is one
of the biggest barriers for the implementation of continuous manufacturing units. It constitutes a high
capital investment to change the facilities to continuous mode [2], [5]. Another barrier to the
implementation of continuous manufacturing is the successes of the batch technology. In order to
achieve a successful transformation, the mind-set has to change [9], [15].
Although the transition to continuous manufacturing has been highly successful in some cases,
resulting in a transformative change through market growth and expansion, the initial implementation is
slow due to the technical challenges and firm mind-sets fixed in the old technology [15].
5
2.2. Types of continuous reactors
Equipment for continuous manufacturing in pharmaceutical industry has advanced significantly in
recent years. Flow reactors commonly used in industrial processing include the Continuous Stirred Tank
Reactor (CSTR), the Plate reactor, the Tubular reactor and the Packed-Bed reactor.
This chapter will focus on two types of continuous reactors: the tubular reactor and the plate reactor.
Their characteristics, advantages and disadvantages will be addressed.
Figure 1 shows the suppliers available for each type and scale of continuous reactors.
2.2.1. Tubular Reactors
The tubular reactor consists of a cylindrical pipe where the feed enters at one end and product
leaves at the other end. Reactants are continually consumed as they flow in the reactor, which operates
at steady-state [16]. No axial mixing, no radial gradients, no temperature gradients and the same
residence time for all flowing rates are assumed in the ideal PFR. Nevertheless, the flow patterns in a
real tubular reactor is characterized by some degree of axial mixing caused by differences in flow
velocities and properties at different radial positions because of temperature gradients. For these
reasons, the control of a tubular reactor can be challenging, since temperature and concentration vary
with length and sometimes with radial position.
The classical idealizations of a plug flow are usually close enough to reality so that they can be used
for studying both steady-state design and the dynamic control of chemical reactors. This assumption is
reasonable for adiabatic reactors. However, for non-adiabatic reactors, radial temperature gradients are
intrinsic features. If tube diameters are small, the plug flow assumption is more correct [17].
Lab scale Capillary systems
(μg to mg)
Lab scale KiloFlow
(mg to Kg’s, 1-2 Kg 1 week)
Pilot Scale(10-50 Kg, 1 week)
Production Scale(>50 Kg)
Micro & Meso Plate reactors
Corning Advanced Flow ReactorsChemtrix
Tubular reactor(PFR)
Agitated Cell Reactor
Spinning Disc reactor
Multireactors systems (microreactor + coil
reactor + PBR)
Cambridge Reactor DesignParr Instrument Company
VapourtecUniqsisAccendo Corporation
FlowID
AM Technology
Future Chemistry Holding
Syrris
DSM Innosyn (3D printed reactors)
Oscillatory Baffled Reactor Nitech Solutions
PFR + Static Mixer solutions
Fluitec|FlowLink
Packed Bed Reactor(PBR)
ThalesNanoNippon Kodoshi Corporation
Iberfluid|PID
Parr Instrument Company
Ehrfeld Mikrotechnik (Lonza & Alfa Laval)
Lab scale Capillary systems
(μg to mg)
Lab scale KiloFlow
(mg to Kg’s, 1-2 Kg 1 week)
Pilot Scale(10-50 Kg, 1 week)
Production Scale(>50 Kg)
Micro & Meso Plate reactors
Corning Advanced Flow ReactorsChemtrix
Tubular reactor(PFR)
Agitated Cell Reactor
Spinning Disc reactor
Multireactors systems (microreactor + coil
reactor + PBR)
Cambridge Reactor DesignParr Instrument Company
VapourtecUniqsisAccendo Corporation
FlowID
AM Technology
Future Chemistry Holding
Syrris
DSM Innosyn (3D printed reactors)
Oscillatory Baffled Reactor Nitech Solutions
PFR + Static Mixer solutions
Fluitec|FlowLink
Packed Bed Reactor(PBR)
ThalesNanoNippon Kodoshi Corporation
Iberfluid|PID
Parr Instrument Company
Ehrfeld Mikrotechnik (Lonza & Alfa Laval)
Figure 1. Types of reactors and the respective supplier.
6
The diameter of tubular reactor can range from a few millimeters to several meters. The choice of
diameter is based on construction cost, pumping cost, the desired residence time, and heat transfer
needs. Typically, long small diameter tubes are used with high reaction rates and larger diameter tubes
are used with slow reaction rates.
When the reactor operates adiabatically there is no external heat transfer along the reactor. Whether
the reaction generates or consumes heat, the temperature of the material flowing in the reactor
increases or decreases, respectively [17].
In tubular reactors, the heat transfer and temperature control are achieved by the use of heat
exchangers which can be concentric tubes or shell and tube. Heat exchangers increase surface area to
volume ratio leading to an improvement of heat transfer rates. They might be used to adjust the
temperature of raw materials and to control the temperature in the mixing zone [1].
Tubular reactors have a wide variety of applications in either gas or liquid phase systems and for
both small and industrial production [16]. They denote a good balance between cost, heat and mass
transfer efficiency, and easy mode of operation [1].
Summarizing, some advantages and disadvantages of this type of continuous reactors are
Results for the minimum flow rate are presented separately because they were different.
The RTDs obtained for coil 1/8 and coil 1/16 using the wavelength of 295 nm are presented in
appendixes B and C, respectively.
0
0.0004
0.0008
0.0012
0.0016
0.002
0 500 1000 1500 2000
E(t
) (s
-1)
Time (s)
A
0
0.0004
0.0008
0.0012
0.0016
0.002
0.0024
0.0028
0.0032
0 200 400 600 800
E(t
) (s
-1)
Time (s)
B
0
0.002
0.004
0.006
0.008
0.01
0 500 1000
E(t
)(s
-1)
Time (s)
C
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0 200
E(t
) (s
-1)
Time (s)
D
30
Figure 10. Experimental Residence Time Distribution for reactor coil 1/8 and coil 1/16 at the wavelength of 520 nm. The green points represent the data for coil 1/8 and the red points for coil 1/16. A=0.2 mL/min (coil 1/8); B=
Figure 11 and Figure 12 illustrate the residence time distributions for all the flow rates for coil 1/8
and coil 1/16, respectively.
0
0.004
0.008
0.012
0.016
0.02
0.024
0.028
0 100 200
E(t
)(s
-1)
Time (s)
E
0
0.008
0.016
0.024
0.032
0.04
0 50 100 150
E(t
)(s
-1)
Time (s)
F
0
0.02
0.04
0.06
0.08
0 50 100
E(t
)(s
-1)
Time (s)
G
31
Figure 11. Residence Time Distribution for all the flow rates for reactor coil 1/8.
Notice that the RTD for the lowest flow rate (0.2 mL/min) is not shown as the residence time was
higher than the others leading to a non clean representation of the remaining results in Figure 11.
Figure 12. Residence Time Distribution for all the flow rates for reactor coil 1/16.
Analyzing Figure 10 it can be concluded that for the coil 1/16 the residence time is always higher.
This is due to the difference in the volumes of the two coils. As reactor coil 1/16 has a higher volume
than coil 1/8, despite its smaller diameter, the fluid remains in the reactor for a longer period.
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0 100 200 300 400 500
E(t
) (s
-1)
Residence Time (s)
Q=1 mL/min
Q=2 mL/min
Q=4 mL/min
Q=6 mL/min
Q=10 mL/min
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 100 200 300 400 500 600 700 800
E(t
) (s
-1)
Time (s)
Q=0.9 mL/min
Q=1 mL/min
Q=2 mL/min
Q=4 mL/min
Q=6 mL/min
Q=10 mL/min
32
As expected, according to Figure 11 and Figure 12 the residence time decreases with the increase
of flow rate (Q). As the flow rates increases, the fluid remains less time in the reactor, and thus the
residence time is shorter.
Table 11 summarizes the residence time for each flow rate for coil 1/8 and coil 1/16.
Table 11. Residence time for each flow rate for coil 1/8 and coil 1/16.
Coil 1/8 Coil 1/16
Q (mL/min)
θ (min)
experimental
(by equation (14))
θ (min)
expected
(by equation (15))
Q (mL/min)
θ (min)
experimental
(by equation (14))
θ (min)
expected
(by equation (15))
0.2 19.9 22.0 0.9 6.0 6.1
1 4.1 4.4 1 5.8 5.5
2 1.9 2.2 2 2.6 2.8
4 1.0 1.1 4 1.4 1.4
6 0.6 0.7 6 0.8 0.9
10 0.4 0.4 10 0.5 0.6
The mean residence times calculated from the RTD measurements (θ experimental), using equation
(14), and the mean residence time calculated based on the reactor volume (θ expected), using equation
(15), for each coil and for each flow rate are shown in Table 11. The mean residence times calculated
from the RTD measurements correspond well to the residence times calculated based on the internal
volume of reactor. With the same purpose of comparing both residence times, Figure 13 and Figure 14
are presented. They represent the residence time as a function of the flow rate. The blue points
represent the experimental residence time for each flow rate and the orange line represents the
residence times calculated by the internal reactor volume.
Figure 13. The blue points represent the experimental residence time for each flow rate for coil 1/8 and the
orange line represents the residence times calculated by the internal volume.
Figure 14. The blue points represent the experimental residence time for each flow rate for coil 1/16 and the orange line represents the residence times calculated
by the internal volume.
0
5
10
15
20
25
0 2 4 6 8 10
θ(m
in)
Flow rate (mL/min)
0
1
2
3
4
5
6
7
0 2 4 6 8 10
θ(m
in)
Flow rate (mL/min)
33
As can be seen from the line in Figures 10 and 11 the data shows a high consistency, which
indicates that there are no significant changes in possible dead-volumes with the changes in flow-rate.
From the data presented, assuming that the measured flow-rates are accurate, the effective volume of
these two reactors are 4.0 mL and 5.3 mL, respectively for coil 1/8 and coil 1/16, which compare well
with the expected values of 4.4 mL and 5.5 mL indicated in Table 7.
However, the residence time distributions present some deviations from plug flow since they have
some width, showing a deviation from the ideality.
4.2. Heat Transfer
In this study, heat transfer rates were investigated at the moment when the outlet temperature
became momentarily stable. The temperature difference, ΔT, between inlet and outlet temperatures at
this moment was taken to calculate the total heat transfer rate Q as follows:
𝑄 = 𝑚 𝐶𝑝 𝛥𝑇 (16)
where m is the mass flow rate of water through the reactor and Cp is the specific heat capacity.
On the other hand, the log mean temperature difference ΔTln is calculated assuming no
temperature change in bath (hot water). Thus the overall heat transfer coefficient, U, is obtained as
follows:
𝑈 =𝑄
𝐴 𝛥𝑇𝑙𝑛 (17)
The log mean temperature difference, ΔTln, is calculated by equation (18).
𝛥𝑇𝑙𝑛 =
𝑇𝑐𝑜𝑖𝑙 𝑜𝑢𝑡 − 𝑇𝑐𝑜𝑖𝑙 𝑖𝑛
𝑙𝑛 (𝑇𝑏𝑎𝑡ℎ − 𝑇𝑐𝑜𝑖𝑙 𝑖𝑛
𝑇𝑏𝑎𝑡ℎ − 𝑇𝑐𝑜𝑖𝑙 𝑜𝑢𝑡)
(18)
In order to calculate the heat transfer coefficient, equations (16) to (18) were used, as discussed
above.
The average temperature of the inlet and outlet was calculated from the moment when the outlet
temperature became constant. These values and the other parameters to calculate the heat transfer
coefficient and the heat transfer itself are listed in Table 12.
Table 12. Parameters for the calculation of UA.
Flow rate= 2 mL/min Flow rate= 4 mL/min
T in (ºC) 26.8 26.0
T out (ºC) 56.7 59.9
T bath (ºC) 92.9 93.5
ΔTln (ºC) 49.7 48.6
Mass (kg/s) 3.33x10-5 6.67x10-5
34
Cp (kJ/(kg K) 4.18 4.18
UA (W/K) 0.08 0.19
The heat transfer coefficients were estimated as 0.08 W/K and 0.19 W/K for the flow rates of
2 mL/min and 4 mL/min, respectively. As expected, the heat transfer coefficient increases with the
increase of the flow rate.
As explained in the previous chapter, the probes were reading the entering and exiting temperature
of the reactor outside of the bath. Concerning the probe at the exit of the coil, reading the temperature
outside of the bath is a source of error because since the reactor comes out of the bath temperature
rapidly decreases. Thus, the temperature read at the exit of the coil reactor may not correspond to the
real temperature of the fluid inside the coil because the probes were measuring the surface temperature
of the coil, which could be affected by the ambient temperature. However, reading the temperature of
the coil reactor inside the bath would have error too because the probe would be reading the bath
temperature. Since the probe is not inside the reactor, temperature gradients inside the coil could not
be detected.
Figure 15 and Figure 16 represent the temperatures at the exit of the reactor. As can be seen,
temperature increases slowly over time until reaching stabilization. A high difference between the final
temperature and the bath temperature (more than 30 ºC) was found. This difference will be critically
discussed later.
Figure 15. Exit temperature of the reactor during the
experiment for a flow rate of 2 mL/min.
Figure 16. Exit temperature of the reactor during the
experiment for a flow rate of 4 mL/min.
0
10
20
30
40
50
60
0 500 1000 1500
Tem
pera
ture
(ºC
)
Time (s)
0
10
20
30
40
50
60
70
0 500 1000
Tem
pera
ture
(ºC
)
Time (s)
35
5. Models Results and Discussion
This chapter describes in detail the development of the models applied to the characterization of the
reactors and for the determination of the kinetic data of the reaction under study and the hypothesis that
were used. Firstly it will be explained the model applied to the batch production data and then the models
for the continuous reactors.
5.1. Batch Reactor
In order to study the reaction carried out in a batch reactor and determine the kinetic parameters,
models based on mass and energy balances were fitted to the batch production data.
Briefly, the process consists in the following steps:
i. Reactant A, reactant B and solvent are charged into the reactor, in this order;
ii. The reaction mixture is heated to the required temperature;
iii. The reaction mixture is stirred for a long time;
iv. The reaction mixture is cooled and a sample is collected for an HPLC analysis and if
the content of the reactant A is within the in-process control limit (IPC), the process
proceeds to the next steps, otherwise, the reaction mixture has to be re-heated and
stirred for additional time until reaching the IPC limit.
Note that during the whole procedure the temperature of the reaction mixture is continuously
monitored and this is the data that is available for the development of the model.
The description of the evolution of the temperature profile within the reactor requires the model to
account both for the heat exchange with the surroundings (which provide heating and cooling) and the
heat involved in the reaction itself.
In order to get the model to adjust to the temperature profile of the reaction mixture and determine
the kinetic parameters, mass and energy balances had to be defined.
To remind, the reaction in study is of the type 𝐴 + 𝐵 → 𝐶 (𝑝𝑟𝑜𝑑𝑢𝑐𝑡) + 𝑠𝑢𝑏𝑝𝑟𝑜𝑑𝑢𝑐𝑡.
Mole Balance:
𝑑𝑛𝐴
𝑑𝑡= −𝑘𝐶𝐴𝐶𝐵𝑉
(19)
𝑑𝑛𝐵
𝑑𝑡= −𝑘𝐶𝐴𝐶𝐵𝑉
(20)
𝑑𝑛𝐶
𝑑𝑡= 𝑘𝐶𝐴𝐶𝐵𝑉
(21)
Energy Balance:
First, the general energy balance is presented in equation (22).
Since nothing is being added to the reactor, equation (22) can be re-written as:
∑ 𝑛𝑖𝐶𝑝𝑖
𝑑𝑇
𝑑𝑡+ ∑
𝑑𝑛𝑖
𝑑𝑡 𝛥𝐻𝑓𝑖 = 𝑈𝐴 (𝑇 − 𝑇𝑐𝑎𝑚𝑖𝑠𝑎)
(23)
which applied to the case study leads to:
∑ 𝑛𝑠𝑜𝑙𝑣𝑒𝑛𝑡 𝐶𝑝𝑠𝑜𝑙𝑣𝑒𝑛𝑡
𝑑𝑇
𝑑𝑡+ (−𝑘𝐶𝐴𝐶𝐵𝑉) 𝛥𝐻𝑓𝐴 + (−𝑘𝐶𝐴𝐶𝐵𝑉)𝛥𝐻𝑓𝐵
+ (𝑘𝐶𝐴𝐶𝐵𝑉𝛥𝐻𝑓𝐶 ) = 𝑈𝐴 (𝑇 − 𝑇𝑐𝑎𝑚𝑖𝑠𝑎)
(24)
Assuming that the variation in the sensible heat is predominantly due to the solvent and using an
average heat capacity of the reaction mixture, the final balance is given by equation (25).
𝑛𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 𝑚𝑖𝑥𝑡𝑢𝑟𝑒 𝐶𝑝𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 𝑚𝑖𝑥𝑡𝑢𝑟𝑒
𝑑𝑇
𝑑𝑡+ (−𝑟𝐴) 𝑉 𝛥𝐻𝑟 = 𝑈𝐴 (𝑇 − 𝑇𝑐𝑎𝑚𝑖𝑠𝑎) (25)
The set of differential equations was integrated using the Euler method with the following
rearrangements:
𝑑𝑇
𝑑𝑡=
𝑈𝐴 (𝑇 − 𝑇𝑐𝑎𝑚𝑖𝑠𝑎) − (−𝑟𝐴) 𝑉 𝛥𝐻𝑟
𝑛𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 𝑚𝑖𝑥𝑡𝑢𝑟𝑒 𝐶𝑝𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 𝑚𝑖𝑥𝑡𝑢𝑟𝑒
(26)
(−𝑟𝐴) = 𝑘0𝑒−𝐸𝑎/(𝑅𝑇)𝐶𝐴𝐶𝐵 (27)
The concentrations of A and B throughout the reaction, necessary for the reaction velocity, are also
given by the Euler method using the mass balance (Equations (28) and (29)).
𝑑𝐶𝐴
𝑑𝑡= −𝑘𝐶𝐴𝐶𝐵
(28)
𝑑𝐶𝐵
𝑑𝑡= −𝑘𝐶𝐴𝐶𝐵
(29)
The temperature that was calculated by the simultaneous integration of the four differential
equations, corresponding to the mass and energy balances, was fitted to the real temperature profile
during the reaction. This temperature profile is given by the temperature inside the reactor over the
whole period of reaction, i.e. heating, reaction time (isothermic conditions) and cooling. On the fitting
parameters that we wanted to obtain are: the pre-exponential factor (k0), the activation energy (Ea) and
the heat transfer coefficient (UA). Solver optimized these three parameters based on an objective
function based on the reactor temperature.
For confidentiality reasons, the temperature profile of the reaction cannot be presented, but Figure
17 shows the percentage error between the real reaction temperature and the temperature obtained
from the model. It can be concluded that the % error between the real reaction temperature and reaction
temperature obtained from the model is at maximum 15% and that model is not tendentious, describing
rather well the whole profile.
Table 13 summarizes the parameters determined from this model (k, Ea and UA).
37
Figure 17. Temperature difference between the experimental temperature and the temperature calculated by the
model for the production in batch mode.
Table 13. Kinetic parameters obtained from the production in batch mode.
k0 3.21x1022 kref
Ea Earef
UA (W/K) 453.59
The pre-exponential factor, k0, corresponds to the reaction rate constant for infinite temperatures.
Based on Arrhenius law, k0 corresponds to k when the exponential is one. For the exponential being
one, the exponent −𝐸𝑎
𝑅𝑇 has to be zero and that only happens when the temperature is infinite, taking
into account the activation energy is a positive value.
For the application of this model, a reaction order had to be admitted. Different orders of reaction
were studied to conclude which had the best adjustment. The following orders were tested:
First order in respect to reactant A and first order in respect to reactant B (overall second
order)
First order in respect to reactant A and zero order in respect to reactant B (overall first order)
Second order in respect to reactant A and first order in respect to reactant B (overall third
order)
It was concluded that first order for each reactant led to the best fitting.
-10
-5
0
5
10
15
20
0.00 10.00 20.00 30.00 40.00 50.00 60.00
% e
rror
Time (h)
38
5.2. Continuous Reactors
The models applied to each continuous reactor and respective purpose (reactor or reaction
characterization) is presented in Table 14.
Table 14. Models applied to each continuous reactor.
Continuous
reactors
Reactor characterization Reaction
characterization Flow (Models for the RTD) Heat
Coil 1/8 Model of N tanks in series
Model of two batteries in
parallel
Dispersion Model
Heat Transfer
Model --
Coil 1/16 -- Segregation Model
Ideal PFR Model
MicR -- -- Ideal PFR Model
The models are presented and discussed in detail in the next subchapters.
5.2.1. Models for the Residence Time Distributions
As summarized in Table 14, the models applied to characterize the reactor were (i) the model of
tanks in series, (ii) the model of two batteries of tanks in parallel and (iii) the dispersion model.
5.2.1.1. Model of Tanks in Series
Figure 18 represents schematically the model of N tanks in series. According to this model, all the
reactors have the same volume and each reactor is a CSTR, i.e, the mixture is perfect.
The number of ideal tanks in series that will give approximately the same RTD as the non-ideal
reactor is determined. In other words, it is going to be found how many reactors of a battery best describe
the behavior of the flow inside the tubular reactor.
Equation (30) represents the exit age function for one battery of N CSTRs in series with a given
average residence time of θ.
N
Figure 18. Representation of the Model of Tanks in Series.
39
𝐸(𝑡) =𝑡𝑁−1
(𝑁 − 1)! (𝜃𝑁
)𝑁 𝑒
−𝑁𝑡𝜃
(30) [43]
Where N represents the number of reactors in the battery, θ is the global residence time in the
battery and t means the time. N and θ were the two parameters estimated by Solver in this model. The
objective function was the sum of the square of the residuals corresponding to E(t).
The results of the application of the model of tanks in series is going to be presented for each
reactor separately.
Coil 1/8
Figure 19 represents the fittings of the model of tanks in series to the residence time distributions
of each flow rate for reactor coil 1/8 and for the wavelength of 520 nm (A=0.2 mL/min; B=1 mL/min; C=2
mL/min; D=4 mL/min; E=6 mL/min; F=10 mL/min). The same model applied to the data obtained at the
wavelength of 295 nm is shown in the Appendix D. The main parameters of this model can be found in
Table 15.
0
0.0004
0.0008
0.0012
0.0016
0.002
0 500 1000 1500 2000
E(t
) (s
-1)
Time (s)
A
0
0.002
0.004
0.006
0.008
0 200 400 600
E(t
) (s
-1)
Time (s)
B
40
Figure 19. Influence of the Flow Rate in the RTD for the Model of Tanks in Series for reactor coil 1/8 at the wavelength of 520 nm. The blue points represent the experimental data and the orange line is the model of tanks
Figure 20. Influence of the Flow Rate in the RTD for the Tanks in Series Model for reactor coil 1/16 at the wavelength of 520 nm. The blue points represent the experimental data and the orange line is the model of tanks
Figure 22. Illustration of the two concentric reactors (side and top view)
The expression of E(t) for two batteries in parallel is given by equation (31).
The parameters required to fit in this model were the number of tanks in each battery (N1 and N2),
the residence time in each battery (θ1 and θ2) and the ratio of flow rates flowing in each battery (Q1/Q
and Q2/Q, where Q= Q1+Q2). The objective function was the sum of the squares of the residuals of E(t).
Coil 1/8
Figure 23 represents the individual fittings of the model of two batteries in parallel to the RTD of each
flow rate for reactor coil 1/8 and for the wavelength of 520 nm (A=0.2 mL/min; B=1 mL/min; C=2 mL/min;
D=4 mL/min; E=6 mL/min; F=10 mL/min). The same model applied to the data obtained at the
wavelength of 295 nm is presented in Appendix F. Table 17 summarizes the main parameters of this
model.
𝐸(𝑡) =𝑡(𝑁1−1)
(𝑁1−1)! (𝜃1𝑁1
)𝑁1
𝑒−𝑁1𝑡
𝜃1 𝑄1
𝑄+
𝑡(𝑁2−1)
(𝑁2−1)! (𝜃2𝑁2
)𝑁2
𝑒−𝑁2𝑡
𝜃2 𝑄2
𝑄 (31) [43]
Q1
Q2
Q2
2
1
r
R
N1
N2
Figure 21. Representation of the Model of two batteries in parallel.
45
Figure 23. Influence of the Flow Rate in the RTD for the Two Batteries in Parallel Model for reactor coil 1/8 at the wavelength of 520 nm. The blue points represent the experimental data and the orange line is the model of two
By the observation of Table 17 it was concluded that the residence time given by the model is in
accordance with the experimental. It is also clear, from the residuals and from Figure 23, that the fittings
have improved significantly in relation to the previous model (a single battery), but this is hardly
surprising as the number of fitting parameters were doubled. The number of tanks per battery is not
significantly different for each flow rate but, nevertheless, a relationship between them is still not clear.
Concerning the ratio of flow rates in the two model streams, they are similar in all cases.
Coil 1/16
Figure 24 represents the individual fittings of the model of two batteries in parallel to the residence
time distributions of each flow rate for reactor coil 1/16 and for the wavelength of 520 nm (A=0.9 mL/min;
B=1 mL/min; C=2 mL/min; D=4 mL/min; E=6 mL/min; F=10 mL/min). The same model applied to the
data obtained at the wavelength of 295 nm is shown in Appendix G. Table 18 summarizes the main
parameters of this model.
47
Figure 24. Influence of the Flow Rate in the RTD for the Two Batteries in Parallel Model for reactor coil 1/16 at the wavelength of 520 nm. The blue points represent the experimental data and the orange line is the model of two
Concerning the residual, it can be seen that the residual is higher for the higher flow rates. However,
observing Figure 25, the adjustments are much better for higher flow rates. The justification for this is
that the values of E(t) are higher for higher flow rates and, therefore, the difference between the model
and the experimental results is bigger.
The parameter β represents the ratio of velocities flowing in reactor 1 and reactor 2. If this
parameter is lower than 1, it means the velocity in the inside reactor is higher. If β is higher than 1, the
velocity of the flow in the swept circle is higher than the flow in the inside reactor. After trying to adjust
parameters for a β higher and lower than 1, it was observed, as expected, that the model is symmetric.
This means that when β is 0.7 with N1=30 and N2=10, the same residual is obtained for a β=1.43 (the
inverse of 0.7) with N1=10 and N2=30. In conclusion, the same results are obtained if the two batteries
are inverted (number of tanks and velocity).
0
0.002
0.004
0.006
0 200 400 600 800
E(t
)(s
-1)
Time (s)
A
0
0.003
0.006
0.009
0 200 400 600 800
E(t
)(s
-1)
Time (s)
B
51
Figure 25. Global adjustment for the Model of the Two Batteries in Parallel for reactor coil 1/16 at the wavelength of 520 nm. The blue points represent the experimental data and the orange line is the model of two batteries in
As observed from the Figure 25 it is noticeable that the global fitting fits best for the higher flow
rates than for the lowest flow rates. Ideally, the model should fit well for all the flow rates to obtain a
model which characterize the reactor, regardless of the flow rate, which means that some additional
work will have to be done to improve the description of the flow in these reactors.
5.2.1.3. Dispersion Model
The dispersion model is used to describe a non-ideal reactor with axial dispersion, i.e., a plug flow
behavior with some degree of backmixing. Dispersion is caused by fluctuations in the velocity profile of
the fluid passing through the reactor due to diffusion. It indicates the displacement of material in the
direction of the flow and results in fluid elements exiting the reactor at different residence times [47].
Figure 26 represents the dispersion model compared with the plug flow [48].
0
0.004
0.008
0.012
0.016
0 200 400
E(t
)(s
-1)
Time (s)
C
0
0.01
0.02
0.03
0 100 200 300
E(t
)(s
-1)
Time (s)
D
0
0.01
0.02
0.03
0.04
0 50 100 150
E(t
)(s
-1)
Time (s)
E
0
0.02
0.04
0.06
0.08
0 50 100
E(t
)(s
-1)
Time (s)
F
52
Figure 26. Representation of the dispersion model [48].
In the residence time distributions tests a pulse of tracer is introduced in the reactor. The pulse
spreads as it passes through the reactor because of non-uniform velocity profiles causing portions of
the tracer to move at different rates. To characterize this spreading process, the dispersion coefficient
D (m2/s) is used. A large D means rapid spreading of the tracer, a small D indicates slow spreading and
a D=0 means no spreading (plug flow) [44], [48].
(𝐷
𝑢𝐿) is the dimensionless group characterising the spread in the whole vessel [44], [48]:
𝐷
𝑢𝐿→ 0 negligible dispersion (plug flow)
𝐷
𝑢𝐿→ ∞ large dispersion (mixed flow)
It was assumed large deviation from Plug Flow (𝐷
𝑢𝐿>0.01). In this case, the pulse response spreads
as it passes through the reactor because of non-uniform velocity profiles causing portions of the tracer
to move at different rates, as shown in Figure 27.
Figure 27. Dispersion in a tubular reactor [46].
In the case of large 𝐷
𝑢𝐿 what happens at the entrance and exit of the vessel strongly affects the
shape of the tracer curve. It is assumed to exist two type of boundary conditions: closed-closed and
open-open vessel. In the closed-closed vessel the flow is plug flow outside the vessel up to the
boundaries and for the open-open vessel, dispersion occurs both upstream and downstream of the
reaction section. Figure 28 represents the two boundary conditions explained before [48].
53
Figure 28. Representation of the two boundary conditions: closed and open vessel [48].
The open vessel condition represents the generally used experimental device and the only situation
where the analytical expression of the exit age distribution is not too complex. The equation of the E(t)
function according to what was previously explained is represented in (39).
𝐸(𝑡) = 1
𝜃
1
√4𝜋 (𝐷𝑢𝐿
)
exp [(1 −
𝑡𝜃
)2
4𝑡𝜃
(𝐷𝑢𝐿
)] (39) [48]
Some authors ([41], [46], [47]) consider that the ratio (𝐷
𝑢𝐿) is the inverse of the Peclet number (Pe)
(equation (40)). Therefore, equation (39) can be written as equation (41).
(𝐷
𝑢𝐿) =
1
𝑃𝑒 (40)
𝐸(𝑡) = 1
𝜃 √𝑃𝑒
√4𝜋exp [
𝑃𝑒 (1 −𝑡𝜃
)2
4𝑡𝜃
] (41)
The Peclet number is the ratio between the rate of transport by convection and the rate of transport
by diffusion or dispersion. According to [46], the dispersion model and the model of tanks in series are
equivalent when the Peclet number and the number of tanks in series can be related by equation (42).
𝑁 = 𝑃𝑒
2+ 1 (42)
Therefore, the number of tanks in series were calculated from the Peclet number (equation (42))
and compared afterwards with the number of tanks obtained in subchapter 5.2.1.1.
The parameters to be adjusted by Solver in this model were the Peclet number and the residence
time.
54
Coil 1/8
Figure 29 represents the fittings of the dispersion model to the residence time distributions of all the
flow rates for reactor coil 1/8 and for a wavelength of 520 nm (A=0.2 mL/min; B=1 mL/min; C=2 mL/min;
D=4 mL/min; E=6 mL/min; F=10 mL/min). The same model applied to the data obtained at the
wavelength of 295 nm is presented in Appendix H. Table 20 summarizes the parameters of the
dispersion model.
0
0.0004
0.0008
0.0012
0.0016
0.002
0 500 1000 1500 2000
E(t
)(s
-1)
Time (s)
A
0
0.002
0.004
0.006
0.008
0 200 400 600
E(t
)(s
-1)
Time (s)
B
0
0.002
0.004
0.006
0.008
0.01
0.012
0 100 200 300
E(t
)(s
-1)
Time (s)
C
0
0.004
0.008
0.012
0.016
0.02
0.024
0 50 100 150 200
E(t
)(s
-1)
Time (s)
D
55
Figure 29. Influence of the Flow Rate in the RTD for the Dispersion Model for reactor coil 1/8 at the wavelength of 520 nm. The blue points represent the experimental data and the orange line is the dispersion model.
Figure 30 represents the fittings of the dispersion model to the residence time distributions of each
flow rate for reactor coil 1/16 and for the wavelength of 520 nm (A=0.9 mL/min; B=1 mL/min; C=2
mL/min; D=4 mL/min; E=6 mL/min; F=10 mL/min). The same model applied to the data obtained at the
wavelength of 295 nm is presented in Appendix I. Table 21 summarizes the parameters of the dispersion
model.
0
0.0008
0.0016
0.0024
0.0032
0 200 400 600 800
E(t
)(s
-1)
Time (s)
A
0
0.002
0.004
0.006
0.008
0.01
0 200 400 600 800
E(t
)(s
-1)
Time (s)
B
0
0.004
0.008
0.012
0.016
0 200 400
E(t
)(s
-1)
Time (s)
C
0
0.004
0.008
0.012
0.016
0.02
0.024
0.028
0 50 100 150 200
E(t
)(s
-1)
Time (s)
D
57
Figure 30. Influence of the Flow Rate in the RTD for the Dispersion Model for reactor coil 1/16 at the wavelength of 520 nm. The blue points represent the experimental data and the orange line is the dispersion model.
[37] G. M. Monsalve, H. M. Moscovo-Vasquez, and H. Alvarez, “Scale-Up Continuous Reactors using
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[42] G. Froment and K. Bischoff, Chemical Reactor Analysis and Design. John Wiley & Sons, 1979.
[43] F. Lemos, J. M. Lopes, and F. R. Ribeiro, Reactores Químicos, 3rd Ed. IST Press, 2014.
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[48] O. Levenspiel, Chemical Reaction Engineering, 3rd Ed. John Wiley & Sons, 1999.
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78
Supplementary information
Appendix A. Deductions of the Scaling factors:
Deduction of equation (4):
𝑆𝑅𝑒 =
𝜌𝑣2𝐷2
𝜇𝜌𝑣1𝐷1
𝜇
=𝐷2
𝐷1
𝑣2
𝑣1
=𝑅2
𝑅1
𝑄2
𝜋𝑅22
𝑄1
𝜋𝑅12
= 𝑆𝑅−1𝑆𝑡ℎ𝑟𝑜𝑢𝑔ℎ𝑝𝑢𝑡 = 𝑆𝑅
−1𝑆
Deduction of equation (7) (Laminar regime):
𝛥𝑃 =8𝜇𝑣𝐿
𝑅2
𝑆𝛥𝑃 =
8𝜇𝑣2𝐿2
𝑅22
8𝜇𝑣1𝐿1
𝑅12
=𝑅1
2𝑣2𝐿2
𝑅22𝑣1𝐿1
= (𝑅1
𝑅2
)2 𝐿2
𝐿1
𝑣2
𝑣1
= 𝑆𝑅−2𝑆𝐿
𝑄2
𝜋𝑅22
𝑄1
𝜋𝑅12
= 𝑆𝑅−4𝑆𝐿𝑆𝑡ℎ𝑟𝑜𝑢𝑔ℎ𝑝𝑢𝑡
Deduction of equation (10) (Turbulent regime):
𝑆𝛥𝑃 =
𝑓𝜌𝑣22𝐿2
𝑅22
𝑓𝜌𝑣12𝐿1
𝑅12
=
0.079
𝑅𝑒21/4
𝜌𝑣22𝐿2
𝑅22
0.079
𝑅𝑒11/4
𝜌𝑣12𝐿1
𝑅12
=𝑣2
7/4𝐿2𝑅1𝐷1
1/4
𝑣17/4
𝐿1𝑅2𝐷21/4
= (
𝑄2
𝜋𝑅22
𝑄1
𝜋𝑅12
)
7/4
𝐿2
𝐿1
𝑅1
𝑅2
(𝐷1
𝐷2
)1/4
= (𝑄2
𝑄1
)7/4
(𝑅1
𝑅2
)7/4 𝐿2
𝐿1
𝑅1
𝑅2
(𝐷1
𝐷2
)1/4
= 𝑆𝑡ℎ𝑟𝑜𝑢𝑔ℎ𝑝𝑢𝑡1.75 𝑆𝐿𝑆𝑅
−4.75
Scaling with geometric similarity:
𝑆𝑅 = 𝑆𝐿
To keep residence time constant: 𝑆 = 𝑆𝑅2 𝑆𝐿
So: 𝑆 = 𝑆𝑅1/3
(55)
Deduction of the scaling factor for the Reynolds number:
𝑅𝑒 =𝜌𝐷𝑣
𝜇
𝑆𝑅𝑒 =
𝜌𝐷2𝑣2
𝜇𝜌𝐷1𝑣1
𝜇
=𝐷2𝑣2
𝐷1𝑣1
=𝑅2
𝑅1
𝑄2𝑅12
𝑄1𝑅22 = 𝑆𝑅
−1𝑆𝑡ℎ𝑜𝑢𝑔ℎ𝑝𝑢𝑡 = 𝑆𝑅−1𝑆
Using equation (55):
𝑆𝑅𝑒 = 𝑆𝑅−1𝑆 = (𝑆1/3)
−1𝑆 = 𝑆2/3
79
Deduction of the scaling factor for the surface area:
𝐴𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙 = 𝜋𝐷𝐿
𝑆𝐴𝑒𝑥𝑡=
𝜋𝐷2𝐿2
𝜋𝐷1𝐿1
= 𝑆𝑅𝑆𝐿
Using equation (55):
𝑆𝐴𝑒𝑥𝑡= 𝑆𝑅𝑆𝐿 = 𝑆2/3
80
Appendix B. Experimental Residence Time Distributions for reactor coil 1/8 (295 nm)
Figure A. Influence of the Flow Rate in the experimental Residence Time Distribution for reactor coil 1/8 at the
wavelength of 295 nm. The blue points represent the experimental data. A=0.2 mL/min; B=1 mL/min; C=2 mL/min; D=4 mL/min; E=6 mL/min; F=10 mL/min.
0
0.0004
0.0008
0.0012
0.0016
0.002
0 500 1000 1500 2000
E(t
) (s
-1)
Time (s)
A
0
0.002
0.004
0.006
0.008
0 200 400 600
E(t
)(s
-1)
Time (s)
B
0
0.002
0.004
0.006
0.008
0.01
0.012
0 100 200 300
E(t
)(s
-1)
Time (s)
C
0
0.004
0.008
0.012
0.016
0.02
0.024
0 50 100 150 200
E(t
)(s
-1)
Time (s)
D
0
0.008
0.016
0.024
0.032
0.04
0 50 100 150
E(t
)(s
-1)
Time (s)
E
0
0.02
0.04
0.06
0 50 100
E(t
)(s
-1)
Time (s)
F
81
Appendix C. Experimental Residence Time Distributions for reactor coil 1/16 (295 nm)
Figure E. Influence of the Flow Rate in the experimental Residence Time Distribution for reactor coil 1/16 at the wavelength of 295 nm. The blue points represent the experimental data.
Appendix D. RTD for the Model of Tanks in Series for reactor coil 1/8 (295 nm)
Figure B. Influence of the Flow Rate in the RTD for the Tanks in Series Model for reactor coil 1/8 at the wavelength of 295 nm. The blue points represent the experimental data and the orange line is the model of tanks
Appendix E. RTD for the Model of Tanks in Series for reactor coil 1/16 (295 nm)
Figure F. Influence of the Flow Rate in the RTD for the Tanks in Series Model for reactor coil 1/16 at the wavelength of 295 nm. The blue points represent the experimental data and the orange line is the model of tanks
Appendix F. RTD for the Model of Two Batteries in Parallel for reactor coil 1/8 (295 nm)
Figure C. Influence of the Flow Rate in the RTD for the Two Batteries in Parallel Model for reactor coil 1/8 at the wavelength of 295 nm. The blue points represent the experimental data and the orange line is the model of two
Appendix G. RTD for the Model of Two Batteries in Parallel for reactor coil 1/16 (295 nm)
Figure G. Influence of the Flow Rate in the RTD for the Two Batteries in Parallel Model for reactor coil 1/16 at the wavelength of 295 nm. The blue points represent the experimental data and the orange line is the model of two
Appendix H. RTD for the Dispersion Model for reactor coil 1/8 (295 nm)
Figure D. Influence of the Flow Rate in the RTD for the Dispersion Model for reactor coil 1/8 at the wavelength of
295 nm. The blue points represent the experimental data and the orange line is the dispersion model. A=0.2 mL/min; B=1 mL/min; C=2 mL/min; D=4 mL/min; E=6 mL/min; F=10 mL/min.
0
0.0004
0.0008
0.0012
0.0016
0.002
0 500 1000 1500 2000
E(t
)(s
-1)
Time (s)
A
0
0.002
0.004
0.006
0.008
0 200 400 600
E(t
)(s
-1)
Time (s)
B
0
0.002
0.004
0.006
0.008
0.01
0.012
0 100 200 300
E(t
)(s
-1)
Time (s)
C
0
0.004
0.008
0.012
0.016
0.02
0.024
0 50 100 150 200
E(t
)(s
-1)
Time (s)
D
0
0.008
0.016
0.024
0.032
0.04
0 50 100 150
E(t
)(s
-1)
Time (s)
E
0
0.02
0.04
0.06
0 50 100
E(t
)(s
-1)
Time (s)
F
87
Appendix I. RTD for the Dispersion Model for reactor coil 1/16 (295 nm)
Figure H. Influence of the Flow Rate in the RTD for the Dispersion Model for reactor coil 1/16 at the wavelength of
295 nm. The blue points represent the experimental data and the orange line is the dispersion model. A=0.9 mL/min; B=1 mL/min; C=2 mL/min; D=4 mL/min; E=6 mL/min; F=10 mL/min.
0
0.0004
0.0008
0.0012
0.0016
0.002
0.0024
0.0028
0.0032
0 200 400 600 800
E(t
)(s
-1)
Time (s)
A
0
0.002
0.004
0.006
0.008
0.01
0 200 400 600 800
E(t
)(s
-1)
Time (s)
B
0
0.004
0.008
0.012
0.016
0 200 400
E(t
)(s
-1)
Time (s)
C
0
0.004
0.008
0.012
0.016
0.02
0.024
0.028
0 50 100 150 200
E(t
)(s
-1)
Time (s)
D
0
0.006
0.012
0.018
0.024
0.03
0.036
0 50 100 150
E(t
)(s
-1)
Time (s)
E
0
0.02
0.04
0.06
0.08
0 50 100
E(t
)(s
-1)
Time (s)
F
88
Appendix J. Deductions of equations present in Chapter 5
Model of two batteries in parallel for reactor coil 1/16 adjusted to all the flow rates