Flow Meso Reactor
Fernando João Dinis Pereira
Master’s Degree in Mechanical Engineering
Supervisor:
Prof. Alexandre Afonso Co-Supervisor:
Dr. António Ferreira
Numerical Optimization of an Oscillatory Flow Meso Reactor
During this thesis, finite volume method software was used to simulate the flow within an oscillatory flow meso reactor with smooth periodic constrictions in order to understand the effects of changing geometric parameters and operative conditions in the mixing of two fluids, in two distinct 3D reactors.
A direct search optimizer algorithm was used in order to optimize the parametrized geometry of two different 3D reactors, as well as an initial 2D geometry, to reduce the recquired fluid mixing time. NOMAD was the chosen software for the optimization process and it used a mesh adaptative direct search algorithm to allocate the geometric variables.
The Gmsh sofware was responsible for the creation of the geometries and non-orthogonal structured meshes used in the numeric calculation of the fluid flow equations. In order to simulate the flow and quantify the mixing phenomenon, OpenFOAM and swak4Foam were used. The indexed mixing time of a particular geometry was communicated with NOMAD as an output to compare with the results obtained from different simulations.
Optima geometries, which minimized the mixing time of each reactor, were found. Moreover, with the data collected from the optimization loops, it was found that the constriction diameter is the geometrical variable which influences the most and has a direct relation with the mixing time. For the 3D cylindrical pipe geometry a ratio of L2/L1 = 1.4, which increases slightly
with Reo, was found to be optimum while in the planar reactor such ratio was not observed
due to the defined range. Furthermore, the height of the planar reactor was found to have an inverse relation with the mixing time up until 28 milimeters, where changes started to become insignificant.
Lastly, a study regarding the influence of the oscillatory flow frequency, center-to-peak amplitude of oscillation and an added net flow onto the oscillatory motion is performed and analyzed in each 3D optimized reactor. It was found that the frequency and amplitude had inverse relations with the mixing time while the net flow caused insignificant change.
Keywords: Fluid Mechanics; CFD; Oscillatory flow; Superimposed flows; Fluid mixing; OFR; SPCs; Optimization; Continuous Reactor.
Optimização Numérico de um Meso Reator de Fluxo Oscilatório
Durante esta tese, um software do método de volumes finitos foi usado para simular o escoa-mento de um meso reator de fluxo oscilatório, com constrições periódicas suaves, de modo a compreender os efeitos dos parâmetros geométricos e condições de operação, na mistura de dois fluidos, em duas geometrias 3D distintas.
Um algoritmo de optimização de pesquisa foi usado para optimizar a geometria 3D de dois reatores, a fim de reduzir o tempo de mistura necessário. O software de optimização escolhido, NOMAD, usa um método de pesquisa direta em malha adaptativa de modo a alocar os diferentes valores das variáveis geométricas.
O Gmsh foi o software responsável pela criação da geometria dos reatores bem como das respetivas malhas estruturadas não ortogonais usadas no cálculo numérico das equações gover-namentativas do escoamento. O OpenFOAM e o swak4Foam foram usados de modo a simular o escoamento e quantificar o fenómeno de mistura no reator.
Geometrias que minimizavam o tempo de mistura em cada reator foram encontradas com sucesso. Com a informação recolhida dos loops de optimização, descobriu-se que o diâmetro de constrição é a variável geométrica que causa o maior impacto no tempo de mistura e possui um relação direta com o mesmo. O reator tubular cilindrico possui um rácio óptimo de L2/L1 =
1.4, que aumenta com Reo, enquanto que no reator planar este rácio não foi observado devido ao
limite estabelecido para as variáveis. Descobriu-se também que a altura do reator planar possui uma relação inversa com o tempo de mistura, até aos 28 milímetros, tornando-se insignificante para valores superiores.
Por fim, um estudo sobre a influência da frequência e amplitude de oscilação, bem como na adição de um caudal, foi efetuado e analizado em cada geometria óptima. Descobriu-se que a frequência e a amplitude possuem uma relação inversa com o tempo de mistura enquanto que a adição de um caudal não causou mudanças significativas.
Palavras-chave: Mecânica de Fluidos; Dinâmica de Fluidos Computacional; Escoamento Os-cilatórios; Mistura de Fluidos; OFR; Constrições Suaves; Optimização;
First of all, I would like to thank my Supervisor, Professor Alexandre Afonso, for all the support, encouragement and availability shown throughout this thesis. The advices were crucial in the most difficult moments and helped in setting the pace and course of this work. Moreover, the ideas and theories, discussed and speculated, were very interesting and enjoyable moments. The amount of time dedicated with the project was truly inspiring and it is seen as a personal role model for future works. These moments will not soon be forgotten.
Furthermore, to my Co-Supervisor, Dr. António Ferreira, for all the expertise and knowledge given in the technical subjects, the help in the documentation and structuring of this thesis and the trust and belief in my work, I would like to express my gratitude.
Additionally, this work was the result of the projects:
(i) Project POCI-01-0145-FEDER-016816 (PTDC/QEQ-PRS/3787/2014) funded by the Project 9471 - Reforçar a Investigação, o Desenvolvimento Tecnológico e a Inovação (Project 9471 - RIDTI), by the European Regional Development Fund (ERDF) and by national funds through Fundação para a Ciência e a Tecnologia I.P. (FCT);
(ii) IF exploratory project [IF/01087/2014] funded by FCT;
(iii) POCI-01-0145-FEDER-006939 (Laboratório de Engenharia de Processos, Ambiente, Biotec-nologia e Energia, UID/EQU/00511/2013) - funded by FEDER through COMPETE2020 - Programa Operacional Competitividade e Internacionalização (POCI) – and by national funds through FCT - Fundação para a Ciência e a Tecnologia;
(iv) NORTE-01-0145-FEDER-000005 – LEPABE-2-ECO-INNOVATION, funded by FEDER - Fundo Europeu de Desenvolvimento Regional, through COMPETE2020 – Programa Operacional Competitividade e Internacionalização (POCI) and Programa Operacional Regional do Norte (NORTE2020)
To my girlfriend, Eng. Inês Mesquita, for taking up with all my stressful moments and all the encouraging words that kept me going in the right direction with the correct focus, a big thank you. There are times when one is so engaged in what he is doing that when things start to go south, one starts doubting himself and all he needs are a couple of boosting words. You managed to fulfill that need. Furthermore, all the belief you demonstrate everyday in my capabilities is something that I am deeply thankful for.
A special word to The LADs for all the years spent together and all the memories. You guys helped in shaping who I am today and it is with you who I spent my most enjoyable times while being my main source of decompression. Thank you for being awsome.
To all my closest friends, which i shall not name in order not to extend this text to much, thank you for contributing, directly or indirectly, towards this work and for being a major part in my personal life. A special mention to João Sousa for being who you are, the experiences and the ability to always being able to lighten up the mood and Eng. André Ramos for the input knowledge in the programming and handling some of the software as well as being able to make me feel smarter everyday, during the lunch hours. You guys rock.
Lastly, to my parents and family, to whom i shall get through in portuguese:
Obrigado por sempre acreditarem em mim e me terem dado asas para chegar onde cheguei. Todos os sacrificios e insistências feitas, bem como as oportunidades agarradas, por vocês ao longo da vida foi o que me trouxe até aqui tanto fisicamente como mentalmente. Estou-vos eternamente grato por tudo e espero trazer-vos ainda mais alegrias no futuro.
My sincere gratitude to all, Fernando João Dinis Pereira
Abstract i Resumo iii Acknowledgements v Nomenclature xiii 1 Introduction 1 1.1 Objectives . . . 2 1.2 Thesis outline . . . 2 2 Literature Review 5 2.1 The Oscillatory Flow Reactor . . . 5
2.1.1 Industrial Applications . . . 6
2.1.2 Oscillatory Flow Mixing . . . 6
2.1.3 Smooth Periodic Constrictions . . . 7
2.1.4 Parameters that govern the phenomena behind the OFR . . . 8
2.1.5 Influence of the geometrical parameters . . . 10
2.1.6 Scale-Up . . . 12
2.1.7 Heat and mass transfer . . . 12
2.2 The Stirred Tank Reactor . . . 14
2.3 Indicators of Performance . . . 15
2.3.1 Measuring Axial Dispersion . . . 15
2.3.2 Residence Time Distribution . . . 16
2.3.3 Power Density . . . 18
3 Problem Description and Governing Equations 21 3.1 Reactor Geometry . . . 21
3.1.1 2D Case . . . 21
3.1.2 3D case . . . 22 ix
3.2 Fluid Flow . . . 24
3.2.1 Scalar Transport . . . 24
4 Numerical Method 27 4.1 Introduction . . . 27
4.2 Governing equation discretization . . . 28
4.3 Numerical Algorithm . . . 30
4.4 Boundary Conditions . . . 31
4.4.1 Velocity . . . 31
4.4.2 Pressure . . . 32
5 Optimization Loop Characterization 33 5.1 Optimization: NOMAD . . . 33
5.1.1 MADS . . . 34
5.1.2 Defining the parameters . . . 35
5.2 Geometrical meshing: Gmsh . . . 35
5.2.1 2D geometry mesh . . . 36
5.2.2 3D geometry mesh . . . 36
5.3 Computational fluid dynamics: OpenFOAM . . . 38
5.3.1 Objective Function . . . 39
5.3.2 Simulation Control . . . 40
5.4 Setup . . . 41
6 Two dimensional reactor optimization 43 6.1 Maximum amplitude, x0 . . . 43
6.2 Conservative amplitude, x0 . . . 44
6.3 Optimum x0 . . . 46
6.4 Increasing the geometric variables range . . . 47
6.5 Exit and entrance effects . . . 48
6.6 New 2D Geometry . . . 51
6.7 Mesh refinement study . . . 52
7 Three dimensional geometry optimization 55 7.1 Cilindrical pipe geometry . . . 55
7.1.1 Mesh refinement study . . . 57
7.1.2 Operative Variables . . . 58
7.2 Planar geometry . . . 66
7.2.1 Mesh refinement study . . . 68
8 Conclusions and Future Work 75 8.1 Future Work . . . 76 Appendix
A Calculation of the optimum amplitude 79
C Dimensionless concentration
C(t) Concentration time response (arbitrary units) CD Coefficient of discharge of the baffles
Co Courant number
D Constant length inner diameter, mm Dimp Impeller diameter, m
d0 Constriction diameter, mm
∆t Time increment, s
E Diffusion coefficient, m2s−1 E(t) Exit age distribution E(θ) Non dimensional response
f Oscillatory flow frequency, Hz h Convenction coefficient, W m−2K−1 kf Fluid thermal conductivity, W m−1K−1
L Length of the reactor, m L1 Baffle length, mm
L2 Baffle spacing, mm
Lc Characteristic length, m
L0 Length of the entrance and exit zones, mm
N Number of stirred tank in series n rotational speed, rpm
Nb Number of baffles per unit length of the tube
N u Nusselt number P0 Scalar coefficient
P e Peclet number Ren Reynolds number
Reo Oscillatory Reynolds number
Sc Schmidt number σ2 Variance
Str Strouhal number t time, s
¯
t Mean residence time U Axial velocity, ms−1 u Mean net flow, m−1 ~
U Velocity field
v Velocity according to the y-axis, ms−1 VL Volume of liquid, m3
x0 Center-to-peak amplitude, mm
w Velocity according to the z-axis, ms−1 Z Dimensionless length
α Scalar coefficient αn Baffle area ratio
θ Dimensionless time µ Fluid viscosity, kgs−1m−1 ν Kinematic viscosity, m2s−1 ρ Fluid density, kgm−3 φ Generic variable ψ Velocity ratio
Introduction
Since the times of ancient Greece with Archimedes, fluid mechanics has been a field of study and interest of many brilliant minds with enumerous discoveries throughout history until today. Nowadays, the study of fluid dynamics is an active field of research with many problems that remain partially, or even utterly, unsolved. Being mathematically complex, the equations that govern the fluid dynamics are usually solved using numerical methods with the use of computers.
Thanks to the continuous development in numerical computation, fluid mechanics problems can be nowadays solved by a common laptop with little effort. Therefore, further complex flows are currently easier to be solved and researched.
With the industrial competitive paradigm that exists in the present days, more efficient and specialized systems are being discovered and researched continuously in order to keep improving production or services to a point of technological excellence.
Studies indicating the use of oscillating flow motion within pipes with periodic baffles date from approximately 1990. Documented increases in mixing efficiency, rises in heat/mass transfer coefficients, reductions in residence times, when compared to convencional systems, and the ability to work in a continuous operating mode can be found in several of these works, such as Brunold et al. 1989; Dickens et al. 1989; Howes et al. 1991, Mackley and Ni, 1991; Stonestreet and van der Veeken, 1999.
These systems, denoted as oscillatory flow reactors, are a recent area of research with ap-plications in a wide variety of fields such as the pharmaceutical and dye industries. This work will aim in studying such reactors a bit further, in order to contribute to its development.
The objective of this work is to present the study of geometric optimization of an oscillatory flow meso reactor with smooth periodic constrictions, aiming at reducing the required mixing time of the flow and the influence of the operative variables.
1.1
Objectives
The main goal of this thesis is to perform a numerical study on the influence of the geometrical parameters, that characterize the novel oscillatory flow meso reactor under study in the present work, in order to find the optimum geometry which minimizes the mixing time of two Newtonian fluids.
Initially optimization loops were conducted on the geometrical parameters of the reactor thus acquiring a geometry which would enhance the mixing efficiency. This was conducted in a 2D, 3D planar and 3D revolution geometries with specific operational conditions.
In order to achieve this, three different open-source softwares named Gmsh, NOMAD and OpenFOAM were used.
NOMAD would initialize the procedure by communicating values for the geometric param-eters which were read by Gmsh. Then, a geometry would be created and discretized in order for OpenFOAM to simulate the flow and calculate the respective mixing time. This mixing time was indexed to the geometry and returned to NOMAD which would output different possible geometries which could minimize the objective function.
In a second stage, operative variables such as the frequency, center-to-peak amplitude and net volumetric flow rate that defined the operational conditions were studied, in each 3D opti-mized geometry, to understand their influence on the reactor.
The limits for the operative variables were limited by some external variables, such as limitations imposed by the experimental setup, since it is planned to perform experimental studies in the laboratory using the optimized geometry.
1.2
Thesis outline
In this section, the overall structure of this thesis is going to be adressed. This thesis is divided into several chapters and, in the end of the document, one appendix, created to clarify a specific calculation, and the references are compiled.
Chapter 2
In this chapter, the oscillatory flow meso reactor will be presented, along with the main applica-tions, phenomena and reviewed literature, crucial to the development of this thesis. Moreover, some conclusions of different studies are also shown in order to examplify and quantify some important indicators regarding the oscillatory flow reactor.
Chapter 3
In this chapter, the reactor geometry is parameterized using different approaches and some boundaries and specifications are defined. Furthermore, the fluid dynamics governing equations are presented along with some assumptions regarding the fluid and its different phases.
Chapter 4
Here, the methods used to numerically solve the governing equations for the fluid flow are explained. The boundary conditions applied to the reactor are also introduced.
Chapter 5
In order to optimize the geometry of the reactor, an optimization loop had to be constructed. This chapter was created to elucidate the different softwares used and their specific contribution. To conclude this chapter, an overview of the designed optimization script is given, to clarify how such softwares interacted with each other.
Chapter 6
In this chapter, the results of the 2D optimization loops are shown along with associated numerical and geometrical accuracy. Also, due to the accuracy obtained, a new geometry with an increased amount of cells is defined and optimized.
Chapter 7
Following the procedures on chapter 6, chapter 7 presents the 3D optimization loops on the 3D pipe and planar, geometries and their respective numerical errors. On both optimized geometries, a study regarding operative variables namely, the oscillatory flow frequency, the center-to-peak amplitude and the net flow is performed.
Chapter 8
Finnaly, conclusions gathered throughout this work are presented, and discussed, along with some suggestions about future works to further develop efficient oscillatory flow reactors.
Literature Review
On this chapter reviewed literature will be presented, which is crucial to the development of this work, about the oscillatory flow meso reactor. The oscillatory flow mixing phenomena, the industrial applications,the effects of geometrical and operational variables and some examples regarding heat and mass transfer as well as indicators of performance will be the focus of this chapter.
2.1
The Oscillatory Flow Reactor
Essentialy, an oscillatory flow reactor (OFR) is a tube with periodic baffles with a superimposed oscillatory flow onto a regular net flow. The complex dynamics of the flow, provided optimum conditions, cause the formation of vortices which gives rise to a highly unsteadyness and chaotic motion throughout the system.
The application of periodic fluid oscillations inside a tube with evenly spaced constrictions in order to increase mixing effects is the physical concept of an OFR. A schemating image of an OFR can be seen in figure 2.1.
Figure 2.1: Representation of an oscillatory flow reactor. Adapted from Zheng and Mackley (2008).
Usually, the OFRs can be operated with one or multiphase fluids, in an horizontal or vertical direction and in batch or continuous mode (without or with net flow, respectively), being the latter a major advantage in the reaction industry. The liquid/multiphase fluid is oscilated in the axial direction by means of a hydraulic or pneumatic piston or a diaphragm.
The ability to work in a continuous mode allows process automization, which not only reduces timings but also increases reliability and supports an increase in production.
2.1.1 Industrial Applications
Areas of the chemical industry such as the pharmaceutical, agrochemical and dye industry are still heavily dependent on batch-type processing at plant scale. Therefore, the use of stirred tank reactors (STRs) remains the standard approach for mass transfer, heat transfer, chemical reaction, particles mixing, crystallization, among others.
Crystallization can be defined as a complex process, involving multiphase unit operations to achieve separation and purification of products. There are several approaches, namely: reactive, evaporative, antisolvent and cooling crystallization.
Adopting continuous processes for the manufacture of high value chemicals, obtained by crystallization, offers numerous advantages:
• Efficient use of materials (less waste) • Improved production and energetic yields • Improved process reliability
• Reductions in energy consumption
• Less time in process development required for scale up operations • Improved handling of hazardous materials
These can all be achieved with an OFR, which allows the process to change from batch to continuous mode.
2.1.2 Oscillatory Flow Mixing
The existence of baffles, in the inner walls of the reactor, creates critical perturbations in the flow, responsible for the increase in the mixing of the fluid. These perturbations, which increase the reverse motion of the flow, potentiates the increase of non-axial velocities with similar magnitudes as the axial velocity, U, which boosts the overall mixing of the fluid within the reactor.
With the sinusoidal inlet velocity condition, it is possible to observe two half-cycles contain-ing flow acceleration and deceleration, respectively,as seen in figure 2.2.
A plug-flow is achieved in each inter-baffle zone as a result of the radial velocities arising from the cyclic vortices formations and development along the length of the reactor. Therefore, uniform mixing in the radial direction is achieved.
Figure 2.2: Phenomenona in an oscillatory flow mixing mechanism. (1) Start of the Up-Stroke. (2) End of Up-Stroke. (3) Start of the Down-Stroke. (4) End of the Down-Stroke. Adapted from Reis (2006).
When considering continuous operations the system should be operated such that the max-imum oscillatory velocity is at least two times higher than the net velocity of the fluid flowing through the tube. This means that the flow is always fully reversing with fluid interaction at the constrictions and therefore the inter-baffle zone can behave as a series of well mixed stirred tanks.
Generally speaking, mixing is independent of the net flow, and so it is possible to have a low net flow velocity (correspondent to laminar regime in the absence of oscillations) but maintain good mixing and plug-flow performance through the control of oscillatory conditions.
For a meso scale reactor, the mixing effect generated by oscillation is achieved generally across typical ranges of 0.5-20Hz (frequency) and 1-100 mm (center-to-peak amplitude) accord-ing to McGlone et al. (2015). Controlaccord-ing these two variables gives full control of the generation of eddies.
2.1.3 Smooth Periodic Constrictions
There are several approaches in the literature for imposing periodic constrictions in an OFR, including single-orifice, multi-orifice baffles and smooth periodic constrictions (SPCs). Single-orifice baffles are the most common, whereas the SPC systems are more development and mainly limited to mesoscale platform. Multi-orifice baffles exhibit a higher degree of similarity in terms of shear rates and mixing intensity when scaling up, when compared to single-orifice baffles platforms.
This work will be focused on the geometrical optimization of an OFR similar to the one rep-resented on figure 2.1. One of the modifications will be that the sharp baffles will be substituted with smoother ones, as seen in figure 2.3, which represents an SPCs reactor.
Figure 2.3: Overview of the smooth periodic constrictions in the reactor.
These geometrical changes are introduced since it is expected that the appearence of dead zones will decrease leading to an increase on the efficiency of the mixing and reducing the area where particles may sediment or become trapped, as well as minimizing high shear regions. The appearence of dead zones in conventional OFRs is mainly due to the existance of sharp baffles, which causes secondary flows to form near these and the reactor walls.
2.1.4 Parameters that govern the phenomena behind the OFR
Due to the highly dynamical nature of the fluid flow in an OFR, it is often easier to characterize it with the help of some dimensionless parameters. The net flow Reynolds number, Ren, the
oscillatory Reynolds number, Reo, the Strouhal number, Str, are enough, according to McGlone
et al. (2015), to fully define the dynamics on an OFR. However the Peclet number, P e, the Schmidt number, Sc, and the Nusselt number, Nu, are indicators used in heat/mass transfer, helpfull in some applications of an OFR.
2.1.4.1 Net flow Reynolds number, Ren
The net flow Reynolds number is a ratio between the inertia and viscous forces, derived from an external axial flow imposed within the reactor. It is of great importance in the majority of fluid dynamics problems due to its extensive use in fluid mechanics literature and because it can fully define the most basic of fluid flow system (axial flow within a plain tube). It is known that turbulent regime boosts mixing phenomena since it gives rise to radial velocities however, the regime in the reactor is laminar, hence the need for oscillations.
It is defined in McGlone et al. (2015) as: Ren=
ρud0
µ (2.1)
where ρ is the fluid density, u is the mean net flow velocity, d0 is the constriction diameter and
µ is the fluid viscosity.
2.1.4.2 Oscillatory Reynolds number, Reo
The oscillatory Reynolds number is also a ratio between the inercia and viscous forces that originate from an oscillatory flow. It describes the intensity of the mixing and it is defined in
Reis et al. (2005) as:
Reo =
2πf x0ρd0
µ (2.2)
where f is the oscillatory flow frequency and x0is the centre-to-peak amplitude of the oscillation.
According to McGlone et al. (2015), flow separation happens at Reo of 50. At values
be-tween 100-300 the system exhibits plug-flow characteristics, where vortices are axi-symmetrical generated within each cavity. At higher values symmetry is broken and flow becomes intensely more turbulent. These critical values, however, depend on the geometry of the reactor as well as the viscosity of the fluid. Moreover, for the novel meso reactor these values can be different due to the change in the shape of the baffles.
It is possible to define a velocity ratio as: ψ = Reo
Ren
= 2πf x0
u (2.3)
This ratio should be greater than 1 so that the maximum oscillatory velocity is always higher than the net flow velocity thus making the oscillatory flow dominate over the net flow. For plug-flow operation in liquids minimum values between the range of 2-10 are recommended in McGlone et al. (2015).
A minimum value for Reo of 100 has been postulated for sufficient mixing.
2.1.4.3 Strouhal number, Str
The Strouhal number is the ratio between the constant length inner diameter, D, and the oscillation amplitude and it represents an evaluation of eddy propagation.
It is defined in Reis et al. (2005) as:
Str = D
4πxo (2.4)
Refering to Reis (2006), this definition was originally used in order to describe the frequency of vortex shedding around objects in a flow. In this work a modified Strouhal number will be used as follows:
Str = d0
πx0 (2.5)
where d0 is the constriction diameter. This modified definition takes into account not only that
the constriction diameter (rather than the constant length inner diameter) is the appropriate length scale but also that two vortices are shed (on each side of the constriction) during each full oscillation since the flow also reverses.
2.1.4.4 Peclet Number, P e
The Peclet number is a ratio between the convective and difusion transport that allows to quantify axial diffusion.
It is defined in McGlone et al. (2015) as:
P e = uL
E (2.6)
where u is the mean net flow, L is the length of the reactor and E is a diffusion coefficient. It is used to estimate the magnitude of experimental errors as well as the effectiveness of a particular reactor, according to Smith (1999).
2.1.4.5 Schmidt Number, Sc
The Schmidt number is a ratio between diffusive momentum and mass difusion and measures the relationship between the thickness of the hydrodynamic and mass transfer boundary layer.
It is defined in McGlone et al. (2015) as: Sc = µ
ρE (2.7)
It is also very helpful when talking about the scale-up of a particular reactor, according to Smith (1999).
2.1.4.6 Nusselt Number, N u
Nusselt is an adimensional parameter which measures a ratio between convection and conduction heat transfer and is widely used in the field of heat transfer. The usual definition is (Incropera et al. (2011)):
N u = hLc
kf (2.8)
where h is the convection coefficient, Lc is a characteristic length and kf is the fluid thermal
conductivity.
2.1.5 Influence of the geometrical parameters
Due to the recent advances in OFR research it is possible to identify some effects caused by the geometrical parameters of the reactor. In short, each baffle cavity can be thought as a continous stirred tank in which radial velocity components are comparable to axial ones and where events at the wall are similar to the events happening in the center.
These effects can be explored experimentally by carrying studies on residence time distri-butions and liquid-liquid dispersions.
Morevover, every studied mentioned below is referred to the pipe reactor, since it is the only geometry found in the literature.
2.1.5.1 The effect of d0
Most of the studies use an adimensional coefficient αn defined as d20/D2 that represents the
Reis (2006) presents several studies from different authors, regarding the influence of this parameter on mixing times and axial dispersions. Generally speaking, a low value of αncauses
higher axial dispersion and consequently shorter mixing times. However, the effects of d0 are
connected with values of f and x0 in a specific situation since it was also reported that for an
αn of 0.26 the vortex rings formed by small symmetrical eddies did not encompass the entire
cross-section of the baffle (nor its length) thus creating stagnant regions and longer mixing times.
On the contrary, having a high value for αn of 0.47, with the same operational variables,
caused axial movement to be predominant hence destroying eddies, which caused low mixing and even longer mixing times.
Nonetheless, these studies are focused not on the smooth constrictions reactor, but on its conventional form, which has a higher tendency to form dead zones. As such, one can say for sure that high values of αnare not beneficial on oscillatory flow mixing and that it is expected,
throught this work, that low values of αnturn up to be the ones which minimize mixing times.
2.1.5.2 The effect of baffle spacing, L2
Reis (2006) presents several studies which focus on the impact of this specific parameter, since for a given x0, it is a key design variable which influences the shape and length of eddies within
each baffle cavity hence defining the flow behaviour.
According to Reis (2006), the optimal value for L2 should ensure a full expansion of vortex
rings so that they spread effectively throughout the entire inter-baffle zone. However, if L2 is
too low, the generation of vortices is supressed, restraining radial motion, whereas if L2happens
to be too high the vortices cannot cover the entire inter-baffle region, which will make them disperse and diminish reducing radial motion.
Regarding minimizing mixing time, a ratio of L2/D between 1.4 to 2 was found to be
optmimum according to Ferreira et al. (2016). Moreover, these effects change with different fluids and more importantly with distinct operative variables, such as f and x0. These operative
variables have the most impact on the fluid behaviour and must not be disregarded when optimizing L2 since its value is deeply connected with x0. After all, one needs to know the
displacement of the fluid caused by an oscillatory motion, x0, before defining the optimum
space for it.
2.1.5.3 The effect of baffle thickness
Given that the geometrical modifications introduced in this work, showed in 2.3, are rather recent, most of the literature that exists focuses on the effects of baffle thickness and as such there are no studies regarding the influence of L1 on its own.
According to the studies presented by Reis (2006) each eddy needs an edge to cling on and possesses an optimal time of development and shedding. Overall, thinner baffles enhance the generation of vortices since being attached to the baffles edges for too long prior to shedding, can distort their shapes, thereby affecting mixing time. Moreover, higher values of baffle thickness
usually result in higher mixing times. Therefore, it is expected that L1 will tend to a minimum
when performing a geometrical optimization aiming in minimizing mixing time.
2.1.6 Scale-Up
The availability of microfluidic and mini/mesofluidic reactors are a major benefit for industries such as the pharmaceutical which benefit greatly from continuous manufacturing and since they usually work with small control volumes and particles. These reactors can be used on laboratory and pilot plant scales for development and implementation of continuous processing.
The reaction parameters, such as temperature, concentration and reactants’ composition established for a small scale flow process can be directly scaled-up or scaled-out. On the other hand, analogous batch-type processes, like the STR, often require significant scale-up design and optimization involving numerous variables, including heat and mass transfer and reactor geometry, according to McGlone et al. (2015).
Moreover, Smith (1999) studied the effectiveness of scale-up in OFRs and found that for similar dynamical flow conditions the mixing efficiency (or axial dispersion) was not a function of the tube diameter and therefore leading a reliabe scale-up in terms of residence time distribution. However, in situations where similar conditions cannot be used when scaling-up the OFR, a solution of switching to multi-orifice tubes was proposed which allowed for a confident scale-up of greater magnitudes.
2.1.7 Heat and mass transfer
Mackley et al. (1990) show the improvement of heat transfer phenomenon in OFRs. The authors, compares the existence of baffles and oscillatory flows with their counterpart and how it affects Nusselt values.
Figure 2.4 presents the curve of Ren vs Nu, presenting the benefits of baffles in the system
without oscillation, which on itself clearly causes an increase in the heat transfer coefficient values, for a range of different Ren values (Mackley et al. (1990)).
The effect of oscillatory flows inside a reactor cell can be observed in figure 2.5, for a fixed value of Ren and Str of 300 and 0.67 respectively, adapted from Mackley et al. (1990).
Figure 2.4: Curve of Nu vs Ren. ( + ) No baffles and no oscillation. ( x ) Baffles but no
oscillation. Adapted from Mackley et al. (1990)
Figure 2.5: Curve of Nu vs Reo for a Ren of 300 and Str of 0.64. ( ) No baffles and
oscillation. ( x ) First run with baffles and oscillation. ( + ) Second run with baffles and oscillation. Adapted from Mackley et al. (1990)
It can be observed that oscillations are insignificant in the heat transfer coefficient, however when combined with baffles it is possible to increase Nu values beyond the ones in figure 2.4. Moreover, according to figure 2.5, for Reo>1000, it can be inferred that Nu is strongly
dependent on the oscillatory flow.
Also in mass transfer phenomenon there is evidence that OFRs can have superior efficiency when comparing them with alternatives systems according to Ni et al. (1995), who conducted experiments regarding the transport of vapor phase oxygen to liquid, in both OFRs and stirred tanks, to boost the growth of cultures of microorganisms. Figure 2.6 shows the relationship between the mass transfer coefficient and the agitational speed of a stirred tank, whereas figure 2.7 shows the variation of the mass transfer coefficient with the frequency of the oscillatory flow for different amplitude values on an OFR.
Figure 2.6: Relationship between the mass transfer coefficient, of vapor oxygen into a liquid solution, and agitation speed inside a stirred tank. Adapted from Ni et al. (1995)
Figure 2.7: Relationship between the mass transfer coefficient, of vapor oxygen into a liquid solution, and oscillatory flow frequency for diferent amplitudes inside an OFR. Adapted from Ni et al. (1995)
Analyzing figure 2.6 we can conclude that, troughout the refered study, the mass transfer coefficient increased, approximatly, linearly with the agitation speed of a stirred tank, up to a maximum value of 180 (l/hour) for 1800 rpm. On the other hand, for an OFR, figure 2.7 shows an increase of the coefficient with the rise of amplitude and frequency values, with the latter being more impactful, reaching a maximum value of 450 (l/hour), which is 250% higher.
Further information, which also supports OFR’s advantages, can be found in Ferreira et al. (2016).
2.2
The Stirred Tank Reactor
A stirred tank reactor (STR), often called agitated vessel, is a traditional reactor.
The conventional STR possesses a cylyndrical shape with a vertical axis, equipped with a centrally positioned vertical shaft which can have several stirrers mounted. Figure 2.8 illustrates a typical STR with two radial flow agitators mounted.
Figure 2.8: Stirred tank reactor. a) Heating or cooling coil; b) Motor drive; c) Baffle; d) Gear box; e) Seal; f) Manhole; g) Ring sparger; h) Agitators; i) Heating or cooling jacket. Adapted from Nienow (2000).
It can operate in either laminar or turbulent conditions, with a wide range of different fluids (even non-Newtonian) and also multiphase fluids. Operating with the torque provided from the eletrical motor, it uses the radial motion provided to the blades to stir and mix the fluid within (Nienow (2000)). One can think of an OFR as several STRs placed in series, where each one represents a cell from the OFR, according to Stonestreet and Van Der Veeken (1999).
2.3
Indicators of Performance
2.3.1 Measuring Axial Dispersion
Axial dispersion is a measure of the rate at which an inert tracer spreads axially along an OFR. It can be an evaluation of macro-mixing, micro-mixing (e.g. molecular diffusion) or of both. The quantification of macro and micro-mixing is of particular interest when trying to predict residence time distributions for larger or smaller scales, respectively.
The dispersion model in dimensionless form can be described as: ∂C ∂θ = 1 P e ∂2C ∂Z2 − ∂C ∂Z (2.9)
where P e is the Peclet number, C is the dimensionless concentration, θ = ut/L is dimensionless time, where t is the time measured since a tracer injection, Z = x/L is the dimensionless length, where u is the mean axial velocity, x the axial position and L the length of the section in testing. One thing to take into account is the fact that the majority of analytical solutions of equation (2.9) are based on a perfect input function (perfect pulse of trace injection) as a boundary condition, which experimentally is not an easy task to do.
Therefore Zheng and Mackley (2008) came up with an explicit correlation between the axial dispersion coefficient and the flow parameters in an OFR with SPCs, based on an imperfect tracer pulse injection and experimental data, presented in equation (2.10).
E ν = 0.026Ren+ 5.91Re 0.058 o Str −1.62 + 2.92Re 2 n 0.026Ren+ 5.91Re0.058o Str−1.62 (2.10) where ν is the kinematyc viscosity of the fluid, which on the experimented done by Zheng and Mackley (2008) was water. Moreover they found that, for regular operating conditions there was an optimum value for Reo which minimized axial dispersion as seen in figure 2.9.
Figure 2.9: Axial dispersion coefficient vs oscillatory reynolds number for different reynolds number with a fixed frequency of 12 Hz. Adapted from Zheng and Mackley (2008).
2.3.2 Residence Time Distribution
The residence time distribution (RTD) is a probability distribution function that describes the amount of time a fluid element is inside the reactor. It is used to characterize mixing and flow within an OFR and also to compare the behaviour between the real reactors and ideal/model reactors. It proves useful for experimental troubleshooting, estimating the efficiency of a specific reaction and designing future reactors.
If one could know the exact velocity distribution profile of all the elements of fluid with respect to time and space within the reactor, onde would possess a complete physical model of an RTD. Even though success has already been achieved in modelling the fluid mechanics using direct numerical solution (DNS) of the Navier-Stokes equation, this is only restricted to a 2D, axisymmetric flow regime (with Reo < 250), which is not enough to be implemented as a full
RTD model.
The alternative approach, widely adopted in the literature, is to use phenomenological models, according to Stonestreet and Van Der Veeken (1999), where two diferent principles can be employed: a dispersion model-type, in which the reactor is seen as a continuous path, or a compartment model-type, where the reactor is visualized as being divided into discrete stages.
Stonestreet and Van Der Veeken (1999) used a compartment model-type, more specifically the tank-in-series model, which treats an OFR as several STRs placed in series, to study the influence of flow parameters on an OFR RTD.
Their experimental study was based on a tracer solution injection at time zero using a pulse input method, where the resulting concentration-time response was recorded and characterised. The data was normalized in order to compare with the analytical solution from the model as seen in equation (2.11)
E(t) = C(t)
C0 (2.11)
where E(t) is the exit age distribution, C(t) is the concentration-time response and C0 is the
total area under the concentration-time curve as defined in equation (2.12). C0 =
Z ∞
0
C(t)dt (2.12)
The time domain normalization is defined in equation (2.13) θ = t
¯
t (2.13)
where ¯t is the mean residence time and is calculated according to equation (2.14). ¯
t = R∞
0 C(t)tdt
C0 (2.14)
Equation (2.15) gives the final coherence to make the results possible to be plotted,
E(θ) = ¯tE(t) (2.15)
where E(θ) is the non dimensional response.
The tank-in-series model assumes that the concentration-time response can be represented by a cascade of equal-size ideal STRs, shown in equation (2.16) where N stands for the number of STRs which gives the best fit to the concentration time data.
E(θ) = N (N θ)
N −1
(N − 1)! e
−N θ (2.16)
The value of N can be calculated through equation 2.17 N = 1
σ2 (2.17)
where σ2 is the variance of the data which is calculated using equation 2.18.
σ2 = R∞ 0 t 2C(t)dt R∞ 0 C(t)dt − ¯t 2 (2.18)
Figure 2.10 shows the comparison between the model and the experimental data obtained by Stonestreet and Van Der Veeken (1999). This is presented here to merely show an example of an RTD on an OFR.
Figure 2.10: Residence time distribution obtained from experimental data followed by a plot of the tank-in-series model. Adapted from Stonestreet and Van Der Veeken (1999).
Stonestreet and Van Der Veeken (1999) found a range for Reo which made the system
operate closest to a plug flow. Even though there was a slight interdependence of Ren and
Reo the latter was more impactful taking into account a range of Ren < 250. Otherwise, flow
oscillations would become less important in improving RTD response (due to shorter residence times at higher values of Ren). An optimum value of ψ around 1.8-2 was also observed.
2.3.3 Power Density
One way to compare performances of OFR systems to STR equivalents (or to compare with different OFRs systems) is to use power density values, P/V (watts per cubic meter), which is an estimate of the amount of power applied per unit volume for each system, or in other words, power consumption.
For an STR power density can be calculated as: P
V =
P0ρn3Dimp5
VL (2.19)
where P0 is a scallar coefficient unique to each system called power coefficient, n is the impeller
rotational speed, Dimp is the impeller diameter and VL is the volume of liquid inside the STR.
and is dependent on the Reynolds number. There are also some corrections for some variations of reactors.
According to McGlone et al. (2015), in an OFR there are two models to estimate the power density: the quasi-steady flow model and the eddy acoustic model.
The eddy acoustic model is used for conditions of low x0 and high f:
P V =
1.5(2πf )3x20L2
L1αn (2.20)
where αnis the cross-sectional area ratio in the constriction, L1 the constriction length and L2
is the constant diameter section lenght.
The quasi-steady model is used for conditions of high x0 and low f:
P V = 2ρNb 3πCD2 1 − α2 n α2 n x30(2πf )3 (2.21) where Nb is the number of baffles per unit length of tube and CD is the coefficient of discharge
of the baffles (depends on αn, has an average value of 0.7).
To present an example, curves comparing values of mass transfer with power density in an STR and in an OFR are shown in figure 2.11. Both curves were calculated using the quasi-steady model with tweaked coefficient values in Ni et al. (1995).
Figure 2.11: Comparison between two reactors on the behaviour of mass transfer coefficient with power density values. Adapted from Ni et al. (1995).
Problem Description and Governing Equations
On this chapter, an explanation regarding the geometric variables which define the reactor is presented as well as the governing equations for the fluid flow, which need to be solved.
3.1
Reactor Geometry
3.1.1 2D Case
A representation of the case geometry can be seen in figure 3.1, in which we can observe the periodic pattern of the reactor. Regarding the two dimensional simulations, the 4 variables
Figure 3.1: Periodic pattern of the reactor and its main variables.
presented in figure 3.1 are enough to fully define the reactor geometry, namelly D is the constant-length diameter, d0 the constriction diameter, L1 the contraction length and L2 the reactor cell
length.
Each variable, except D which will be fixed throughout this work, also has its own upper and lower limit previously defined as seen in table 3.1. These limits are used as guidelines, since they were established in order to reduce dead zones or stagnant regions identified in conventional OFRs, according to Ferreira et al. (2016).
Table 3.1: Lower and upper limits of the reactor’s geometric variables. D [mm] d0 [mm] L1 [mm] L2 [mm]
8 0.41-0.60 D 1.4-2 D 3-4.5 D
Furthermore, taking into account the fact that the flow within the reactor has an oscillatory motion, there will be volumes of fluid that are going to leave the control volume defined in figure 3.1.
In order to minimize the effects caused by the loss of fluid volume (due to the oscillatory motion), entrance and exit zones will be created and added to the control volume. These zones will have a lenghth of roughly 10D, defined as L0 and will be assembled with the reactor cells
as shown by figure 3.2.
Figure 3.2: Overview of the two dimensional reactor geometry.
3.1.2 3D case
In this work, two different three-dimensional geometries were used. These 3D geometries have two mainly different patterns, one shaped like a canal (an extruded and planar version of the 2D) presented in figure 3.3 and another one cilindrical, displayed in figure 3.4.
The 3D planar version introduces a new geometric variable, ω which defines the height of the reactor and will have a defined range of 2.5 to 15 mm. Table 3.2 summarizes the adopted variable range for this geometry. These limits arise from the optimal values refered in Ferreira et al. (2017) as well as from the limitations imposed from the computational resources for the optimization loops and mesh constructions.
Table 3.2: Variable range for the 3D planar geometry. D[mm] d0 [mm] L1 [mm] L2 [mm] ω [mm]
Figure 3.3: Overview of the three dimensional planar reactor geometry and its parametric variables.
The pipe version does not require additional variables to fully parametrize it, since it is a 360 degrees revolution of the 2D geometry. Moreover, the variable range to be analyzed is the same as the 2D.
Figure 3.4: Overview of the three dimensional pipe reactor geometry and its parametric vari-ables.
One can easily foresee higher velocity gradients in the pipe reactor due to the area reduction in the constriction zone which is a function of d2
0/D2 whereas the planar reactor possesses an
area reduction function of d0/D. This means faster velocities (assuming the same amount of net
flow and oscillatory conditions) in both reactors thus, a lower mixing time in the pipe reactor can be expected.
However, the planar reactor has other added benefits. Despite the fact that it is a more compact geometry, particles deposition is a much lesser problem. Even though this work does not study suspension or transport of particles/sediments, since this is a major application for an OFR, it is something to take into account.
3.2
Fluid Flow
In the context of this work, it is assumed that the working fluid is water with no change in the temperature. Thus, the flow can be described as incompressible, isothermal, newtonian and in laminar regime.
Therefore, the continuity and Navier-Stokes equations suffice to describe the flow behaviour. These equations can be represented in a simplified form and in all spacial directions, Munson et al. (2013), as Continuity equation: ∂u ∂x + ∂v ∂y+ ∂w ∂z = 0 (3.1) Navier-Stokes equations: ρ ∂u ∂t + u ∂u ∂x+ v ∂u ∂y + w ∂u ∂z = −∂p ∂x+ ρgx+ µ ∂2u ∂x2 + ∂2u ∂y2 + ∂2u ∂z2 (3.2) ρ ∂v ∂t + u ∂v ∂x+ v ∂v ∂y+ w ∂v ∂z = −∂p ∂y + ρgy+ µ ∂2v ∂x2 + ∂2v ∂y2 + ∂2v ∂z2 (3.3) ρ ∂w ∂t + u ∂w ∂x + v ∂w ∂y + w ∂w ∂z = −∂p ∂z + ρgz+ µ ∂2w ∂x2 + ∂2w ∂y2 + ∂2w ∂z2 (3.4) On the context of this work the terms regarding the gravity (gx, gy and gz) are negligable.
Furthermore, everytime the analysis is focused on 2D it is the same as negleting equation (3.4) as well as any term regarding ∂z.
3.2.1 Scalar Transport
In order to evaluate the mixing power of the reactor, a scalar α, has been used to divide the flow into 2 initial phases as seen in figure 3.5. The darker area represents a phase where α takes the value of 1 where in the lighter part α is 0. The time it takes for α to reach the value of 0.5 in the entire domain is also the time the reactor takes to reach a full mixed state.
This scalar allows the calculation of the mean value of the properties of the fluid, in the regions of the domain that are filled with both phases, as shown by equations (3.5) and (3.6)
ρ = αρ1+ (1 − α)ρ2 (3.5)
µ = αµ1+ (1 − α)µ2 (3.6)
Since both of those phases are water (ρ1=ρ2 and µ1=µ2), both equations (3.5) and (3.6) end
up being redundant. Despite this, they are presented here to show how to cope with a situation where the phases were representing fluids with diferent properties.
Figure 3.5: The two diferent phases in the reactor, the darker area α = 1 (phase 1) and the lighter area α=0 (phase 2).
Regarding the actual calculation of α in each timestep as well as each point in the domain, the generic scalar conservation equation (3.7), adapted from Versteeg et al. (1995), is enough to define it,
∂(ρα)
∂t + div(ρα ~U ) = div(Γ grad α) + qα (3.7) Looking at each parcel individually:
∂(ρα)
∂t : Transient term that takes into account the change of α in the control volume.
div(ρα ~U ): Convection term responsible for the transport of α due to the existance of the velocity field ~U.
div(Γ grad α): Diffusion term accountable for the transport of α due to its gradients, where Γ is 10−6m2s−1 in the context of this work.
qα: Source term that defines phenomenom that creates or destroys α in the control volume
(or any other term that doesn’t fit in the previously ones above). This term is zero in the entire domain of this case.
In chapter 4 an explanation will be made on how the diferent terms were discretized, in order to obtain a full algebric system of equations which allows to calculate a proper solution.
Numerical Method
In this chapter, a brief description of the Finite Volume Method used by OpenFOAM to solve the governing equations of the flow, will be presented. It is expected with this explanation that one can understand the overall steps in calculating the equations in order to get numerical results.
4.1
Introduction
According to Istvan Farago (2013) the Finite Volume Method is a numerical technique that transforms parcial derivative equations, representative of conservative laws, in discrete algebric equations over a finite number of non overlapping control volumes. These are the result of discretizing the geometry in cells which together, in this case, form a non orthogonal structured mesh.
Each cell posesses six faces named Top (T), Bottom (B), North (N), South (S), West (W) and East (E), as shown by figure 4.1. The governing equations are integrated in each face of the cell and the values of the variables (φ) are calculated inside it, on point P.
Figure 4.1: Cell representative of a Control Volume. (J.H. Ferziger 2002)
4.2
Governing equation discretization
According to Versteeg et al. (1995) all the conservation equations can be written in the same generic differential form, as seen in equation (4.1),
∂(ρφ) ∂t + ∂ ∂xi ρU φ − Γ∂φ ∂xi = Sφ (4.1)
where φ is a generic variable, xi is the direction vector of the 3D Euclidean space, Γ is the
diffusion coefficient for the diffusive transport of a generic variable φ and Sφ is the respective
source term.
Different discretization schemes were used to discretize the integrated equation (4.1) over each cell volume, VP, hence, a brief remark will be given to each one. Every scheme was
consulted from Moukalled et al. (2016). (i) Transient Term
The transient term is discretized using the first order Euler method, Z VP ∂(ρφ) ∂t dV = ρVP ∆t (φP − φ n P) (4.2) where φn
P represents the stored variable φ inside cell P, in the prior timestep n. Since
this method is an implicit one, every other variable which does not possess a temporal index are refered to the new timestep.
(ii) Convective Term
Integrating the convective term, one gets the sum of the convective fluxes of φ across all the control-volume faces f in a specific cell,
Z VP ∂(ρU φ) ∂xi = 6 X f =1 Ffφi,f (4.3)
where Ff is the mass flow rate across face f of the control volume and φi,f is the value
of φ at face f in the i direciton. The interpolation scheme used in the estimation of φi,f is a very important part of the calculation process hence, there are a number of
different methods available.
The default interpolation scheme used is the central differencing scheme, φi,f = φi,G xf − xP xG− xP + φi,P 1 − xf− xP xG− xP (4.4) where the index G is refered to the neighbour point of P which has the face f separating them. This scheme has a second order accuracy and also a good convergence rate (when comparing with other schemes) however, it is also known to be a bit dissipative which causes sharp changes in solution gradients to look smoother.
Thus, when solving for α a different scheme was used. A higher resolution, non linear and Fromm based scheme, developed by van Leer was used.
The calculation of φi,f, being the index A and B the neighbourly downstream nearest
points (where B is the furthest one away from P ), is as follows:
φi,f = φi,A+ 0.5ϕ(r)(φi,A− φi,B) (4.5)
where ϕ(r) is defined as a limiter function, with r being the gradient ratio, where M is the upstream nearest point from P , as seen in equation 4.6.
r = φi,M − φi,A
φi,A− φi,B (4.6)
The van Leer scheme, according to van Leer (1974), dictates a non linear function for ϕ(r), seen in equation 4.7.
ϕ(r) = r + |r|
r + 1 (4.7)
The calculation of Ff comes from the integration of the continuity equation:
Z V ∂ ∂xi (ρU )dV = 3 X i=0 ((ρU A)Bf − (ρU Ac)M f) (4.8)
where Acstands for the cell face area and the indexes Bf and Mf stand for upstream
and downstream faces respectively in the i direction. (iii) Diffusion Term
The integration, along the volume of cell P , of the diffusion term results in: Z VP ∂ ∂xi Γ∂φ ∂xi dV = 6 X f =1 Γf ∂φf ∂xi · ~Sf (4.9)
where Sf is a surface vector and denotes the area of face f from the control volume
and Γf the value of Γ at face f.
For non-orthogonal meshes a correction is applied to the surface vector so that the gradient is calculated along a line which joins the centroids of two neighbour control volumes.
Moukalled et al. (2016) assumes a linear profile for the remaining term which is possible to see in equation 4.10:
∂φf
∂xi
= φN − φP
∆N P (4.10)
(iv) Source Term
The source term is zero except when solving the conservation of momentum equations (Navier-Stokes), which has the pressure gradient ∂p
∂xi. This term is discretized
assum-ing a linear profile between the adjacent cells middle points’, similarly to equation (4.10).Integrating this term results in the following equation:
Z VP ∂p ∂xi = 3 X i=1 [∆p]i,PS~P (4.11)
4.3
Numerical Algorithm
The numerical algorithm used to calculate the solutions for the Navier-Stokes equations is the Pressure Implicit solution by Split Operator method (PISO), which will be briefly explained following Moukalled et al. (2016).
Firstly, initial and boundary conditions are set, followed by an initial computation of an intermediate velocity field by solving the discretized momentum equations. The discretized momentum equation in the i direction, has the following form:
aPui,P − 6 X f =1 afui,f = Sui+ ρVP ∆t u n i,P (4.12)
where the index n is refered to previous timestep, the af represents the convective and diffusion
terms and is calculated according to (4.13).
af = Ff+ µfS~f (4.13)
Since Sui is calculated through equation (4.11), the remaining term, aP, is calculted using the
following expression: aP = ρVP ∆t + 6 X f =1 af (4.14)
In order to solve equation (4.12) in every direction an initial guess to the pressure field is needed (based on the initial and boundary conditions).
With the initial ui known, a correction is added, u∗i in order to satisfy mass conservation.
As such, a new velocity field un+1
i , which satisfies both momentum and mass conservation is
calculated, followed by an updated pressure field. The calculation of the α field is also updated along the pressure and velocity fields. This step is repeated for a certain amount of times or until the discrepancy between iterations is below a certain tolerance. Finally, the timestep is increased and the entire process starts again.
4.4
Boundary Conditions
In order to obtain a solution for the equations presented in (4.12) boundary and initial condi-tions for the velocity, pressure and α need to be defined. Furthermore, since the couple pres-sure/velocity are calculated in a very strongly dependent way, any missassumption or overly-defined boundary conditions can lead to non physical solutions.
4.4.1 Velocity
The fluid within the reactor starts with zero velocity as an initial condition. Moreover, since the fluid moves within the reactor with an oscillatory motion the velocity value on the inlet will continuosly change along time similar to the function shown in figure 4.2.
Figure 4.2: Function representative of the inlet velocity boundary condition along time. A mathematical expression that defines it is presented in equation 4.15.
U = Unet+ 2πf x0sin(2πf t) (4.15)
where U is the axial velocity at the inlet, Unet is a constant velocity calculated through the net
flow imposed on the reactor divided by the inlet area, f is the oscillatory flow frequency and x0 is the center-to-peak amplitude.
The outlet velocity boundary condition is a Neumann type boundary condition defined as, ∂U
A zero velocity gradient at the outlet means that the flow is assumed to be fully developed and therefore, it does not change along space.
The velocity value at the walls is zero since the reactor is not moving.
As a side note, the calculation of α relies solely on the velocity field and it doesn’t affect the values of the former and as such the boundary conditions for the calculation of α are of zero gradient in the outlet, inlet and walls. Regarding the initial conditions, half of each analyzed reactor has an α of 1 while the rest is set at 0.
4.4.2 Pressure
Regarding the pressure boundary conditions it is going to be assumed an atmospheric pressure within the reactor as initial conditions. At the outlet, a fixed value of zero relative pressure is set and since the remaining boundaries are already strongly specified by the velocity, it is going to be assumed a zero pressure gradient (similarly to equation 4.16) in all the other boundaries.
Optimization Loop Characterization
The geometrical optimization of a reactor is a rather complex and time-consuming process. First one needs to define one objective function which is capable of evaluating the efficiency of the reactor. Then, after generating a mesh for a specific geometry, simulate the flow behaviour to provide data as input for the objective function. All this needs to be done in a loop to make it feasible within a welcomed amount of time.
Three diferent main open-source softwares were used in order to design the loop mentioned above. The core communications between them are illustrated in figure 5.1.
Figure 5.1: Core diagram of the optimization loop.
5.1
Optimization: NOMAD
NOMAD is an optimization software which uses the mesh adaptative direct search (MADS) algorithm, according to Audet et al. (2009).
Instead of manually selecting enumerous different values for the geometric parameters to later compare their results, NOMAD follows an algorithm which makes this selection faster and smaller. This software sequencially find the optmimum geometric parameters based on the results of the mixing time, which are an input in the software.
5.1.1 MADS
Consulting Le Digabel (2011), the MADS algorithm is an iterative method where an objective function is evaluated at specific trial points lying on a mesh. In other words, given an initial y0 ∈ Ω, where Ω is the feasible region of possible solutions, it attempts to locate a minimizer
of the function f over Ω by evaluating fΩ at trial points. Morever, it does not require any
derivative information from f.
At each iteration, k, a finite number of trial points are generated and its function values are compared with the previous current value fΩ(yk), i.e., the best feasible objective function value
found so far. Each of the trial points lies on a mesh, which is constructed from a finite set of nD directions D ⊂ Rn and is defined in Le Digabel (2011) at a current iteration k as:
Mk=
[
y∈Vk
{y + ∆mkDz : z ∈ NnD} (5.1)
where ∆m
k ∈ R+ is the mesh size parameter, Vk are the set of points on which the objective
function has been evaluated at the start of iteration k (V0 contains the starting points manually
inserted), D is called the set of mesh directions and is constructed so that D = Gzj, where G
is a nonsingular n × n matriz and zj ∈ Zn is an integer vector, according to Le Digabel (2011).
Also refering to Le Digabel (2011), each iteration is composed of three steps: the poll, the search and updates. The search step is very flexible and allows the creation of trial points anywhere on the mesh. If an improved mesh point is generated, then the current iteration is stoped and the next one is initiated with a new incumbent solution yk+1∈ Ω with fΩ(yk+1) <
fΩ(yk) and a new mesh size parameter ∆mk+1, as dictated by equation (5.2)
∆mk+1= τwk∆m
k+1 (5.2)
where τ > 1 is a fixed rational number and wk is a binary value that represents wether an
iteration was a success or not.
Whenever the search step fails in generating an improved mesh point, then the poll step is invoked before the end of the iteration. The poll step is a local exploration of the space of optimization variables near the current incumbent solution yk. The generation of trial points
now is more rigidly defined as seen in equation (5.3)
Pk= {yk+ ∆mkd : d ∈ Dk} ⊂ Mk (5.3)
where Dk is called the set of poll directions, and is an integer combination from the columns of
D.
Points of Pk are generated so that their distance to the poll center xk is bounded by the
poll size parameter ∆p
k ∈ R+ which is always bigger than the mesh size parameter. Both can
be linked, for example, as ∆p
If the poll step fails in generating an improved mesh point then ∆m
k+1 is reduced in order to
increase the mesh resolution to allow the evaluation of f even closer to the incumbent solution. The update step is the finalization of an iteration which simply updates values for ∆m
k+1,
∆pk+1 and Vk+1.
The algorithm stops either because it reached a pre-defined maximum value for k or an established minimum value for ∆m
k or ∆ p k.
5.1.2 Defining the parameters
NOMAD recquires a text file where the user defines every controlable variable from the MADS algorithm as well as intermediate shell commands, starting points and creation of input/output files.
The optimization loops were conducted using, usually, the same values in each case. A maximum value for k of 100 was established as well as initial mesh size and minimal poll size parameters for each variable. The initial mesh size was set as being roughly 10% of the highest possible value for each specific geometric parameter (which meant that each variable can have diferent mesh sizes). The minimal poll size was set has being roughly 0,1% of the maximum value of each geometric parameter.
Moreover, the creation of Dk was chosen to follow a method called OrthoMADS, since it
was the default one, with n + 1 directions. Which meant that for every case (except the 3D planar), during the poll fase 4 trial points were created, which were orthogonal to each other.
Furthermore, in order to reduce computational time, it was also defined that a specific variation of the algorithm called p-MADS was to be adopted. This specific method handled the evaluation of the trial points in parallel, according to Le Digabel (2011). In order to do that, a master and several slave processors had to be defined. Also, since parallel computation means that the trial points were going to be evaluated at the same time, unique specifiers called Seed and Tag were generated by the master central processing unit (CPU), which made possible the creation of intermediate files between the slave CPUs without mixing information amongst them.
5.2
Geometrical meshing: Gmsh
Gmsh is an open-source 3D generator. It was chosen due to its great flexibility in generating geometries and meshes while presenting an intuitive working method.
First, one needs to define the position of points, which characterize vertices of the geometry, in a 3D coordinate system. The information is all stored in a text file which will be read by Gmshonce the creation of a mesh is desired. Followed by the creation of points is the definition of lines which intersect the generated points. This idea is maintained when creating surfaces and volumes. Then, the lines are discretized in a certain amount of nodes to further create quadrangles in a surface and hexahedrons in a volume.
5.2.1 2D geometry mesh
In order to create the constriction region of the reactor, the mathematic expression in equation (5.4) was designed, so to create 21 points, equally spaced, which were later used to define the curve of the region which best aproximated the real design.
y(x) = D 2 + 1 2 d0 2 − D 2 " 1 +cos 2πx − L1 2 L1 !# (5.4) This equation is also used in the 3D geometries.
Altough the mesh created is structured it is not uniform, in other words, the computational cells can present different volumes. Moreover, even though it is a 2D case, OpenFOAM does not allow absence of volume so a thickness of 0.1 mm in the reactor was set. Since every line of the geometry needs to be discretized, 4 variables were created, where each one would define the number of nodes included in L0, L1, L2 and D would possess. L1 and L2 were divided
into 12 equally spaced nodes, D was divided into 9 equally spaced nodes while L0 was divided
into 12 nodes with a positive reduction ratio in length of 1.2 in each adjacent segment. L0 was
discretized this way because it is not going to affect directly the objective function analysis, however it needs enough resolution in order to converge and that is achieved by maintaining the aspect ratio of adjacent cells as low as possible. This is achieved with the existance of the sequencial reduction in volume along L0 length. The overall aspect is presented in figure 5.2.
Figure 5.2: View of a particular zone of the two dimensional mesh of the reactor.
5.2.2 3D geometry mesh
The 3D planar mesh is just an extruded version of the 2D with w being discretized in 13 equally spaced segments, as seen in figure 5.3.
Figure 5.3: View of a particular zone of the three dimensional planar mesh of the reactor. The method which creates quadrangles in the geometry surfaces is called the transfinite algorithm and it requires opposite lines within a surface to have the same number of discretized nodes, according to Geuzaine and Remacle (2009). This can be a problem for the pipe geometry, since it is a revolution object the algorithm cannot find adjacent lines within a circle. Therefore, it cannot generate hexahedric cells.
To overcome this issue, an o-grid, exemplified in figure 5.4, was created.
Figure 5.4: Representation of an o-grid in a cylinder.
Using the o-grid approach, one can sucessfully create an hexahedric mesh in an object made by revolution. The only downside is the amount of extra work recquired to fully define it, e.g.: the inlet area is now defined by 5 different geometric surfaces, as seen in figure 5.5, whereas the planar reactor needs only one.
