Two-phase flow in netmix reactors

(1)

TWO-PHASE FLOW IN NETMIX REACTORS

ISABEL SOUSA FERNANDES

MASTER’S DISSERTATION PRESENTED

TO THE FACULTY OF ENGINEERING OF THE UNIVERSITY OF PORTO IN CHEMICAL ENGINEERING

(2)

Two-Phase Flow in NETmix Reactors

A Master’s dissertation

of

Isabel Sousa Fernandes

Developed within the course of dissertation

held in

Associate Laboratory LSRE-LCM

Supervisor: Ricardo Jorge Santos, PhD

Departamento de Engenharia Química

(3)

Agradecimentos

Foram várias as pessoas que contribuíram para a realização deste trabalho e que tornaram possível chegar até esta etapa, pelo que gostaria de deixar os meus sinceros agradecimentos por tudo o que me proporcionaram.

Primeiramente, gostaria de agradecer ao meu orientador, Doutor Ricardo Santos, por toda a ajuda, disponibilidade e conhecimento transmitido ao longo do trabalho.

À Professora Madalena Dias e ao Professor José Carlos Lopes pela oportunidade de pertencer ao Mixing group, ao qual deixo também o meu agradecimento.

Aos meus amigos e colegas, por toda a partilha de experiências e conhecimentos, por enfrentarmos as dificuldades em conjunto o que ajudou a superá-las mais facilmente. Em particular, à Cláudia uma companheira e conselheira sempre presente.

Ao Pedro, por todo o apoio, ajuda, paciência e lealdade mostrada, sem ti tudo seria muito mais complicado.

Por último gostaria de agradecer à minha família que sempre me incentivou a alcançar os meus objetivos. Um agradecimento especial aos meus pais, pelo apoio incondicional ao longo de toda a minha vida, por toda a confiança transmitida nos momentos mais difíceis. À minha irmã, Inês, que está comigo desde o começo da minha existência, agradeço a cumplicidade e fiel amizade. Ao meu irmão, Daniel, pela referência que foi para mim de responsabilidade e perseverança. À minha sobrinha, Matilde, por todas as nossas brincadeiras e sorrisos partilhados. À minha cunhada, Ana, pela sua ajuda sempre pronta. E, por fim, à minha avó, pela preocupação e orgulho que sempre demonstrou pelos meus feitos.

_____________________

Este trabalho foi financiado por: Projeto POCI-01-0145-FEDER-006984 - Laboratório Associado LSRE-LCM - financiado pelo Fundo Europeu de Desenvolvimento Regional (FEDER), através do COMPETE2020 – Programa Operacional Competitividade e Internacionalização (POCI) e por fundos nacionais através da Fundação para a Ciência e a Tecnologia I.P. O Dr. Ricardo Santos, orientador desta dissertação, é membro integrado do Laboratório Associado LSRE-LCM financiado pelo Projeto POCI-01-0145-FEDER-006984, Fundo Europeu de Desenvolvimento Regional (FEDER), através do COMPETE2020 - Programa Operacional Competitividade e Internacionalização (POCI) e por fundos nacionais através da Fundação para a Ciência e a Tecnologia.

(4)

Abstract

The purpose of this work is to characterize the mixing mechanisms between two phases (immiscible-liquids and gas-liquid systems) in NETmix reactors using Computational Fluid Dynamics (CFD) simulation. For that purpose, the interfacial area between the two phases was measured. The surface tension was considered and it is clear it plays a key role on multiphase flow. Thus, simulations with different values of this physical parameter were carried out. The interaction of the fluids with the walls of the reactor was also studied by assigning a contact angle. Moreover, the mass transfer mechanism of a species from the gas phase to the liquid phase was studied.

Due to the symmetric and quasi-periodic structure of the NETmix reactor, only a small representative part of the geometry was modelled in the commercial CFD software ANSYS/Fluent, with periodic translation zones applied to the boundaries.

Given the instability observed in the NETmix flow for the Reynolds numbers studied, simulations were performed to achieve an initial steady-state symmetric flow for all dynamic simulations. A two-dimensional two-phase model was developed based on Euler-Euler approach for liquid-liquid and gas-liquid-liquid systems at transient state. The Volume Of Fluid (VOF) method was employed to capture the flow interface between the two phases. The interfacial forces were added through a source term in the momentum equation, being used the Continuum Surface Force (CSF) model to simulate the surface tension. Regarding mass transfer, it was studied through the Multi-Fluid VOF and species of transport models. Furthermore, a three-dimensional two-phase model was developed for a gas-liquid system.

From the CFD results, it is seen that NETmix creates a mixture of two phases with a fine lamellar structure, but the surface tension causes a decrease of the interfacial area between the phases. Regarding the contact angle, it is seen that this parameter affects the dynamics of droplets/bubbles formation. The gas-liquid simulations showed that the phase density is a determining factor for the hold-up.

This work shows that NETmix technology has great potential to be used in operations involving interfacial mass transfer between two phases.

Keywords (theme): Multiphase flow, NETmix, CFD, interfacial area, surface

(5)

Resumo

O objetivo desta dissertação é caraterizar os mecanismos de mistura entre duas fases (sistemas líquidos imiscíveis e gás-líquido) dos reatores NETmix usando simulação de Dinâmica de Fluidos Computacionais (CFD). Para este fim, mediu-se a área interfacial entre as duas fases. Foi considerado o efeito da tensão superficial no escoamento, dada a sua notória influência em escoamento multifásico. Assim, foram realizadas simulações em que se atribuíram valores diferentes a este parâmetro físico. A interação dos fluidos com as paredes foi estudada através da atribuição de um ângulo de contacto. Além disso, estudou-se a transferência de massa de um componente da fase gasosa para a fase líquida.

Dada a simetria e a estrutura quase periódica do reator NETmix, apenas uma pequena parte da geometria foi modelada no programa informático ANSYS Fluent, com aplicação de zonas periódicas translacionais nas fronteiras.

Atendendo à instabilidade observada no escoamento no NETmix para os números de Reynolds estudados, foram realizadas simulações de modo a obter um escoamento simétrico estacionário, para ser usado como solução inicial das simulações dinâmicas.

Foi desenvolvido um modelo bidimensional de duas fases com base na abordagem Euler-Euler para um sistema líquido-líquido e gás-líquido em estado transiente. O método Volume de Fluido (VOF) foi usado para determinar a interface de escoamento. As forças interfaciais foram consideradas na equação de movimento através de um termo fonte, tendo-se usado o modelo Força Contínua de Superfície (CSF) para simular a tensão superficial. Relativamente à transferência de massa, esta foi estudada através dos modelos Multi-Fluido VOF e espécies de transporte. Além disso, desenvolveu-se um modelo tridimensional para um sistema gás-líquido. Da análise dos resultados de CFD, pode ver-se que o NETmix induz a mistura das duas fases criando uma estrutura lamelar fina, no entanto, a ação da tensão superficial leva a uma diminuição da área interfacial entre as fases. Relativamente ao efeito do ângulo de contacto, este parâmetro afeta a dinâmica de formação de gotas/bolhas. As simulações gás/líquido mostraram que a diferença de densidade entre fases apresenta uma grande contribuição para o hold-up.

A presente dissertação demonstra que a tecnologia NETmix tem um grande potencial para ser aplicada em operações que envolvam a transferência de massa entre duas fases.

Palavras Chave (Tema): Escoamento multifásico, NETmix, CFD, área interfacial, tensão superficial, ângulo de contacto, hold-up, transferência de massa.

(6)

Declaration

I hereby declare, on my word of honour, that this work is original and that all non-original contributions were properly referenced with source identification.

Isabel Sousa Fernandes

(7)

List of Figures

Figure 1 - NETmix reactor [5]. ...1

Figure 2 - (a) Representation of NETmix network with 𝑛𝑥 = 9 and 𝑛𝑦= 5; (b) 2D unit cell; (c) 3D unit cell [2, 6]. ...4

Figure 3 - Contact angle for a nonwetting gas–liquid–solid interface [13]. ...8

Figure 4 - NETmix Unit Block and surfaces identification: (a) 2D model; (b) 3D model. ... 14

Figure 5 - Contours of steady-state symmetric flow for liquid-liquid simulations: (a) volume fraction of phase 1; (b) velocity magnitude; (c) pressure. ... 21

Figure 6 - Contours of volume fraction of phase 1 for 𝜎 = 0 𝑚𝑁/𝑚 at different times. ... 22

Figure 7 - Volume fraction of phase 1 and phase 2 for 𝜎 = 0 𝑚𝑁/𝑚. ... 22

Figure 8 - Contours of volume fraction of phase 1 for 𝜎 = 30 𝑚𝑁/𝑚 and 𝜃 = 60° at different times... 23

Figure 9 – Volume fraction of phase 1 and phase 2 for 𝜎 = 30 𝑚𝑁/𝑚 and 𝜃 = 60°. ... 23

Figure 10 - Contours of volume fraction of phase 1 for 𝜎 = 73 𝑚𝑁/𝑚 and 𝜃 = 30° at different times. 24 Figure 11 - Volme fraction of phase 1 and phase 2 for 𝜎 = 73 𝑚𝑁/𝑚 and 𝜃 = 30°. ... 24

Figure 12 - Contours of volume fraction of phase 1 for 𝜎 = 73 𝑚𝑁/𝑚 and 𝜃 = 60° at different times. 25 Figure 13 - Volume fraction of phase 1 and phase 2 for 𝜎 = 73 𝑚𝑁/𝑚 and 𝜃 = 60°. ... 25

Figure 14 - Contours of volume fraction of phase 1 for 𝜎 = 73 𝑚𝑁/𝑚 and 𝜃 = 90° at different times. 26 Figure 15 - Volume fraction of phase 1 and phase 2 for 𝜎 = 73 𝑚𝑁/𝑚 and 𝜃 = 90°. ... 26

Figure 16 - Contours of volume fraction of phase 1 for 𝜎 = 73 𝑚𝑁/𝑚 and 𝜃 = 120° at different times. ... 27

Figure 17 - Volume fraction of phase 1 and phase 2 for 𝜎 = 73 𝑚𝑁/𝑚 and 𝜃 = 120°. ... 27

Figure 18 - Interfacial area for 𝜎 = 0 𝑚𝑁/𝑚. ... 28

Figure 19 - Interfacial area for 𝜃 = 60°. ... 28

Figure 20 - Interfacial area for 𝜎 = 73 𝑚𝑁/𝑚. ... 28

Figure 21 - Contours of steady-state symmetric flow for the startup of the gas-liquid simulation: (a) volume fraction of phase 1; (b) velocity magnitude; (c) pressure. ... 32

Figure 22 - Contours of volume fraction of phase 1 for a 𝑔/𝑙𝑞 ratio of 40/60 at different times. ... 33

Figure 23 - Velocity vectors for a 𝑔/𝑙𝑞 ratio of 40/60 at different times. ... 33

(10)

iv

Figure 27 - Volume fraction of phase 1 and phase 2 for a 𝑔/𝑙𝑞 ratio of 50/50. ... 34

Figure 30 - Volume fraction of phase 1 and phase 2 for 𝑔/𝑙𝑞 ratio of 60/40... 35

Figure 31 - Interfacial area for the three 𝑔/𝑙𝑞 ratios. ... 36

Figure 32 - Volume fraction of phase 1 at different times. ... 39

Figure 33 - Mass fraction of air in phase 2 at different times. ... 39

Figure 34 - Maps of steady-state symmetric flow for 3D simulation: (a) volume fraction of phase 1; (b) velocity pathlines; (c) pressure. ... 42

Figure 35 - Gas phase flow in the NUB model at different moments. ... 43

Figure 36 - Contours of volume fraction of phase 1 in the middle plane XY of the 3D grid at different times. ... 44

Figure 37 - Velocity streamlines in the middle plane XY of the 3D grid at different times. ... 44

Figure 38 - Volume fraction of phase 1 and phase 2 for the 3D simulation. ... 44

Figure 39 - Interfacial area for the 3D simulation. ... 45

Figure 40 - Net mass flow rate for simulation performed with 𝜎 = 0 𝑚𝑁/𝑚. ... 54

Figure 41 - Net mass flow rate for simulation performed with 𝜎 = 30 𝑚𝑁/𝑚 and 𝜃 = 60°. ... 55

Figure 46 - Net mass flow rate for simulation performed with a 𝑔/𝑙𝑞 ratio of 40/60. ... 60

Figure 49 - Net mass flow rate for the 3D simulation. ... 62

Figure 50 - Contours of steady-state symmetric flow for the startup of the mass transfer simulation: (a) volume fraction of phase 1; (b) velocity magnitude; (c) pressure. ... 63

(11)

List of Tables

Table 1 - Surface tension, Weber number and contact angle for immiscible-liquid simulations. ... 20

Table 2 - Mean and percent deviation of volume fraction of phase 1 and phase 2. ... 28

Table 3 - Mean and standard deviation of the interfacial area ∈ [2.5𝜏, 3.5𝜏]. ... 29

Table 4 - Mean and percent deviation of volume fraction of phase 1 and phase 2 ∈ [1.5𝜏, 2.5𝜏]. ... 36

Table 5 - Mean and standard deviation of the interfacial area ∈ [1.5𝜏, 2.5𝜏]. ... 36

Table 6 - Mass flow rate at the inlets and outlets for 𝜎 = 0 𝑚𝑁/𝑚. ... 53

Table 7 - Mass flow rate at the inlets and outlets for 𝜎 = 30 𝑚𝑁/𝑚 and 𝜃 = 60°. ... 54

Table 12 - Mass flow rate at the inlets and outlets for a 𝑔/𝑙𝑞 ratio of 40/60. ... 59

(12)

vi

Nomenclature

Upper-case Roman 𝐴 𝐴2D Ca Co 𝐷 𝐷q 𝐹vol 𝐿 𝑀 𝑃

𝑅⃗

_pq Re Re2D Re3D Rec Sh 𝑈 𝑉 𝑉3D 𝑉c We 𝑌𝑖 Area

Area of the 2D NUB model Capillary number Courant number Chamber diameter Diffusivity of phase q Volume force Characteristic length Momentum exchange terms Wet perimeter

Interaction force between phases Reynolds number

Reynolds number for 2D simulations Reynolds number for 3D simulations Critical Reynolds number

Sherwood number Volume flux Volume

Volume of the 3D NUB model Volume of a cell

Weber number

Local mass fraction of species 𝑖

m2 m2 m m2_/s N/m3 m N/m3 m N/m3 m3_/s m3 m3 m3 Lower-case Roman 𝑎𝑣∗ 𝑑 𝑑h 𝑑p 𝑔 𝑔 𝑘 𝑙 𝑙q 𝑚̇ 𝑚̇pq 𝑛̂ 𝑛̂𝑤 𝑛𝑥 𝑛𝑦 𝑝 𝑡 𝑡̂𝑤 𝑣 𝑣 𝑣̅ 𝑣 pq 𝜔

Dimensionless interfacial area Width of the channel

Hydraulic diameter Bubble diameter

Volumetric flow rate of gas Gravitational acceleration vector Mass transfer coefficient

Length of the channel

Volumetric flow rate of liquid Mixture mass flow rate

Mass transfer rate Surface normal

Unit vectors normal to the wall Number of the rows

Number of the columns Pressure

Time

Unit vectors tangential to the wall Velocity

Velocity vector

Maximum fluid velocity Interphase velocity Network thickness m m m m3_/s m/s2 m/s m m3_/s kg/s kg/(m3_∙s) Pa s m/s m/s m/s m/s m

(13)

𝑦 𝑎𝑥𝑖𝑠 Cartesian vertical coordinate Lower-case Greek 𝛼 𝛼1 𝛼2 ∆𝑡 ∆𝑥 ̅̅̅̅ 𝜃 𝜅 𝜇 𝜌 𝜎 𝜏 𝜏2D 𝜏3D

𝜏

q 𝜏𝑥𝑦 Volume fraction

Volume fraction of phase 1 Volume fraction of phase 2 Time step size

Mesh size Contact angle Interface curvature Viscosity Density Surface tension Passage time

Passsage time for 2D simulations Passage time for 3D simulations Stress-strain tensor of phase q Shear stress s m ° m-1 Pa∙s kg/m3 N/m s s s Pa Pa Indices 𝑓 in inlet_1_1_2 inlet_1_2_1 𝑛 𝑛 + 1 outlet_1_1_2 outlet_1_2_1 p q t w 𝑥 𝑦 𝑧

refers to cell face refers to inlet

Left inlet of the NUB model Right inlet of the NUB model refers to the previous time step refers to the current time step Left outlet of the NUB model Right outlet of the NUB model Phase p

Phase q Total Wall

denotes component in the 𝑥 direction denotes component in the 𝑦 direction denotes component in the 𝑧 direction

Abbreviations 2D Two-Dimensional 3D API CFD CSF CSTR E-E E-L g/l l/l LCM LSRE Three-Dimensional

Active Pharmaceutical Ingredients Computational Fluid Dynamics Continuum Surface Force

Continuous Stirred Tank Reactor Euler-Euler

Euler-Lagrange Gas-liquid systems Liquid-liquid systems

Laboratory of Catalysis and Materials

(14)

viii MUSCL

NUB PFR PRESTO!

Monotone Upstream-Centered Schemes for Conservation Laws NETmix Unit Block

Plug Flow Reactor

Pressure Staggering Option SIMPLEC

(15)

1 Introduction

1.1 Motivation and Relevance

NETmix is a meso-structured static mixer, consisting of a network of transport channels and mixing chambers (Figure 1) [1, 2]. This novel mixing technology was introduced and patented at LSRE-LCM [3], and already demonstrated a good performance in the production of phosphate nanoparticles, magnetic and metal organic nanoparticles, metal-organic frameworks, polymeric microcapsules and API (Active Pharmaceutical Ingredients) encapsulation. Industrially this technology is being applied in Fluidinova a spin-off from the Faculty of Engineering of the University of Porto, created in 2015. Fluidinova is a Portuguese company specializing in the production of high quality synthetic nanocrystalline hydroxyapatite [4].

Figure 1 – NETmix reactor [5].

The NETmix technology is particularly suitable for mixing liquids and/or gases having several advantages over conventional reactors, and its good control of the fluid mixture is an extremely important characteristic especially in the control of complex reactions [6].

So far, studies on these mixers have been mainly single-phase, however, multiphase flow is a common phenomenon in many industrial processes and energy related industries. The simplest case of multiphase flow is two-phase flow. Two-phase ﬂow takes place when only two phases are present, a phase is defined as one of the states of the matter [7]. The intimate contact between a continuous phase and a dispersed phase is required in several processes such as fermentation, hydrogenation, oxidation, water treatment, petrochemical, nuclear and aerospace [8].

(16)

Introduction 2 Given the complex nature of multiphase flow, the use of computational simulation is being extensively used to better understand and predict the multiphase phenomenon [9, 10].

Recent laboratory experiments in NETmix have demonstrated a great potential for this technology for the production of hydrates of carbon dioxide (CO2) [11], so the use of CFD

techniques becomes of high interest to better understand the processes involved.

1.2 Thesis Objectives and Layout

This thesis has the objective of study the mixing mechanisms between two phases that occur in NETmix. The interfacial area between the two phases was measured since the knowledge of this variable is paramount for the accurate simulation of several processes such as heat transfer and mass transfer.

Then the layout of the thesis will be presented.

Chapter 2 begins with an introduction of static mixers in Section 2.1 explaining their operation and the main advantages and applications of these reactors. In Section 2.2, the meso-reactor NETmix is presented, through a brief description of its geometry, and in Subsection 2.2.1, a review of CFD studies performed on these mixers is made. In Section 2.3, some characteristics of the multiphase flow are pointed out, and an introduction to surface tension and mass transfer is given in Subsections 2.3.1 and 2.3.2, respectively.

In Chapter 3, the CFD models for the simulation of the two-phase flow in NETmix are presented, namely the integrated equations to know the flow field (Section 3.1), the grid geometry used in the simulations (Section 3.2), the boundary and initial conditions assigned (Section 3.3) and the solution methods used in the calculations (Section 3.4). Sections 3.5 and 3.6 describe the procedures for data treatment of results obtained from CFD simulations.

In Chapter 4, the characteristics of the fluids used and the parameters attributed to the variables are described, for the 2D simulations performed, followed by the description of the results obtained and their analysis. In Section 4.1, the influence of surface tension and contact angle on flow hydrodynamics for multiphase liquid-liquid flow is studied. In Section 4.2 for multiphase gas-liquid flow for different gas-liquid flow rate ratios are studied (Subsection 4.2.2) and the transfer of a component of the gas phase to the liquid phase is simulated (Subsection 4.2.3).

Chapter 5, is an analysis similar to the previous chapter, for a gas-liquid flow in a three-dimensional space.

In Chapter 6, the conclusions of the work are presented, as well as the limitations and the future work.

(17)

2 State of the Art

2.1 Static Mixers

Static mixers were designed to obtain fluid mixing, even under low Reynolds number laminar flow regimes. Later, they began to be used in applications of heat transfer, turbulence, and multiphase systems [12, 13]. Regarding their configuration, static mixers can consist of a hollow tube or channel with a certain geometry and another known type is composed of a series of flow reorientation devices inserted into a pipe, channel or ducts. The mixing mechanism is based on changes in the fluid streamlines, the homogenization is obtained by redistributing the fluid in the radial and tangential directions. Static mixers are used to perform continuous operation and present several advantages compared to batch processing and mechanically stirred vessels. The low equipment cost, no moving parts, small space required, a narrow distribution of residence times, and no mechanical moving parts, (energy is injected from pumping) are some of the features that make static mixers an attractive alternative [14, 15]. While the energy needed to accomplish mixing in the continuous stirred tank reactor (CSTR) is achieved through mechanical agitation, in static mixers it is obtained through a loss in pressure as fluids flow through the device [16]. Therefore, good mixing efficiency is obtained at the cost of higher pressure drop. A common problem in static mixers is the potential of fouling, which translates into greater cleaning costs. Static mixers can be used for different purposes such as blending of miscible fluids (the most common use), interface generation between immiscible phases, heat transfer operation and thermal homogenization, and axial mixing [14, 15]. The first use of static mixers was in the 1970s and since then the range of areas of application is vast: pharmaceutical, food processing, polymer synthesis, pulp and paper, paint and resin, water treatment, and petrochemical industries are some examples [15].

2.2 The NETmix Reactor

The NETmix reactor is a meso-structured static mixer that consists of a network of transport channels and mixing chambers (Figure 2a). Each chamber of this network has two inlet channels and two outlet channels. In this way, a mixing chamber contacts fluids from two chambers in the previous row and splits them feeding two chambers in the next row. NETmix operation is based on this pattern of fluid contacting and separation with successive expansions and contractions from channels to chamber, which has a large potential to generate interfacial area between two fluids even immiscible fluids. The network is generated by the repetition of a unit cell, which is composed of a chamber and four half-channels, oriented at a 45° angle from the main flow direction. The unit cell can assume two geometries: cylindrical chambers and

(18)

State of the Art 4 rectangular cross-section area known as a 2D unit cell (Figure 2b), or spherical chambers and cylindrical channels called the 3D unit cell (Figure 2c) [1, 6].

Figure 2 – (a) Representation of NETmix network with 𝑛𝑥 = 9 and 𝑛𝑦= 5; (b) 2D unit cell; (c)

3D unit cell [2, 6].

The dimensions of unit cells are the chamber diameter, 𝐷, the length of the channel, 𝑙, the channel width or diameter, 𝑑, for 2D and 3D unit cells, respectively, and for 2D unit cell the network thickness, 𝜔. The network size is given by the number of unit cells in the 𝑥 and 𝑦 directions, that is, the number of rows and columns, 𝑛𝑥 and 𝑛𝑦, respectively. It should be

emphasized that in NETmix structure each row alternates between 𝑛_𝑦 e 𝑛_𝑦− 1 chambers, starting the first row with the smallest number [11, 17].

2.2.1 Previous Works

NETmix is a patented technology [3] that was developed by Lopes et al. in 2005 at LSRE- Laboratory of Separation and Reaction Engineering, and since then several studies have been performed in these reactors.

The first studies were performed by Laranjeira et al. [1], who studied the network hydrodynamics having demonstrated that the chambers are zones of complete mixing and channels are zones of total segregation. Being that the case, the NETmix was modeled as a network of continuous flow reactors, associating in series continuous stirred tank reactors (CSTRs) to represent the chambers, and plug flow reactors (PFRs) to characterize the behavior of the channels. Furthermore, Laranjeira [2, 5] performed tracer flow visualization experiments in a NETmix 3D prototype and used Computational Fluid Dynamics (CFD) techniques to simulate the flow field in a network structure and validate the experimental results. A 2D grid composed of 16 chambers was used. This grid had the singularity of being made of less one bounding

(19)

chamber and its respective channels on each odd-numbered row than the NETmix network that was presented in this section. From the first studies made by Laranjeira et al. [2], he concluded that mixing in the NETmix reactor can only be achieved over a critical channel's Reynolds number. At Reynolds numbers (based on the channel diameter) greater than this value it was noticed a significant change in the flow dynamics, from a fully developed laminar steady and symmetric flow, where no advective mixing was observed, to a self-sustained dynamic laminar flow regime with flow oscillation that induced a strong local laminar mixing. The flow instabilities result from the geometric characteristics of NETmix network that promotes local hydrodynamic instabilities induced by the opposed jets interactions at the chambers [6]. Moreover, the initial studies on NETmix established some of its advantages over other static mixers, such as the simplicity of the NETmix structure, a high surface contact area with the reactor walls particularly suited to exothermic or catalytic reactions, the ability to control the mixture and the residence time of the reagents, and the possibility of scale-up through the combination of the mixers in parallel or in series [5].

A second Ph.D. thesis from Gomes [17] characterized mixing mechanisms that occurred in NETmix, from CFD simulations. Focusing on liquid-liquid systems, Gomes [17] developed a two-dimensional two-phase model. Both phases had the same physical properties, having defined two species with the properties of liquid water. The CFD simulations were based on a 2D model composed of 27 chambers, 4 of which were half-chambers, this grid had the peculiarity of being rotated 45° relative to the usual NETmix reactor orientation. Moreover, the 2D NETmix prototype was built, replacing the spherical chambers and cylindrical channels of the existing 3D NETmix prototype by cylindrical chambers connected by channels with rectangular cross-section. NETmix technology was used on the continuous production of nanoparticles of hydroxyapatite with a very narrow size distribution, evidencing the good mixing capabilities of this reactor.

Fonte in his Ph.D. thesis [18] presented a model for prediction of the pressure drop throughout the mixer and developed a 3D CFD model introducing the concept of NETmix Unit Block (NUB). The NUB concept consists of modeling only one representative section of the NETmix reactor, thus reducing the simulation domain and providing a good grid refinement. Moreover, Fonte [18] studied the applicability of NETmix to heterogeneous liquid-liquid reactions notably the adiabatic heterogeneous reaction of nitration of benzene, having concluded that the reactor is suitable for large-scale production of nitrobenzene. For this application it was achieved a high degree of mixing in laminar regime with large throughputs. Besides that, Fonte [18] used NETmix technology to produce a metal-organic framework material: MIL-88A.

In Costa’s Ph.D. thesis [11] it was analyzed the potential of NETmix technology for the production of gas hydrates namely carbon dioxide hydrates for the capture and storage of

(20)

State of the Art 6 carbon dioxide. A 3D CFD model was developed to evaluate the heat transfer capacity of NETmix using the NUB model. This study showed the large specific capacity of NETmix to remove heat, evidencing that this mixer is particularly suitable for fast reactions where heat transfer is crucial [11, 19]. It should be noted that both Fonte [18] and Costa [11] used the 3D NUB model to simulate the flow of the 2D NETmix prototype. The grid used consists of 2 complete chambers and 6 half chambers, comprising 5 rows and 3 columns.

Lastly, Torres [20] in his Master thesis studied turbulent flow through statistic analyses in NETmix reactors. 2D simulations were performed for several geometries using the NUB model, in order to determine how the number of chambers influences the distribution of velocity statistics. Several NUB geometries and grids with different numbers of chambers were simulated, with the following names: NormalNUB (2 chambers), DoubleNUB (4 chambers), SideNUB (4 chambers with an extra column at each side) and BigNUB (8 chambers). With his studies, he concluded that the DoubleNUB is the geometry that allows more uniform distribution of velocity fluctuations, being the most appropriate geometry to describe NETmix technology. Focusing on the CFD simulations carried out to date, it is verified that the flows studied are mostly single-phase liquid mixing, except the case of the study conducted by Gomes [17] in which was considered a liquid-liquid two-phase system. However, Gomes’ [17] studies neglected the effect of surface tension on the flow hydrodynamics. Thus, the present work intends to extend the study of NETmix capabilities in multiphase flow through CFD simulations.

2.3 Multiphase Flow

Multiphase flow exists in a wide variety of forms, which integrate our natural environment such as rain, clouds, tornadoes, typhoons, volcanic activities etc. In industrial technology, fluidized beds, distillation columns, electric power generation, aerosols, catalytic oil cracking, are some examples. The understanding of multiphase systems is crucial in a vast number of engineering applications and requires often analytical and numerical strategies in order to predict their behavior [21-23].

While a phase refers to the state of matter (solid, liquid or vapor), the designation of multiphase flow is applied to any fluid flow systems composed by more than one phase flowing simultaneously in mixture [24]. It should be noticed that the term mixture is applied to denote the presence of two or more phases and does not necessarily meaning that these are intimately mixed [25]. Moreover, the definition of multiphase flow comprises flows that have some level of phase separation, excluding the circumstances in which the components are well mixed above the molecular level. One possible way to classify multiphase flow is according to the state of the different phases, in the case of systems composed of two phases, they can be divided into gas-liquid mixture, gas-solid mixture, liquid-solid mixture and immiscible-liquid

(21)

mixture [26]. The most important feature of two-phase flow when compared to single-phase flow, is the presence of deformable interfaces, making its analysis more complex [25].

When these systems flow within a duct, the two phases are distributed in particular ways, the typical configurations define flow regimes. The flow regimes can be distinguished into two main classes: dispersed phase and separated flows. In dispersed phase flow, one phase consists of discrete elements that are not connected, such as droplets in a gas or bubbles in a liquid, while in separated flow both phases are considered as continuous and are separated by a line of contact. The phase distribution influences heat transfer, pressure drop, mass transfer, etc. In this way, it is crucial to determine the geometry of the interfaces when multiphase flow modelling is desired. However, the topology of the flow depends on many parameters such as the inclination of the duct, the geometry, volumetric flow ratio, pressure, physical properties of fluids, etc. Thus, it is not always possible to know a priori the shape of interfaces so that CFD techniques are often used in order to predict them [25].

This work focus on immiscible-liquids and gas-liquid mixtures as they are the possible applications of NETmix technology. In case of liquids, the two flowing media can be two chemically different species and the fluids can be immiscible due to their polarity, that is, they present different intermolecular interactions. On the other hand, the gas-liquid mixture consists of a dispersion of gas bubbles within in liquid medium [17, 25].

The behavior of these systems depends mainly on the interfacial forces that result from the interaction between the phases, in this work the surface tension was considered as the dominant interface force. After the hydrodynamics of the two-phase flow in the NUB model is known, the phenomenon of mass transfer between a gaseous phase and surrounding liquid was studied.

2.3.1 Surface Tension

Surface tension (𝜎) results from attractive forces between the molecules of a fluid [27]. Considering, for example, a bubble dispersed in a liquid medium, within a bubble a molecule of gas is completely surrounded by many others, so the net force on a molecule is zero. On the other hand, near the surface an imbalance of the net force is verified due to the nonuniformity in the number of neighboring molecules, leading to a strong inwardly direct attractive force. Given this, the energy state of the molecules on the interior is much lower than that of the molecules at the surface, whereby the surface will tend to contract, leading to a pressure increase on the concave side of the bubble. Surface tension is a force that acts at the free surface interface to balance the inward attractive force with outward pressure force, in order to maintain the equilibrium. In case of the separated flow, the surface tension acts to minimize free energy by decreasing the area of the interface [13, 26].

(22)

State of the Art 8 In this way, the surface tension can be defined as the amount of Gibbs free energy at constant temperature, pressure and composition required to change the interface of a given system [28]. Quantitatively, surface tension is the work per unit area, Nm/m2_{or force per unit length of}

interface in N/m. Thus, for a given interfacial composition, the surface tension property is a function of both pressure and temperature, but it is noted a much stronger dependence of temperature [13].

A consequence of the pressure difference resulting from surface tension is the phenomenon of wall adhesion. Wall adhesion force results from the cohesive forces between two fluids and a solid surface. The energies between the fluids and solid surfaces determine the contact angle, 𝜃. This parameter is used to quantify the wettability of solid surfaces defined as shown in Figure 3. The term wetting is applied when a liquid contacts a solid and liquid spreading occurs along the surface of the solid. The contact angle may have any value between 0 and 180°,for contact angles of less than 90° the solid surface is classified as hydrophilic (wetting fluids) while for contact angles of more than 90° it is called hydrophobic (nonwetting fluids). A contact angle of 90° corresponds to a no wall adhesion effects, that is, the interface is normal to the adjacent wall [24, 26, 29].

Figure 3 – Contact angle for a nonwetting gas–liquid–solid interface [13].

To quantify the importance of surface tension forces, the value of two dimensionless numbers is often used: the capillary number, Ca, and the Weber number, We. The capillary number relates the viscous forces in a system to the surface tension forces, and Weber number is the ratio between the inertial force and the surface tension force [30]. These numbers are expressed as follows Ca =𝜇𝑣 𝜎 (2.1) We =𝜌𝐿𝑣 2 𝜎 (2.2)

where 𝜇, 𝜌 and 𝑣 are the fluid viscosity, density and velocity, respectively, and 𝐿 is the characteristic length.

(23)

The capillary number is used for Reynolds numbers, Re, << 1, otherwise, for Re >> 1, the Weber number is used. The influence of surface tension can be neglected when Ca >> 1 and We >> 1.

2.3.2 Mass Transfer

The phenomenon of diffusional mass transfer is related to the transport of a chemical species from a region of relatively high concentration to regions of lower concentration. The primary driving force for diffusional mass transfer it is the concentration difference which is attenuated as a result of random movement of molecules. Thus, it is necessary the presence of two regions at different chemical composition for the phenomenon of mass transfer to occur.

Mass transfer can occur through a phase or between two phases, always requiring a deviation from equilibrium concentration conditions. Once the equilibrium within the system is established, the concentration gradient and the net diffusion rate of the diffusing species becomes zero.

Regarding the mass transfer mechanism, mass can be transferred by random molecular motion in quiescent fluids, being called the molecular diffusion mass transfer. Or it can be transferred from a surface into a moving fluid, defined as the convective mass transfer. It should be noted that these two mechanisms often act simultaneously, however one mechanism can dominate quantitatively relatively to the other. In these cases, it is possible to roughly describe the behavior of the systems using only the dominant model [13, 31].

The dimensionless Sherwood number, Sh, is generally used to quantify mass transfer phenomena, it represents the convective to diffusive mass transport ratio and is defined, for the phase q, as follows

Shq=

𝑘𝑑p

𝐷q

(2.3)

where 𝑘 is the mass transfer coefficient, 𝑑p is the bubble/droplet diameter and 𝐷q is the

diffusivity of the phase q.

Several numerical methods have been used in CFD simulations to study multiphase flow and related interfacial phenomenon. The most common numerical technique for tracking and locating the two-phase interface is the Volume of Fluid (VOF) approachdeveloped by Hirts and Nichols [32, 33]. This approach is characterized by enabling an easy implementation and small computational complexity and high precision [34]. The implementation of VOF for mass transfer with coupled mass balance to provide the shrinkage of the gaseous discrete phase in CFD is still not very developed, not many studies have been reported in this respect [35]. Thus, this work intends to study mass transfer in the NUB model using the VOF approach.

(24)

CFD Models 10

3 CFD Models

3.1 Governing Equations

The CFD simulation process is based on solving the continuity, momentum, and energy equations. These equations, which result from the application of the fundamental laws of mechanics, are applied to the fluids in space and time domain. Thus, a set of partial differential equations are solved to predict the hydrodynamic behavior of the flows [36, 37]. As the two-phase system was considered isothermal, the energy equation will not be solved.

For this purpose, the commercial CFD software ANSYS/Fluent 18.2 was used. Since the flow under study involves two phases it was necessary to evaluate the multiphase models available in Fluent. Two different approaches are available for the modelling of multiphase flows: Euler-Lagrange (E-L) and Euler-Euler (E-E). With the E-L approach, the fluid phase is considered as a continuum and the dispersed phase is tracked in the Lagrangian reference frame. It is, therefore, more appropriate to use when the flows containing a low volume fraction of dispersed phase. On the other hand, in the E-E approach, both phases are treated as a continuous phase and they can interfere with each other [38, 39]. Since the two phases to be modelled in this work have an identical volume fraction, none can be neglected, so the E-L approach is not applicable. Taking into account these considerations, the next step was to choose between the three E-E multiphase models available in Fluent: the volume of fluid (VOF) model, the mixture model, and Eulerian model. The available models differ essentially in the way the dispersed phase is coupled in the continuous phase and, consequently, in the type of interface(s) formed. The VOF model it is usually used to describe fluid flows that contain a free surface (separated or stratified flows), or when the dispersed phase is well separated from the continuous phase. This model is particularly suitable when the position and shape of the interface are of interest [40, 41]. The mixture model is used to model two or more phases fluid or particle. The Eulerian model is the most complex of the multiphase models, it solves a set of momentum and continuity equations for each phase [7].

In this thesis, for the hydrodynamics simulation of the multiphase flow the NUB was used with the VOF model. For the study of mass transfer between the two phases, the Eulerian model was chosen.

3.1.1 VOF Model

The VOF method is a tracking interfaces technique characterized by solving a single momentum equation using the properties of the mixture of both phases, being the resulting velocity field shared by both phases. The interface tracking is done considering the volume fraction occupied

(25)

by a particular phase within one cell mesh that can range from 0 to 1, where 1 corresponds to a cell filled with the phase considered and 0 to an empty cell. Cells with values between these limits correspond to cells containing the interface. The VOF formulation assumes that most of the computational cells have a volume fraction of 0 or 1, i.e., the phases are not interpenetrating [40, 42].

Next, the governing equations with VOF method for two-phase systems are presented. The continuity equation (conservation of mass) for the mixture can be express as

𝜕𝜌

𝜕𝑡

+ ∇ ∙ (𝜌𝑣 ) = 0

(3.1)

where 𝜌 is the mixture density and 𝑣 is the velocity vector.

The mixture density, viscosity and velocity are determined, respectively, by the following laws:

𝜌 = 𝜌q𝛼q+ 𝜌p𝛼p (3.2)

𝜇 = 𝜇q𝛼q+ 𝜇p𝛼p (3.3)

𝑣 =1

𝜌[𝜌q𝛼q𝑣⃗⃗⃗⃗ + 𝜌q p𝛼p𝑣⃗⃗⃗⃗ ] p

(3.4)

where 𝛼 is the volume fraction. The subscripts q and p denote, respectively, phase q and phase p.

In addition, the interface tracking is done by solving the phase continuity equation for convection of volume fraction field of secondary phase and can be express as

𝜕(𝛼

_q𝜌_q)

𝜕𝑡

+ ∇ ∙ (𝛼

q𝜌q

𝑣 ) = 0

(3.5)

It should be referred that 𝛼_q is subjected to two constraints: it can only take values between 0 and 1 and the volume fractions of two phases must sum to unity, being the volume fraction of primary phase determined by this last constraint.

The phase continuity equation, also called volume fraction equation, can be solved either through explicit or implicit time discretization. The explicit scheme is the most suitable to use to calculate time dependent solutions, and besides that is recommended for simulations where surface tension is important providing an accurate curvature calculation. In addition, it has the advantage of using the geo-reconstruct discretization function, which enables a clear and sharp interface without numerical diffusion, which is not available in the implicit scheme [26]. In this way, the explicit time discretization was used, the volume fraction being discretized as follows

𝛼q𝑛+1𝜌q𝑛+1− 𝛼q𝑛𝜌q𝑛 ∆𝑡 𝑉c+ ∑(𝜌q𝑈𝑓 𝑛_𝛼 q,𝑓𝑛 ) 𝑓 = 0 (3.6)

(26)

CFD Models 12 where ∆𝑡 represents the time step size, 𝑉_c is the cell volume and the index 𝑛 and 𝑛 + 1 denotes the previous and the current time values of the variables, respectively; 𝛼q,𝑓 is the volume

fraction of phase q on the face, and 𝑈𝑓 is the volume flux through the face, based on normal

velocity.

Regarding the equation of motion (conservation of momentum), a single equation is solved and can be written as

𝜕

𝜕𝑡

(𝜌𝑣 ) + ∇ ∙ (𝜌𝑣 𝑣 ) = −∇𝑝 + ∇ ∙ [𝜇(∇𝑣 + ∇𝑣

𝑇

_{)] + 𝜌𝑔 + 𝑀}

(3.7)

where 𝑝 and 𝑔 represents the pressure that is shared by the phases and gravitational acceleration vector, respectively. 𝑀 are the interphase momentum exchange terms, which includes the interfacial forces that result of the interaction between the phases.

In Fluent, different types of interaction are available, which enable to set the contribution of drag, lift force, wall lubrication, surface tension, reactions etc. The surface tension was considered as the dominant interphase force on the multiphase flow and the contribution of the remaining effects was neglected. The 2D simulations were performed with and without surface tension, in order to assess the influence of this physical effect.

The approach used in Fluent to model surface tension was the continuum surface force (CSF) proposed by Brackbill et al. [43]. This formulation replaces the stress given by Laplace equation by an equivalent volume force 𝐹vol [44]. The volume force, for multiphase systems, is given by

𝐹vol= ∑ 𝜎𝑖𝑗 𝑝𝑎𝑖𝑟𝑠 𝑖𝑗, 𝑖<𝑗

𝛼𝑖𝜌𝑖𝜅𝑗

∇𝛼

𝑗

+

𝛼𝑗𝜌𝑗𝜅𝑖

∇𝛼

𝑖

0.5(𝜌𝑖+ 𝜌𝑗)

(3.8)

where 𝜅 is the interface curvature. For systems containing only two phases, the previous expression can be simplified considering that 𝜅q= − 𝜅p and ∇𝛼q = −∇𝛼p, resulting in

𝐹vol= 𝜎

𝜌𝜅q

∇𝛼

q

0.5(𝜌q+ 𝜌p)

(3.9)

In this model, the surface curvature is calculated from local gradients in the normal vector to the interface assuming the form

𝜅q

= ∇ ∙ 𝑛

̂

= −∇ ∙

(

∇𝛼

_q |∇𝛼_q|)

(3.10)

where 𝑛̂ is the unit normal vector.

Thus, for the simulations in which a surface tension value was set between phases, the term 𝑀 in the equation of motion was no longer zero and was replaced by the expression shown in Equation 3.9.

(27)

3.1.2 Eulerian Model

To simulate mass transfer in the NUB model, the Eulerian Multi-Fluid VOF model must be used. This is a hybrid model that enables to couple the VOF and Eulerian multiphase models, combining the interface modelling options offered with the VOF model with the advantage of the velocity flow being solved for each phase by the Eulerian model. Besides that, it was necessary to activate the transport model of the species.

The equations for conservation of mass, species, momentum, with these models are presented in Equations 3.11-3.13.

Conservation of mass for phase q: 𝜕(𝛼q𝜌q)

𝜕𝑡 + ∇ ∙ (𝛼q𝜌q𝑣 q) = ∑ 𝑚̇pq

𝑛

p=1

(3.11)

Conservation of chemical species for species 𝑖 in phase q: 𝜕

𝜕𝑡(𝛼q𝜌q𝑌𝑖,q) + ∇ ∙ (𝛼q𝜌q𝑣 q𝑌𝑖,q) = ∇ ∙ (𝛼q𝜌q𝐷q∇𝑌𝑖,q) + ∑ (𝑚̇p𝑖q𝑗− 𝑚̇q𝑗p𝑖)

𝑛

p=1,p≠q

(3.12)

Conservation of momentum for phase q:

𝜕 𝜕𝑡(𝛼q𝜌q𝑣 q) + ∇ ∙ (𝛼q𝜌q𝑣 q𝑣 q) = −𝛼q∇p + ∇ ∙ 𝜏q+ 𝛼q𝜌q𝑔 q+ ∑(𝑅⃗ pq+ 𝑛 p=1 𝑚̇pq𝑣 pq) + 𝐹vol (3.13)

where 𝑚̇pq is the mass transfer source from phase p to q; 𝑌𝑖,q is the local mass fraction of

species 𝑖 in phase q; 𝐷q is the diffusion coefficient of phase q; 𝜏q is the stress-strain tensor of

phase q; 𝑅⃗ _pq is an interaction force between phases, and 𝑣 _pq is the interphase velocity obtained as:

•

𝑣

_pq

=

𝑣

⃗

p for 𝑚̇pq> 0 , mass transfer occurs from phase p to q;

•

𝑣

_pq

=

𝑣

⃗

q for 𝑚̇pq< 0 , mass transfer occurs from phase q to p.

As with the VOF model, an explicit time discretization was used and the CSF model was applied to model the surface tension.

3.2 Geometry

As previously mentioned the NETmix structure is composed of a repetition of unit cells, each cell consists of a chamber and four half-channels (two inlets and two outlets). Thus, given the network symmetry and quasiperiodicity of the flow, only a small part of the geometry was modelled in ANSYS Fluent, with periodic translation zones applied to the boundaries. The mesh

(28)

CFD Models 14 periodicity was assigned by coupling a pair of faces, being the flow through the periodic boundary calculated as if these zones were direct neighbors to the adjacent cells [45].

The decrease of the domain size to be simulated enables a significant reduction of the computational effort, which translates into an increase of the speed of the simulation and in a better management of available computer memory. In this way, the NETmix Unit Block was used to simulate the flow hydrodynamics of the two phases. According to the above statement, Torres [20] concludes that the DoubleNUB configuration is best suited to describe flow in the NETmix mixer, and so this was the configuration adopted to carry out the simulations. The NUB model used comprises 9 rows and 3 columns, the middle column consisting of 4 complete chambers and the remaining columns contain 5 half chambers. The chambers have a diameter, 𝐷, of 6.5 mm, and the channels have a length, 𝑙, of 2 mm and a width, 𝑑, of 1 mm. It should be noticed that the length of the inlet and outlet channels of the NUB model is half the length of the remaining channels. The geometry used to perform the 2D simulation is represented in Figure 4a. The grid to simulate this geometry in CFD has 92 541 mixed cells. The 3D geometry, which can be seen in Figure 4b, was obtained from the extrusion of the 2D mesh by specifying a depth (𝜔) equal to the width of the channels. Both grids have a mesh element size of 62.5 μm.

(a) (b)

Figure 4 - NETmix Unit Block and surfaces identification: (a) 2D model; (b) 3D model.

x y

z x

(29)

3.3 Boundary and Initial Conditions

In order to know the flow field, it is necessary to integrate numerically the set of differential equations previously described as governing equations. For this, it is necessary to establish boundary conditions and initial conditions, these are, respectively, a set of constraints in the spatial domain and at the initial time on the domain.

Regarding boundary conditions, the velocity magnitude was defined at the input sections to set the Reynolds number to 600. This dimensionless number is defined as the ratio of inertial to viscous forces and can be express as

Re =𝜌𝑣in𝑑h 𝜇

(3.14)

where 𝜌 and 𝜇 are the density and viscosity of the fluid, respectively; 𝑣in is the velocity in the

input channel and 𝑑h is the hydraulic diameter. The Reynolds number is defined in terms of the

hydraulic diameter, taking as a reference the NETmix 2D prototype since the geometry of the NUB is based on this reactor, the hydraulic diameter is defined as

𝑑h= 4𝐴 𝑃 = 2𝜔𝑑 𝜔 + 𝑑 (3.15)

where 𝐴 is the cross-sectional area of the flow and P is wet perimeter. When the grid used in fluent is two-dimensional, the thickness of the channels is not defined, so it is assumed that this value tends to infinity: 𝜔 → ∞. Taking into account the considerations made, applying the limit to Equation 3.15, the hydraulic diameter is given by 2𝑑. Then, the Reynolds number is given by

Re2D= 2

𝜌𝑣in𝑑

𝜇

(3.16)

On the other hand, the Reynolds for 3D geometry is defined as

Re3D= 𝜌𝑣in 𝜇 2𝜔𝑑 (𝜔 + 𝑑) (3.17)

Considering that 𝑑 = 𝜔 in the geometry used for 3D simulations, the above expression becomes:

Re3D=

𝜌𝑣in𝑑

𝜇

(3.18)

As for the outlet sections, a gauge pressure of 0 Pa was defined.

Since the Reynolds number is greater than the critical value marking the onset of chaotic flow regimes (Rec= 200 [17]), to improve stability, a first simulation was performed in which

(30)

CFD Models 16 direction of the y-axis, to achieve a steady-state symmetric flow between the phases. At the outer walls, no-slip condition (𝑣 = 0) was established. These conditions were used to simulate a steady-state flow field, which is used as initial condition. After, these wall boundary conditions were eliminated, and the fluids were allowed to cross the chamber axis. Both simulations were performed in transient state.

Moreover, in the simulations in which the surface tension model was enabled, wall adhesion effect and contact angle on the solid walls was defined. Surface tension and wall adhesion are both bulk properties that result from intermolecular interactions. Surface tension is a force that results from attractive forces between the molecules of a fluid, whereas the wall adhesion force describes the cohesive forces acting between the fluid and walls [46].

Using Fluent to model wall adhesion, a constant contact angle was specified through a boundary condition. The equation solved by Fluent to describe the phenomenon of wall adhesion was proposed by Brackbill et al. [43]. The contact angle, 𝜃, is defined as the angle formed between the wall and the tangent to the interface at the wall, measured inside of one of the phases [26].In this model, the surface normal at the live cell next to the wall is given by the following expression

𝑛̂ = 𝑛̂wcos𝜃 + 𝑡̂wcos𝜃 (3.19)

where 𝑛̂_w and 𝑡̂_w are the unit vectors normal and tangential to the wall, respectively

.

At last, the flow field was initialized in the entire domain by the standard initiation method.

3.4 Solution Methods

Regarding the method of calculating the solution, a first order implicit scheme for time discretization and the SIMPLEC scheme for the pressure-velocity coupling were used. For spatial discretization, the following options were used: least squares cell based for the gradient, PRESTO! for the pressure, third-order MUSCL for the moment and geo-reconstruct for the volume fraction.

When a transient calculation is performing, it is important to select a time step size consistent with the dimensions of the problem, which should be small enough to obtain a valid dynamic flow solution and to maintain solver stability. The Courant, Co, number is usually used to estimate the time step size. This dimensionless number relates the time step in a calculation to the characteristic time of transit of a fluid element across a control volume [26] and is expressed as

Co =𝑣̅∆𝑡 ∆𝑥 ̅̅̅̅

(31)

where 𝑣̅ is the maximum fluid velocity and ∆𝑥̅̅̅̅ is the mesh size. The mesh size is the same for the 2D and 3D grid, having the value of 62.5 μm.

So, this gives the indication about the number of mesh elements the fluid passes through in one time step. For an explicit solver to stably converge, the Courant number must be smaller than 1.

As for the number of time steps to be performed, it is important to evaluate the passage time, which can be defined, for the two-dimensional simulations carried out, as

𝜏2D=

𝐴2D

(𝑣in,1+ 𝑣in,2) 𝑑

(3.21)

where 𝐴_2D is the area of the 2D NUB model being this value 335 mm2_{, 𝑣}

in,1 and 𝑣in,2 are the

velocities in the two channels.

As for three-dimensional simulations the passage time is defined as

𝜏3D=

𝑉3D

(𝑣in,1+ 𝑣in,2) 𝐴in

(3.22)

where 𝑉3𝐷 is the volume of 3D NUB model being this value 335 mm3, and 𝐴in is the area of inlet

channels, that is 1 mm2_.

3.5 Interfacial Area

One way to evaluate the mixture between two phases is through the length of the interface between the phases for 2D simulations, or through the interfacial area in the case of 3D simulations. It is known that at the interface the volume fraction of each phase is 0.5. Thus, in order to determine the interfacial area, an isosurface was created for a volume fraction of one of the phases equal to 0.5, and then the surface integral was calculated on the created surface. It should be noted that the results obtained for 2D and 3D dimensional spaces are in terms of the interfacial area thus in the case of 2D simulations a depth of 1 m was considered.

The area obtained corresponds to the sum of the areas of the facets that define the surface. Thus, the interfacial area was calculated by the following integral

where 𝐴𝑖 is the area of each facets.

3.6 Numerical Validation of CFD Models

In order to obtain the numerical validation of the Fluent results, reports in terms of the mass flow rates in the inlet and outlet sections of the NUB (to verify the mass conservation), as well

Interfacial area = ∫ 𝑑𝐴 = ∑ 𝐴𝑖 𝑛

𝑖=1

(32)

CFD Models 18 as the volume fractions of both phases were collected for different times steps. The volume fraction of the phases was calculated by dividing the summation of the volume fraction and cell volume by the total volume of the cells:

𝛼t= 1 𝑉∫ 𝛼 𝑑𝑉 = ∑ 𝛼𝑖𝑉𝑖 𝑛 𝑖=1 (3.24)

where 𝛼_𝑖 and 𝑉_𝑖 are the volume fraction and the volume of each cell.

The volume fractions collected correspond to the volume ratio occupied by each phase in the entire spatial domain of the reactor, so the sum of the fractions of the two phases must be 1. In Appendix A, the procedure for gathering the required data is explained.

(33)

4 Multiphase 2D Simulations

Multiphase flow in NETmix is studied from 2D CFD simulations with the NUB. In order to ensure the CFD solution converged residuals less than 10-5 were set and a time step size of 10 μs.

To define the VOF model it is necessary to specify a primary phase and a secondary phase, in this sense it was chosen to define that the primary phase was the phase 1 and the secondary phase was the phase 2. As for the interface modelling, the sharp option was chosen, this option is particularly suitable when a distinct interface is present between the phases.

In dynamics simulations, it was defined that in the left input of the NUB model only entered phase 1 (𝛼₁= 1) and in the right only phase 2 (𝛼1= 0).

The mixing between the two phases is assessed from the value of the interfacial area. The simulations were performed in a 2D space and so the interface between the two phases is a length, which is normalized by the length of the axis where the two phases contact at the initial symmetric flow: 𝑎_𝑣∗_{. It should be noted that an axis length of 58.5 mm was considered, which}

corresponds to the number of rows of the NUB model multiplied by the diameter of the chamber.

4.1 Multiphase Liquid-Liquid Flow

The fluid used in liquid-liquid simulations was liquid water at 20 ºC with a density of 𝜌 = 998.2 kg/m3_{and a viscosity of 𝜇 = 1.003 × 10}−3_{Pa ∙ s. It should be noted that both phases}

have the same physical properties. Since the two fluids have an equal density, it was not necessary to consider the effects of gravity.

Considering the velocity in the input channels, 𝑣_in, for a Reynolds number of 600, solving the Equation 3.16 in order to 𝑣_in, a velocity of 0.301 m/s is obtained. This value of 𝑣_in= 0.301 m/s corresponds to a Courant number (Equation 3.20) of 0.05.

To study the mixing of two immiscible-liquids in NETmix, simulations were performed for three different values of surface tension, it was considered a case without surface tension, a characteristic value of water-in-oil emulsions (𝜎 = 30 mN/m [47]) and a characteristic value for water-air system (𝜎 = 73 mN/m [13]). Moreover, the influence of the contact angle on the flow hydrodynamics and the interfacial area generation was studied, and for that purpose was considered a no wall adhesion effect (𝜃 = 90°), and two cases in which the wall is hydrophilic in relation to phase 2 (𝜃 = 30°; 𝜃 = 60°) and a case in which it is hydrophobic (𝜃 = 120°). It should be noted that the values of the contact angle shown are measured inside the phase 2.

(34)

Multiphase 2D Simulations 20 Table 1 summarizes the values of surface tension, Weber number and contact angle used in each simulation.

Table 1 - Surface tension, Weber number and contact angle for immiscible-liquid simulations.

Furthermore, all simulations were performed up to 3.5 times the passage time in the NETmix. The dynamic results are presented with the time normalized by the passage time (𝑡/𝜏). The passage time for a velocity in the inlet channels of 0.301 m/s is, from Equation 3.21, 556 ms.

4.1.1 Initial Solution

The initial conditions for multiphase flow were set by patching mass fraction variables in a given zone over the steady flow field. Two zones with the same area were defined, this division was performed in the direction of the y-axis, taking half of the width of the mesh. The zone on the left side of the mesh was filled with phase 1 and the zone on the right side with phase 2. In order to achieve a steady-state symmetric flow between the phases, it was defined that in the left input only entered phase 1 and in the right only phase 2. Both phases enter with the same velocity of 0.301 m/s. The phases were initially separated by walls, not being foreseen interactions between both. Figure 5 shows the results from the 2D CFD simulation of the flow in the NETmix, using the NUB model, of the two liquid phases. Figure 5a shows the contours of volume fraction of phase 1. Figure 5b shows the contours of velocity magnitude. Figure 5c shows the contours of pressure. These results were obtained after 100 000 times steps.

Surface tension (mN/m) We Contact angle (°)

0 ∞ - 30 3.0 60 73 1.2 30 60 90 120

(35)

From the analysis of the volume fraction distribution of the phases in the reactor, it is possible to see a state of total segregation between the two phases. The left side of the NUB model is filled with phase 1 (in red) and the right side with phase 2 (in blue). On the other hand, from the velocity and pressure maps, it is possible to see that the flow field at steady state is symmetrical, denoting a decrease in pressure from the inlet to the outlet of the reactor. This solution was used for all liquid-liquid simulations.

4.1.2 Study of the Influence of Surface Tension and Contact Angle

The results obtained for the conditions listed in Table 1 are shown below. Figures 6, 8, 10, 12, 14 and 16 show the contours of volume fraction of phase 1 for 16 different times. Figures 7, 9, 11, 13, 15 and 17 show the volume fraction fluctuation of phase 1, 𝛼_1,t, and 2, 𝛼_2,t, during the simulations. The evolution of interfacial area during the simulations is shown in Figure 18, 19 and 20. Table 2 summarizes the arithmetic mean of volume fraction of phase 1 and phase 2 considering the entire time domain of the simulations, and presents a percent deviation of the values obtained. The percent deviation was calculated by dividing the difference between the volume fraction of each phase in the whole spacial domain and its fraction in the inlet by the latter, that is:

% Deviation =|𝛼t− 0.5|

0.5 × 100 (4.1)

Table 3 presents the mean and standard deviation of the interfacial area for the 2D liquid-liquid simulations, it is read the value of interface length, 𝑎𝑣∗ and specific area, the latter defined as

the interface length normalized by the area of the NUB model. These mean values were obtained considering the time interval between 2.5𝜏 and 3.5𝜏.

Figure 5 – Contours of steady-state symmetric flow for liquid-liquid simulations: (a) volume fraction of phase 1; (b) velocity magnitude; (c) pressure.

V ol ume F ra cti on P ha se1 V el oci ty [ m s -1 ] Pr ess ur e [ Pa ] (a) (b) (c)

(36)

Multiphase 2D Simulations 22

𝑡/ 𝜏 = 0.00 𝑡/ 𝜏 = 0.05 𝑡/ 𝜏 = 0.25 𝑡/ 𝜏 = 0.30 𝑡/ 𝜏 = 0.40 𝑡/ 𝜏 = 0.50 𝑡/ 𝜏 = 0.60 𝑡/ 𝜏 = 0.70

Figure 6 - Contours of volume fraction of phase 1 for 𝜎 = 0 𝑚𝑁/𝑚 at different times.

V ol ume F ra cti on P ha se1 0.48 0.49 0.50 0.51 0.52 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Vo lu m e fr ac tio n t·𝜏-1 Phase 1 Phase 2

(37)

Figure 8 - Contours of volume fraction of phase 1 for 𝜎 = 30 𝑚𝑁/𝑚 and 𝜃 = 60° at different times. V ol ume F ra cti on P ha se1 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 V olu me fra ction t·𝜏-1 Phase 1 Phase 2