Yang / Mao Numerical Simulation of Multiphase Reactors with Continuous Liquid Phase

1. Auflage 2014
ISBN: 978-0-12-411579-8
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 320 Seiten

Numerical simulation of multiphase reactors with continuous liquid phase provides current research and findings in multiphase problems, which will assist researchers and engineers to advance this field. This is an ideal reference book for readers who are interested in design and scale-up of multiphase reactors and crystallizers, and using mathematical model and numerical simulation as tools. Yang and Mao's book focuses on modeling and numerical applications directly in the chemical, petrochemical, and hydrometallurgical industries, rather than theories of multiphase flow. The content will help you to solve reacting flow problems and/or system design/optimization problems. The fundamentals and principles of flow and mass transfer in multiphase reactors with continuous liquid phase are covered, which will aid the reader's understanding of multiphase reaction engineering. - Provides practical applications for using multiphase stirred tanks, reactors, and microreactors, with detailed explanation of investigation methods - Presents the most recent research efforts in this highly active field on multiphase reactors and crystallizers - Covers mathematical models, numerical methods and experimental techniques for multiphase flow and mass transfer in reactors and crystallizers

Institute of Process Engineering, Chinese Academy of Sciences

Autoren/Hrsg.

Yang, Chao

Mao, Zai-Sha

Leseproben

Chapter 2

Fluid flow and mass transfer on particle scale

Abstract

A single particle (bubble, drop, or solid particle) in an infinite continuous phase is a simplified model used to probe the law of multiphase flow and transport processes in complex multiphase systems, and it has been studied extensively by both experimental and numerical simulation. Interfacial instability such as the Marangoni effect on a sub-drop scale, which plays a significant role in heat and mass transfer, is being approached numerically with encouraging success. Different numerical methods, such as orthogonal boundary-fitted coordinate system-based simulation and the level set method, are adopted to simulate the motion and interphase mass transfer of a drop or a bubble and investigate the respective effect of particle size, deformation, surface active agent, etc. on the simplified model for summarizing transport rules. Also, the mirror fluid method and the cell model are respectively used to study the motion and transport processes of a solid particle and particle swarm, which is conducive to expanding the research of a single particle to that of particle swarm even on a reactor scale.

Keywords

Mass transfer

Marangoni effect

drop

bubble

solid particle

numerical simulation

2.1. Introduction

Fluid flow of and mass transfer from/to drops, bubbles, and solid particles are often observed in nature and various areas of engineering. Chemical and metallurgical engineers rely on bubbles and drops for unit operations such as distillation, absorption, flotation and spray drying, while solid particles are used as catalysts or chemical reactants. In these processes, there is relative motion between bubbles, drops or particles on one hand, and a surrounding fluid on the other. In many cases, transfer of mass and/or heat is also of importance. Owing to rapid progress of computer techniques and numerical methods in fluid mechanics and transport phenomena, the application of numerical simulation has recently become increasingly popular in understanding multiphase flow and transport on a particle (a generic term including drops and bubbles) scale.

In this chapter, this topic is discussed in detail in the following six sections. Firstly, the theoretical basis and numerical methods frequently adopted are summarized in Sections 2.2 and 2.3 respectively. We choose to focus mostly on three methods: simulation on orthogonal boundary-fitted coordinates, an improved level set method, and a mirror fluid method. This choice reflects our own background, as well as the fact that these methods are deemed successful and reliable for computing the motion and mass transfer of fluid particles (bubbles and drops) or solid particles. The validity of these methods is demonstrated and compared with the reported experimental data in Section 2.4. Also, considering the trace quantities of surfactants unavoidable in most industrial systems, study of the motion and mass transfer of a solute to/from a single drop with a surfactant adsorbed on the interface is carried out to better understand the liquid extraction processes and for the scientific design of relevant equipment. The Marangoni effect, one of the most sophisticated interphase transport phenomena, interests researchers due to its influence on transport rates and it has been mathematically formulated and numerically simulated to shed light on these mechanisms. Recent studies relating to the Marangoni effect are presented in Section 2.5. In Section 2.6, numerical simulation methods on particle swarms are discussed briefly and modified cell models are introduced to examine the flow and transport behaviors of particle swarms. Section 2.7 incorporates related progress on particle motion controlled by fluid shear or extension.

2.2. Theoretical basis

The mathematical formulation of two-phase particle flow may be exemplified using two-fluid systems in which a liquid drop or a gas bubble moves in another continuous liquid as it follows in this section. The fundamental physical laws governing the motion of and mass transfer from/to a single particle immersed in another fluid are Newton’s second law, the principle of mass conservation, and Fick’s diffusion law. So the flow field and solute transport in both fluid phases must be formulated using the first principles of fluid mechanics and transport phenomena. When a solid particle is involved, the flow in the solid domain is usually not necessary and the particle is tracked mechanically as a rigid body. In this context, two-phase flow with a solid particle is a simplified case of general two-phase systems.

2.2.1. Fluid mechanics

The motion of a small particle (drop, bubble or solid particle) of around 1 mm size under gravity through an immiscible continuous fluid phase can be resolved using the following assumptions: (1) the fluid is viscous and incompressible; (2) the physical properties of the fluid and the particle are constant; (3) the two-phase flow is axisymmetric or two-dimensional; (4) the flow is laminar at low Reynolds numbers.

The flow in each fluid phase is governed by the continuity and Navier–Stokes equations:

·u=0

(2.1)

?u?t+u·?u=-?p+?g+?·t+S

(2.2)

where t is the stress tensor defined as

=µ(?u+(?u)T)

(2.3)

and the source term S is formulated differently in different cases.

Boundary conditions for the governing equations are essential when an interface exists between the two phases. For a bubble or a drop, the normal velocity in each phase is equal at the interface. If the gas in a bubble is taken as inviscid, the bubble surface is mobile and not subject to any shear force. However, if the gas is taken as a viscous fluid, both the velocity vector and shear stress should be continuous across the interface. For a solid particle, both the normal and tangential velocity components of the continuous phase must be zero at the particle surface; that is, the solid surface should satisfy the “no-slip” condition.

For the case with constant physical properties of both fluid phases, including that on the interface, the solution for mass transfer will be decoupled from the problem of fluid flow. Thus, the information of the flow field, required for solution of convective diffusion problems, whether for steady or unsteady mass transfer, can be provided directly from numerical simulation of steady-state fluid flow only once.

2.2.2. Mass transfer

In general, the transient mass transfer to/from a drop (or a bubble) is governed by the convective diffusion equation in vector form:

c?t+u·?c=D?2c

(2.4)

in each phase subject to two interfacial conditions:

1?c1?n1=D2?c2?n2 (flux continuity at the interface)

(2.5)

2=mc1 (interfacial dissolution equilibrium)

(2.6)

In the above equations, subscript 1 indicates the continuous phase and 2 the dispersed phase. The solution of Eq. (2.5) is reliant on the resolved fluid flow both in the dispersed and the continuous phases, as addressed by Li and Mao (2001). In accordance with Fick’s first law, for steady external mass transfer the local diffusive flux across the interface is calculated by

loc=-D2?c2?n2=kloc(c¯2-mc18)

(2.7)

where the remote boundary concentration 1 8 and the only available measurement of the bubble/drop concentration ¯2 (averaged over the whole drop, taking a drop as an example) are used to define the driving force and the mass transfer coefficient. The latter may be expressed in terms of dimensional concentration gradient as

loc=-D2(c¯2-mc18)?c2?n2

(2.8)

Then, the local Sherwood number is

hloc=dklocD2=-d(c¯2-mc18)?c2?n2

(2.9)

and the drop area averaged Shod is

hod=?Shlocds?ds

(2.10)

On the other hand, the overall mass transfer coefficient kod may be evaluated from the overall solute conservation based on the drop as follows:

od(c2*-c¯2)A=Vddc¯2dt

(2.11)

where ¯2 is the average concentration of the drop at any time instant, which is almost the only available measure of solute concentrations of drops in conventional experiments. If the time interval tout–tin is chosen small enough, kod may be evaluated approximately from integration of the above equation as

od=-VdA1tout-tinlnc2*-c¯2,outc2*-c¯2,in

(2.12)

where A and Vd are the volume and the surface area of the drop, and for a spherical drop Vd/A =...