Fluid Mechanics for Chemical Engineering

Mathieu Mory

• Publisher: Wiley-ISTE
• Year: 2013
• ISBN:
  9781118616918
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Mechanics

The book aims at providing to master and PhD students the basic knowledge in fluid mechanics for chemical engineers. Applications to mixing and reaction and to mechanical separation processes are addressed.

The first part of the book presents the principles of fluid mechanics used by chemical engineers, with a focus on global theorems for describing the behavior of hydraulic systems. The second part deals with turbulence and its application for stirring, mixing and chemical reaction. The third part addresses mechanical separation processes by considering the dynamics of particles in a flow and the processes of filtration, fluidization and centrifugation. The mechanics of granular media is finally discussed.

Part I

Elements in Fluid Mechanics

Chapter 1

Local Equations of Fluid Mechanics

In this chapter, to begin with, we recall the Navier–Stokes equations that govern the flow of a Newtonian fluid. These equations explain the behavior of common fluids such as water or air. For a given force field and boundary conditions, the solution of Navier–Stokes equations controls both the flow velocity and pressure at any point and at any time in the domain under consideration. The Navier–Stokes equations are the most commonly used equations in fluid mechanics; they provide the knowledge of the flow of Newtonian fluids at the local level.

The solutions to Navier–Stokes equations are typically very difficult to arrive at. This fact is attested to by the extraordinary development of numerical computation in fluid mechanics. Only a few exact analytical solutions are known for Navier– Stokes equations. We present in this chapter some laminar flow solutions whose interpretation per se is essential in this regard. We then introduce the boundary layer concept. We conclude the chapter with a discussion on the uniqueness of solutions to Navier–Stokes equations, with special reference to the phenomenon of turbulence.

This being the introductory chapter, we have not included a prolonged discussion on continuum mechanics. The derivation of Navier–Stokes equations is available in other continuum mechanics or fluid mechanics books.¹ We have consciously avoided concentrating on the derivational aspects of Navier–Stokes equations as we are convinced that it is far more important to understand the meaning of the different terms of these equations and to hence interpret the way they are applied in the study of fluid mechanics in general. In addition, we limit ourselves to introducing the only classical concept from continuum mechanics to be used in this book, namely, the ability to calculate the force acting through a surface passing through a point that lies inside a continuum, using the stress tensor. Hence, Chapter 1 partly serves as a collection of formulae, while proper physical principles are discussed in the remainder of this book. The reader might wish to read this chapter without pondering on it for long, and then may refer to it later, if necessary, for it may be insightful in such a case.

1.1. Forces, stress tensor, and pressure

Consider a domain, V, containing a fluid. The fluid’s flow is controlled by various forces acting on it. The laws of mechanics help us to distinguish two types of forces:

– Body forces, which are exerted at every point in a domain. Weight is the most common body force.

– Forces that are transferred from one particle to another, at the boundary of and within the domain. These forces are expressed using the stress tensor. This is where the continuum concept intervenes.

The force at a point M in the continuum is associated with surface element ds whose orientation is given by the unit normal vector

(Figure 1.1). The force

, which is proportional to the surface ds, varies when the orientation of the surface changes. It is determined at the point M using the stress tensor [Σ], which is a symmetric,² 3 × 3 matrix:

[1.1]

Figure 1.1. Forces

and

exerted at a point M through two surface elements ds₁ and ds₂, whose orientations are given by normals

and

. Both ds₁and ds₂ are elements of closed surfaces S₁(solid line) and S₂(dashed line), respectively, surrounding volumes V₁ and V₂

The force through the surface element ds whose normal is

is written as:

[1.2]

The force applied to a closed surface S surrounding an arbitrary volume V in the continuum (Figure 1.1) can be derived using the surface integral:

[1.3]

The concept of the stress tensor is inseparable from the mechanical principle of action and reaction. The normal vector

is oriented toward the exterior of the domain on which the force is applied. The direction of the force is reversed if one considers the force exerted by the domain V on the exterior. Therefore, the domain under consideration should always be specified. The force is exerted by the external environment through the surface of separation.

In equation [1.1] and [1.2], the stress tensor is expressed in a Cartesian coordinate system (O, x, y, z). In this coordinate system, the single-column matrices define the normal vector

and force

. We only have to multiply matrix [Σ] by

to calculate the force.

The stress tensor embodies two notions: pressure and friction forces. For a Newtonian fluid, pressure is introduced by adding together the diagonal terms of the stress tensor:

[1.4]

We also introduce the stress deviator tensor [Σ′].

[1.5]

such that σ′_xx + σ′_yy + σ′_zz = 0. The stress deviator is associated with friction forces, whereas the pressure produces a force that is perpendicular to the surface element under consideration. Alternatively, the stress tensor can be expressed as:

[1.6]

Figure 1.1 depicts two volumes V₁ and V₂ surrounded by closed surfaces S₁ and S₂. At a point M belonging to both surfaces, the normals are different, and therefore the forces exerted on the surface elements, ds₁ and ds₂, are also different. The orientation of the force vectors shown in Figure 1.1 satisfies two properties. The forces are not perpendicular to the two surfaces onto which they are applied, if friction forces exist. However, they are not far from being perpendicular to the surfaces, and they are oriented toward the interior of the domain; this i...

Table of contents

• Cover
• Title Page
• Copyright
• Preface
• Part I: Elements in Fluid Mechanics
• Part II: Mixing and Chemical Reactions
• Part III: Mechanical Separation
• Physical Properties of Common Fluids
• Index