Fluid flow conservation principles

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

The fundamental principles that apply to the analysis of fluid flows are few and can be described by the conservation laws ... 

The fuel flow in the gas channels is modeled by applying the equation of state and the principles of mass and momentum conservation. From Equation (3.27), considering that no reaction takes place within the gas channel (at the anode side only humidified hydrogen is provided), and that the fluid flow is regarded as incompressible (assumption (3)), the mass conservation equation becomes ... 

The engineering science of transport phenomena as formulated by Bird, Stewart, and Lightfoot (1) deals with the transfer of momentum, energy, and mass, and provides the tools for solving problems involving fluid flow, heat transfer, and diffusion. It is founded on the great principles of conservation of mass, momentum (Newton s second law), and energy (the first law of thermodynamics).1 These conservation principles can be expressed in mathematical equations in either macroscopic form or microscopic form. 

A large part of the present book will be concerned with a discussion of some of the analytical and numerical methods that can be used to try to predict h for various flow situations. Basically, these methods involve the simultaneous application of the principles governing viscous fluid flow, i.e., the principles of conservation of mass and momentum and the principle of conservation of energy. Although considerable success has been achieved with these methods, there still remain many cases in which experimental results have to be used to arrive at working relations for the prediction of h. 

In order to determine the distributions of pressure, velocity, and temperature the principles of conservation of mass, conservation of momentum (Newton s Law) and conservation of energy (first law of Thermodynamics) are applied. These conservation principles represent empirical models of the behavior of the physical world. They do not, of course, always apply, e.g., there can be a conversion of mass into energy in some circumstances, but they are adequate for the analysis of the vast majority of engineering problems. These conservation principles lead to the so-called Continuity, Navier-Stokes and Energy equations respectively. These equations involve, beside the basic variables mentioned above, certain fluid properties, e.g., density, p viscosity, p conductivity, k and specific heat, cp. Therefore, to obtain the solution to the equations, the relations between these properties and the pressure and temperature have to be known. (Non-Newtonian fluids in which p depends on the velocity field are not considered here.) As discussed in the previous chapter, there are, however, many practical problems in which the variation of these properties across the flow field can be ignored, i.e., in which the fluid properties can be assumed to be constant in obtaining fire solution. Such solutions are termed constant... 

As mentioned above, the continuity equation is obtained by the application of the principle of conservation of mass to the fluid flow. A differentially small rectangular control volume with sides parallel to the three coordinate directions is introduced. This is shown in Fig. 2.2. 

As was mentioned above, the Navier-Stokes equations are obtained by the appli-cation of the conservation of momentum principle to the fluid flow. The same control volume that was introduced above in the discussion of the continuity equation is considered and the conservation of momentum in each of the three coordinate directions is separately considered. The net force acting on the control volume in any of these directions is then set equal to the difference between the rate at which momentum leaves the control volume in this direction and the rate at which it enters in this direction. The net force arises from the pressure forces and the shearing forces acting on the faces of the control volume. The viscous shearing forces for two-dimensional flow (see later) are shown in Fig. 2.3. They are expressed in terms of the velocity field by assuming the fluid to be Newtonian and are then given by [4],[5] ... 

A flow is completely defined if the values of the velocity vector, the pressure, and the temperature are known at every point in the flow. The distributions of these variables can be described by applying the principles of conservation of mass, momentum, and energy, these conservation principles leading to the continuity, the Navier-Stokes, and the energy equations, respectively. If the fluid properties can be assumed constant, which is very frequently an adequate assumption, the first two of these equations can be simultaneously solved to give the velocity vector and pressure distributions. The energy equation can then be solved to give the temperature distribution. Fourier s law can then be applied at the surface to get the heat transfer rates. 

Most equipment used in the chemical, petroleum, and related industries is designed for the movement of fluids, and an understanding of fluid flow is essential to a chemical engineer. The underlying discipline is fluid mechanics,t which is based on the law of mass conservation, the linear momentum principle (Newton s second law), and the first and second laws of thermodynamics. 

This section will first deal with the phases in particle-fluid two-phase flow by developing a mathematical model to quantify local hydrodynamic states. This analysis will reveal the insufficiency of the conditions for the conservation of mass and momentum alone in determining the hydrodynamic states of heterogeneous particle-fluid systems, and calls for a methodology different from what is used in analyzing dilute uniform flow. For this purpose the concept of multi-scale interaction between particles and fluid and the principle of energy minimization are proposed. 

Due to the advent of CFD the aforementioned approach can still be followed but now the E t) and F(t) functions can in principle be computed from the computed velocity distribution. Alternatively, the species conservation equations can be solved simultaneously with the fluid flow equations and thereby the extent of chemical conversion can also be obtained directly without invoking the concept of residence time distributions. 

The equations of fluid mechanics originate from the momentum and mass conservation principles. The overall mass conservation or continuity equation for laminar flows is... 

We see that application of the angular acceleration principle does reduce, somewhat, the imbalance between the number of unknowns and equations that derive from the basic principles of mass and momentum conservation. In particular, we have shown that the stress tensor must be symmetric. Complete specification of a symmetric tensor requires only six independent components rather than the full nine that would be required in general for a second-order tensor. Nevertheless, for an incompressible fluid we still have nine apparently independent unknowns and only four independent relationships between them. It is clear that the equations derived up to now - namely, the equation of continuity and Cauchy s equation of motion do not provide enough information to uniquely describe a flow system. Additional relations need to be derived or otherwise obtained. These are the so-called constitutive equations. We shall return to the problem of specifying constitutive equations shortly. First, however, we wish to consider the last available conservation principle, namely, conservation of energy. 

Conservation Equations. In the above section, the material functions of nonnewtonian fluids and their measurements were introduced. The material functions are defined under a simple shear flow or a simple shear-free flow condition. The measurements are also performed under or nearly under the same conditions. In most engineering practice the flow is far more complicated, but in general the measured material functions are assumed to hold. Moreover, the conservation principles still apply, that is, the conservation of mass, momentum, and energy principles are still valid. Assuming that the fluid is incompressible and that viscous heating is negligible, the basic conservation equations for newtonian and nonnewtonian fluids under steady flow conditions are given by... 

These four ideas are applied to fluid mechanics problems as follows In Chap. 1 we discuss some of the measurable properties of fluids and some definitions. In Chap. 2 we apply Newton s law of motion to the particularly simple case of a fluid which is not moving. In Chap. 3 we explain and apply the principle of the conservation of mass. In Chap. 4 we consider the principle of the conservation of energy. In Chaps. 5 and 6 we apply the principle of the conservation of energy to a class of relatively simple fluid flows which includes many flows of great practical importance. In Chap. 7 we recast Newton s laws... 

In order to simulate fluid flow, heat transfer, and other related physical phenomena over various length scales, it is necessary to describe the associated physics in mathematical terms. Nearly all the physical phenomena of interest to the fluid dynamics research community are governed by the principles of continuum conservation and are expressed in terms of first- or second-order partial differential equations that mathematically represent these principles (within the restrictions of a continuum-based firamework). However, in case the requirements of continuum hypothesis are violated altogether for certain physical problems (for instance, in case of high Knudsen number rarefied gas flows), alternative formulations in terms of the particle-based statistical tools or the atomistic simulation techniques need to be resorted to. In this entry, we shall only focus our attention to situations in which the governing differential equations physically originate out of continuum conservation requirements and can be expressed in the form of a general differential equation that incorporates the unsteady term, the advection term, the diffusion term, and the source term to be elucidated as follows. 

The behavior of isothermal pressure-driven single-phase fluid flows in microchannels can be smdied by determining the velocity distribution in the fluid region with the aid of the mass conservation principle continuity equation) and the equations of conservation of fluid momentum [3] ... 

As opposed to the described MEIS with variable parameters and the mechanisms of physicochemical processes in this case we will try to determine the objective function of applied model for a dissipative system based on the equilibrium principle of conservative systems, i.e. the Lagrange principle of virtual works. Derivation will be given on the example of the closed (not exchanging the fluid flows with the environment) active (with sources of motive pressures) circuit. The simplest scheme of such a circuit is presented in Fig. 3,a. A common character of the chosen example is explained by the easiness of passing to other possible schemes. For example, if at the modeled network nodes there are external... 

The Volume of Fluid (VOF) method, as introduced by Hirt and Nichols (1981), is based on the mass conservation principle. Similar to the control-volume method, in the VOF method, the whole domain can also be divided into control volumes, each of which is associated to an element node, and the volume fraction of fluid in each control volume is defined. The flow front is advanced by solving the following transport equation ... 

This conservation principle applied to a general control volume is illustrated in Figure 1.1. The control volume is located in the fluid flow field. The velocity at any point on the surface is given by v and the vector normal to the surface is given by n. The angle between the velocity vector v and the normal vector n is a. 

The conservation of mass, momentum and energy govern all fluid flows. These basic principles are expressed in the so-called Navier-Stokes equations. CFD involves the use of a numerical code on a suitable computer to solve these relationships as a series of partial differential equations. 

The physical aspects of any fluid flow are governed by three principles mass is conserved, Newton s second law is fulfilled (also referred as momentum equation) and energy is conserved these principles are expressed in integral equations or partial differential equations (continuity, momentum and energy equations), being the most common form the Navier-Stokes equations for viscous flows and the Euler equations for inviscid flows. 

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