Fluid dynamics

First we give a brief introduction to the historical development of the science of fluid mechanics, and subsequently the development and present areas of application of CFD are highlighted. 

The equations that form the theoretical foundation for the whole science of fluid mechanics were derived more than one century ago by Navier (1827) and Poisson (1831) on the basis of molecular hypotheses. Later the same equations were derived by de Saint Venant (1843) and Stokes (1845) without using such hypotheses. These equations are commonly referred to as the Navier-Stokes equations. Despite the fact that these equations have been known of for more than a century, no general analytical solution of the Navier-Stokes equations is known. This state of the art is due to the complex mathematical (i.e., nonlinearity) nature of these equations. 

Toward the end of the nineteenth century the science of fluid mechanics began to develop in two branches, namely theoretical hydrodynamics and hydraulics. The first branch evolved from Euler s equations of motion for a frictionless, non- 

The study of fire in a compartment primarily involves three elements (a) fluid dynamics, (b) heat transfer and (c) combustion. All can theoretically be resolved in finite difference solutions of the fundamental conservation equations, but issues of turbulence, reaction chemistry and sufficient grid elements preclude perfect solutions. However, flow features of compartment fires allow for approximate portrayals of these three elements through global approaches for prediction. The ability to visualize the dynamics of compartment fires in global terms of discrete, but coupled, phenomena follow from the flow features. 

Because of their practical importance, water drops in air and air bubbles in water have received more attention than other systems. The properties of water drops and air bubbles illustrate many of the important features of the ellipsoidal regime. 

Drops larger than about 1 mm in diameter are significantly nonspherical the mean height to width ratio is approximated (P5) by  

Water drops become unstable and tend to break up before they reach 1 cm in diameter (see Chapter 12). Drops approaching this size show periodic shape fluctuations of relatively low amplitude (J3, M4). 

As for other types of fluid particle, the internal circulation of water drops in air depends on the accumulation of surface-active impurities at the interface (H9). Observed internal velocities are of order 1% of the terminal velocity (G4, P5), too small to affect drag detectably. Ryan (R6) examined the effect of surface tension reduction by surface-active agents on falling water drops. 

Aybers and Tapucu (A4, A5) measured trajectories of air bubbles in water. When surface-active agents continue to accumulate during rise, the terminal velocity may never reach steady state (A4, Bl) and may pass through a maximum (W4). Five types of motion were observed, listed in Table 7.1 with Re based on the maximum instantaneous velocity. Secondary motion of fluid par- 

We limit our discussion here to laminar flows governed by the steady or unsteady, incompressible Navier-Stokes equations. In addition, we restrict ourselves to flows where the solution to the energy or the concentration equation does not influence the flow field, a circumstance not uncommon to isothermal constant viscosity liquid flows of relevance for many electrochemical systems. The incompressible, constant-property, Navier-Stokes equations are given below, with summation over repeated indices  

Equations (1) and (2) are said to be in primitive Ux, Uy, u, p) variable form. While there are other forms that rely on derived variables, such as stream function vorticity, vorticity-velocity potential, and dual-potential methods, we restrict ourselves to the primitive variable form because of their popularity and ease of interpretation. For a discussion of these methods, as well as for further details of most aspects of CFD discussed here, see Refs. 73, 74, and 77. 

The governing equations (1) and (2) are of a mixed parabolic-elliptic nature. A key feature of incompressible flow is that that the time derivative of pressure vanishes from the equations. Hence the equations do not transmit any pressure history directly, and it is as if a new pressure field is established at each step. This situation does not arise for compressible flow where, owing to the presence of the time derivative of the pressure term in the continuity equation, one can solve the coupled hyperbolic system by advancing in time. In the absence of such a term, the algebraic system of equations becomes singular. This is also why attempts to solve the incompressible flow problem as a low Mach-number, compressible-flow problem lead to ill-conditioned algebraic systems with poor algorithmic efficiency and accuracy. For a detailed discussion of these issues, see Ref. 74, p. 642. 

The second class of methods, which seem to be more popular among the commercial CFD codes, relies on the pressure-correction approach. Here, the velocity components are solved in an uncoupled manner without using the continuity equation as a constraint. An equation for pressure or a change in pressure (hence the term pressure correction ) is derived that will alter the velocity field so as to better satisfy the continuity equation. The precise formulation and the iterative procedures may differ as long as iterative application of the momentum and the pressure-correction equations produces a solution that satisfies both the momentum and the continuity equations. Algorithms that differ in this fashion are employed 

PLE or SIMPLER methods of Patankary, and the PISO method of Issa. 

Eigure 2.8 shows that the rectangular pulse, which is introduced at the column inlet (x = 0), is symmetrically broadened as it travels along the column. As a consequence of the band broadening, the maximum concentration of the solute is decreased. This causes an unfavorable dilution of the target component fraction. Main factors that influence axial dispersion are discussed below. 

Every centimeter of tubing, as well as any detector, between the point of solute injection and the point of fraction withdrawal contributes to the axial dispersion of 

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Computational Fluid Dynamics Applied to Process Engineering. 

Dey B D, Askar A and Rabitz H 1998 Multidimensional wave packet dynamics within the fluid dynamical formulation of the Schrddinger equation J. Chem. Phys. 109 8770-82... 

If a fluid is placed between two concentric cylinders, and the inner cylinder rotated, a complex fluid dynamical motion known as Taylor-Couette flow is established. Mass transport is then by exchange between eddy vortices which can, under some conditions, be imagmed as a substantially enlranced diflfiisivity (typically with effective diflfiision coefficients several orders of magnitude above molecular difhision coefficients) that can be altered by varying the rotation rate, and with all species having the same diffusivity. Studies of the BZ and CIMA/CDIMA systems in such a Couette reactor [45] have revealed bifiircation tlirough a complex sequence of front patterns, see figure A3.14.16. 

This method has been devised as an effective numerical teclmique of computational fluid dynamics. The basic variables are the time-dependent probability distributions f x, f) of a velocity class a on a lattice site x. This probability distribution is then updated in discrete time steps using a detenninistic local rule. A carefiil choice of the lattice and the set of velocity vectors minimizes the effects of lattice anisotropy. This scheme has recently been applied to study the fomiation of lamellar phases in amphiphilic systems [92, 93]. 

C. Similarities Between Potentiai Fluid Dynamics and Quantum Mechanics... 

In writing the Lagrangean density of quantum mechanics in the modulus-phase representation, Eq. (140), one notices a striking similarity between this Lagrangean density and that of potential fluid dynamics (fluid dynamics without vorticity) as represented in the work of Seliger and Whitham [325]. We recall briefly some parts of their work that are relevant, and then discuss the connections with quantum mechanics. The connection between fluid dynamics and quantum mechanics of an electron was already discussed by Madelung [326] and in Holland s book [324]. However, the discussion by Madelung refers to the equations only and does not address the variational formalism which we discuss here. 

Unlike classical systems in which the Lagrangean is quadratic in the time derivatives of the degrees of freedom, the Lagrangeans of both quantum and fluid dynamics are linear in the time derivatives of the degrees of freedom. 

Computer modelling provides powerful and convenient tools for the quantitative analysis of fluid dynamics and heat transfer in non-Newtonian polymer flow systems. Therefore these techniques arc routmely used in the modern polymer industry to design and develop better and more efficient process equipment and operations. The main steps in the development of a computer model for a physical process, such as the flow and deformation of polymeric materials, can be summarized as ... 

Hughes, T. J. R., Franca, L. P. and Balestra, M., 1986. A new finite-element formulation for computational fluid dynamics. 5. Circumventing the Babuska-Brezzi condition - a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal order interpolations. Cornput. Methods Appl. Meek Eng. 59, 85-99. 

Diffusion in flowing fluids can be orders of magnitude faster than in nonfiowing fluids. This is generally estimated from continuum fluid dynamics simulations. 

Although the Arrhenius equation does not predict rate constants without parameters obtained from another source, it does predict the temperature dependence of reaction rates. The Arrhenius parameters are often obtained from experimental kinetics results since these are an easy way to compare reaction kinetics. The Arrhenius equation is also often used to describe chemical kinetics in computational fluid dynamics programs for the purposes of designing chemical manufacturing equipment, such as flow reactors. Many computational predictions are based on computing the Arrhenius parameters. 

The simplest case of fluid modeling is the technique known as computational fluid dynamics. These calculations model the fluid as a continuum that has various properties of viscosity, Reynolds number, and so on. The flow of that fluid is then modeled by using numerical techniques, such as a finite element calculation, to determine the properties of the system as predicted by the Navier-Stokes equation. These techniques are generally the realm of the engineering community and will not be discussed further here. 

Fluid dynamics Fluid energy mills Fluidization... 

K. Schuged and co-workers. Proceedings of the 2nd International Conference on Bioreactor Fluid Dynamics, British Hydromechanics Research Association, Cranfield, UK, 1988, p. 229. 

P. J. Roache, Computational Fluid Dynamics, Hermosa Pubhshers, Albuquerque, N.M., 1982. 

It has become quite popular to optimize the manifold design using computational fluid dynamic codes, ie, FID AP, Phoenix, Fluent, etc, which solve the full Navier-Stokes equations for Newtonian fluids. The effect of the area ratio, on the flow distribution has been studied numerically and the flow distribution was reported to improve with decreasing yiR. 

A numerical study of the effect of area ratio on the flow distribution in parallel flow manifolds used in a Hquid cooling module for electronic packaging demonstrate the useflilness of such a computational fluid dynamic code. The manifolds have rectangular headers and channels divided with thin baffles, as shown in Figure 12. Because the flow is laminar in small heat exchangers designed for electronic packaging or biochemical process, the inlet Reynolds numbers of 5, 50, and 250 were used for three different area ratio cases, ie, AR = 4, 8, and 16. 

Computer Models, The actual residence time for waste destmction can be quite different from the superficial value calculated by dividing the chamber volume by the volumetric flow rate. The large activation energies for chemical reaction, and the sensitivity of reaction rates to oxidant concentration, mean that the presence of cold spots or oxidant deficient zones render such subvolumes ineffective. Poor flow patterns, ie, dead zones and bypassing, can also contribute to loss of effective volume. The tools of computational fluid dynamics (qv) are useful in assessing the extent to which the actual profiles of velocity, temperature, and oxidant concentration deviate from the ideal (40). 

Third, design constraints are imposed by the requirement for controlled cooling rates for NO reduction. The 1.5—2 s residence time required increases furnace volume and surface area. The physical processes involved in NO control, including the kinetics of NO chemistry, radiative heat transfer and gas cooling rates, fluid dynamics and boundary layer effects in the boiler, and final combustion of fuel-rich MHD generator exhaust gases, must be considered. 

In cases where a large reactor operates similarly to a CSTR, fluid dynamics sometimes can be estabflshed in a smaller reactor by external recycle of product. For example, the extent of soflds back-mixing and Hquid recirculation increases with reactor diameter in a gas—Hquid—soflds reactor. Consequently, if gas and Hquid velocities are maintained constant when scaling and the same space velocities are used, then the smaller pilot unit should be of the same overall height. The net result is that the large-diameter reactor is well mixed and no temperature gradients occur even with a highly exothermic reaction. 

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