Fluid flow mathematical analysis

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. 

To constitute the We number, characteristic values such as the drop diameter, d, and particularly the interfacial tension, w, must be experimentally determined. However, the We number can also be obtained by deduction from mathematical analysis of droplet deforma-tional properties assuming a realistic model of the system. For a shear flow that is still dominant in the case of injection molding, Cox [25] derived an expression that for Newtonian fluids at not too high deformation has been proven to be valid ... 

It follows from the above facts that fluids can be treated as continuous media with continuous distributions of properties such as the pressure, density, temperature and velocity. Not only does this imply that it is unnecessary to consider the molecular nature of the fluid but also that meaning can be attached to spatial derivatives, such as the pressure gradient dP/dx, allowing the standard tools of mathematical analysis to be used in solving fluid flow problems. 

In order to understand the mechanisms for fluid flow in the metering channel, it is important to understand the reference frames used in the mathematical analysis of the section. As discussed in many chapters leading up to this point, the extruder is a cylindrical structure, as shown in Fig. 7.1(a), that has a helical channel formed... 

Meskat (M8) has presented a mathematical analysis of the effect of fluctuations in pressure and other variables on the comparative fluctuations in extrusion rates of Newtonian and non-Newtonian fluids. This work indicates the possibility of amplification of such fluctuations under certain circumstances with non-Newtonians rather than the uniform damping predicted for Newtonian behavior. If the validity of this analysis can be proved, it would warrant major attention being given to the problem of unsteady flow of non-Newtonian materials. 

As in the case of capillary-tube units, the shear rate (rotational speed) should be variable over wide ranges (10- to 1000-fold) and baffles or other obstructions which could interfere with the laminar-flow pattern must be absent. Since the fluid is sheared for long periods of time in these instruments, temperature control is much more critical, especially in the case of high-consistency materials, for which temperature rises of over 20°C. (W2) have been recorded. Weltmann and Kuhns (W5) subsequently presented an erudite mathematical analysis of the temperature distribution within the layers of sheared fluid. 

The mathematical analysis of flow in ducts of noncircular cross section is vastly more complex in laminar flow than for circular pipes and is impossible for turbulent flow. As a result, relatively little theoretical base has been developed for the flow of fluids in noncircular ducts. In order to deal with such flows practically, empirical methods have been developed. 

These fluctuations maybe caused by rapid variations in pressure or velocity producing random vortices and flow instabilities within the fluid. A complete mathematical analysis of turbulent flow remains elusive due to the erratic nature of the flow. Often used to promote mixing or enhance transport to surfaces, turbulent flow has been studied using electrochemical techniques [i]. 

Computational fluid dynamics involves the analysis of fluid flow and related phenomena such as heat and/or mass transfer, mixing, and chemical reaction using numerical solution methods. Usually the domain of interest is divided into a large number of control volumes (or computational cells or elements) which have a relatively small size in comparison with the macroscopic volume of the domain of interest. For each control volume a discrete representation of the relevant conservation equations is made after which an iterative solution procedure is invoked to obtain the solution of the nonlinear equations. Due to the advent of high-speed digital computers and the availability of powerful numerical algorithms the CFD approach has become feasible. CFD can be seen as a hybrid branch of mechanics and mathematics. CFD is based on the conservation laws for mass, momentum, and (thermal) energy, which can be expressed as follows ... 

If a discrete inf-sup inequality is satisfied by the velocity and pressure elements, then the discrete problem has a unique solution converging to the solution of the continuous one (see equation (27) of the chapter Mathematical Analysis of Differential models for Viscoelastic Fluids). This condition is referred to as the Brezzi-Babuska condition and can be checked element by element. Finite element methods for viscous flows are now well established and pairs of elements satisfying the Brezzi-Babuska condition are referenced (see [6]). Two strategies are used to compute the numerical solution of these equations involving velocity and pressure. 

In addition to these impediments to rheological measurements, some complex fluids exhibit wall slip, yield, or a material instability, so that the actual fluid deformation fails to comply with the intended one. A material instability is distinguished from a hydrodynamic instability in that the former can in principle be predicted from the constitutive relationship for the material alone, while prediction of a flow instability requires a mathematical analysis that involves not only the constitutive equation, but also the equations of motion (i.e., momentum and mass conservation). 

The theoretical and numerical basis of computational flow modeling (CFM) is described in detail in Part II. The three major tasks involved in CFD, namely, mathematical modeling of fluid flows, numerical solution of model equations and computer implementation of numerical techniques are discussed. The discussion on mathematical modeling of fluid flows has been divided into four chapters (2 to 5). Basic governing equations (of mass, momentum and energy), ways of analysis and possible simplifications of these equations are discussed in Chapter 2. Formulation of different boundary conditions (inlet, outlet, walls, periodic/cyclic and so on) is also discussed. Most of the discussion is restricted to the modeling of Newtonian fluids (fluids exhibiting the linear dependence between strain rate and stress). In most cases, industrial... 

We saw above that the concentration gradient at an electrode will be linear with respect to the spatial coordinate perpendicular to the electrode surface if the anode/cathode cell were operated at a constant current density and if the fluid velocity were zero. In actuality, there will always be some bulk liquid electrolyte stirring during current flow, either an imposed forced convection velocity or a natural convection fluid motion due to changes in the reacting species concentration and fluid density near the electrode surface. In electrochemical systems with fluid flow, the mass transfer and hydrodynamic fluid flow equations are coupled and the solution of the relevant differential equations is often a formidable task, involving complex mathematical and/or numerical solution techniques. The concept of a stagnant diffusion layer or Nemst layer parallel and adjacent to the electrode surface is often used to simplify the analysis of convective mass transfer in... 

A dimensional analysis cannot be made unless enough is known about the physics of the situation to decide what variables are important in the problem and what physical laws would be involved in a mathematical solution if one were possible. Sometimes this condition is fairly easily met the fundamental differential equations of fluid flow, for example, combined with the laws of heat conduction and diffusion, suffice to establish the dimensions and dimensionless groups appropriate to a large number of chemical engineering problems. Dimensional analysis, however, does not yield a numerical equation, and experiment is needed to complete the solution to the problem. 

In laminar flow, heat transfer occurs only by conduction, as there are no eddies to carry heat by convection across an isothermal surface. The problem is amenable to mathematical analysis based on the partial differential equations for continuity, momentum, and energy. Such treatments are beyond the scope of this book and are given in standard treatises on heat transfer, Mathematical solutions depend on the boundary conditions established to define the conditions of fluid flow and heat transfer. When the fluid approaches the heating surface, it may have an already completed hydrodynamic boundary layer or a partially developed one. Or the fluid may approach the heating surface at a uniform velocity, and both boimdary layers may be initiated at the same time. A simple flow situation where the velocity is assumed constant in all cross sections and tube lengths is called... 

Controlled-release polymer implants are useful for delivering drugs directly to the brain interstitium. This approach may improve the therapy of brain tumors or other neurologieal disorders. The mathematical models described in this section—which are based on methods of analysis developed in earlier chapters—provide a useful framework for analyzing mechanisms of drug distribution after delivery. These models describe the behavior of chemotherapy compounds very well and allow prediction of the effect of changing properties of the implant or the drug. More complex models are needed to describe the behavior of macromolecules, which encounter multiple modes of elimination and metabolism and are subject to the effects of fluid flow. 

Sometimes in fluid mechanics we may start with these four ideas and the measured physical properties of the materials under consideration and proceed directly to solve mathematically for the desired forces, velocities, and so on. This is generally possible only in the case of very simple flows. The observed behavior of a great many fluid flows is too complex to be solved directly from these four principles, so we must resort to experimental tests. Through the use of techniques called dimensional analysis (Chap. 13) often we can use the results of one experiment to predict the results of a much different experiment. Thus, careful experimental work is very important in fluid mechanics. With the development of supercomputers, we are now able to solve many complex problems mathematically by using the methods outlined in Chaps. 10 and 11, which previously would have required experimental tests. As computers become faster and cheaper, we will probably see additional complex fluid mechanics problems solved on supercomputers. Ultimately, the computer solutions must be tested experimentally. 

Computer flow-analysis programs used throughout the plastics industry worldwide utilize two- and three-dimension models in conjunction with rheology equations. Models range from a simple Poiseuille s equation for fluid flow to more complex mathematical models involving differential calculus. These models are only approximations. Their relational techniques, coupled with the user s assumptions, determine whether the findings of the flow analysis have any real validity. What actually happens is determined after processing the plastic. See flow, Poi-seuille. 

No exact mathematical analysis of turbulent flow has yet been developed for power-law fluids, though a number of semi-theoretical modifications of the expressions for the shear stress in Newtonian fluids at the walls of a pipe have been proposed. 

The present entry briefly discusses two of the basic discretization techniques commonly used for computational fluid dymamics (CFD) analysis of systems involving fluid flow, heat transfer, and associated phenomena such as chemical reactions. These mathematical tools, however, are generalized enough to address other transport processes occurring in electrical, magnetic, or electromagnetic systems as well. 

The lattice Boltzmann method is a mesoscopic simulation method for complex fluid systems. The fluid is modeled as fictitious particles, and they propagate and coUide over a discrete lattice domain at discrete time steps. Macroscopic continuum equations can be obtained from this propagation-colhsion dynamics through a mathematical analysis. The particulate nature and local d3mamics also provide advantages for complex boundaries, multiphase/multicomponent flows, and parallel computation. 

The mathematical analysis is not covered in this introductory text but for both true co-current and counter-current flow the temperature driving force when properly averaged is found to be the logarithmic mean temperature difference between the fluids ... 

Mathematical modeling and determination of the characteristic parameters to predict the performance of membrane solvent extraction processes has been studied widely in the literature. The analysis of mass transfer in hollow fiber modules has been carried out using two different approaches. The first approach to the modeling of solvent extraction in hollow fiber modules consists of considering the velocity and concentration profiles developed along the hollow fibers by means of the mass conservation equation and the associated boundary conditions for the solute in the inner fluid. The second approach consists of considering that the mass flux of a solute can be related to a mass transfer coefficient that gathers both mass transport properties and hydrodynamic conditions of the systan (fluid flow and hydrodynamic characteristics of the manbrane module). 

Briefly we treated the perfectly mixed reactor RTD in the mathematical analysis provided above. It is important to note from Example 3.1 that the RTD theory is not fully capable of explaining the behavior of the reactors, especially when the fluid elements are interacting. Thus, we give examples for a few other models here and refer the reader to an excellent text by Fox (2003) for a more in-depth analysis of these models in the turbulent flow regime. Four broad classes of micromixing models are sketched in Figure 110. 

An exact comparison of the present results with those from zhu Dong and Wen Shi-zhu [25] cannot be made, since some differences exist between the mathematical descriptions of the problem and in the methods by which they were solved (The work by [25] was probably not formulated with a conservative fluid flow solution, used a higher order deflection analysis, neglected conduction of heat In the x-direction within the solids, but Included heat convection within the fluid). 

---