Other flow equations

At low fluid velocities through packed beds of powders the laminar flow term predominates, whereas at higher velocities both viscous and kinetic effects are important. Er n and Oming [14] found that in the transitional region between laminar and turbulent flow, the equation relating pressure gradient and superflcial fluid velocity uf was  

It has been found that some variation between specific surface and porosity occurs. Carman [15] suggested a correction to the porosity function to eliminate this variation. This correction may be written  

Harris [17] discussed the role of adsorbed fluid in permeametry but prefers the term immobile fluid. He stated that discrepancies usually attributed to errors in the porosity function or non-uniform packing are, in truth, due to the assumption of incorrect values for e and S. Associated with the particles is an immobile layer of fluid that does not take part in the flow process. The particles have a true volume vj and an effective volume v a true surface S and an effective surface 5 a true density p, and an effective density p. The true values can be 

Equation (1.10) is assumed correct but the true values are replaced with effective values yielding  

Combining equations (1.15) and (1.16), the Carman-Kozeny equation take the form  

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Due to the very low volumetric concentration of the dispersed particles involved in the fluid flow for most cyclones, the presence of the particles does not have a significant effect on the fluid flow itself. In these circumstances, the fluid and the particle flows may be considered separately in the numerical simulation. A common approach is to first solve the fluid flow equations without considering the presence of particles, and then simulate the particle flow based on the solution of the fluid flow to compute the drag and other interactive forces that act on the particles. 

This unit consists of two pipes or tubes, the smaller centered inside the larger as shown in Figure 10-92. One fluid flows in the annulus between the tubes the other flows inside the smaller tube. The heat transfer surface is considered as the outside surface of the inner pipe. The fluid film coefficient for the fluid inside the inner tube is determined the same as for any straight tube using Figures 10-46-10-52 or by the applicable relations correcting to the O.D. of the inner tube. For the fluid in the annulus, the same relations apply (Equation 10-47), except that the diameter, D, must be the equivalent diameter, D,.. The value of h obtained is applicable directly to the point desired — that is, the outer surface of the inner tube. ... 

The Rf values for a substance measured in linear, circular and anticircular chromatograms, for which the flow conditions vary, can be related to each other by equation 7.18... 

For other discussions of two-phase models and numerical solutions, the reader is referred to the following references thermofluid dynamic theory of two-phase flow (Ishii, 1975) formulation of the one-dimensional, six-equation, two-phase flow models (Le Coq et al., 1978) lumped-parameter modeling of one-dimensional, two-phase flow (Wulff, 1978) two-fluid models for two-phase flow and their numerical solutions (Agee et al., 1978) and numerical methods for solving two-phase flow equations (Latrobe, 1978 Agee, 1978 Patanakar, 1980). 

For pipe fittings, valves, and other flow obstructions the traditional method has been to use an equivalent pipe length Lequiv in Equation 4-30. The problem with this method is that the specified length is coupled to the friction factor. An improved approach is to use the 2-K method,s-6 which uses the actual flow path length in Equation 4-30 — equivalent lengths are not used — and provides a more detailed approach for pipe fittings, inlets, and outlets. The 2-K method defines the excess head loss in terms of two constants, the Reynolds number and the pipe internal diameter ... 

This expression does not relate to a true cell because the two electrode potentials are not measured with electrodes, nor can we relate AGr to the emf, because electrons do not flow from one half-cell via an external circuit to the other. Nevertheless, Equation (7.40) is a kind of proof that the overall value of AGr relates to the constituent half-cells. 

The interaction of forced and natural convective flow between cathodes and anodes may produce unusual circulation patterns whose description via deterministic flow equations may prove to be rather unwieldy, if possible at all. The Markovian approach would approximate the true flow pattern by subdividing the flow volume into several zones, and characterize flow in terms of transition probabilities from one zone to others. Under steady operating conditions, they are independent of stage n, and the evolution pattern is determined by the initial probability distribution. In a similar fashion, the travel of solid pieces of impurity in the cell can be monitored, provided that the size, shape and density of the solids allow the pieces to be swept freely by electrolyte flow. 

The step-strain experiments discussed above furnish the simplest example of a strong flow. Many other flows are of experimental importance transient and steady shear, transient extensional flow and reversing step strains, to give a few examples. Indeed the development of phenomenological constitutive equations to systematise the wealth of behaviour of polymeric liquids in general flows has been something of an industry over the past 40 years [9]. It is important to note that it is not possible to derive a constitutive equation from the tube model in... 

We particularly like these three flow or reacting patterns because they are easy to treat (it is simple to find their performance equations) and because one of them often is the best pattern possible (it will give the most of whatever it is we want). Later we will consider recycle reactors, staged reactors, and other flow pattern combinations, as well as deviations of real reactors from these ideals. 

Section IV, D, also belongs in this category. While the equations of Nicklin et al. for void fractions are firmly based on theory for slug or bubble flow, the extension of these equations to other flow patterns, or to void-fractions above 0.8, is empirical. 

It would seem that no theoretical calculations have been made for shapes other than spheroids. In addition, no experimental measurements have been reported for shapes other than spheres or circular cylinders in creeping flow. Equation (4-60) is useful for cases in which Pe is small. 

Regardless of what other conservation equations may be appropriate, a bulk-fluid mass-conservation equation is invariably required in any fluid-flow situation. When N is the mass m, the associated intensive variable (extensive variable per unit mass) is r) = 1. That is, r) is the mass per unit mass is unity. For the circumstances considered here, there is no mass created or destroyed within a control volume. Chemical reaction, for example, may produce or consume individual species, but overall no mass is created or destroyed. Furthermore the only way that net mass can be transported across the control surfaces is by convection. While individual species may diffuse across the control surfaces by molecular actions, there can be no net transport by such processes. This fact will be developed in much depth in subsequent sections where mass transport is discussed. 

A vorticity-transport equation can be derived by taking the taking the vector curl of the full Navier-Stokes equations. For incompressible flows with constant viscosity, the vorticity-transport equation can be expressed in a form that is quite similar to the other transport equations. Begin with the full Navier-Stokes equations, which for constant viscosity can be written in compact vector form as (Eq. 3.61)... 

In this form one sees an analogy in the vorticity equation to the other transport equations— a substantial-derivative description of advective transport, a Laplacian describing the diffusive transport, and possibly a source term. It is interesting to observe that the vorticity equation does not involve the pressure. Since pressure always exerts a normal force that acts through the center of mass of a fluid packet (control volume), it cannot alter the rotation rate of the fluid. That is, pressure variations cannot cause a change in the vorticity of a flow field. 

Stagnation flows can be viewed either as a similarity reduction of the flow equations in a boundary-layer region or as an exact reduction of the Navier-Stokes equations under certain simplifying assumptions. Depending on the circumstances of a particular problem of interest, one or the other view may be more natural. In either case, the same governing equations emerge, with the differences being in boundary conditions. The alternatives are explored in later sections, where particular problems and boundary conditions are discussed. 

With the brief discussion of index, it is now possible to identify and compare some aspects of the high-index behavior of the constant-pressure and the compressible stagnation-flow equations. To understand the structure of the DAE system, it is first necessary to identify all variables that are not time differentiated (i.e., the x vector). In the constant-pressure formulation, neither the axial velocity u nor the pressure curvature A has time derivatives. By introducing the axial momentum equation, the compressible formulation introduces du/dt. To be of value in reducing the index, however, the momentum equation must be coupled to the other equations. The coupling is accomplished through pressure, which is included as a dependent variable. The variable A is not time differentiated in either formulation. 

In order for a process to be controllable by machine, it must represented by a mathematical model. Ideally, each element of a dynamic process, for example, a reflux drum or an individual tray of a fractionator, is represented by differential equations based on material and energy balances, transfer rates, stage efficiencies, phase equilibrium relations, etc., as well as the parameters of sensing devices, control valves, and control instruments. The process as a whole then is equivalent to a system of ordinary and partial differential equations involving certain independent and dependent variables. When the values of the independent variables are specified or measured, corresponding values of the others are found by computation, and the information is transmitted to the control instruments. For example, if the temperature, composition, and flow rate of the feed to a fractionator are perturbed, the computer will determine the other flows and the heat balance required to maintain constant overhead purity. Economic factors also can be incorporated in process models then the computer can be made to optimize the operation continually. 

When motion of the fluid consists of only small fluctuations about a state of near-rest, Lhe continuum equations are linearized by neglecting nonlinear terms and they become lhe equalions of acoustics. A large variety of fluid motions are described as sound waves when the small-motion or acoustic description can be used, the principle of superposition is valid. This powerful principle allows addition of simple simultaneous motions to represent a more complex motion, such as the sound reaching lhe audience from the instruments of a symphony orchestra. The superposition principle does not apply to large-scale (nonacoustical) motions, and the subject of fluid dynamics (in distinction from acoustics) treats nonlinear flows. i.e.. those that cannot be described as superpositions of other flows. 

Abstract Unsteady liquid flow and chemical reaction characterize hydrodynamic dispersion in soils and other porous materials and flow equations are complicated by the need to account for advection of the solute with the water, and competitive adsorption of solute components. Advection of the water and adsorbed species with the solid phase in swelling systems is an additional complication. Computers facilitate solution of these equations but it is often physically more revealing when we discriminate between flow of the solute with and relative to, the water and the flow of solution with and relative to, the solid phase. Spacelike coordinates that satisfy material balance of the water, or of the solid, achieve this separation. Advection terms are implicit in the space-like coordinate and the flow equations are focused on solute movement relative to the water and water relative to soil solid. This paper illustrates some of these issues. 

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