Combined momentum flow

A combined momentum flow tensor is defined when a convective effect exits... 

Equation (2.16) consists of two contributions the molecular momentum flow tensor, it, and the convective momentum flow tensor, pvv. The term p8 represents the pressure effect, while the contribution t, for a Newtonian fluid, is related to the velocity gradient linearly through the viscosity. The convective momentum flow tensor pw contains the density and the products of the velocity components. A component of the combined momentum flow tensor of x-momentum across a surface normal to the x-direction is... 

The numerical solution of the energy balance and momentum balance equations can be combined with flow equations to describe heat transfer and chemical reactions in flow situations. The simulation results can be in various forms numerical, graphical, or pictorial. CFD codes are structured around the numerical algorithms and, to provide easy assess to their solving power, CFD commercial packages incorporate user interfaces to input parameters and observe the results. CFD... 

The three-dimensional, fully parabolic flow approximation for momentum and heat- and mass-transfer equations has been used to demonstrate the occurrence of these longitudinal roll cells and their effect on growth rate uniformity in Si CVD from SiH4 (87) and GaAs MOCVD from Ga(CH3)3 and AsH3 (189). However, gas expansion in the entrance zone combined with flow obstructions, such as a steeply sloped susceptor, can also produce... 

Of course, in RTM process modelling one must combine the above kinetic and chemoviscosity models into mass, momentum and energy balances within a flow simulation. Specifically, the momentum balance must combine any flows induced by pressure and any flows into porous media (as characterized by Darcy s law). Simple onedimensional RTM flow modelling and two- and three-dimensional RTM simulations have been summarized by Rudd et al. (1997) and show the importance of kinetic, rheological and permeability coefficients to the simulation of pressure-profile and flow-front predictions. 

On pages 92 to 98 tliie distribution of velocity and its accompanying momentum flux in a flowing stream in turbulent flow through a pipe was described. Three rather ill-defined zones in the cross section of the pipe were identified. In the first, immediately next to the wall, eddies are rare, and momentum flow occurs almost entirely by viscosity in the second, a mixed regime of combined viscous and turbulent momentum transfer occurs in the main part of the stream, which occupies the bulk of the cross section of the stream, only the momentum flow generated by the Reynolds stresses of turbulent flow is important. The three zones are called the viscous sublayer, the buffer zone, and the turbulent core, respectively. 

In this hydrodynamic entrance region, the apparent friction factor /app is employed to incorporate the combined effects of wall shear and the change in momentum flow rate due to the developing velocity profile. Based on the total axial pressure drop from the duct inlet (x = 0) to the point of interest, the apparent friction factor is defined as follows ... 

The stability analysis is concerned with the stability of the interface between the phases, presumably smooth, under fully developed stratified flow conditions. In this case the LHS of the combined momentum equation for the two-phases. Equation 7 vanishes ... 

The relation of the general dispersion Equations 34 to dynamic waves is derived here by recalling that a pure dynamic wave occurs whenever the net force on the flowing fluids is produced only by concentration gradients (and is independent of the insitu concentration, Wallis [74]). In this case, the quasi-steady shear stress terms on the rhs of the combined momentum equation, which are functions of the insitu concentration, are ignored, whereby AF is considered as identically zero. However, the dynamic interfacial shear stress term, which is proportional to the concentration gradient, evolves from the Reynolds shear stresses in the turbulent field and is retained. The general dispersion Equation 34, with V = 0, becomes ... 

What we have are two seemingly incompatible ideas. One is that the behavior of an electron is described by a wavefunction. The other is that the uncertainty principle limits the certainty with which one can measure various combinations of observables, like position and momentum. Flow can we discuss the motion of electrons in any detail at all ... 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

The improvement of the flow distribution by increasing the value of DJD results from the decrease of momentum gain in the combining header. 

Flow and Performance Calculations. Electro dynamic equations are usehil when local gas conditions (, a, B) are known. In order to describe the behavior of the dow as a whole, however, it is necessary to combine these equations with the appropriate dow conservation and state equations. These last are the mass, momentum, and energy conservation equations, an equation of state for the working duid, an expression for the electrical conductivity, and the generalized Ohm s law. 

For turbulent flow, with roughly uniform distribution, assuming a constant fricdion factor, the combined effect of friction and inerrtal (momentum) pressure recovery is given by... 

The discussion of the interaction of air jets supplied at some angle to each other shows that application of the method of superposition of the interacting jets momentums and surplus heat to predict velocity and temperatures in the combined flow results in inaccuracy when two unequal jets are supplied at a right angle. A different approach was undertaken in the studies of interaction of the main stream with vertical directing jets. Ti i... 

In addition to momentum, both heat and mass can be transferred either by molecular diffusion alone or by molecular diffusion combined with eddy diffusion. Because the effects of eddy diffusion are generally far greater than those of the molecular diffusion, the main resistance to transfer will lie in the regions where only molecular diffusion is occurring. Thus the main resistance to the flow of heat or mass to a surface lies within the laminar sub-layer. It is shown in Chapter 11 that the thickness of the laminar sub-layer is almost inversely proportional to the Reynolds number for fully developed turbulent flow in a pipe. Thus the heat and mass transfer coefficients are much higher at high Reynolds numbers. 

Numerical computations of reacting flows are often difficult owing to the different time-scales involved and the highly non-linear dependence of the reaction rate on concentrations and temperature. The solution of the species concentration equations in combination with the momentum and the enthalpy equation generally requires an iterative procedure such as the one outlined in Section 1.3.4. A rough sketch of the numerical structure of a stationary reacting-flow problem is given as... 

CFD may be loosely thought of as computational methods applied to the study of quantities that flow. This would include both methods that solve differential equations and finite automata methods that simulate the motion of fluid particles. We shall include both of these in our discussions of the applications of CFD to packed-tube simulation in Sections III and IV. For our purposes in the present section, we consider CFD to imply the numerical solution of the Navier-Stokes momentum equations and the energy and species balances. The differential forms of these balances are solved over a large number of control volumes. These small control volumes when properly combined form the entire flow geometry. The size and number of control volumes (mesh density) are user determined and together with the chosen discretization will influence the accuracy of the solutions. After boundary conditions have been implemented, the flow and energy balances are solved numerically an iteration process decreases the error in the solution until a satisfactory result has been reached. 

The CRE approach for modeling chemical reactors is based on mole and energy balances, chemical rate laws, and idealized flow models.2 The latter are usually constructed (Wen and Fan 1975) using some combination of plug-flow reactors (PFRs) and continuous-stirred-tank reactors (CSTRs). (We review both types of reactors below.) The CRE approach thus avoids solving a detailed flow model based on the momentum balance equation. However, this simplification comes at the cost of introducing unknown model parameters to describe the flow rates between various sub-regions inside the reactor. The choice of a particular model is far from unique,3 but can result in very different predictions for product yields with complex chemistry. 

Theoretical investigations of the problem were carried out on the base of the mathematical model, combining both deterministic and stochastic approaches to turbulent combustion of organic dust-air mixtures modeling. To simulate the gas-phase flow, the k-e model is used with account of mass, momentum, and energy fluxes from the particles phase. The equations of motion for particles take into account random turbulent pulsations in the gas flow. The mean characteristics of those pulsations and the probability distribution functions are determined with the help of solutions obtained within the frame of the k-e model. 

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