Fluid flow equations

As with the two-dimensional workbench problem, the numerical solution of this problem can be found by solving the full turbulent fluid flow equations using the methods described in Chapter 13. 

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. 

Assuming steady-state, frictionless (due to the short length of the nozzles) drilling fluid flow, Equation 4-102 is written... 

For a fan the changes in pressure and temperature are small enough that the incompressible fluid flow equations given for pumps may be used. In fact, for most gas systems, if the pressure drop is less than 40% of absolute upstream pressure the fluids can be treated as incompressible.30... 

In order to determine the required reactor volume one must relate the temperature (and thus k) and the local pressure P to the fraction conversion using an energy balance and conventional fluid flow equations. 

If the volume change of the gas is neglected, and hence accelerative effects are also neglected, then for single-phase fluid flow. Equation (3) becomes... 

The porous media model (8) is used to simulate flows through the biomass in a packed bed where biomass is the porous media. In this model, a momentum source term S, is added to the standard fluid flow equation ... 

The hydrodynamic conditions influence the concentration distribution explicity through the velocity term present in the convective diffusion equation. For certain well-defined systems the fluid flow equations have been solved, but for many systems, especially those with turbulent flow, explicit solutions have not been obtained. Consequently, approximate techniques must frequently be used in treating mass transfer. 

In addition to the cross-sectional area and shape of the hole, the flow of a liquid is controlled by the available pressure (vapor space pad pressures, pump head, and liquid head) and the pressure drop caused by the presence of fittings and valves in the fluid flow path. Flow rates can be estimated using standard fluid flow equations when the opening is properly characterized. 

In problems of heat convection, the most complex equations to solve are the fluid flow equations. Often times, the governing equations for the fluid flow are the Navier-Stokes equations. It is useful, therefore, to study a model equation that has similar characteristics to the Navier-Stokes equations. This model equation has to be time-dependent and include both convection and diffusion terms. The viscous Burgers equation is an appropriate model equation. In the first few sections of this chapter, several important numerical schemes for the Burgers equation will be discussed. A simple physical heat convection problem is solved as a demonstration. 

The first feature concerns the structure of the terms in Eq. (A.33). Each term can be viewed as the product between a generalized (driving) force Xk and a generalized flux Jk. The first term in Eq. (A.33) has the temperature gradient as a force and heat transfer rate as a flux. The second term has a composition gradient and a mass transfer flux. The third term has affinity as a force (indicative of the distance away from chemical equilibrium) and reaction rate as the flux. The fourth term is already a composite related to pressure drop and fluid flow. Equation (A.33) can therefore be written compactly as... 

The Froude number described above is frequently used for the description of radial and axial flotvs in liquid media when the pressure difference along a mixing device is important. When cavitation problems are present, the dimensionless group (Pj — p,) /pw - called the Euler number - is commonly used. Here p is the liquid vapour saturation pressure and p is a reference pressure. This number is named after the Swiss mathematician Leonhard Euler (1707-1783) who performed the pioneering work showing the relationship between pressure and flow (basic static fluid equations and ideal fluid flow equations, which are recognized as Euler equations). 

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. 

FORTRAN computer program that predicts the species, temperature, and velocity profiles in two-dimensional (planar or axisymmetric) channels. The model uses the boundary layer approximations for the fluid flow equations, coupled to gas-phase and surface species continuity equations. The program runs in conjunction with CHEMKIN preprocessors (CHEMKIN, SURFACE CHEMKIN, and TRAN-FIT) for the gas-phase and surface chemical reaction mechanisms and transport properties. The finite difference representation of the defining equations forms a set of differential algebraic equations which are solved using the computer program DASSL (dassal.f, L. R. Petzold, Sandia National Laboratories Report, SAND 82-8637, 1982). 

Issa, R. I. (1986) Solution of the Implicitly Discretised Fluid Flow Equations by Operator-Splitting, Journal of Computational Physics, Vol. 62(1), pp. 40-65. 

Issa, R.I. (1986), Solution of the implicitly discretized fluid flow equations by operator splitting, J. Computat. Phys., 62, 40-65. 

Introductory work has however been made to analyze the mathematical properties of the equations. In their pioneer work Gidaspow [84] and Ly-ckowski et al [144] consider the ID, incompressible, in-viscid two-fluid flow equations with no added mass or lift effects given by ... 

Hutchinson BR, Galpin PF, Raithby GD (1988) Apphcation of Additive Correction Multigrid to the Coupled Fluid Flow Equations. Numerical Heat Transfer 13 133-147... 

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... 

For turbulent fluid flow, equation 28 is used, and the incremental... 

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