Laminar modelling

The quasi-steady laminar model is now employed to describe the heat transfer near the wall. Note that while the shear stress at the wall can be related easily to the pressure drop for the flow in a tube, it is more difficult to establish a relation between these two quantities for a packed or fluidized bed. However, while for the flow in a tube the dissipated energy is not uniform over the section... 

Assuming that the motion of the liquid is turbulent and using the quasisteady laminar model to describe turbulence, the heat transfer coefficient is given by the expression... 

Recently, Hanratty presented a comprehensive review of the attempts to account for the interfacial waviness in modelling the interfacial shear stress for the stability analysis of gas-liquid two-phase flows [53]. Basically, the approach taken was to implement the models obtained for the surface stresses in air flow over a solid wavy boundary as a boundary condition for the momentum equation of the liquid layer over its it mobile wavy interface. Craik [98] adopted the interfacial stresses components which evolve from the quasi-laminar model by Benjamin [84]. Jurman and McCready [99], Jurman et al. [100], and Asali and Hanratty [101] used correlated experimental values of shear stress components (phase and amplitude) based on turbulent models which consider relaxation effects in the Van Driest mixing length. Since the characteristics of the predicted surface stresses are dependent on the wave number, Asali and Hanratty picked the phase and amplitude values which correspond to the wave lengths of the capillary ripples observed in their experiments of thin liquid layers sheared by high gas velocities [101]. It was shown that the growth of these ripples is controlled by the interfacial shear stress component in phase with the wave slope. 

Garcaa-Sema, J., Garcia-Verdugo, E., Hyde, J., et al. (2007). Modelling residence time distribution in chemical reactors A novel generalised n-laminar model Application to supercritical CO2 and subcritical water tubular reactors, J. Supercrit. Fluid, 41, pp. 82—91. 

The mesh geometry file is imported into Fluent 3D solver the imported grid is checked and scaled to actual units of measurements. Segregated solver (default) was selected for the incompressible resin flow through fabric dining RTM process (low velocities of the fluid-low Reynolds number). order implicit, physical velocity porous formulation for 3D unsteady flow was opted in the model-solver options. Viscous laminar model was selected for physical model (laminar flow). 

The solution flow is nomially maintained under laminar conditions and the velocity profile across the chaimel is therefore parabolic with a maximum velocity occurring at the chaimel centre. Thanks to the well defined hydrodynamic flow regime and to the accurately detemiinable dimensions of the cell, the system lends itself well to theoretical modelling. The convective-diffiision equation for mass transport within the rectangular duct may be described by... 

Modelling plasma chemical systems is a complex task, because these system are far from thennodynamical equilibrium. A complete model includes the external electric circuit, the various physical volume and surface reactions, the space charges and the internal electric fields, the electron kinetics, the homogeneous chemical reactions in the plasma volume as well as the heterogeneous reactions at the walls or electrodes. These reactions are initiated primarily by the electrons. In most cases, plasma chemical reactors work with a flowing gas so that the flow conditions, laminar or turbulent, must be taken into account. As discussed before, the electron gas is not in thennodynamic equilibrium... 

In this chapter the general equations of laminar, non-Newtonian, non-isothermal, incompressible flow, commonly used to model polymer processing operations, are presented. Throughout this chapter, for the simplicity of presentation, vector notations are used and all of the equations are given in a fixed (stationary or Eulerian) coordinate system. 

Petera,. 1. and Nassehi, V., 1993. Flow modelling using isoparametric Hermite elements. In Taylor C. (ed.), Numerical Methods in Laminar and Turbulent Flow, Vol. VIII, Part 2, Pineridge Press, Swansea. 

Hannart, B. and Hoplinger, E.J., 1998. Laminar flow in a rectangular diffuser near Hele-Sliaw conditions - a two dinien.sioiial numerical simulation. In Bush, A. W., Lewis, B. A. and Warren, M.D. (eds), Flow Modelling in Industrial Processes, cli. 9, Ellis Horwood, Chichester, pp. 110-118. 

The phenomenon of concentration polarization, which is observed frequently in membrane separation processes, can be described in mathematical terms, as shown in Figure 30 (71). The usual model, which is weU founded in fluid hydrodynamics, assumes the bulk solution to be turbulent, but adjacent to the membrane surface there exists a stagnant laminar boundary layer of thickness (5) typically 50—200 p.m, in which there is no turbulent mixing. The concentration of the macromolecules in the bulk solution concentration is c,. and the concentration of macromolecules at the membrane surface is c. 

The criterion of maintaining equal power per unit volume has been commonly used for dupHcating dispersion qualities on the two scales of mixing. However, this criterion would be conservative if only dispersion homogeneity is desired. The scale-up criterion based on laminar shear mechanism (9) consists of constant > typical for suspension polymerization. The turbulence model gives constant tip speed %ND for scale-up. 

Thickness of the laminar layer is deterrnined both by the need to reproduce fine detail in the object and by the penetration depth of the actinic laser light into the monomer bath (21,76). There is thus a trade-off between precision of detail in the model and time required for stereohthography, ie, the number of layers that have to be written, and an optimum Light-absorbing initiator concentration in the monomer bath corresponding to the chosen layer thickness. Titanocene-based initiators, eg, bis-perfluorophenyltitanocene has been recommended for this apphcation (77). Mechanistic aspects of the photochemistry of titanocenes and mechanisms of photoinitiation have been reviewed (76). 

Computer simulation of the reactor kinetic hydrodynamic and transport characteristics reduces dependence on phenomenological representations and idealized models and provides visual representations of reactor performance. Modem quantitative representations of laminar and turbulent flows are combined with finite difference algorithms and other advanced mathematical methods to solve coupled nonlinear differential equations. The speed and reduced cost of computation, and the increased cost of laboratory experimentation, make the former increasingly usehil. 

The discussion of laminar diffusion flame theory addresses both the gaseous diffusion flames and the single-drop evaporation and combustion, as there are some similarities between gaseous and Hquid diffusion flame theories (2). A frequentiy used model of diffusion flames has been developed (34), and despite some of the restrictive assumptions of the model, it gives a good description of diffusion flame behavior. 

In addition to the Burke and Schumann model (34) and the Displacement Distance theory, a comprehensive laminar diffusion flame theory can be written using the equations of conservation of species, energy, and momentum, including diffusion, heat transfer, and chemical reaction. 

The stagnant-film model discussed previously assumes a steady state in which the local flux across each element of area is constant i.e., there is no accumulation of the diffusing species within the film. Higbie [Trans. Am. Jn.st. Chem. Eng., 31,365 (1935)] pointed out that industrial contactors often operate with repeated brief contacts between phases in which the contact times are too short for the steady state to be achieved. For example, Higbie advanced the theory that in a packed tower the liquid flows across each packing piece in laminar flow and is remixed at the points of discontinuity between the packing elements. Thus, a fresh liquid surface is formed at the top of each piece, and as it moves downward, it absorbs gas at a decreasing rate until it is mixed at the next discontinuity. This is the basis of penetration theoiy. 

David W. Taylor Model Basin, Washington, September 1953 Jackson, loc. cit. Valentin, op. cit.. Chap. 2 Soo, op. cit.. Chap. 3 Calderbank, loc. cit., p. CE220 and Levich, op. cit.. Chap. 8). A comprehensive and apparently accurate predictive method has been publisned [Jami-alahamadi et al., Trans ICE, 72, part A, 119-122 (1994)]. Small bubbles (below 0.2 mm in diameter) are essentially rigid spheres and rise at terminal velocities that place them clearly in the laminar-flow region hence their rising velocity may be calculated from Stokes law. As bubble size increases to about 2 mm, the spherical shape is retained, and the Reynolds number is still sufficiently small (<10) that Stokes law should be nearly obeyed. 

A model of a reaction process is a set of data and equations that is believed to represent the performance of a specific vessel configuration (mixed, plug flow, laminar, dispersed, and so on). The equations include the stoichiometric relations, rate equations, heat and material balances, and auxihaiy relations such as those of mass transfer, pressure variation, contac ting efficiency, residence time distribution, and so on. The data describe physical and thermodynamic properties and, in the ultimate analysis, economic factors. 

Different processes like eddy turbulence, bottom current, stagnation of flows, and storm-water events can be simulated, using either laminar or turbulent flow model for simulation. All processes are displayed in real-time graphical mode (history, contour graph, surface, etc.) you can also record them to data files. Thanks to innovative sparse matrix technology, calculation process is fast and stable a large number of layers in vertical and horizontal directions can be used, as well as a small time step. You can hunt for these on the Web. 

In a steady-state situation when gas flows through a porous material at a low velocity (laminar flow), the following empirical formula, Darcy s model, is valid ... 

In displacement ventilation, there are regions with very low turbulence, and the flow can even be laminar. Hence it is important to use a turbulence model which can handle these regions. The k-f model gives rise to large numerical problems in regions of low turbulence. The reason is thar as k goes to zero, the destruction term in the e equation goes to infinity. The c equation is... 

The general form of the population balance including aggregation and rupture terms was solved numerically to model the experimental particle size distributions. While excellent agreement was obtained using semi-empirical two-particle aggregation and disruption models (see Figure 6.15), PSD predictions of theoretical models based on laminar and turbulent flow considerations... 

Over the years, this concept was refined in several ways. A scale dependency was modeled by the introduction of scale-dependent quenching of combustion. The first stage of the process was simulated by quasi-laminar flame propagation. In addition, three-dimensional versions of the code were developed (Hjertager 1985 Bakke 1986 Bakke and Hjertager 1987). Satisfactory agreement with experimental data was obtained. 

Bakke, J. R., and B. H. Hjertager. 1986a. Quasi-laminar/turbulent combustion modeling, real cloud generation and boundary conditions in the FLACS-ICE code. CMI No. 865402-2. Chr. Michelsen Institute, 1986. Also in Bakke s Ph.D. thesis Numerical simulation of gas explosions in two-dimensional geometries. University of Bergen, Bergen, 1986. 

What will be the pressure drop, when the suspension is flowing under laminar conditions in a pipe 200 m long and 40 mm diameter, when the centre line velocity is 1 m/s, according to the power-law model Calculate the centre-line velocity for this pressure drop for the Bingham-plastic model. 

Equation 5.2, with the modified parameter X used in place of X, may be used for laminar flow of shear-thinning fluids whose behaviour can be described by the power-taw model. 

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