Modeling laminar flow

Catalytic Non-permselective Membrane Multiphase Reactor (CNMMR) Model - Laminar Flow Liquid Stream... 

SEGREGATED FLOW MODEL ( LAMINAR FLOW MODEL ( n ) ... 

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). 

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 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. 

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 ... 

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. 

Shah RK, London AL (1978) Laminar flow forced convection in ducts. Academic, New York Sher I, Hetsroni G (2002) An analytical model for nucleate pool boiling with surfactant additives. Int J Multiphase Elow 28 699-706... 

The quasi-one-dimensional model of laminar flow in a heated capillary is presented. In the frame of this model the effect of channel size, initial temperature of the working fluid, wall heat flux and gravity on two-phase capillary flow is studied. It is shown that hydrodynamical and thermal characteristics of laminar flow in a heated capillary are determined by the physical properties of the liquid and its vapor, as well as the heat flux on the wall. 

Peles el al. (2000) elaborated on a quasi-one-dimensional model of two-phase laminar flow in a heated capillary slot due to liquid evaporation from the meniscus. Subsequently this model was used for analysis of steady and unsteady flow in heated micro-channels (Peles et al. 2001 Yarin et al. 2002), as well as the study of the onset of flow instability in heated capillary flow (Hetsroni et al. 2004). 

Below we consider a quasi-one-dimensional model of flow and heat transfer in a heated capillary, with hydrodynamic, thermal and capillarity effects. We estimate the influence of heat transfer on steady-state laminar flow in a heated capillary, on the shape of the interface surface and the velocity and temperature distribution along the capillary axis. 

Weislogel MM, Lichter S (1998) Capillary flow in an interior corner. 1 Eluid Mech 373 349-378 Wu PY, Little WA (1984) Measurement of the heat transfer characteristics of gas flow a fine channels heat exchangers used for microminiature refrigerators. Cryogenics 24 415 20 Xu X, Carey VP (1990) Film evaporation from a micro-grooved surface an approximate heat transfer model and its comparison with experimental data. J Thermophys 4(4) 512-520 Yarin LP, Ekelchik LA, Hetsroni G (2002) Two-phase laminar flow in a heated micro-channels. Int J Multiphase Flow 28 1589-1616... 

Peles et al. (1998) and Khrustalev and Faghri (1996) considered two-phase laminar flow in a heated micro-channel with distinct evaporating meniscus in the frame of quasi-one-dimensional and two-dimensional models. 

The capillary flow with distinct evaporative meniscus is described in the frame of the quasi-dimensional model. The effect of heat flux and capillary pressure oscillations on the stability of laminar flow at small and moderate Peclet number is estimated. It is shown that the stable stationary flow with fixed meniscus position occurs at low wall heat fluxes (Pe -Cl), whereas at high wall heat fluxes Pe > 1, the exponential increase of small disturbances takes place. The latter leads to the transition from stable stationary to an unstable regime of flow with oscillating meniscus. 

Chapter 3 introduced the basic concepts of scaleup for tubular reactors. The theory developed in this chapter allows scaleup of laminar flow reactors on a more substantive basis. Model-based scaleup supposes that the reactor is reasonably well understood at the pilot scale and that a model of the proposed plant-scale reactor predicts performance that is acceptable, although possibly worse than that achieved in the pilot reactor. So be it. If you trust the model, go for it. The alternative is blind scaleup, where the pilot reactor produces good product and where the scaleup is based on general principles and high hopes. There are situations where blind scaleup is the best choice based on business considerations but given your druthers, go for model-based scaleup. 

The temperature counterpart of Q>aVR ccj-F/R and if ccj-F/R is low enough, then the reactor will be adiabatic. Since aj 3>a, the situation of an adiabatic, laminar flow reactor is rare. Should it occur, then T i, will be the same in the small and large reactors, and blind scaleup is possible. More commonly, ari/R wiU be so large that radial diffusion of heat will be significant in the small reactor. The extent of radial diffusion will lessen upon scaleup, leading to the possibility of thermal runaway. If model-based scaleup predicts a reasonable outcome, go for it. Otherwise, consider scaling in series or parallel. 

McLaughlin, H. S., Mallikarjun, R., and Nauman, E. B., The Effect of Radial Velocities on Laminar Flow, Tubular Reactor Models, AIChE J., 32, 419-425 (1986). 

The models of Chapter 9 contain at least one empirical parameter. This parameter is used to account for complex flow fields that are not deterministic, time-invariant, and calculable. We are specifically concerned with packed-bed reactors, turbulent-flow reactors, and static mixers (also known as motionless mixers). We begin with packed-bed reactors because they are ubiquitous within the petrochemical industry and because their mathematical treatment closely parallels that of the laminar flow reactors in Chapter 8. 

Turbulent flow reactors are modeled quite differently from laminar flow reactors. In a turbulent flow field, nonzero velocity components exist in all three coordinate directions, and they fluctuate with time. Statistical methods must be used to obtain time average values for the various components and to characterize the instantaneous fluctuations about these averages. We divide the velocity into time average and fluctuating parts ... 

Chapters 8 and Section 9.1 gave preferred models for laminar flow and packed-bed reactors. The axial dispersion model can also be used for these reactors but is generally less accurate. Proper roles for the axial dispersion model are the following. 

Laminar Pipeline Flows. The axial dispersion model can be used for laminar flow reactors if the reactor is so long that At/R > 0.125. With this high value for the initial radial position of a molecule becomes unimportant. 

The molecule diffuses across the tube and samples many streamlines, some with high velocity and some with low velocity, during its stay in the reactor. It will travel with an average velocity near u and will emerge from the long reactor with a residence time close to F. The axial dispersion model is a reasonable approximation for overall dispersion in a long, laminar flow reactor. The appropriate value for D is known from theory ... 

When two or more phases are present, it is rarely possible to design a reactor on a strictly first-principles basis. Rather than starting with the mass, energy, and momentum transport equations, as was done for the laminar flow systems in Chapter 8, we tend to use simplified flow models with empirical correlations for mass transfer coefficients and interfacial areas. The approach is conceptually similar to that used for friction factors and heat transfer coefficients in turbulent flow systems. It usually provides an adequate basis for design and scaleup, although extra care must be taken that the correlations are appropriate. 

The dimensionless variance has been used extensively, perhaps excessively, to characterize mixing. For piston flow, a = 0 and for a CSTR, a = l. Most turbulent flow systems have dimensionless variances that lie between zero and 1, and cr can then be used to fit a variety of residence time models as will be discussed in Section 15.2. The dimensionless variance is generally unsatisfactory for characterizing laminar flows where > 1 is normal in liquid systems. 

In the absence of diffusion, all hydrodynamic models show infinite variances. This is a consequence of the zero-slip condition of hydrodynamics that forces Vz = 0 at the walls of a vessel. In real systems, molecular diffusion will ultimately remove molecules from the stagnant regions near walls. For real systems, W t) will asymptotically approach an exponential distribution and will have finite moments of all orders. However, molecular diffusivities are low for liquids, and may be large indeed. This fact suggests the general inappropriateness of using to characterize the residence time distribution in a laminar flow system. Turbulent flow is less of a problem due to eddy diffusion that typically results in an exponentially decreasing tail at fairly low multiples of the mean residence time. 

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