Laminar convective models

For laminar flow in short tubes or laminar flow of viscous materials these models may not apply, and it may be that the parabolic velocity profile is the main cause of deviation from plug flow. We treat this situation, called the pure convection model, in Chapter 15. 

When a tube or pipe is long enough and the fluid is not very viscous, then the dispersion or tanks-in-series model can be used to represent the flow in these vessels. For a viscous fluid, one has laminar flow with its characteristic parabolic velocity profile. Also, because of the high viscosity there is but slight radial diffusion between faster and slower fluid elements. In the extreme we have the pure convection model. This assumes that each element of fluid slides past its neighbor with no interaction by molecular diffusion. Thus the spread in residence times is caused only by velocity variations. This flow is shown in Fig. 15.1. This chapter deals with this model. 

The important result is that the two-mode models for a turbulent flow tubular reactor are the same as those for laminar flow tubular reactors. The two-mode axial dispersion model for turbulent flow tubular reactors is again given by Eqs. (130)—(134), while the two-mode convection model for the same is given by Eqs. (137)—(139), where the reaction rate term r((c)) is replaced by the Reynolds-averaged reaction rate term rav((c)). The local mixing time for turbulent flows is given by... 

Now the dimensionless ratio hu/K is known as the Nusselt number Nu(ro), and for systems with convection it takes values of about 5 if the flow is not turbulent. (In the absence of convection /i, the heat transfer at the walls is determined by the temperature gradient at the walls, which in turn is proportional to K/tq,) It is interesting to note that the simple model which permits laminar convection gives values of 8c of about the order of 0, which is reasonably close to the value of 3.32 calculated for pure conduction. 

Symmetrical growth and collapse curves of the sort pictured herein are difficult to obtain by a laminar-flow heat-convection model. This is because the heat equation is not symmetric with respect to time reversal. On the other hand, such symmetry is observed in the growth and collapse of cavitation bubbles, and indeed, is implied by the extended Rayleigh equation (Bll) ... 

The model most firequendy used for describing flow through pores is based on the well-known Hagen—PoiseuiUe equation for laminar convective flow through straight capillaries with... 

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... 

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... 

S Neervannan, LS Dias, MZ Southard, VJ Stella. A convective-diffusion model for dissolution of two non-interacting drug mixtures from co-compressed slabs under laminar hydrodynamic conditions. Pharm Res 11 1228-1295, 1994. 

When fluid is pumped through a cell such as that shown in Fig. 12, transport of dissolved molecules from the cell inlet to the IRE by convection and diffusion is an important issue. The ATR method probes only the volume just above the IRE, which is well within the stagnant boundary layer where diffusion prevails. Figure 13 shows this situation schematically for a diffusion model and a convection-diffusion model (65). The former model assumes that a stagnant boundary layer exists above the IRE, within which mass transport occurs solely by diffusion and that there are no concentration gradients in the convection flow. A more realistic model of the flow-through cell accounts for both convection and diffusion. As a consequence of the relatively narrow gap between the cell walls, the convection leads to a laminar flow profile and consequently to concentration gradients between the cell walls. 

In laminar flows through porous media, the pressure is proportional to velocity and C2 can be taken as zero. Ignoring convective acceleration and diffusion, the porous media model can be changed into Darcy s Law ... 

In order to. illustrate how natural convection in a vertical channel can be analyzed, attention will be given to flow through a wide rectangular channel, i.e., to laminar, two-dimensional flow in a plane channel as shown in Fig. 8.15. This type of flow is a good model of a number of flows of practical importance. 

We can extend the hyperbolic model to cases in which the solute diffuses in more than one phase. A common case is that of a monolith channel in which the flow is laminar and the walls are coated with a washcoat layer into which the solute can diffuse (Fig. 4). The complete model for a non-reacting solute here is described by the convection-diffusion equation for the fluid phase coupled with the unsteady-state diffusion equation in the solid phase with continuity of concentration and flux at the fluid-solid interface. Transverse averaging of such a model gives the following hyperbolic model for the cup-mixing concentration in the fluid phase ... 

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