The Convection Model for Laminar Flow

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. 

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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. 

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 ... 

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... 

Process models based on the convective diffusion equation have an inherent level of micromixing. Examples of such models include laminar flow with or without radial diffusion and the axial dispersion model. The models can be used to predict a RTD. With that distribution comes a specific extent of micromixing, and the model contains no adjustable parameter to vary the extent of micromixing that does not also vary the RTD. Predictions from such models are used directly without explicit concern for micromixing. The RTD corresponding to the models could be associated with a range of micromixing, but this would be inconsistent with the physical model. 

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... 

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. 

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. 

The use of thermal conductivity, heat capacity and rheological properties for [C4mim][NTf2] was also shown by Chen et al. [123] to correlate with Shah s equation for forced convective heat transfer in the laminar flow regime, indicating that knowledge of these parameters can successfully be used to model heat transfer behavior of ionic liquid systems at the larger scale. 

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