Mechanical laminar flow

Eddy diffusion as a transport mechanism dominates turbulent flow at a planar electrode ia a duct. Close to the electrode, however, transport is by diffusion across a laminar sublayer. Because this sublayer is much thinner than the layer under laminar flow, higher mass-transfer rates under turbulent conditions result. Assuming an essentially constant reactant concentration, the limiting current under turbulent flow is expected to be iadependent of distance ia the direction of electrolyte flow. 

For laminar flow of power law fluids in channels of noncircular cross section, see Schecter AIChE J., 7, 445 48 [1961]), Wheeler and Wissler (AJChE J., 11, 207-212 [1965]), Bird, Armstrong, and Hassager Dynamics of Polymeric Liquids, vol. 1 Fluid Mechanics, Wiley, New York, 1977), and Skelland Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967). 

Concentration and temperature differences are reduced by bulk flow or circulation in a vessel. Fluid regions of different composition or temperature are reduced in thickness by bulk motion in which velocity gradients exist. This process is called bulk diffusion or Taylor diffusion (Brodkey, in Uhl and Gray, op. cit., vol. 1, p. 48). The turbulent and molecular diffusion reduces the difference between these regions. In laminar flow, Taylor diffusion and molecular diffusion are the mechanisms of concentration- and temperature-difference reduction. 

Essentially, except for once-through boilers, steam generation primarily involves two-phase nucleate boiling and convective boiling mechanisms (see Section 1.1). Any deposition at the heat transfer surfaces may disturb the thermal gradient resulting from the initial conduction of heat from the metal surface to the adjacent layer of slower and more laminar flow, inner-wall water and on to the higher velocity and more turbulent flow bulk water. 

The water evaporation takes place below the evaporator in a substantial laminar flow. Because the entire phenomenon is driven by the heat transfer rate, in order to preserve the quality of the end product it is of the utmost importance to avoid any buildup of product on the wall of the evaporator this action is assured by the internal mechanical features of the evaporator, consisting of a rotating shaft holding a series of movable blades supported by appropriate frames. These rotating blades provide for a strong eddy effect on the fed prod-... 

Figure 7.27. Twisted-blade type of static mixer operating in the laminar flow regime (a) Distributive mixing mechanism showing, in principle, the reduction in striation thickness produced (f>) Radial mixing contribution...

Sedov LI (1993) Similarity and dimensional methods in mechanics, 10th edn. CRC, Boca Raton Shah RK, London AL (1978) Laminar flow forced convection in duct. Academic, New York Shapiro AK (1953) The dynamics and thermodynamics of compressible fluid flow. Wiley, New York... 

These observations are consistent with the proposed mechanism of the reaction being diffusion controlled in the laminar flow regime. The mass transport is aided by the velocity gradient and thus the reaction rate increases as the Reynolds number is increased. 

Many experimental results have been published, which deal with shear stress in biological systems. Most of them use laminar flow systems such as viscosimeters, flow channels or flasks and very small agitated vessels which are not relevant to technical reactor systems with fully developed turbulent flow. On the other hand the geometric and technical parameters are often not sufficiently described. Therefore it is not possible to explain the complex mechanism of force in bioreactors only on the basis of existing results from biological systems. 

Cherry and Papoutsakis [33] refer to the exposure to the collision between microcarriers and influence of turbulent eddies. Three different flow regions were defined bulk turbulent flow, bulk laminar flow and boundary-layer flow. They postulate the primary mechanism coming from direct interactions between microcarriers and turbulent eddies. Microcarriers are small beads of several hundred micrometers diameter. Eddies of the size of the microcarrier or smaller may cause high shear stresses on the cells. The size of the smallest eddies can be estimated by the Kolmogorov length scale L, as given by... 

Confined flows typically exhibit laminar-flow regimes, i.e. rely on a diffusion mixing mechanism, and consequently are only slowly mixed when the diffusion distance is set too large. For this reason, in view of the potential of microfabrication, many authors pointed to the enhancement of mass transfer that can be achieved on further decreasing the diffusional length scales. By simple correlations based on Fick s law, it is evident that short liquid mixing times in the order of milliseconds should result on decreasing the diffusion distance to a few micrometers. 

This expression applies to the transport of any conserved quantity Q, e.g., mass, energy, momentum, or charge. The rate of transport of Q per unit area normal to the direction of transport is called the flux of Q. This transport equation can be applied on a microscopic or molecular scale to a stationary medium or a fluid in laminar flow, in which the mechanism for the transport of Q is the intermolecular forces of attraction between molecules or groups of molecules. It also applies to fluids in turbulent flow, on a turbulent convective scale, in which the mechanism for transport is the result of the motion of turbulent eddies in the fluid that move in three directions and carry Q with them. 

Fig. 6. Examples of types of meshes developed to resolve laminar flow around particles (a) Chimera grid. Reprinted, with permission, from the Annual Review of Fluid Mechanics, Volume 31 1999 by Annual Reviews www.annualreviews.org (b) Unstructured grid with layers of prismatic cells on particle surfaces. Reprinted from Chemical Engineering Science, Vol. 56, Calis et al., CFD Modeling and Experimental Validation of Pressure Drop and Flow Profile in a Novel Structured Catalytic Reactor Packing, pp. 1713-1720, Copyright (2001), with permission from Elsevier.

Of these three mechanisms, i.e. molecular diffusion, laminar flow and Knudsen diffusion, only two are important in pressure-driven separations. These are laminar flow and Knudsen diffusion. These can be qualitatively understood as follows. If the molecules "see each other much more than they see the pore wall (which means the mean free path of the molecules is much smaller than the mean pore radius), laminar ow a molecular diffusion are important. The laminar flow is much larger, howcver, nd thelhdlecular flow can be neglected (Present and de Bethune 1949). If the molecules see the pore wall much more than they see each other, only Knudsen diffusion will occur. Thus, the molecular diffusion can be neglected in all circumstances. From now on it will be assumed, that only laminar flow and Knudsen diffusion occur. 

For pure gases, a good indication of which mechanism is dominant, Knudsen diffusion or laminar flow, is given by the Knudsen number... 

When the objective of the modeling effort is to develop and validate a reaction mechanism, the major uncertainty in the model must reside in the detailed chemical kinetic mechanism. Under these conditions, the process must be studied either under transport-free conditions, e.g., in plug-flow or stirred-tank reactors, or under conditions in which the transport phenomena can be modeled very precisely, e.g., under laminar flow conditions. This way. 

If flow is laminar with no radial or axial mixing (diffusion coefficient zero), we can write pit) finding p at each annulus that flows along the streamline in plug flow, as shown in Figure 8-4. From fluid mechanics we know that the velocity profile u(R) [really u,(R)] in laminar flow is given by the expression... 

For any more complex flow pattern we must solve the fluid mechanics to describe the fluid flow in each phase, along with the mass balances. The cases where we can still attempt to find descriptions are the nonideal reactor models considered previously in Chapter 8, where laminar flow, a series of CSTRs, a recycle TR, and dispersion in a TR allow us to modify the ideal mass-balance equations. 

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