Hydrodynamics laminar flow

Chapter 7 deals with the practical problems. It contains the results of the general hydrodynamical and thermal characteristics corresponding to laminar flows in micro-channels of different geometry. The overall correlations for drag and heat transfer coefficients in micro-channels at single- and two-phase flows, as well as data on physical properties of selected working fluids are presented. The correlation for boiling heat transfer is also considered. 

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. 

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. 

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. 

Micromixing Models. Hydrodynamic models have intrinsic levels of micromixing. Examples include laminar flow with or without diffusion and the axial dispersion model. Predictions from such models are used directly without explicit concern for micromixing. The residence time distribution corresponding to the models could be associated with a range of micromixing, but this would be inconsistent with the physical model. 

Hydrodynamic stress exists independently of this. In the case of laminar flow, the stress is given by Newton s law (1) ... 

Of the methodologies considered, the Lewis cell, employs direct contact of the two liquids, but does not have well-defined hydrodynamics. The constant interface cell with laminar flow has better-defined hydrodynamics, but the interfacial flux is not measured... 

The lack of hydrodynamic definition was recognized by Eucken (E7), who considered convective diffusion transverse to a parallel flow, and obtained an expression analogous to the Leveque equation of heat transfer (L5b, B4c, p. 404). Experiments with Couette flow between a rotating inner cylinder and a stationary outer cylinder did not confirm his predictions (see also Section VI,D). At very low rotation rates laminar flow is stable, and does not contribute to the diffusion process since there is no velocity component in the radial direction. At higher rotation rates, secondary flow patterns form (Taylor vortices), and finally the flow becomes turbulent. Neither of the two flow regimes satisfies the conditions of the Leveque equation. 

Just as for laminar flow, a minimum hydrodynamic entry length (Le) is required for the flow profile to become fully developed in turbulent flow. This length depends on the exact nature of the flow conditions at the tube entrance but has been shown to be on the order of Le/D = 0.623/VRe5. For example, if /VRe = 50,000 then Le/D = 10 (approximately). 

Figure 5. Exact (numerical solution, continuous line) and linearised (equation (24), dotted line) velocity profile (i.e. vy of the fluid at different distances x from the surface) at y = 10-5 m in the case of laminar flow parallel to an active plane (Section 4.1). Parameters Dt = 10 9m2 s-1, v = 10-3ms-1, and v = 10-6m2s-1. The hydrodynamic boundary layer thickness (<50 = 5 x 10 4 m), equation (26), where 99% of v is reached is shown with a horizontal double arrow line. For comparison, the normalised concentration profile of species i, ct/ithe linear profile of the diffusion layer approach (continuous line) and its thickness (<5, = 3 x 10 5m, equation (34)) have been added. Notice that the linearisation of the exact velocity profile requires that <5,

Figure 8. Variation of the hydrodynamic boundary layer thickness (So, equation (26), continuous line), the diffusion layer thickness (<5,-, equation (34), dotted line) and the ensuing local flux (/, equation (32), dashed line) with respect to the distance from the leading edge (y) in the case of laminar flow parallel to an active plane (the surface is a sink for species i). Parameters /), = 10-9nrs, v= 10 3ms, c = lmolm-3, and v — 10-6 m2 s 1. Notice that c5, o (as required for the derivation of the flux equation (32)), and that the flux decreases when <5, increases...

Concerning the hydrodynamics and the dimensioning of the test reactor, some rules of thumb are a valuable aid for the experimentalist. It is important that the reactor is operated under plug-flow conditions in order to avoid axial dispersion and diffusion limitation phenomena. Again, it has to be made clear that in many cases testing of monolithic bodies such as metal gauzes, foam ceramics, or monoliths used for environmental catalysis, often needs to be performed in the laminar flow regime. 

All measurements, of course, have to be made at a finite concentration. This implies that interparticle interactions cannot be fully neglected. However, in very dilute solutions we can safely assume that more than two particles have only an extremely small chance to meet [72]. Thus only the interaction between two particles has to be considered. There are two types of interaction between particles in solution. One results from thermodynamic interactions (repulsion or attraction), and the other is caused by the distortion of the laminar fiow due to the presence of the macromolecules. If the particles are isolated only the laminar flow field is perturbed, and this determines the intrinsic viscosity but when the particles come closer together the distorted flow fields start to overlap and cause a further increase of the viscosity. The latter is called the hydrodynamic interaction and was calculated by Oseen to various approximations [3,73]. Figure 7 elucidates the effect. 

Fig. 7. Schematic representation of laminar flow distortion due to the presence of isolated particles (left) and the corresponding effect at higher concentration when the perturbed flow flelds start to overlap (right). The latter effect causes the hydrodynamic interaction...

Figure 7.4 Representations of hydrodynamic flow, showing (a) laminar flow through a smooth pipe and (b) turbulent flow, e.g. as caused by an obstruction to movement in the pipe. The length of each arrow represents the velocity of the increment of solution. Notice in (a) how the flow front is curved (known as Poiseuille flow ), and in (b) how a solution can have both laminar and turbulent portions, with the greater pressure of solution flow adjacent to the obstruction.

Turbulent mass transfer near a wall can be represented by various physical models. In one such model the turbulent flow is assumed to be composed of a succession of short, steady, laminar motions along a plate. The length scale of the laminar path is denoted by x0 and the velocity of the liquid element just arrived at the wall by u0. Along each path of length x0, the motion is approximated by the quasi-steady laminar flow of a semiinfinite fluid along a plate. This implies that the hydrodynamic and diffusion boundary layers which develop in each of the paths are assumed to be smaller than the thickness of the fluid elements brought to the wall by turbulent fluctuations. Since the diffusion coefficient is small in liquids, the depth of penetration by diffusion in the liquid element is also small. Therefore one can use the first terms in the Taylor expansion of the Blasius expressions for the velocity components. The rate of mass transfer in the laminar microstructure can be obtained by solving the equation... 

Fig. 7.93. Influence of location on boundary layer thickness in laminar flow along an electrode 8,8p and 8V are the thicknesses of the diffusion, thermal, and hydrodynamic boundary layers, respectively.

The laminar flow hydrodynamics for a radial wall-jet were first considered by Glauert [41] and subsequently by Scholtz and Trass [42]. A more complete evaluation for electrochemical purposes has recently appeared [43]. In Fig. 5 are shown the velocity profiles and schematic streamlines. Close to the wall we find... 

Diffusion-limited currents at hydrodynamic electrodes under laminar flow conditions3... 

Certain criteria have to be met in the construction of hydrodynamic electrodes, such that the laminar flow pattern, which is used in the derivation of the theoretical equations, is conformed to. Thus edge effects, which are due to the fact that electrode and surrounding mantle are not of infinite size and which are also dependent on cell dimensions, must be minimised. The shape of the electrode and mantle is important the surfaces must be smooth and there must be no discontinuities or electrolyte penetration at the electrode/mantle junction. 

Amperometric detectors can operate over a range of conversion efficiencies from nearly 0% to nearly 100%. From a mathematical point of view, a classical amperometric determination (conversion of analyte is negligible) is one where the current output is dependent on the cube root of the linear velocity across the electrode surface as described by Levich s hydrodynamic equations for laminar flow. Conversely, the current response for a cell with 100% conversion is directly proportional to the velocity of the flowing solution. While the mathematics describing intermediate cases is quite interesting, it is beyond the scope of this chapter. 

These simple and mostly intuitive arguments apply mainly to chemical and biological reaction engineering systems. For other systems such as fluid flow systems, the sources and causes of bifurcation and chaos can be quite different. It is well established that the transition from laminar flow to turbulent flow is a transition from nonchaotic to chaotic behavior. The synergetic interaction between hydrodynamically induced bifurcation and chaos and that resulting from chemical and biological reaction/diffusion is not well studied and calls for an extensive multidisciplinary research effort. 

The intense heat dissipated by viscous flow near the walls of a tubular reactor leads to an increase in local temperature and acceleration of the chemical reaction, which also promotes an increase in temperature the local situation then propagates to the axis of the tubular reactor. This effect, which was discovered theoretically, may occur in practice in the flow of a highly viscous liquid with relatively weak dependence of viscosity on degree of conversion. However, it is questionable whether this approach could be applied to the flow of ethylene in a tubular reactor as was proposed in the original publication.199 In turbulent flow of a monomer, the near-wall zone is not physically distinct in a hydrodynamic sense, while for a laminar flow the growth of viscosity leads to a directly opposite tendency - a slowing-down of the flow near the walls. In addition, the nature of the viscosity-versus-conversion dependence rj(P) also influences the results of theoretical calculations. For example, although this factor was included in the calculations in Ref.,200 it did not affect the flow patterns because of the rather weak q(P) dependence for the system that was analyzed. 

Core stream The core stream is the stream-within-a-stream that has been injected into the center of the sheath stream and is maintained there by the hydrodynamic considerations of laminar flow at increasing velocity. The core contains the sample particles that are to be analyzed in the flow cytometer. If the sample is injected too rapidly, the core stream widens and particles may be unequally illuminated. In addition, with a wide core stream, coincidence events are more likely. 

Hydrodynamic focusing Hydrodynamic focusing is the property of laminar flow in a stream of increasing velocity that maintains particles in the narrowing central core of a column of fluid. 

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