Hydrodynamic flow paths

One of the fundamental tasks required to achieve the ultimate goal of hydrogeology is to understand the controls on energy distribution and transformation within an aquifer system. If this is accepted, it then becomes the hydrologists role to bring together into one concept the fluxes and forces of the chemical reactions, of the hydrodynamic flow paths, and of heat. This idea was clearly articulated and developed by G. B. Maxey (7, p. 145), who stated in part ... 

Figure 6,1, Ternary diagram of normalized varimax factor components for Q-mode on crude oils from Alberta, Canada, showing relation of factor loadings to hydrodynamic flow path for three selected series of crude oils.

Figure 18 illustrates the difference between normal hydrodynamic flow and slip flow when a gas sample is confined between two surfaces in motion relative to each other. In each case, the top surface moves with speed ua relative to the bottom surface. The circles represent gas molecules, and the length of an arrow is proportional to the drift velocity for that molecule. The drift velocity variation with distance is illustrated by the plots on the right. When the ratio of the mean free path to the separation distance between surfaces is much less than unity (Fig. 18a), collisions between gas molecules are much more frequent than collisions of the gas molecules with the surfaces. Here, we have classical fluid flow or viscous flow. If the flow were flow in tubes, Poiseuille s law would be obeyed. The velocity of gas molecules at the surface is the same as the velocity of the surface, and in the case of the stationary surface the mean tangential velocity of the gas at the surface is zero. 

Figure 1. Block diagram of commercial hydrodynamic chromatograph. Solid lines indicate fluid flow path. Broken lines indicate data communication. Arrows indicate fluid flow direction.

A dilute solution of polymer in the GPC solvent is injected into the (lowing eluant. In the column, the molecules with smaller hydrodynamic volumes can diffuse into and out of pores in the packing, while larger solute molecules are excluded from many pores and travel more in the interstitial volume between the porous beads. As a result, smaller molecules have longer effective flow paths than larger molecules and their exit from the GPC column set is relatively delayed. 

Most synthetic polymers are analyzed in organic solvents, using appropriate SEC column packings in which the only interaction between the macromolecular solute and the packing is steric. Separation of the polymeric species is inversely related to their hydrodynamic volumes because the flow paths of the larger species... 

Ferry and Dipple (1991) derived a more formal one-dimensional, local equilibrium, steady-state expression based on Equation (14) that neglects hydrodynamic dispersion and explicitly accounts for changes in concentration along the flow path due to T and P ... 

In flow that is neither hydrodynamically nor thermally fully developed the velocity and temperature profiles change along the flow path. Fig. 3.34 shows qualitatively some velocity and temperature profiles, under the assumption that the fluid flows into the tube at constant velocity and temperature. The wall temperature of the tube is lower than the inlet temperature of the fluid. 

An important improvement in Smoluchowski s approach was to consider hydrodynamic interactions between two particles as they approach each other. These interactions are of two types and result in curvilinear models. First are deviations from rectilinear flow paths that occur as two particles approach each other. Second is the increasing hydrodynamic drag that occurs as two particles come into close proximity. 

There is not only mixing by diffusion, but there is also a mechanical dispersion. Mechanical dispersion occurs when the two solutions of differing chemical compositions meet. Mixing along the direction of flow path is called longitudinal dispersion, while dispersion perpendicular to the flow path is called transverse dispersion (Scott 2000). The mathematical equation for the flux density of solutes by hydrodynamic dispersion is... 

The most noticeable dynamical features were the tendency of the wavepacket to move to the outside of the reaction path near the col, climbing the wall as it moved into the products valley, and the appearance of vortices in the flux stream around the col. A very strong resemblance was found between wavepacket motion and hydrodynamical flow. A comparison with classical flux lines showed very good agreement in cases where classical motion was allowed. 

Brady and Kamke [26] investigated the penetration of PF resins into thin wood flakes using fluorescence microscopy and showed that resin penetration was influenced rather by the natural variability of the wood material than by pressing conditions. Also, it was about three times greater in Douglas Fir earlywood than in latewood. Cell wall fractures enhanced penetration by providing additional paths for hydrodynamic flow. 

Application Laminar flow was established using flow rates such that the Reynolds Number, Re < 10. Thus, after sufficient lead-in (specifically 0.1 x Re x h), which in this case is negligible, a parabolic velocity profile develops across the flow-path. In this manner, the hydrodynamics of this electrochemical setup are equivalent to that of the channel electrode flow system the mass transport-limiting current is therefore given by the Levich equation [86],... 

Internal calandrias are not without problems, however. Hydrodynamics of internal calandrias are difficult to predict. Downcomers must be adequately provided in most systems apparent liquid levels are actually controlled in the downcomers. A froth exists above the top tubesheet this froth can build to appreciable heights. Liquid circulation must migrate across the tubesheet to the nearest downcomer. In doing so, it is agitated by the two-phase mixture leaving the tube ends. Froth can be reduced somewhat by extending the tube ends beyond the tubesheet to provide an undisturbed liquid flow path. However, the two-phase mixture must be separated before liquid can accumulate in this path this is often not accomplished satisfactorily. 

Minor modifications have been made to the parallel-plate flow chamber geometry in order to produce a spatially dependent range of shear stresses across a single substrate. For example, a variable gap height device has been constructed in which /i is a linear function of the distance from the inlet, x (Figure 34.5b) [ 103 ]. In another device the width of the flow path varies inversely with distance from the inlet, w oc 1/x, to produce a Hele-Shaw flow pattern and a hydrodynamic shear stress that increases linearly with distance, t OCX [104,105]. 

Equations 15.5a through 15.5c have been shown to accurately model dispersion in saturated porous media and for a stationary flow in unsaturated media. In transient conditions, however, the relationship between hydrodynamic dispersion coefficients and velocity becomes more complicated. In unsaturated media, the water content of the soil changes with the water flux. Hence, the structure of the water-filled pore space also changes with the water flux. The flow field, and therefore the distribution of pore velocities, depends on the saturation of the medium (Flury et al., 1994). As a consequence, dispersivity coefficients are strongly impacted by the volumetric water content. Usually, dispersivity is found to increase when the water content decreases as a result of the larger tortuosity of solute trajectories and a disconnection of continuous flow paths (Vanclooster et al., 2006). In some cases, especially when the activation of macropores significantly enhances pore-water variability, dispersivity is found to increase with volumetric water content. Currently, there is no unique validated theoretical model available for dispersivity in transient unsaturated flow. 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

In chemical micro process technology with porous catalyst layers attached to the channel walls, convection through the porous medium can often be neglected. When the reactor geometry allows the flow to bypass the porous medium it will follow the path of smaller hydrodynamic resistance and will not penetrate the pore space. Thus, in micro reactors with channels coated with a catalyst medium, the flow velocity inside the medium is usually zero and heat and mass transfer occur by diffusion alone. 

---