Zero bulk fluid velocity

Fig. 3.14 Effect of increasing solution velocity in increasing the limiting diffusion current density. At zero bulk fluid velocity, density changes and gas evolution can produce interface turbulence, which increases the current density.

A generalization of the condition (2-112) is required if there is an active phase transformation occurring at S, i.e., if the liquid is vaporizing or the solid is melting. In this case, we must distinguish between the bulk fluid velocities in the limit as we approach the interface, and the velocity of the interface itself, u1 n (where the interface is specified still by the criteria of zero excess mass discussed earlier). The condition of conservation of mass then requires that... 

In Fig. 3.6 the dashed line OL is so drawn that the velocity changes are confined between this line and the trace of the wall. Because the velocity lines are asymptotic with respect to distance from the plate, it is assumed, in order to locate the dashed line definitely, that the line passes through all points where the velocity is 99 percent of the bulk fluid velocity Line OL represents an imaginary surface that separates the fluid stream into two parts one in which the fluid velocity is constant and the other in which the velocity varies from zero at the wall to a velocity substantially equal to that of the undisturbed fluid. This imaginary surface separates the fluid that is directly affected by the plate from that in which the local velocity is constant and equal to the initial velocity of the approach fluid. The zone, or layer, between the dashed line and the plate constitutes the boundary layer. 

Assuming constant viscosity and permittivity, and negligible pressure gradients as a result of the fluid flow, integration of equation (19.1) from the hydrodynamic plane of shear where the fluid velocity is zero (v = 0), to a point in the bulk where the potential is zero and fluid velocity is constant (Ueo) gives the following ... 

The most fundamental process in dispersion is molecular diffusion, which is a special case of dispersion when the velocity of the fluid is zero. Molecules in the liquid state are not stationary, even if the bulk fluid velocity is zero, because the molecules are in continuous motion. The flux due to the random molecular motion is given by Eq. 3.29 ... 

Initially it was assumed that no solution movement occurs within the diffusion layer. Actually, a velocity gradient exists in a layer, termed the hydrodynamic boundary layer (or the Prandtl layer), where the fluid velocity increases from zero at the interface to the constant bulk value (U). The thickness of the hydrodynamic layer, dH, is related to that of the diffusion layer ... 

The void fraction should be the total void fraction including the pore volume. We now distinguish Stotai from the superficial void fraction used in the Ergun equation and in the packed-bed correlations of Chapter 9. The pore volume is accessible to gas molecules and can constitute a substantial fraction of the gas-phase volume. It is included in reaction rate calculations through the use of the total void fraction. The superficial void fraction ignores the pore volume. It is the appropriate parameter for the hydrodynamic calculations because fluid velocities go to zero at the external surface of the catalyst particles. The pore volume is accessible by diffusion, not bulk flow. 

Fig. 5.1.5 Quantitative data on the correlation of biofilm and velocity for a slice perpendicular to the flow axis. The images on the left are from top to bottom T2 map, z velocity component, x velocity component and y velocity component. One dimensional profiles through lines A, in bulk fluid, and B, intersecting biofilm fluid interface, are shown on the right. The biofilm signal indicator, dotted grey line, has been normalized so that zero corresponds to no biomass and 1 corresponds to the highest...

The kinetic-energy terms of the various energy balances developed h include the velocity u, which is the bulk-mean velocity as defined by the equati u = m/pA Fluids flowing in pipes exhibit a velocity profile, as shown in Fi 7.1, which rises from zero at the wall (the no-slip condition) to a maximum the center of the pipe. The kinetic energy of a fluid in a pipe depends on actual velocity profile. For the case of laminar flow, the velocity profile parabolic, and integration across the pipe shows that the kinetic-ertergy should properly be u2. In fully developed turbulent flow, the more common in practice, the velocity across the major portion of the pipe is not far fro... 

The special case V - 0 corresponds to a stationary medium, which can now be defined more precisely as a medium whose mass-average velocity is zero. Therefore, mass transport in a stationary medium is by diffusion only, aud zero mass-average velocity indicates that there is no bulk fluid motion. 

For systems operating under transitionary and turbulent conditions, it is frequently necessary to introduce a friction factor, which arises from the fact that the fluid velocity is not zero at the surface, as assumed in an ideal hydrodynamic model. The friction factor (/) is a function of the velocity of the fluid in the bulk (V), the shear... 

A difference between the perturbation considered here and that in the section on LRT considered earlier is the term involving qo, the position at which the drift velocity (i.e., the velocity contribution from the external field) of the fluid is zero. This term was chosen to be zero in the treatment for bulk fluids for simplicity it must be used here because confinement has broken the translational invariance of the system. The perturbation generates a planar Couette flow in the fluid between two surfaces ... 

Let us denote the bulk-phase densities on the two sides of the interface as p and p and the fluid velocities as u and u. The orientation of surface S is specified in terms of a unit normal n. In general, the surface S is not a material surface. For example, if there is a phase transition occurring between the two bulk phases (e.g., a solid phase is melting or a liquid phase is evaporating), mass will be transferred across S. However, the surface S is not a source or sink for mass, and thus mass conservation requires that the net flux of mass to (or from) the surface must be zero. 

We saw above that the concentration gradient at an electrode will be linear with respect to the spatial coordinate perpendicular to the electrode surface if the anode/cathode cell were operated at a constant current density and if the fluid velocity were zero. In actuality, there will always be some bulk liquid electrolyte stirring during current flow, either an imposed forced convection velocity or a natural convection fluid motion due to changes in the reacting species concentration and fluid density near the electrode surface. In electrochemical systems with fluid flow, the mass transfer and hydrodynamic fluid flow equations are coupled and the solution of the relevant differential equations is often a formidable task, involving complex mathematical and/or numerical solution techniques. The concept of a stagnant diffusion layer or Nemst layer parallel and adjacent to the electrode surface is often used to simplify the analysis of convective mass transfer in... 

Mass transfer coefficients are the basis for models where the dissolved species are transported by a combination of diffusive and advective processes. The diffusive mass transfer coefficient ko, m/sec) is based on boundary layer theory. The basic premise of boundary layer theory is that, for laminar ffow, the ffuid velocity adjacent to a solid surface is zero (the no slip condition ) and the velocity increases as a parabolic function of distance away from the surface until it matches the velocity of the bulk fluid (Figure 7.5). This means that there is a thin layer of fluid with a thickness of 5d (m) adjacent to the surface that is effectively static. The rate of mass transport through this layer is limited by the diffusion rate of the dissolved species. The diffusional boundary layer is much thinner than the velocity boundary layer. For laminar flow past a flat surface, the thickness of the diffusional boundary layer is related to the thickness of the velocity boundary layer (Sy) by the Schmidt number, which compares the fluid viscosity to the diffusivity (Probstein, 1989). 

In order to help explain boundary layers, an example of boundary-layer formation in the steady-state flow of a fluid past a flat plate is given in Fig. 3.10-1. The velocity of the fluid upstream of the leading edge at x = 0 of the plate is uniform across the entire fluid stream and has the value. The velocity of the fluid at the interface is zero and the velocity in the x direction increases as one goes farther from the plate. The velocity, approaches asymptotically the velocity v of the bulk of the stream. 

A fluid passing over a solid surface develops a boundary layer in which the velocity parallel to the surface varies rapidly over a very short distance normal to the direction of flow. The velocity is zero at the solid surface but approaches the bulk-stream velocity at a distance less than a millimeter from the surface. Mixing occurs in the main fluid stream where reactants and products are transported at rates that depend on the nature of flow. The fluid velocity near the surface is low with little mixing therefore, mass transport perpendicular to the surface is by molecular diffusion. Although mass transport in the main stream is essentially independent of the molecular diffusion coefficient, it is proportional to DA-mix very... 

At the edge of the Debye layer, the potential drops to zero. As a result the bulk fluid is moving at a velocity... 

If there is no bulk fluid motion, the mass average velocity components are zero and... 

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