Hydrodynamics fluid velocity

A number of analytical solutions have been derived for iC as a function of channel dimensions and fluid velocity (30). In practice, the fit between theory and data for K is poor except in idealized cases. Most processes exhibit either higher fluxes, presumably caused by physical dismption of the gel layer from the nonideal hydrodynamic conditions, or lower fluxes caused by fouling (31). In addition, iCis a function of the fluid composition. 

Decoupled Driving Force and Depolarization Needs for improved fractionation motivate designers to reduce autofiltration. Using fluid velocity for depolarization means that hydrodynamic pressure drop will be additive to the transmembrane pressure driving force. Schemes to hmit this effeci confront a harsh economic reahty. Two novel schemes decouple the driving from the depolarizing force. 

To permit a more general discussion, we can replace the snowplow with a piston, and replace the snow with any fluid (Fig. 2,3), We consider the example shown in a reference frame in which the undisturbed fluid has zero velocity. When the piston moves, it applies a planar stress, a, to the fluid. For a non-viscous, hydrodynamic fluid, the stress is numerically equal to the pressure, P, The pressure induces a shock discontinuity, denoted by which propagates through the fluid with velocity U. The velocity u of the piston, and the shocked material carried with it (with respect to the stationary frame of reference), is called the particle velocity, since that would be the velocity of a particle caught up in the flow, or of a particle of the fluid. 

The hydrodynamic forces are usually described by a linear relation between drag resistance and relative fluid velocity ... 

The non-free draining character of flexible polymer chains was considered in the Zimm model [48], In this model, the effect of hydrodynamic interaction at the location of bead i is taken into account by an additional fluid velocity term vj ... 

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. 

Velocity profile elongation. Low fluid velocities near the tube wall give rise to high extents of pol5merization, high viscosities, and yet lower velocities. The velocity profile elongates, possibly to the point of hydrodynamic instability. 

Recently, studies were performed to quantitatively examine the hydrodynamics of the two most common in vitro dissolution testers. Rotational (tangential) fluid velocities were corre-... 

Rotational fluid velocities are calculated since horizontal (rotational) flow prevails in the hydrodynamic regime within the dissolution vessels. Thus, the overall hydrodynamics and hence dissolution is dominated by the substantially higher rotational (tangential) fluid velocities. 

When considering boundary conditions, a useful dimensionless hydrodynamic number is the Knudsen number, Kn = X/L, the ratio of the mean free path length to the characteristic dimension of the flow. In the case of a small Knudsen number, continuum mechanics will apply, and the no-slip boundary condition assumption is valid. In this formulation of classical fluid dynamics, the fluid velocity vanishes at the wall, so fluid particles directly adjacent to the wall are stationary, with respect to the wall. This also ensures that there is a continuity of stress across the boundary (i.e., the stress at the lower surface—the wall—is equal to the stress in the surface-adjacent liquid). Although this is an approximation, it is valid in many cases, and greatly simplifies the solution of the equations of motion. Additionally, it eliminates the need to include an extra parameter, which must be determined on a theoretical or experimental basis. 

Note The fluid velocity v in pipes—or the superficial gas velocity vq in mixing vessels or in bubble columns—presents a well-known process parameter which combines the fluid throughput q and the diameter of the device D V q/T>. Nevertheless this parameter is not an intermediate quantity. It cannot replace the diameter of the device it is simply another expression for the fluid throughput. Reference The kinematic process numbers like the Reynolds and Froude numbers, which govern the hydrodynamics, necessarily contain the linear dimension of the device. 

McCarthy LG, Kosiol C, Healy AM, Bradley G, Sexton JC, Corrigan OI. Simulating the hydrodynamic conditions in the United States Pharmacopeia paddle dissolution apparatus. AAPS Pharm Sci Tech 2003 4(2) Article 22. McCarthy LG, Bradley G, Sexton JC, Corrigan OI, Healy AM. Computational fluid dynamics modeling of the paddle dissolution apparatus agitation rate, mixing patterns, and fluid velocities. AAPS Pharm Sci Tech 2004 5(2) Article 31. 

When a fluid flow of constant velocity, u, impinges parallel to the edge of a plate, the boundary condition is such that the fluid velocity is zero on the surface of the plate. This results in the formation of a hydrodynamic boundary layer in which the flow velocity parallel to the surface varies with distance normal to the surface. The hydrodynamic boundary layer thickness increases with distance, jc, from the upstream edge of the plate as given by equation (10.5) [7] ... 

When the flow is constrained to a channel bounded on both faces, then at a sufficient distance from the entry point, xe, the two hydrodynamic boundary layers associated with each wall overlap and the fluid velocity varies parabolically with position across the channel. 

The information required to predict electrochemical reaction rates (i.e., experimentally determined by Evans diagrams, electrochemical impedance, etc.) depends upon whether the reaction is controlled by the rate of charge transfer or by mass transport. Charge transfer controlled processes are usually not affected by solution velocity or agitation. On the other hand, mass transport controlled processes are strongly influenced by the solution velocity and agitation. The influence of fluid velocity on corrosion rates and/or the rates of electrochemical reactions is complex. To understand these effects requires an understanding of mixed potential theory in combination with hydrodynamic concepts. 

Either anodic or cathodic mass transport limited corrosion may be observed in numerous corrosion systems. Such phenomena may be simulated and investigated in the laboratory by establishing experimental conditions that match those in the field application. This is accomplished by equating z L or 8d in the laboratory to the same values present in the field. In this way the effect of fluid velocity or mass flow rate on the corrosion rate may be investigated. Similarly, the hydrodynamic conditions in the field must be matched by those in the laboratory. Procedures for establishing such correlations between field and laboratory measurements are described below. 

It was shown above that the limiting c.d. increases with velocity raised to the 0.8 power and the pipe diameter raised to the -0.2 power for piping corrosion rates that are controlled by mass transport. In contrast, it is evident that the shear stress increases with the fluid velocity raised to the 1.75 power and the pipe diameter raised to the -0.2 power. Thus equality of shear stress does not give equality of mass transfer rates. In both cases corrosion is enhanced in pipes of smaller diameter for the same solution velocity. Such a relationship can be rationalized based on the effect of pipe diameter on the thickness of the mass transport and hydrodynamic boundary layers for a given fixed geometry. Cameron and Chiu (19) have derived similar expressions for defining the rotating cylinder rotation rate required to match the shear stress in a pipe for the case of velocity-... 

Hydrodynamic boundary layer — is the region of fluid flow at or near a solid surface where the shear stresses are significantly different to those observed in bulk. The interaction between fluid and solid results in a retardation of the fluid flow which gives rise to a boundary layer of slower moving material. As the distance from the surface increases the fluid becomes less affected by these forces and the fluid velocity approaches the freestream velocity. The thickness of the boundary layer is commonly defined as the distance from the surface where the velocity is 99% of the freestream velocity. The hydrodynamic boundary layer is significant in electrochemical measurements whether the convection is forced or natural the effect of the size of the boundary layer has been studied using hydrodynamic measurements such as the rotating disk electrode [i] and - flow-cells [ii]. 

Designs based on such a value of Re may well not adequately describe the true hydrodynamics of the system. Re is defined as pvA1 2/pij where p and p are the density and viscosity of the fluid phase, and v and A1 2 are a characteristic fluid velocity and length-scale, respectively, of the system under study. Reproduced with permission from Johns et al. (2000). Copyright (2000), A.I.Ch.E. 

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