Profile viscous flow, velocity

In his analysis of the open tube distillation column Westhaver (1942) goes into a detailed consideration of radial concentration gradients which is very similar to Taylor s approach. His final formula, however, is the same as if he had assumed a constant velocity profile and an effective diffusion coefficient (Dt + llU2r2l48Dt). This is just the diffusion coefficient that we have found for viscous flow in the presence of a film on the tube wall in which the solute concentration is infinitely greater than in the fluid. This is clearly the case for... 

Along with limiting flow, our analysis has shown that viscous effects control the detailed form of flow. This form, which exhibits itself in the various flow velocity profiles already noted, is important to many methods of separation. 

The photo in Fig. 6-3 obtained from a video clip clearly shows the evolution of a velocity gradient as a result of the fluid sticking to the surface of a blunt nose. The layer that slicks lo the surface slows the adjacent fluid layer because of viscous forces between the fluid layers, which slows the next layer, and so on. Therefore, the no-slip condition is responsible for (he development of the velocity profile. The flow region adjacent lo the wall in which the viscous effects (and thus the velocity gradients) are significant is called the boundary layer. The fluid propeity responsible for the no-slip condition and the development of the boundary layer is viscosity and is discussed briefly in Section 6-2. 

The turbulent flow velocity profile for Newtonian fluids is arbitrarily divided into three regions the viscous sublayer, the buffer layer, and the turbulent core. To represent velocity profiles in pipe flow, friction velocity defined as... 

When a nondeformable object is implanted in the flow field and the streamlines and equipotentials are distorted, the nature of the interface does not affect the potential flow velocity profiles. However, the results should not be used with confidence near high-shear no-slip solid-liquid interfaces because the theory neglects viscous shear stress and predicts no hydrodynamic drag force. In the absence of accurate momentum boundary layer solutions adjacent to gas-liquid interfaces, potential flow results provide a reasonable estimate for liquid-phase velocity profiles in Ihe laminar flow regime. Hence, potential flow around gas bubbles has some validity, even though an exact treatment of gas-Uquid interfaces reveals that normal viscous stress is important (i.e., see equation 8-190). Unfortunately, there are no naturally occurring zero-shear perfect-slip interfaces with cylindrical symmetry. 

Figure 23-1 One-dimensional velocity profile for laminar viscous flow in a straight channel with square cross section.

Large Re numbers are equivalent to weak viscosity, as a consequence, the fluid may be considered as an ideal one and the velocity profile may be assumed to be flat However, such an approximation is not valid in the neighbourhood of the wall. For viscous flow fluid velocity must be zero at the wall, for ideal flow, only the normal velocity component must satisfy this condition. Thus, for large Reynolds numbers, velocity cancellation occurs in a thin layer, close to the rigid surface the so called dynamic boundary layer . 

In Eig. 5.13 profiles of axial and radial velocity are shown at three axial positions for the annulus, the membrane and the packed bed. The profiles correspond to an a-alumina membrane with a relatively low permeability (Bo = 9.5 x 10" m ) and a mean pore diameter of dp = 3pm (compare with Table 5.1). The Knudsen number is approximately 10 for this membrane at the given conditions, so that only viscous flow occurs. Because the annulus is empty and the flow within it is laminar, the resulting profile for the axial velocity component is parabolic. Compared to the annulus flow, the axial velocity component in the membrane is... 

Viscous flow This mode of transport is due to a total pressure gradient of a continuum fluid mixture (Figure 7.3-2). Hence, there is no separation of species due to the viscous flow. The driving force is the total pressure gradient and the parameter characterizing the transport is the mixture viscosity, p, and the viscous flow parameter, B, which is a function of solid properties only. The flow inside the pore is assumed laminar, hence the velocity profile is parabolic in shape. 

In Chapter 2, the design of the so-called ideal reactors was discussed. The reactor ideahty was based on defined hydrodynamic behavior. We had assumedtwo flow patterns plug flow (piston type) where axial dispersion is excluded and completely mixed flow achieved in ideal stirred tank reactors. These flow patterns are often used for reactor design because the mass and heat balances are relatively simple to treat. But real equipment often deviates from that of the ideal flow pattern. In tubular reactors radial velocity and concentration profiles may develop in laminar flow. In turbulent flow, velocity fluctuations can lead to an axial dispersion. In catalytic packed bed reactors, irregular flow with the formation of channels may occur while stagnant fluid zones (dead zones) may develop in other parts of the reactor. Incompletely mixed zones and thus inhomogeneity can also be observed in CSTR, especially in the cases of viscous media. 

The solution for the velocity profile and flow rate is correct as long as the viscous dissipation can be neglected. The solution for the temperature profile is approximate because of the assumption of a linear temperature profile for viscosity determination. if the calculated temperature profile deviates sharply from the linear profile the error in the temperature profile can be significant. This issue will be addressed in the following section. 

Fig. 2.4. Velocity profile in a gas undergoing viscous flow. The lower plate is stationary while the upper one has a velocity v = dU.

Hennk Soeberg Viscous flow in curved Tubes-1 Velocity profiles Chemical Engineering Science Vol 43, No 4 (1988) 855-862... 

VELOCITY PROFILE ANALYTIC SOLUTION FOR NONUNIFORM VISCOUS FLOW (A Method Derived by R. L. Turner and E. D. Wohlsen)... 

Because of microbial conversion and mass-transfer resistance effects, substrate and product profiles develop inside biofilms. If the biofilm is impermeable, difiusion is the only transport mechanism inside the matrix. The turbulent bulk liquid is usually well mixed by advective transport (transport by liquid flow). Adjacent to the matrix is a viscous boundary layer in which the mixing and flow velocity gradually decrease as the surface is approached. Consequently, the mode of transport changes gradually from advectional in the bulk liquid to diflusional in the laminar boundary layer. Diffusional transport is driven by the concentration differences as expressed in Pick s law ... 

In turbulent flow, the velocity profile is much more blunt, with most of the velocity gradient being in a region near the wall, described by a universal velocity profile. It is characterized by a viscous sublayer, a turbulent core, and a buffer zone in between. 

---