Fluid flow, kinematics velocity

The centrifugal flows considered in this chapter are those dominated by rotatioa The azimuthal component of the velocity is preponderant, that is, Ur ue and Uz Ue in the cylindrical coordinate system. In such a configuration, the flow tends to become two-dimensional in a plane perpendicular to the Oz axis. The Ur and Ue components of the velocity are quasi-independent from coordinate z. The proof of this property goes beyond the scope of this chapter. Our goal is to describe the centrifugation of solid particles in a rotating flow. We simply choose to consider steady-state axisymmetric fluid flows, whose velocity and pressure fields possess the following kinematic characteristics ... 

The unit of viscosity, the poise, is defined as the force in dynes cm-2 required to maintain a relative velocity of 1 cm/sec between two parallel planes 1 cm apart. The unit commonly used for milk is the centi-poise (10 2 poise). A useful quantity in fluid flow calculations is the kinematic viscosity, or viscosity divided by density. 

Now that we have discussed the geometric interpretation of the rate of strain tensor, we can proceed with a somewhat more formal mathematical presentation. We noted earlier that the (deviatoric) stress tensor t related to the flow and deformation of the fluid. The kinematic quantity that expresses fluid flow is the velocity gradient. Velocity is a vector and in a general flow field each of its three components can change in any of the three... 

Assuming that the flow kinematics of CEF and Newtonian fluids are identical, the velocity profile in steady torsional disk flow is... 

For laminar flow, the characteristic time of the fluid phase Tf can be deflned as the ratio between a characteristic velocity Uf and a characteristic dimension L. For example, in the case of channel flows confined within two parallel plates, L can be taken equal to the distance between the plates, whereas Uf can be the friction velocity. Another common choice is to base this calculation on the viscous scale, by dividing the kinematic viscosity of the fluid phase by the friction velocity squared. For turbulent flow, Tf is usually assumed to be the Kolmogorov time scale in the fluid phase. The dusty-gas model can be applied only when the particle relaxation time tends to zero (i.e. Stp 1). Under these conditions, Eq. (5.105) yields fluid flow. This typically happens when particles are very small and/or the continuous phase is highly viscous and/or the disperse-to-primary-phase density ratio is very small. The dusty-gas model assumes that there is only one particle velocity field, which is identical to that of the fluid. With this approach, preferential accumulation and segregation effects are clearly not predicted since particles are transported as scalars in the continuous phase. If the system is very dilute (one-way coupling), the properties of the continuous phase (i.e. density and viscosity) are assumed to be equal to those of the fluid. If the solid-particle concentration starts to have an influence on the fluid phase (two-way coupling), a modified density and viscosity for the continuous phase are generally introduced in Eq. (4.92). 

Kinematic similarity is the similarity of fluid flow behavior in terms of time within the similar geometries. Kinematic similarity requires that the motion of fluids of both the scale model and prototype undergo similar rate of change (velocity, acceleration, etc.). This similarity criterion ensures that streamlines in both the scale model and prototype are geometrically similar and spatial distributions of velocity are also similar. 

The presence of droplets also introduces new kinematic and dynamic boundary conditions on the fluid flow. Since the immiscible fluids cannot cross the interface, boundary condition states that the local normal component of the velocities in each fluid must be equal to the interface velocity, the velocity tangents to the interface must be also equal inside and outside the droplet, and the tangential shear stresses must be balanced at the interface when it is clean of surfactants. 

The side wall of the confining container is assumed to be a macroscopically homogeneous membrane with permeability Bq (in particular, Bq = 0 corresponds to the impermeable membrane i.e. the FBR). The flow rate through the membrane depends on both, permeability and the local pressure drop across the membrane (by Darcy s law). Local gas density (and pressure) is assumed to be constant at the membrane shell-side. In addition, fluid flow through the packed bed is assumed to be isothermal, nonreactive and nonturbulent, while gas kinematic viscosity is assumed to be independent on gas density. The following boundary conditions have been imposed to simulate the pressure and velocity fields in the fixed bed ... 

Reynolds Number Dimensionless number defined by the product of the pipe diameter and the flow velocity, divided by the fluid s kinematic viscosity. 

Removal of the corrosion product or oxide layer by excessive flow velocities leads to increased corrosion rates of the metallic material. Corrosion rates 2ire often dependent on fluid flow and the availability of appropriate species required to drive electrochemical reactions. Surface shear stress is a measure of the force applied by fluid flow to the corrosion product film. For seawater, this takes into account changes in seawater density and kinematic viscosity with temperature and salinity [33]. Accelerated corrosion of copper-based alloys under velocity conditions occurs when the shear surface stress exceeds the binding force of the corrosion product film. Alloying elements such as chromium improve the adherence of the corrosion product film on copper alloys in seawater based on measurements of the surface shear stress. The critical shear stress for C72200 (297 N/m, 6.2 Ibf/ft ) far exceeds the critical shear stresses of both C70600 (43 N/m, 0.9 Ibf/ft ) and C71500 (48 N/m, 1.0 Ibf/ft ) copper-nickel alloys [33]. 

The control parameters for the experiment described in Figure 15.6 are the mass of particles per unit surface, Mp / S, the area-averaged empty-bed velocity of the fluid in the column, U = Q/S, the density ps of the particles, and the density pf of the fluid. The diameter/) of the particles, as will be seen in section 15.6.3, chiefly influences the fluid flow in a porous medium, jointly determining with the kinematic viscosity v of the fluid the minimum fluidization velocity U f and the terminal entrainment velocity Ut, between which a steady-state fluidization regime is obtained. Furthermore, for a given velocity U of the fluid, the thickness of the fluidized bed increases if the diameter of the particles decreases, for the same mass of particles in the colunrn. 

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