Parallel plates, average velocity

Equation (4) applies to the steady flow of a fluid between parallel plates. The velocity is the time average for the flow at the point in question. A corresponding definition of eddy viscosity in a circular conduit may be evolved in which the radial deficiency is substituted for the distance from the wall in Eq. (4). 

For simple flow geometries, as in the case of stratified two-phase flow between two parallel plates, the velocity field varies only in the perpendicular direction and expressions for the shear stresses, in terms of the phases average velocities, U, U, and the layer depth, H, can be analytically derived by solving the Navier-Stokes equations in the two-phases domains (see, for example, Hanratty and Hershman [61]). Coutris et al. showed that even for the simplest case of fully developed laminar two-layer flow the closure relations which evolve for the wall and inteifacial shear stresses are quite complicated [62]. 

It should be recognized that the boundary conditions of the problem will establish the value of the hydrodynamic velocity, u. In the case of most turbulent flows the indirect influence of molecular diffusion on the hydro-dynamic velocity can be neglected. It should be emphasized that the hydrodynamic velocity is the time-average point velocity in Reynolds sense (R2). Under unsteady, nonuniform conditions of flow between parallel plates the material balance may be expressed for turbulent flow in the following form ... 

Mathematica package is developed that computes the eigenvalues, the eigenfunctions, the eigenintegrals, the dimensionless temperature, the average dimensionless temperature, and the Nusselt number for steady state and periodic heat transfer in micro parallel plate channel and micro tube taking into account the velocity slip and the temperature jump. Some results in form of tables and plots are given bellow. 

The peak velocity is 1.5 times the average velocity for flow between parallel plates. These formulas are provided here because they provide a good benchmark against which to check any numerical solution by integrating over boundaries. 

Spurk (1997) derived closed-form solutions for the average flow velocity v between parallel plates and in full ducts. For laminar flow between parallel plates relationships between viscosity, geometry, pressure gradient and mean flow velocity are given as ... 

Consider the steady flow between parallel planes [24]. Two planes are a distance 2b apart in the y direction and extend to infinity in the z direction. Assume that the fluid flow between them is steady. Derive the average velocity, u, in terms of the impressed pressure gradient, the distance between the plates, and the fluid viscosity. Also write down the ratio of the velocity to that of the average. 

THE PROBLEM An electrochemical reactor with vertical parallel plates is used for the generation of a gaseous product involving a two-electron change at temperature 300 K and pressure 10 Nm. The electrolyte, which can be considered stationary, has a conductivity of 50 mho m . The reaction is to be tested initially on a unit with electrodes 0.25 m long before the electrodes are scaled up to 0.75 m. Average current density on both units is 5000 Am You can assume the bubbles to rise with a mean velocity of 1.55 X 10 ms ... 

Equation (39) gives the electro viscosity in a microtube with an inner charged surface at the potential f(,. The reduced forward volumic flow rate observed experimentally is interpreted as an increased apparent viscosity. Using the expressions of Ki, K2, s and K3 (respectively Eqs. (15b), (15c), (29) and (30)) it is possible to evidence that iJis liJi is a function of neither the average fluid velocity (C/q) nor the pressure gradient (Pz) but only a function of the microtube diameter and the electrokinetic parameters. The case of two parallel plates has been studied by Mala et al. [6] and will not be presented here. The electroviscosity equation obtained differs slightly from Eq. (39) due to the geometry. 

An electrostatic precipitator consists of two parallel plates separated by a distance of 0.06 m. The flow of gas through the system at 121°C and 207 X 10 N/m2 pressure has a Reynolds number of lO. Combustion products are anticipated to have log normal distribution with the average log normal length diameter of 6.0 Mm. The solid combustion product has a density of 1282 kg/m3. The electric field is 10,000 N/C with a particle charge of 10" ih c. What is the drift velocity of the average particle, the length of the precipitator needed for capture of this particle, and the efficiency of the precipitator ... 

Figure 7.2.5. (a) Concentration profile of species i and axial velocity profile in a crossflow UP membrane channel, (b) Parallel-plate crossflow UP membrane channel, (c) Reduction of averaged membrane module filtration flux with the extent of solute concentration, (d) Process schematic for batch UP with a crossflow membrane module, (e) Process schematic for membrane diafiltration. (f) Variation of the yield of purified product species 1 in the filtrate with its purification factor for different values of the parameters ("DFjAS. 

Similarly, the velocity gradient is constant across the gap as in flow between parallel plates (eq. 5.2.2), and thus the shear rate is the average in the gap... 

FIGURE 10.6 Flow pattern in the advancing front for flow between two parallel plates as observed relative to the average velocity. A fluid element approaching the front is compressed along the flow direction and stretched along the y direction before being laid up on the cold wall. (Reprinted by permission of the publisher from Tadmor, 1974.)... 

Determine TE for a l.O-fxm-diameter sphere of standard density positioned between two parallel plates 1 cm apart maintained at a 9000-V potential dif> ference. Assume that the particle is charged to the average Boltzmaim equi librium charge. What is the ratio of this electrostatic velocity to its gravitational settling velocity ... 

Figure 1.25 shows the boundary layer that develops over a flat plate placed in, and aligned parallel to, the fluid having a uniform velocity upstream of the plate. Flow over the wall of a pipe or tube is similar but eventually the boundary layer reaches the centre-line. Although most of the change in the velocity component vx parallel to the wall takes place over a short distance from the wall, it does continue to rise and tends gradually to the value vx in the fluid distant from the wall (the free stream). Consequently, if a boundary layer thickness is to be defined it has to be done in some arbitrary but useful way. The normal definition of the boundary layer thickness is that it is the distance from the solid boundary to the location where vx has risen to 99 per cent of the free stream velocity v . The locus of such points is shown in Figure 1.25. It should be appreciated that this is a time averaged distance the thickness of the boundary layer fluctuates owing to the velocity fluctuations.

A 2-m X 3-m flat plate is suspended in a room, and is subjected to air flow parallel to its surfaces along its 3-m-long side. The free stream temperature and velocity of air are 20°C and 7 m/s. The total drag force acting on the plate is measured to be 0.86 N. Determine the average convection heal transfer coefficient for the plate (Fig. 5-36). 

The function q(x, x) is the average over the film thickness of the (parallel to the solid vertical plate z = 0) component u of the velocity. Due to the fact that q(x, x), as a general rule, cannot be expressed in terms of H(x, x) or some of its spatial derivatives, equation (5.2) is not a closed form evolution equation for the thickness H(x, x). Our main goal, in this 5 is to examine a particular situation, arising mainly from averaging process, from which a closed system of equations may be obtained. 

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