Velocity, mean

This expression corresponds to the Arrhenius equation with an exponential dependence on the tlireshold energy and the temperature T. The factor in front of the exponential function contains the collision cross section and implicitly also the mean velocity of the electrons. 

Now suppose, as before, that distance is measured on a scale whose unit is comparable with the dimensions of che flow channel, and let v° be some characteristic mass mean velocity in the system, like introduced above... 

For our present purpose it is convenient to reformulate equation (4.11) as a condition on the mass mean velocity. Let us write the mean axial components of molecular velocities in the form... 

This is Che required boundary condition for the mass mean velocity, Co be applied at the tube surface r = a. With a non-vanishing value for v (a), Che Poiseuille solution (4.5) must now be replaced by the simple modification. 

Equipped with a proper boundary condition and a complete solution for the mass mean velocity, let us now turn attention to the diffusion equations (4.1) which must be satisfied everywhere. Since all the vectors must... 

Kramers and Kistemaker evaluated the momentum transfer Co the wall by molecular impacts in a frame of reference moving with the mole mean velocity This has some algebraic advantage over working in the rest frame of the tube... 

The quantity k is related to the intensity of the turbulent fluctuations in the three directions, k = 0.5 u u. Equation 41 is derived from the Navier-Stokes equations and relates the rate of change of k to the advective transport by the mean motion, turbulent transport by diffusion, generation by interaction of turbulent stresses and mean velocity gradients, and destmction by the dissipation S. One-equation models retain an algebraic length scale, which is dependent only on local parameters. The Kohnogorov-Prandtl model (21) is a one-dimensional model in which the eddy viscosity is given by... 

MUNICIPAL WATER treatment). Scale-up of orthokinetic flocculators, generally in the form of paddle devices, is based on the product of mean velocity gradient and time, for a constant volume concentration of the flocculating particles. 

The effect of pulsating flow on pitot-tube accuracy is treated by Ower et al., op. cit., pp. 310-312. For sinusoidal velocity fluctuations, the ratio of indicated velocity to actual mean velocity is given by the factor /l + AV2, where X is the velocity excursion as a fraction of the mean velocity. Thus, the indicated velocity would be about 6 percent high for velocity fluctuations of 50 percent, and pulsations greater than 20 percent should be damped to avoid errors greater than 1 percent. Tne error increases as the frequency of flow oscillations approaches the natural frequency of the pitot tube and the density of the measuring fluid approaches the density of the process fluid [see Horlock and Daneshyar, y. Mech. Eng. Sci, 15, 144-152 (1973)]. 

Traversing for Mean Velocity Mean velocity in a duct can be obtained by dividing the cross section into a number of equal areas, finding the local velocity at a representative point in each, and averaging the results. In the case of rectangular passages, the cross section is usually divided into small squares or rectangles and the velocity is found at the center of each. In circular pipes, the cross section is divided into several equal annular areas as shown in Fig. 10-7. Read-... 

For normal velocity distribution in straight circular pipes at locations preceded by runs of at least 50 diameters without pipe fittings or other obstructions, the graph in Fig. 10-7 shows the ratio of mean velocity V to velocity at the center plotted against the Reynolds number, where D = inside pipe diameter, p = flmd density, and [L = fluid viscosity, all in consistent units. Mean velocity is readily determined from this graph and a pitot reading at the center of the pipe if the quantity Du p/ I is less than 2000 or greater than 5000. The method is unreliable at intermediate values of the Reynolds number. 

The ] atio of the I rns velocity fluctuation to the avei age velocity in the irnpelle] zone is about 50 pei cent with many open irnpellei s. If the ] rns velocity fluctuation is divided bv the avei age velocity iji the I est of the vessel, howevei the I atio is on the oi dei of 5 pei cent. This is also the level of I rns velocity fluctuation to the mean velocity in pipeline flow, Thei e ai e phenomena in rnici o-scale mixing that can occiu in mixing tanks that do not occiu in pipeline I eactoi s, Whethei this is good or bad depends upon the process requirements,... 

Using the expressions given earlier for the mean velocity and the mean free patlr... 

On average, a molecule will remain in the mobile phase a time (ta) before it is adsorbed. During this time, it will be moving at the mean velocity of the mobile phase (u) and will, thus, move a distance (uta). Thus, in moving a distance (l), the total... 

It should be noted that the velocity employed in equation (7) will be the exit velocity (uq) and not the mean velocity (umean)- Differentiating equation (3) and equating to zero, to obtain an expression for the optimum velocity (uopt). 

Piping systems should be designed for an economic flow velocity. For relatively clean fluids, a recommended velocity range where minimum corrosion can be expected is 2 to 10 fps. If piping bores exist, maximum fluid velocities may have a mean velocity of 3 fps for a 3/8-in. bore to 10 fps for an 8-in.-diameter bore. Higher flow velocities are not uncommon in situations that require uniform, constant oxygen supply to form protective films on active/passive metals. 

Instead of the mean velocity weighted with the molar fractions, a velocity weiglited with the mass fractions—the mass center velocity —can be... 

For small downdraft tables used for chemical work, a flow rate of 0.28 m s and m- table is used. This gives a mean velocity immediately above the surface of approximately 0.3 m s h This is a very low velocity, which can not capture moving contaminants. When these values are used, a maximum use height of 0.15 m is recommended, which should result in a small leakage from the source to the surrounding. This presumes there is at least 0.1 m of uncovered surface between the worker and the source and that the surface is covered to less than 30%. For these tables the pressure difference is between 50 and 100 Pa, depending on the density of holes. ... 

The probability density function of u is shown for four points in Fig. 11.16, two points in the wall jet and two points in the boundary layer close to the floor. For the points in the wall jet (Fig. 11.16<2) the probability (unction shows a preferred value of u showing that the flow has a well-defined mean velocity and that the velocity is fluctuating around this mean value. Close to the floor near the separation at x/H = I (Fig. 11.16f ) it is hard to find any preferred value of u, which shows that the flow is irregular and unstable with no well-defined mean velocity and large turbulent intensity. From Figs. 11.15 and 11.16 we can see that LES gives us information about the nature of the turbulent fluctuations that can be important for thermal comfort. This type of information is not available from traditional CFD using models. 

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