Traversing for Mean Velocity

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-... 

The mean velocity field is found to be self-similar for all the nozzle exit velocities, distances between nozzles and turbulence generators tested. This similarity allows the mean axial velocity traverses to be normalized so that all the measured data lie on the same curve. The shape of the curve is similar to that given in Fig. 1.8. 

Figure 12 shows the velocity distribution in front of the monolith inlet for an industrially housed automobile catalyst [14]. Since the flow cannot follow the sudden widening of the inlet funnel, one third of the total cross section is traversed at a velocity that is roughly three times the mean velocity. It can be estimated that, with uniform flow through the catalyst, half the catalyst volume would be sufficient for the same mean conver-... 

It is well known that the particle current density Sjj. in the direction i — k is given by the product of the homogeneous volume concentration n of the migrating particles and their mean velocity caused by the i k transitions. This velocity, however, is equal to the number wjfc of i - k transitions per second multiplied by the path r traversed during each transition. For the reverse transitions k — i one must substitute wjci The resultant particle current density in the direction i k is thus... 

Equations (5.34) and (5.35) can be used to obtain expressions for streamline flow of time-independent fluids through beds of particles, in which case V must be replaced by the mean velocity in the pores or interstices, i.e. V, and the length L is replaced by the average length of the tortuous path, L , traversed by the fluid elements. 

Example 1.17 Assume that the two laser beams Li and L2 for excitation and ionization have the diameter D = 1 cm and traverse a collimated molecular beam with 1 cm cross section perpendicularly. During the pulse time AT = 10 s the distance traveled by the molecules at the mean velocity... 

Assume that the two laser beams LI and L2 for excitation and ionization have the diameter D = 1 cm and traverse a collimated molecular beam with 1-cm cross section perpendicularly. During the pulse time AT = 10" s the distance traveled by the molecules at the mean velocity V = 500m/s is = ATT 5 X 10" cm. This means that all molecules in the excitation volume of Icm can be ionized during the time AT. During the dark time T = 1//l, however, the molecules travel the distance d vT 500 cm at /l = lO s " Therefore, only the fraction 1/500 = 2 X 10 of all molecules in the absorbing level (/ are ionized in a continuous molecular beam. 

Once these traverse points have been determined, velocity measurements are made to determine gas flow. The stack-gas velocity is usually determined by means of a pitot tube and differential-pressure gauge. When velocities are very low (less than 3 m/s [10 ft/s]) and when great accuracy is not required, an anemometer may be used. For gases moving in small pipes at relatively high velocities or pressures, orifice-disk meters or venturi meters may be used. These are valuable as continuous or permanent measuring devices. 

To determine the mean angular velocity of this spiraling motion, we first calculate the time required for the particle to traverse one turn of the coil. We have from Eq. 5... 

Molecular dynamics with periodic boundary conditions is presently the most widely used approach for studying the equilibrium and dynamic properties of pure bulk solvent,97 as well as solvated systems. However, periodic boundary conditions have their limitations. They introduce errors in the time development of equilibrium properties for times greater than that required for a sound wave to traverse the central cell. This is because the periodicity of information flow across the boundaries interferes with the time development of other processes. The velocity of sound through water at a density of 1 g/cm3 and 300 K is 15 A/ps for a cubic cell with a dimension of 45 A, the cycle time is only 3 ps and the time development of all properties beyond this time may be affected. Also, conventional periodic boundary methods are of less use for studies of chemical reactions involving enzyme and substrate molecules because there is no means for such a system to relax back to thermal equilibrium. This is not the case when alternative ensembles of the constant-temperature variety are employed. However, in these models it is not clear that the somewhat arbitrary coupling to a constant temperature heat bath does not influence the rate of reequilibration from a thermally perturbed... 

The average velocity of a gas molecule is determined by the molecular weight and the absolute temperature of the gas. Air molecules, like many other molecules at room temperature, travel with velocities of about 500 m s"1 but there is a distribution of molecular velocities. This distribution of velocities is explained by assuming that the particles do not travel unimpeded but experience many collisions. The constant occurrence of such collisions produces the wide distribution of velocities. The quantitative treatment was carried out by Maxwell in 1859, and somewhat later by Boltzmann. The phenomenon of collisions leads to the concept of a free path, that is the distance traversed by a molecule between two successive collisions with other molecules of that gas. For a large number of molecules, this concept must be modified to a mean free path which is the average distance travelled by all molecules between collisions. For molecules of air at 25°C, the mean free path X at 1 mbar is 0.00625 cm. It is convenient therefore to use the following relation as a scaling function ... 

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