Reduced relative velocity

It should be noted, however, that some of the tendencies described above may become invalid for very small droplets (for example, smaller than 10 pm under conditions in Ref. 156). Such small droplets may require a longer flight time to a given axial distance far from the atomizer due to the high deceleration, and their cooling rates may decrease as a result of the reduced relative velocity and temperature. In addition, the two-way coupling 576] may affect the momentum and heat transfer between atomization gas and droplets so that the droplet behavior may be different from that discussed above, particularly the radial distributions of droplet sizes and velocities. 

This dissipative force is proportional to the relative velocity of the two beads and acts so as tc reduce their relative momentum, v is tire difference between the two velocities (Vy = v, — v ) and vP rjj) is a weight function that depends upon the distemce and disappears for interbead distances greater than unity (i.e. r ). 

Impulse diagram. For the impulse rotor, the reaction is zero, so the relative velocity of the gas is constant, or = W/. If the work factor is less than 2.0, the exit swirl is positive, which reduces the stage work. For this reason, an impulse diagram should be used only if the work factor is 2.0 or... 

The centrifugal force produced increases the impeller static pressure [the first term of the right-hand side of Eq. (9.79)] and reduces the relative velocity U2 To increase the static pressure by a change in the relative velocity, the following relationship is necessary ... 

The proposed technique will be used here to illustrate the case of interfacial heat and multicomponent mass transfer in a perfectly mixed gas-liquid disperser. Since in this case the holding time is also the average residence time, the gas and liquid phases spend the same time on the average. If xc = zd = f, then for small values of t, the local residence times tc and td of adjacent elements of the continuous and dispersed phases are nearly of the same order of magnitude, and hence these two elements remain in the disperser for nearly equal times. One may conclude from this that the local relative velocity between them is negligibly small, at least for small average residence times. Gal-Or and Walatka (G9) have recently shown that this is justified especially in dispersions of high <6 values and relatively small bubbles in actual practice where surfactants are present. Under this domain, Eqs. (66), (68), (69) show that as the bubble size decreases, the quantity of surfactants necessary to make a bubble behave like a solid particle becomes smaller. Under these circumstances (pd + y) - oo and Eq. (69) reduces to... 

For example, for equal volumes of gas and liquid ( =0.5), Eq. (266) predicts that the Stokes velocity (which is already very small for relatively fine dispersions) should be reduced further by a factor of 38 due to hindering effects of its neighbor bubbles in the ensemble. Hence in the domain of high values and relatively fine dispersions, one can assume that the particles are completely entrained by the continuous-phase eddies, resulting in a negligible convective transfer, although this does not preclude the existence of finite relative velocities between the eddies themselves. 

The same is true for the distribution of relative velocities, provided one replaces the mass with the reduced mass fi (defined in Eq. 55) of the two molecules ... 

The relative collision energy in a CMB experiment is given by Ec = gv , where g is the reduced mass of the system and vr is the relative velocity. In general... 

Because the batch flux data are obtained in a closed system with no outflow, the net solids flux is zero in the batch system and Eq. (14-40) reduces to FL = —(pV%/ 1 — cp). Note that FL and Vs are of opposite sign, because the displaced liquid moves upward as the solids settle. The relative velocity between the solids and liquid is Vr = Vs — VL which, from Eq. (14-20), is Vr = Fs/(1 — relative velocity that controls the dynamics in the thickener. If the underflow draw-off rate from the thickener is gu, the additional solids flux in the thickener due to superimposition of this underflow is qu = Qu/A = Vu. Thus, the total solids flux at any point in the thickener (qs) is equal to the settling flux relative to the suspension (i.e., the batch flux qsb) at that point, plus the bulk flux due... 

For two particles having masses mi, m2, the distribution function of relative velocity is obtained by substituting for m the reduced mass m mil m + m2) (see Problem 6 at the end of this chapter). 

Prove the assertion in the text that the relative velocity of two sets of particles having individual Maxwellian velocity distribution functions also has a Maxwellian distribution with the masses replaced by the reduced mass. 

It is now found that (22) is indeed invariant under (24), which is known as the Lorentz5 transformation of Special Relativity. It is important to note that in the limit v/c —> 0 the Lorentz formulae reduce to the Galilean transformation, suggesting that Lorentzian (relativistic) effects only become significant at relative velocities that approach c. The condition t = t which... 

The influence of liquid density on the mean droplet size is relatively small but complex. An increase in liquid density may reduce the mean droplet size due to a decrease in sheet thickness at the atomizing lip of a prefilming atomizer, or due to an increase in the relative velocity between liquid and air for a plain-jet atomizer. However, increasing liquid density may also increase the mean droplet size because a liquid sheet may extend further downstream of the atomizing lip of a prefilming atomizer so that the sheet breakup may take place at lower relative velocity between liquid and air. 

In addition to controlling the standard process parameters such as down force and the relative velocity, it is also important to have random access capability to route wafers through a CMP tool to optimize both performance and throughput. Low-down-force processes and special CMP pads are likely to be necessary to reduce copper dishing just as they improve oxide planarization. Furthermore, a balance between high relative velocity to reduce copper dishing and moderate relative velocity to minimize the sheering of small oxide feature may be necessary. 

Before deriving the expressions for gas-phase collision frequency, we need to discuss the relative velocity that is important in collisions. The reduced mass mn is also obtained from this analysis, and will be the appropriate mass to use in the Maxwell-Boltzmann expression for collision velocities. 

The formulas that we have derived in this chapter and in Chapter 8 to describe energy and velocity distributions also apply to the center of mass and relative velocities. In particular, the distribution of relative velocities obeys the Maxwell-Boltzmann distribution of Eq. 10.27, with the mass replaced by the reduced mass /W 2 ... 

In the impinging streams of gas-liquid systems, high relative velocity between phases and collision between droplets favor surface renewing of droplets, resulting in reduced liquid film resistance and thus increased overall mass transfer coefficient. 

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