Relative velocity, definition

Here, p is the density of the fluid, V is the relative velocity between the fluid and the solid body, and A is the cross sectional area of the body normal to the velocity vector V, e.g., nd1/4 for a sphere. Note that the definition of the drag coefficient from Eq. (11-1) is analogous to that of the friction factor for flow in a conduit, i.e.,... 

This result can also be applied directly to coarse particle swarms. For fine particle systems, the suspending fluid properties are assumed to be modified by the fines in suspension, which necessitates modifying the fluid properties in the definitions of the Reynolds and Archimedes numbers accordingly. Furthermore, because the particle drag is a direct function of the local relative velocity between the fluid and the solid (the interstitial relative velocity, Fr), it is this velocity that must be used in the drag equations (e.g., the modified Dallavalle equation). Since Vr = Vs/(1 — Reynolds number and drag coefficient for the suspension (e.g., the particle swarm ) are (after Barnea and Mizrahi, 1973) ... 

In a 3D system, however, it becomes more complicated. The particle-particle contact now occurs in a plane. The tangential component of the relative velocity is always in this plane and vertical to the normal unit vector according to the definition. Since the normal unit vector is not necessarily situated in the same plane at any time, it is desirable to transfer the old tangential displacement to the new contact plane before we calculate the new tangential displacement. To this end, a 3D rotation of the old tangential displacement should be applied. As... 

The reaction rate coefficients in the above equations may be related to reaction rates per pair of particles 2/, in nuclear physics (e.g., Fowler et al., 1975 Harris et al., 1983) by k = Xj/(1 + 5/ ), where 8 = 0 except for i= , for which 5/ = 1. That is, for Reactions 2-145 and 2-147 in which two identical particles collide to react, the definition of k is half of defined by nuclear physicists and for reactions in which different particles collide, the definition of k is the same as Xij. The reaction rate coefficients depend on temperature in a complicated way (Table 2-3) and may be calculated as the average value of the product of relative velocity times cross section. The concentrations of the intermediate species can be derived as follows. From Equation 2-155, 145 [ H] = ki4e[ H]pH]. That is. 

POISE (P). A unit of dynamic viscosity. The unit is expressed in dyne second per square centimeter The centipoise (cP) is more commonly used The formal definition of viscosity arises from the concept put forward by Newton that under conditions of parallel flow, the shearing stress is proportional to the velocity giadieut. If lire force acting on each of two planes of aiea A parallel to each oilier, moving parallel lo each other with a relative velocity V, and separated by a perpendicular distance X, be denoted by F. the shearing stress is F/A and the velocity gradient, which will be linear for a true liquid, is V/X. Thus, Ft A = q V/X, where the constant if is the viscosity coefficient or dynamic viscosity of the liquid. The poise is the CGS unit of dynamic viscosity. 

Since the scope of this article is purely theoretical, we just outline below the state of the experimental situation. The ideal experiment in Chemical Dynamics would be that in which starting with reactants in definite intramolecular quantum-states and running towards each other in a definite way (relative velocity and orbital angular momentum) the distribution of the products over the various intramolecular quantum-states and the state of the relative motion (direction and velocity) would be measured. Such an experiment would show whether there is a preferential molecular orientation at the heart of the collision, what the lifetime of the intermediate complex is, how the excess energy is distributed over the various degrees of freedom of... 

Darton and Harrison (1975) derived a criterion for the point of transition to predict whether a solid-liquid fluidized bed will expand or contract when the gas is first introduced. The definition of Pa used by Darton and Harrison was the ratio of upper clear (particle-free) wake volume to the bubble volume. But since they did not consider the circulation of sohds associated with the lower nonclear portion of the wake, their Pa was effectively the same as that of Bhatia and Epstein (1974). The use of the Wallis drift flux approach by Darton and Harrison (1975) also represents no real difference from the relative velocity approach taken by Bhatia and Epstein (1974), since the two methods are rigorously interrelated. It is therefore not surprising that the final criteria of Bhatia and Epstein (1974) and Darton and Harrison (1975) are identical. 

By Newton s definition the viscosity or, more appropriately, the viscosity coefficient, jj, of a fluid in a laminar steady-state flow is expressed as the tangential force, F, per unit area. A, required to maintain a unit rate of shear (or velocity gradient), G, in the liquid. If the liquid fills the space between two parallel planes of area. A, one of which moves at a constant distance, r, from the other with a relative velocity, u, then we have... 

The experiment measures the rate of transition into a solid angle dSl subtended by the detector at scattering angles 9, energy channels i, 0 are defined by energy resolution. In atomic units the relative velocity V is ko. We use the notation dai 9,4>) for the differential cross section in this experiment. The definition (6.41) becomes... 

In (3.427) the dispersed phase velocity occurs as an undetermined variable. The phasic velocities are related to the mixture velocity through the mixture velocity definition (3.421). The dispersed phase velocity is computed from the continuous phase velocity Vc and a relative (slip) velocity v fc, in accordance with the definitions ... 

The relationship between holdup and slip (relative) velocity between the two phases is, by definition,... 

The actual composition of the ternary compound is not known definitely, and has, therefore, been given in the simplest form, but even if different numbers of the reacting molecules were contained in it, the same principles and relations may be applied although the equations would be much more complicated. The relative velocities of the dissociation of the intermediate compound would determine which products are observed at any time, if the complete reaction has not been allowed to come to equilibrium. It is perhaps more usual to speak of the velocities of the various reactions than of the values of their equilibrium constants. 

The olefinates of water differ from those of the hydrogen halides in that the tertiary alcohols appear to lose water to form the olefins more readily than do the secondary, and the secondary more readily than do the primary. This is the opposite to the order observed with the halides, but whether this difference refers to the actual thermodynamic stability or to the relative velocities under certain conditions is not definite. 

When a flame traverses a pre-formed combustible mixture which is at rest in a horizontal tube, a zone of intense reaction moves along the tube with a definite velocity. If the gas mixture is streamed through the tube at an appropriate rate, the flame front itself may be maintained stationary at the mouth. In either case the relative velocity of flame front and gas mixture is the same. If the speed of the stream is less than the rate of travel of the combustion zone, the flame performs the operation known as striking back. If, on the other hand, the gas stream emerges from the tube faster than the combustion is propagated, the flame spreads forth from the mouth... 

The definitions (30,11) and (31.11) also hold for the cross sections of inelastic and reactive scattering in which the particles before and after the collision have different internal states and/or different compositions. Then, the letters i and j may be used to represent two sets of qusmttmi numbers for the initial state (separated reactants) and for the final state (separated products), respectively, in which also the reduced masses i and j ) and the relative velocities (v and v ) are generally different. 

The orthokinetic definition sketch of Figure 4.12 will still apply, with relative velocity u replaced by the differential settling velocity (v, — y,) in the direction of gravity (represented by the x-direction). The flux of i-particles through the cylinder radius Ry ( = r, + / ), centred on a y-particle, is... 

We have to be careful in the definition of gas and particle velocities and in the relative velocity between them, the slip velocity. The terms are often used loosely in the literature and are defined below. 

Solids can be transferred by gravity from a low pressure to a higher pressure if gas flows upward relative to the downward flowing solids. This relative gas-solids flow will then generate the sealing pressure drop required for the system. The direction of the actual gas flow in the standpipe relative to the standpipe wall can be either up or down and still have the relative gas-solids velocity, v, directed upwards. This is sometimes difficult to understand, but it can be explained with the aid of Fig. 2, and the definition of the relative velocity. The relative gas-solids velocity, v,., is defined as... 

The vector diagram for a typical scattering collision in the laboratory system of coordinates is shown in P ig. 4.26. The unprimed quantities refer to the velocities before collision, the primed, after collision. As indicated in the definition above for is the angle between the relative velocity vector before collision p and the relative velocity vector after collision p ... 

In an ideal situation both the trapped ions and the cooling neutrals reach a thermal equilibrium at a common temperature T which is usually defined by the surrounding walls. Under such conditions the distribution of the relative velocity, f g) (definition see Eq. (3.10) of Chapter 3 and also Ref. 18) is given by... 

A pure liquid means that this material is one single component and at room temperature it is in liquid form. Polymer solutions are obviously excluded from this section and will be discussed in section 3 of this chapter. Viscosity of liquid was treated theoretically by Andrade [6] and Glasstonc [7]. Consider two layers of molecules in a liquid having a distance d apart and relative velocity between these two layers is v, under a shear force f, the viscosity, r, according to the definition, can be expressed as... 

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