Transfer of momentum

Now encounters between molecules, or between a molecule and the wall are accompanied by momentuin transfer. Thus if the wall acts as a diffuse reflector, molecules colliding wlch it lose all their axial momentum on average, so such encounters directly change the axial momentum of each species. In an intermolecuLar collision there is a lateral transfer of momentum to a different location in the cross-section, but there is also a net change in total momentum for species r if the molecule encountered belongs to a different species. Furthermore, chough the total momentum of a particular species is conserved in collisions between pairs of molecules of this same species, the successive lateral transfers of momentum associated with a sequence of collisions may terminate in momentum transfer to the wall. Thus there are three mechanisms by which a given species may lose momentum in the axial direction ... 

Transfer of Momentum Deceleration of one fluid (motivating fluid) in order to transfer its momentum to a second fluid (pumped fluid) is a principle commonly used in the handhng of corrosive materials, in pumping from inaccessible depths, or for evacuation. Jets and eductors are in this categoiy. 

Since the dislocation drag coefficient B represents the transfer of momentum per unit area, we assume that B/m remains constant as the velocity increases and hence... 

Mixing by agitation of liquids normally involves the transfer of momentum from an impeller to the liquid. In some cases, mixing is achieved by gas injection or circulation via a pump loop. An impeller, which is mounted on a shaft driven by an electric motor, is divided into two operation categories ... 

Mucus gel is propelled toward the epiglottis by a two-phase ciliary beat cycle. Forward mucus movement occurs during the effective or power phase of the cycle, when cilia fully extend and traverse an arc perpendicular to the epithelial surface (Fig. 5.24). Claw-like structures, 25-35 nm long, project from each cilia tip and appear to assist in the mechanical transfer of momentum from cilia to mucus gel. Maximum mucus velocity depends on the extent cilia penetrate the epiphase during the power phase, periciliary and mucus gel viscosity, and cilia density. 

A number of three-phase processes (processes in which contact is established between a gaseous phase, a liquid phase, and a solid-particle phase in order to promote chemical conversion and the transfer of momentum, heat, and mass) are becoming increasingly important in the process industries. 

The gas-liquid-particle processes considered in this paper may be grouped into two major classes. In the first, components of all three phases participate in the chemical reaction. In the second, components of only the gaseous and the solid phases participate in the chemical reaction, the liquid phase functioning as a chemically inactive medium for the transfer of momentum, heat, and mass. Important examples of these two types of processes are described, respectively, in Sections II,A and II,B. 

The rate of transfer of momentum through the elementary strip... 

In die streamline boundary layer the only forces acting within the fluid are pure viscous forces and no transfer of momentum takes place by eddy motion. 

The shear stress Ry within the fluid, at a distance y from the boundary surface, is a measure of the rate of transfer of momentum per unit area at right angles to the surface. 

Since (pux) is the momentum per unit volume of the fluid, the rate of transfer of momentum per unit area is proportional to the gradient in the Y-direction of the momentum per unit volume. The negative sign indicates that momentum is transferred from the fast-to the slow-moving fluid and the shear stress acts in such a direction as to oppose the motion of the fluid. 

The essential similarity between the three processes is that the rates of transfer of momentum, heat, and mass are all proportional to the concentration gradients of these quantities. In the case of gases the proportionality constants p/p, Dh, and D, all of which have the dimensions length2/time, all have a physical significance. For liquids the... 

The net rate of transfer of momentum away from the surface... 

Thus the rate of transfer of momentum per unit area which can be written as ... 

It is thus seen that the kinematic viscosity, the thermal diffusivity, and the diffusivity for mass transfer are all proportional to the product of the mean free path and the root mean square velocity of the molecules, and that the expressions for the transfer of momentum, heat, and mass are of the same form. 

If there is a temperature gradient within the fluid, the eddies will be responsible for heat transfer and an eddy thermal diffusivity Ep may be defined in a similar way. It is suggested that, since the mechanism of transfer of heat by eddies is essentially the same as that for transfer of momentum, Eh is related to mixing length and velocity gradient in a similar manner. 

Consider the equilibrium set up when an element of fluid moves from a region at high temperature, lying outside the boundary layer, to a solid surface at a lower temperature if no mixing with the intermediate fluid takes place. Turbulence is therefore assumed to persist right up to the surface. The relationship between the rates of transfer of momentum and heat can then be deduced as follows (Figure 12.5). 

Close to the ground, in the constant flux layer, u. is a measure of the turbulent transfer of momentum. The friction velocity is defined by... 

Fig. 9. Reduction in kt by spread monolayer of protein, due to prevention of transfer of momentum from ethylacetate, and to reduction in turbulence in the aqueous phase near the interface. Experimental data used here are taken from reference (60).

The observed rates of transfer are lower than those calculated by the correlation of Eq. 26 for organic molecules which themselves are surface-active, without specifically added long-chain molecules thus in the transference of (C4H9)4NI from water to nitrobenzene, of benzoic acid from toluene to water and the reverse, of diethylamine between butyl acetate and water, of n-butanol from water to benzene, and of propionic acid between toluene and water, the rates (44, 4 ) are of the order one-quarter to one-half those calculated by Eqs. (25) and (26). Since with these systems the solute itself is interfacially active, and therefore its monolayers should reduce the transfer of momentum, we interpret these findings as indicative that Ri and R2 are increased in this way. This is... 

This natural circulation occurs by a direct transfer of momentum across the interface, and the presence of a monolayer at the interface will affect it in two ways. Firstly, the surface viscosity of the monolayer may cause a dissipation of energy and momentum at the surface, so that the drop behaves rather more as a solid than as a liquid, i.e., the internal circulation is reduced. Secondly, momentum transfer across the surface is reduced by the incompressibility of the film, which the moving stream of gas will tend to sweep to the rear of the drop (Fig. 14b) whence, by its back-spreading pressure n, it resists further compression and so damps the movement of the surface and hence the transfer of momentum into the drop. This is discussed quantitatively below, where Eq. (32) should apply equally well to drops of liquid in a gas. 

The problems of two-phase gas-liquid flow in pipe lines have become of greater concern to engineers in recent years. This type of flow is encountered in an increasing number of important situations, and a clear understanding of the rates of transfer of momentum, heat, and material will be required for logical and careful design and operation of a very wide variety of chemical engineering equipment and processes. 

The primary means of transfer of energy to the fluid that causes flow are gravity, displacement, centrifugal force, electromagnetic force, transfer of momentum, mechanical impulse, and a combination of these energy-transfer mechanisms. Displacement and centrifugal force are the most common energy-transfer mechanisms in use. 

In chemical engineering, the terms transfer of heat, mass, and momentum are referred to as the transport phenomena. The heating or cooling of fluids is a case of heat transfer, a good example of mass transfer being the transfer of oxygen from air into the culture media in an aerobic fermentor. When a fluid flows through a conduit, its pressure drops because of friction due to transfer of momentum, as shown later. 

---