Radial flow outwards, velocity

It should he pointed out, however, that Eq. (11.3) assumes Newtonian behavior, which the complex polymeric resists and B ARC fluids do not necessarily exhibit. In particular, mass is not lost, neither from the radial flow of material nor from evaporation of solvent. Meyerhofer considered the effects of evaporation on the final film thickness. He reported that the final solid film thickness is inversely proportional to the square root of the rotational velocity. He also developed a model similar to that considered above, but allowed the solvent to evaporate during the spinning process. His central assumption was that the thinning process could be divided into two major stages, namely, one dominated by radial flow outward and another by evaporation of solvent. Effectively, he assumed a constant rate of evaporation and the viscosity concentration relationship expressed as... 

Cyclone collectors utilize the principle of centrifugal force to separate particulates from a gas stream vortex flow is induced by the design of the gas inlet duct (Figure 23.4). The main vortex is characterized by axial flow away from the gas inlet and radial flow outward from the edge of the cyclone body. The central core has the same rotary direction, but the axial and radial velocity components are in the opposite direction to that of the main vortex. 

As illustrated in Fig. 5.2, the classic Jeffery-Hamel flow concerns two-dimensional radial flow in a wedge-shaped region between flat inclined walls. The flow may be directed radially outward (as illustrated) or radially inward. The flow is assumed to originate in a line source or terminate in a line sink. Velocity at the solid walls obeys a no-slip condition. In practice, there must be an entry region where the flow adjusts from the line source to the channel-confined flow with no-slip walls. The Jeffery-Hamel analysis applies to the channel after this initial adjustment is accomplished. 

Spiral vortex. So far the discussion has been confined to the rotation of all particles in concentric circles. Suppose there is now superimposed a flow with a velocity having radial components, either outward or inward. If the height of the walls of the open vessel were less than that of a liquid surface spread out by some means of centrifugal force, and if liquid were supplied to the center at the proper rate by some means, then it is obvious that liquid would flow outward, over the vessel walls. If, on the other hand, liquid flowed into the tank over the rim from some source at a higher elevation and were drawn out at the center, the flow would be inward. The combination of this approximately radial flow with the circular flow will result in path lines that are some form of spirals. 

Spiral vortex. If a radial flow is superimposed upon the concentric flow previously described, the path lines will then be spirals. If the flow goes out through a circular hole in the bottom of a shallow vessel, the surface of liquid takes the form of an empty hole, with an air core sucked down the hole. If an outlet symmetrical with the axis is provided, as in a pump impeller, we might have a flow either radially inward or radially outward. If the two plates are a constant distance B apart, the radial flow with a velocity Vr is then across a series of concentric cylindrical surfaces whose area is 0.2nrB. Thus Q = 2nrBVr is a constant, from which it is seen that rVr is a constant. Thus the radial velocity varies in the same way with r that the circumferential velocity did in the preceding discussion. Hence the pressure variation with the radial velocity is just the same as for pure rotation. Therefore the pressure gradient of flow applies exactly to the case of spiral flow, as well as to pure rotation. 

In contrast, a vaned wheel directs the flow of the liquid feed across the surface of an inner liquid distributor in which liquid slippage over the surface of the distributor occurs until there is contact with the vane or channel. The feed then flows outward due to centrifugal force and forms a thin film across the surface of the vane. As the liquid film leaves the edge of the vane, droplet formation occurs as a result of the radial and tangential velocities experienced. Atomizer wheel characteristics that influence droplet size include speed of rotation, wheel diameter, and wheel design, e.g., the number and geometry of the vanes. 

Equation 26 is accurate only when the Hquids rotate at the same angular velocity as the bowl. As the Hquids move radially inward or outward these must be accelerated or decelerated as needed to maintain soHd-body rotation. The radius of the interface, r, is also affected by the radial height of the Hquid crest as it passes over the discharge dams, and these crests must be considered at higher flow rates. 

If, in an infinite plane flame, the flame is regarded as stationary and a particular flow tube of gas is considered, the area of the flame enclosed by the tube does not depend on how the term flame surface or wave surface in which the area is measured is defined. The areas of all parallel surfaces are the same, whatever property (particularly temperature) is chosen to define the surface and these areas are all equal to each other and to that of the inner surface of the luminous part of the flame. The definition is more difficult in any other geometric system. Consider, for example, an experiment in which gas is supplied at the center of a sphere and flows radially outward in a laminar manner to a stationary spherical flame. The inward movement of the flame is balanced by the outward flow of gas. The experiment takes place in an infinite volume at constant pressure. The area of the surface of the wave will depend on where the surface is located. The area of the sphere for which T = 500°C will be less than that of one for which T = 1500°C. So if the burning velocity is defined as the volume of unbumed gas consumed per second divided by the surface area of the flame, the result obtained will depend on the particular surface selected. The only quantity that does remain constant in this system is the product u,fj,An where ur is the velocity of flow at the radius r, where the surface area is An and the gas density is (>,. This product equals mr, the mass flowing through the layer at r per unit time, and must be constant for all values of r. Thus, u, varies with r the distance from the center in the manner shown in Fig. 4.14. 

The pressures developed in the deton reaction zone in condensed expls are of the order of 103 to 10 atm. Material at such pressures cannot in general be contained, so that the flow behind the front has a component radially outward. Gases, which develop much lower deton pressures (of the order of 10 atm), can be confined in a tube, and for them the one- dimensional approximation is good. The diverging flow is expected and is found experimentally to result in lower pressures and densities within the steady wave, and consequently in lower detonation velocities. 

Air flows radially outward from the surface of a porous spherical shell. The sphere has diameter D = 0.02 m. Initially the surrounding gas is at rest and the sphere s surface temperature and the surrounding-gas temperature is T oo = 300 K. Abruptly, at the start of the problem, the surface temperature is raised to Tsm = 600 K, and the gas velocity at the spherical surface is v = 10 m/s. 

Fig. 5.3 Nondimensional velocity distribution for the Jeffery-Hamel flow between two plates inclined at a = 10°. A negative Re indicates radially inward flow. Separation occurs at approximately Re 4 for the outward flow.

---