Flow, cylindrical duct

For the fluid to expand, the channel must be convergent (Dz < 0) under subsonic conditions (Ma < 1) and divergent (Dz > 0) under supersonic conditions (Ma > 1). Note that Eq. (31) has a singularity at Ma = 1 and changes its sign when the flow speed u z) crosses the speed of sound Msnd- Because Dz = 0 in a cylindrical channel, an adiabatic expansion must be driven by friction (or else the trivial solution dp/dz = 0 is obtained). The friction factor / in Eq. (30) is always positive thus, if one starts at subsonic conditions, flow can never be accelerated beyond the speed of sound inside a cylindrical duct. Flow that attains exactly the speed of sound at the exit of the duct is called choked flow. As we... 

Figure 7.9 shows a cylindrical duct whose radius varies as a function of position, R(z). As long as the radius varies smoothly and relatively smoothly, the channel flow may be treated as a boundary-layer problem. Discuss what, if any, changes must be made to the boundary-layer equations and the boundary-condition specifications to solve the variable-area boundary-layer problem. 

In separation processes and chemical reactors, flow through cylindrical ducts filled with granular materials is important. In such systems conduction, convection, and radiation all contribute to the heat flow, and thermal conduction in axial ke x and radial ke r directions may be quite different, leading to highly anisotropic thermal conductivity. For a bed of uniform spheres, the axial and radial elements are approximated by... 

S. L. Hagen, and D. A. Ratkowsky, Laminar Flow in Cylindrical Ducts Having Regular Polygonal Shaped Cores, Can. J. Chem. Eng., (46) 387-388,1968. 

The Stokes-layer excitation in a cylindrical duct is one of the effective techniques for dynamic calibration of shear stress sensors [8]. The flow inside the duct is driven by an oscillating pressure gradient generated by a loud speaker (see Fig. 10). The loud speaker driven by an amplifier generates an acoustic wave. The amplifier receives sinusoidal input from a function generator. The microphone and shear stress sensor are mounted at opposite locations of the tube. The data acquisition system records the signal from the microphone and shear stress sensor. 

Sherwood numbers for laminar flow in a cylindrical duct. 

Knaff and Schlunder [9] studied the evaporation of naphthalene and caffeine from a cylindrical surface (a sintered metallic rod impregnated with the solute) to high-pressure carbon dioxide flowing over an annular space around the rod. They studied the diffusion flux within the bar and in the boundary layer. The mass-transfer coefficient owing to forced convection from cylinder to the gas flowing in the annular duct was correlated, using the standard correlation due to Stephan [7]. For caffeine, it does not require a free-convection correction, as the Reynolds dependence is that expected by a transfer by forced convection. This is... 

Although it is probably true that flames per se are inherently unstable as pointed up by the work of Markstein (59, 60), nevertheless perfectly stable flames on burners or in ducts can be obtained. The stability arises from interactions among the flame, the flow, and the nearby solid surfaces. But given a stable flame, a condition may be changed —e.g., flow velocity—and the flame caused to flash back or blow off. This section considers flash-back and blow-off of open flames on cylindrical burners or nozzles, and of confined flames stabilized on simple flameholders in small ducts. 

Fully developed flow in a pipe, i.e., a duct with a circular cross-sectional shape, will first be considered [l],[2],[3]. The analysis is, of course, carried out using the governing equations written in cylindrical coordinates. The z-axis is chosen to lie along the center line of the pipe and the velocity components are defined in the same way that they were in Chapter 2, i.e., as shown in Fig. 4.3. 

In axial-flow viscometers, the sample is made to flow through a duct of regular cross-section. Capillary (circular cross-section) and slit (rectangular cross-section) viscometers are controlled stress instruments a known pressure difference (which causes shear stress in the sample) is applied over the duct length, and the resulting volumetric flow rate measured. In the extrusion viscometer, a controlled shear rate instrument, the sample is extruded through a capillary tube by the action of a constant speed piston, acting on the sample in a cylindrical reservoir to which the capillary is attached. The pressure difference between the ends of the capillary is measured. 

Now the experiments not only cover cases in which the assumptions of the theory are fairly well fulfilled but also such in which the duct differs very much from a flat channel placed with its largest side perpendicular to the magnetic field. They even comprise investigations on the flow in cylindrical pipes. Obviously in such cases the theory as given by the equations (1)—(3) cannot be expected to hold good directly. It must be modified in some way or other and it is with this modification or adjustment we are concerned in the following. We may divide our problem into two. The pressure drop in cm. Hg, the quality directly observed, may according to (1) and (2) be written... 

Characteristic length scales chosen depend on system geometry. Eor example, diameter is commonly chosen for cylindrical systems, while the hydraulic diameter, = AAIP, may be chosen for Cartesian geometries (such as a microfluidic duct) in which A denotes cross-sectional area and P is the perimeter of the cross section. As mentioned earlier, the lack of turbulence in microfluidic devices indicates inertial effects are minimal. Consequently, viscous forces dominate. Reynolds numbers characteristic of microfluidic devices are generally on the order of 0(10) to 0(10) [1]. Eurthermore, the transient time required to achieve this laminar flow goes according to t plAlp. Consequently, one can see that flow in microfluidic devices tends to be rather devoid of turbulence. 

Cylindrical surfaces may be placed not only in the center of the stream, but also adjacent to the wall of the duct. For example, in reinforcing the walls of certain air ducts such as in mine drifts, timber sets are used, usually with cylindrical surfaces. The air flow passes around these surfaces. The particular features of flow around cylindrical surfaces in contact with flat surfaces determine the specific features of deposition and adhesion of aerosol particles. The reduction in dust concentration because of adhesion as the dust-laden stream passes through a series timber set can be determined from the formula [248]... 

Fig. X.5. Adhesion number in detachment of loess particles with a diameter of 40-100 jum, by an air stream, from a cylindrical porcelain surface located vertically in a duct as a function of incident angle ip of the stream on the surface, with different flow velocities (m/sec) (1) 0 (2) 5 (3) 7 (4) 10 (5) 15.

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