Annular axial flow

Pressure-driven axial flow through a narrow annulus is essentially the same as flow through a slit, but without the side walls. The lack of side walls may be helpful for studying slip phenomena and for reducing residence time distribution. Furthermore, the pressure difference between the outer and inner walls of the annulus gives the second normal stress difference. 

Schematic of axial annular flow die with wall pressure difference measurement 

The shear rate here is actually an average. Curvature causes the shear rate on the inner cylinder to be higher than on the outer wall. Equation 6.4.2 is valid for narrow annuli. Hanks and Larsen (1979) give a full solution for a power law fluid, while McEachem (1966) uses the Ellis model (Bird et al., 1987, p. 233). 

Okubo and Hori use a change of variables and an estimate of the location of the radial position for zero shear stress to derive an explicit relation for N2 

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Volumetric Flow Rate of a Power Law Fluid in Axial Annular Flow... 

The SDF, like the RTD functions, can be calculated from the velocity distribution in the system that is, a certain flow pattern determines both functions. The reverse, however, does not necessarily apply. The calculation of the SDF requires a complete description of the flow pattern, whereas RTD functions often can be calculated from a less than complete flow pattern. For example, the RTD of axial annular flow between two rotating concentric cylinders (helical flow) of a Newtonian fluid depends only on the axial velocity, whereas the SDF depends on both the axial and the tangential velocity... 

Axial annular flow Slit with no edges Ap can give N2 Difficult construction and cleaning... 

Hydrolysis of lactose to glucose and P-galactose (P-galactosidase) Axial-annular flow reactor Delactosization of milk or whey for human consumption... 

In axial annular flow there is curvature across the flow streamlines, that is, curvature in the plane perpendicular to the flow direction x (see Figure 5.1.2). This is similar to the pressure hole error for flow over a parallel slot, eq.6.3.14. Both geometries measure the second normal stress difference. The pressure difference between the outer and inner cylinder gives... 

Normal stress measurements versus wall shear stress for a 1 % solution of polyacrylamide at 25°C in annular flows —N2 from axial annular flow, N] from (solid line) tangential annular flow, N from (dashed line) cone and plate, and Ph the hole pressure error. Adapted from Osmers and Lobo (1976),... 

When a viscoelastic fluid flows down a tilted, open channel, the free surface bulges slightly as indicated in Figive 6.4.4. This curvature is across the streamlines of the flow thus, as with parallel pressure slot and axial annular flow, we expect the bulge to be proportional... 

Pressure oscillations with RMS value up to 10 kPa in two models of lean-burn gas turbine combustors, with heat release around 100 kW, have been actively controlled by the oscillation of fuel flow. The flames were stabilized behind an annular ring and a step in one arrangement, and downstream of an expansion and aided by swirl in the other. Control was sensitive to the location of addition of oscillated fuel. Oscillations in the annular flow were attenuated by 12 dB for an overall equivalence ratio of 0.7 by the oscillation of fuel in the core flow and comprising 10% of the total fuel flow, but negligibly for equivalence ratios greater than 0.75. Oscillation of less than 4% of the total fuel in the annulus flow led to attenuation by 6 dB for all values of equivalence ratio considered. In the swirling flow, control was more effective with oscillations imposed on the flow of fuel in a central axial jet than in the main flow, and oscillations were ameliorated by 10 dB for equivalence ratio up to 0.75, above which the flame moved downstream so that the effectiveness of the actuator declined. The amelioration of pressure oscillations resulted in an increase in NOj, emissions by between 5% and 15%. 

Figure 19.3 Influence of equivalence ratio on antinodal RMS pressure fluctuation annular flow arrangement, bulk mean velocity in main flow, Um = 7-5 m/s bulk mean velocity in pilot stream. Up = 8 m/s Re j = UmD/v = 40,000, axial separation between annular ring and step, A = 0.513. 1 — 4 m = 0.62 2 — 0.70 3 — 0.76 dashed line corresponds to flame detachment...

As a partial check on the derivations in the conical coordinates, it should be possible to recover two, easily identified, special cases—the radial flow between parallel disks and the axial Poiseuille flow in an cylindrical annular gap. The parallel-disk flow (Section 5.5) is the case where 0 = 0, with x taking the role of r and y taking the role of z. In this case, h = De/2 + x = r. The momentum equations become... 

One important aspect of wire-coating is the thickness distribution of the polymer on the surface of the wire as well as the velocity distribution within the die. A simplified wire coating process is presented in the Fig. 6.37, where the wire radius is defined by R and the annulus radius by kR. This type of flow is often referred to as an axial annular Couette flow. 

Axial Pressure Flow between Concentric Cylinders Solve the problem of flow in the horizontal concentric annular space formed by two long cylinders of length L and radii Rt and Ra, caused by an entrance pressure Pp, which is higher than the exit (atmospheric) pressure. Consider the limit as (Rp — R,) / (Rp + Rt) approaches zero. 

Helical Annular Flow Consider the helical annular flow between concentric cylinders with an axial pressure gradient and rotating outer cylinder as shown in the accompanying figure. Specify the equations of continuity and motion (z and 6 components) and show that, if a Newtonian fluid is used, the equations can be solved independently, whereas if tj = t](y), where y is the magnitude of y, the equations are coupled. 

When the radius of the flowing preform reaches the value of Ra, the fluid is forced to flow in the annular space Ra — Rt. For a constant squeeze rate, the rate of increase of the axial annular distance l occupied by the melt is... 

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