Axial flow, between cylinders

PIG. S-S. Identification of normal stresses in different experimental geometries, (a), rotation between coaxial cylinders (b), torsion between cone and plate or parallel plates (c) annular axial flow between coaxial cylinders. 

Axial flow in the annulus between two concentric cylinders, as illustrated in Fig. 7-3, is frequently encountered in heat exchangers. For this geometry the hydraulic diameter is Dh = (D0—Di, and the Newtonian laminar flow solution is... 

A publication by B. Colleman, H. Markwitz and W. Noll27 describes in detail the theory of viscosimetric flows which was further used by a number of investigators (see, for example, 23,24,28-29) to analyze axial flow in a clearance between motionless and rotating cylinders. The authors of 27,30) have demonstrated theoretically that the spiral flow can be considered as a mutual overlapping (superimposing) of two strains occurring in a simple shear (see Sect. 1.2). 

Flow in a circular channel with a significant relative length 1/H (here 1 — is the length of circular head H = R2 — Rt is the width of clearance, i.e., the difference between the inner radius of the outer cylinder, the tip, and the outer radius of the inner cylinder, the core) was simulated by the flow of a polymer between two parallel plates removed from one another to a distance H (see Fig. la). The resultant flow occurs due to the pressure difference AP = P, — P2 and motion of the upper plate with velocity U0 in the direction transverse to the axial flow. In this case boundary conditions in the Cartesian system of coordinates are ... 

One of the major results Berman et al. obtained is that the mole ratio of Ca in the absorbent to S in the flue gas has the most important effect. The experimental results for the influence of the Ca/S ratio on the sulfur-removal efficiency, jjs, at different concentrations of C02 in the flue gas are shown in Fig. 7.7. These data were obtained in a reactor with two co-axial cylinders the experimental conditions were flue gas flow rate Vo = 0.001 m3-s 1, diameter of Cylinder I in the reactor D, = 0.06 m, diameter of flue gas exit of the nozzle d = 10 mm, clearance between Cylinders I and II A2 = 5 mm. The results in Fig. 7.7 show the significant influence of C02 in flue gas on the sulfur-removal efficiency %. The reason for this is clear C02 reacts with the absorbent Ca(OH)2, too ... 

Example 2.6 Axial Drag and Pressure Flow between Concentric Cylinders The... 

Axial Drag Flow between Concentric Cylinders Consider the drag flow created in the space formed by two concentric nonrotating cylinders of radii R and Rt, with the inner cylinder moving with an axial velocity V. The system is open to the atmosphere at both ends, (a) Derive the velocity profile, (b) Also obtain the result by making a force balance on a thin fluid shell previously discussed. 

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. 

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... 

To describe the flow between the inner and outer cylinders, we use cylindrical coordinates (r, 6, z) and note that the fluid moves in a circular motion the velocities in the radial the axial directions are zero = 0, Vz = 0, and due to symmetry... 

O. Miyatake, and H. Iwashita, Laminar-Flow Heat Transfer to a Fluid Flowing Axially Between Cylinders with a Uniform Surface Temperature, Int. 1 Heat Mass Transfer, (33) 417-425,1990. 

It has been found that in the case of viscometers with co-axial cylinders, the flow is not homogenous if the distance between cylinders is too wide [1, 18]. The maximum distance should be not higher than 1 mm [ 1]. At higher distance the flow curves correspond to the pseudoplastic fluids with no clearly marked yield stress value. For low shear rate the flow of paste in the gap between the cylinders is not uniform as long as the stress does not exceed the yield stress value this can be derived from the Reiner-Rivhn equation ... 

Baier, G. and Graham, M.D., Two-fluid Taylor-Couette flow with countercurrent axial flow linear theory for immiscible liquids between co-rotating cylinders, Phys. Fluids, 12 (2), 294-303, 2000. 

If a fluid can be considered incompressible, the first principal invariant of the rate of deformation tensor will be zero, Ij = 0. The third principal invariant vanishes in many simple flow situations, like axial flow in pipe, tangential flow between concentric cylinders, etc. In more general terms, the third invariant is zero in rectilinear... 

FIGURE 2.7. Annulus between two concentric cylinders with axial flow. r,. = Radius of inner cylinder = radius of outer cylinder. 

YAMADA, Y "Torque resistance of a flow between rotating coaxial cylinders having axial flow". Bull. JSME, 1962, 5, p 635... 

Fan stack Cylindrical or modified cylindrical structure in which the fan operates. Fan stacks are used on both induced-draft and forced-draft axial-flow propeller fans. Also known as/on cylinder. Fill The section of the cooling tower consisting of materials which are placed within the coohng tower to effect heat and mass transfer between the circulating water and the air flowing through the tower. 

The above equation is the result of Prasanth and Shenoy [11] and may be used with their table for A(n, k). Their tabulated values of A (denoted by in Fig. 5) are simply obtained by solving I, = 0 [where /, is the integral in Eq. (19)]. The value of A increases with increasing n. Prasanth and Shenoy [11] argue that the plane of zero shear necessarily lies between the inner cylinder and the plane of maximum velocity in the case of tangential flows (see Fig. 7), whereas it coincides with the maximum velocity plane in the case of axial flows. 

As the annular flow models have complex solutions in some cases, they may be approximated by flows in an equivalent parallel-plate geometry (where the gap B between the parallel plates corresponds to the diflerence in cylinder radii). The width W of the parallel plates corresponds to the mean circumference of the cylinders (in the axial flow case) and to the length of the cylinders (in the tangential flow case). Worth [14] has provided flow-rate solutions for the equivalent parallel-plate geometries in the following four simple flow situations ... 

Most rheological measurements measure quantities associated with simple shear shear viscosity, primary and secondary normal stress differences. There are several test geometries and deformation modes, e.g. parallel-plate simple shear, torsion between parallel plates, torsion between a cone and a plate, rotation between two coaxial cylinders (Couette flow), and axial flow through a capillary (Poiseuille flow). The viscosity can be obtained by simultaneous measurement of the angular velocity of the plate (cylinder, cone) and the torque. The measurements can be carried out at different shear rates under steady-state conditions. A transient experiment is another option from which both y q and ]° can be obtained from creep data (constant stress) or stress relaxation experiment which is often measured after cessation of the steady-state flow (Fig. 6.10). 

The plots in Figure 10.17 show the axial velocity profiles for various positions along the channel for a Deborah number, De = 1.87. The vertical, dashed lines indicate the locations of each scan of the channel. The initial profile, which is furthest upstream of the cylinder, shows an expected, parabolic shape. As the cylinder is approached, the velocity field becomes distorted, with a larger portion of the flow being diverted toward the lower region where the gap between the cylinder and the boundary of the channel is the largest. 

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