Annulus, fluid flow velocity profile

Determine the shear stress distribution and velocity profile for steady, fully developed, laminar flow of an incompressible Newtonian fluid in a horizontal pipe. Use a cylindrical shell element and consider both sign conventions. How should the analysis be modified for flow in an annulus ... 

If flow is laminar with no radial or axial mixing (diffusion coefficient zero), we can write pit) finding p at each annulus that flows along the streamline in plug flow, as shown in Figure 8-4. From fluid mechanics we know that the velocity profile u(R) [really u,(R)] in laminar flow is given by the expression... 

The laminar velocity profile in Figure 8-2la is approximated by a series of annuli, within each of which the velocity is constant as illustrated in Figure 8-2lb. Each annulus is considered to be a plug flow tubular reactor having its own space velocity. The velocities of the fluid elements at different radii are given by the parabolic velocity profile for fully developed laminar flow. The velocity is expressed as... 

Example 4.5 Entropy production in a flow through an annular packed bed The introduction of suitable packing into a fluid flow passage considerably enhances wall-to-fluid heat transfer, and hence reduces the entropy production due to heat transfer but increases the entropy production due to fluid-flow friction. Heat transfer to a fluid flowing in an annulus has a technical importance because we can heat or cool either or both of the surfaces independently. Entropy production provides a new criterion in analyzing such processes. In terms of the velocity and temperature profiles, the local rate of entropy production per unit volume of an incompressible Newtonian fluid for a two-dimensional annular flow is... 

A coal-in-oil slurry which behaves as a power-law fluid is to be heated in a double-pipe heat exchanger with steam condensing on the annulus side. The inlet and outlet bulk temperatures of the slurry are 291 K and 308 K respectively. The heating section (inner copper tube of 40 mm inside diameter) is 3 m long and is preceded by a section sufficiently long for the velocity profile to be fully estabhshed. The flow rate of the slurry is 400kg/h and its thermo-physical properties are as follows density = 900 kg/m heat capacity = 2800 J/kg K thermal conductivity = 0.75 W/mK. In the temperature interval 293 < T < 368 K, the flow behaviour index is nearly constant and is equal to 0.52. 

Profiles are all extruded articles having a cross-sectional shape that differs from that of a circle, an annulus, or a very wide and thin rectangle (flat film or sheet). The cross-sectional shapes are usually complex, which, in terms of solving the flow problem in profile dies, means complex boundary conditions. Furthermore, profile dies are of nonuniform thickness, raising the possibility of transverse pressure drops and velocity components, and making the prediction of extrudate swelling for viscoelastic fluids very difficult. For these reasons, profile dies are built today on a trial-and-error basis, and final product shape is achieved with sizing devices that act on the extrudate after it leaves the profile die. 

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