Pipe flow analysis

The phenomena are quite complex even for pipe flow. Efforts to predict the onset of instabiHty have been made using linear stabiHty theory. The analysis predicts that laminar flow in pipes is stable at all values of the Reynolds number. In practice, the laminar—turbulent transition is found to occur at a Reynolds number of about 2000, although by careful design of the pipe inlet it can be postponed to as high as 40,000. It appears that linear stabiHty analysis is not appHcable in this situation. 

Many developers of software for finite-element analysis (18) provide drafting of pipelines and associated flow analysis. These companies include Algor, McAuto, MacNeal-Schwindler, and ElowDesign. In software, in-house developed pipe fittings are modularized and isometric views of the piping systems with three-dimensional detailing are now commonplace. 

The Lapple charts for compressible fluid flow are a good example for this operation. Assumptions of the gas obeying the ideal gas law, a horizontal pipe, and constant friction factor over the pipe length were used. Compressible flow analysis is normally used where pressure drop produces a change in density of more than 10%. 

The scope of coverage includes internal flows of Newtonian and non-Newtonian incompressible fluids, adiabatic and isothermal compressible flows (up to sonic or choking conditions), two-phase (gas-liquid, solid-liquid, and gas-solid) flows, external flows (e.g., drag), and flow in porous media. Applications include dimensional analysis and scale-up, piping systems with fittings for Newtonian and non-Newtonian fluids (for unknown driving force, unknown flow rate, unknown diameter, or most economical diameter), compressible pipe flows up to choked flow, flow measurement and control, pumps, compressors, fluid-particle separation methods (e.g.,... 

Figure 4-13 Sonic pressure drop for adiabatic pipe flow for various heat capacity ratios. From AICHE/CCPS, Guidelines for Consequence Analysis of Chemical Releases (New York American Institute of Chemical Engineers, 1999).

Heat flow analysis, 10 163 Heat flow tables, 23 284 Heat-flux transformation, in heat pipes, 13 228... 

To simplify the analysis of wave motions in stratified pipe flows, several assumptions can be made ... 

A stratified horizontal pipe flow is schematically illustrated in Fig. 6.16. In addition to the assumptions given in 6.5.1.1, other assumptions are introduced in the analysis of the stratified horizontal pipe flow ... 

To analyze the dynamic behavior of gas-solid pipe flows, the most common and easiest system to consider is a dilute gas-solid pipe flow which is fully developed and is subject to the effects of electrostatic force and gravitational force. The fully developed flow here refers to the situation where the velocity profiles of both gas and particles are unchanged along the axial direction. The system of this nature was analyzed by Soo and Tung (1971). In this section, the analysis of Soo and Tung (1971) is presented. It is assumed that no particles are deposited on the wall surface of the pipe (or the particle deposition rate is zero). Moreover, the pipe flow is considered to be turbulent, as is true for most flow conditions. 

Figure 11.11. General coordinates for the analysis of fully developed dilute pipe flows.

The preceding analysis is based on Tien s model of heat transfer by a gas-solid suspension in turbulent pipe flow [Tien, 1961]. However, nonuniform distribution of solids, slip between solids and gas, and effect of thermal radiation were excluded in Tien s work. 

Velocity measurements made in a trapezoidal canal, reported by O Brien, yield the distribution contours, with the accompanying values of the correction factors for kinetic energy and momentum. The filament of maximum velocity is seen to lie beneath the surface, and the correction factors for kinetic energy and momentum are greater than in the corresponding case of pipe flow. Despite the added importance of these factors, however, the treatment in this section will follow the earlier procedure of assuming the values of a and p to be unity, unless stated otherwise. Any thoroughgoing analysis would, of course, have to take account of their true values. 

Recall that, using dimensional analysis for pipe flow, we saw (see Section 3.2) ... 

Coordinate system used in analysis of pipe flow. 

Fully developed flow in a very wide duct is considered in this section, the flow situation considered being shown in Fig. 4.5. Because the duct is wide, changes in the flow properties in the jc-direction are negligible, i.e., flow between two large parallel plates is effectively being considered. The analysis is, of course, similar to that adopted in dealing with pipe flow. 

The Modifications to the above analysis to deal with the uniform wall heat flux case are basically the same as that required for pipe flow and will not be discussed here. 

In order to illustrate the main features of the analysis of turbulent flow, attention will be restricted to two-dimensional boundary layer flows and to axially symmetric pipe flows. It will also be assumed that the fluid properties are constant and that the mean flow is steady. 

Eqs. (5.3), (5.16), and (5.20) are basically the form of the governing equations that will be used in the analysis of turbulent boundary layer flows. As mentioned before, attention will also be given to turbulent pipe flows. If the same coordinate system that was used in the discussion of laminar pipe flows is adopted, i.e., if the coordinate system shown in Fig. 5.2 is used, the equations governing turbulent pipe flow are, if assumptions similar to those used in dealing with boundary layer flows are adopted and if it is assumed that there is no swirl, as follows ... 

FIGURE 5.2 Coordinate system used in the analysis of turbulent pipe flow. 

Linear stability theory results match quite well with controlled laboratory experiment for thermal and centrifugal instabilities. But, instabilities dictated by shear force do not match so well, e.g. linear stability theory applied to plane Poiseuille flow gives a critical Reynolds number of 5772, while experimentally such flows have been observed to become turbulent even at Re = 1000- as shown in Davies and White (1928). Couette and pipe flows are also found to be linearly stable for all Reynolds numbers, the former was found to suffer transition in a computational exercise at Re = 350 (Lundbladh Johansson, 1991) and the latter found to be unstable in experiments for Re > 1950. Interestingly, according to Trefethen et al. (1993) the other example for which linear analysis fails include to a lesser degree, Blasius boundary layer flow. This is the flow which many cite as the success story of linear stability theory. 

A quantity of interest in the analysis of pipe flow is the pressure drop AP since it is directly related to the power requirements of the fan or pump to maintain flow. We note that dP/dx = constant, and integrating from x =, ri where the pressure is A, to, r . r, -f L where the pressure is gives... 

The molecules of these other species get in the way of the molecules of species 1 (say) and, in effect, exert a drag on them in much the same way that a pipe exerts a frictional drag on the fluid flowing through it. The analogy with pipe-flow does not end here an analysis of diffusion may be carried out in essentially the same way that we may derive, for example, Poiseuille s equation for the rate of fluid flow in a pipe—through the application of Newton s second law. 

Small particles in a turbulent gas dilfuse from one point to another as a result of the eddy motion. The eddy diffusion coefficient of the particles will in general differ from that of the carrier gas. An expression for the particle eddy diffusivity can be derived for a Stokesian particle, neglecting the Brownian motion. In carrying out the analysis, it is assumed that the turbulence is homogeneous and that there is no mean gas velocity. The statistical properties of the system do not change with time. Essentially what we have is a stationary, uniform turbulence in a large box. This is an approximate representation of the core of a turbulent pipe flow, if we move with the mean velocity of the flow. 

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