Pipe flow dimensional analysis

For heat transfer for a fluid flowing through a circular pipe, the dimensional analysis is detailed in Section 9.4.2 and, for forced convection, the heat transfer coefficient at the wall is given by equations 9.64 and 9.58 which may be written as ... 

Obtain by dimensional analysis a functional relationship for the wall heat transfer coefficient for a fluid flowing through a straight pipe of circular cross-section. Assume that the effects of natural convection can be neglected in comparison with those of forced convection. 

The procedure for performing a dimensional analysis will be illustrated by means of an example concerning the flow of a liquid through a circular pipe. In this example we will determine an appropriate set of dimensionless groups that can be used to represent the relationship between the flow rate of an incompressible fluid in a pipeline, the properties of the fluid, the dimensions of the pipeline, and the driving force for moving the fluid, as illustrated in Fig. 2-1. The procedure is as follows. 

This can be compared with the results of the dimensional analysis for the laminar flow of a Newtonian fluid in a pipe (Chapter 2, Section V), for which we deduced that JNRe = constant. In this case, we have determined the value of the constant analytically, using first principles rather than by experiment. 

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.,... 

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

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. 

In order to explain dimensional analysis in chemical engineering, we present a typical problem of chemical engineering that requires an experimental approach. Consider the steady flow of an incompressible Newtonian fluid through a long, smooth-walled, horizontal and circular pipe which is heated from the outside. 

DIMENSIOM ANALYSIS METHOD. Dimensional analysis of the heat flow to a fluid in tur ulait flow in a long, straight pipe yields the dimensionless relationship ... 

Besides clarifying the strange shape of Fig. 6.2, Reynolds made the most celebrated application of dimensional analysis (Chap. 13) in the history of fluid mechanics. He showed that for smooth, circular pipes, for all newtonian fluids, and for all pipe diameters, the transition from laminar to turbulent flow occurs when the dimensionless group DVpIfjt, has a value of about 2000. Here D is the pipe diameter, V is the average fluid velocity in the pipe, p is the fluid density, and fi is the fluid viscosity. This dimensionless group is now called the Reynolds number For flows other than pipe flow, some other appropriate length is substituted for the pipe diameter in the Reynolds number, as discussed later. 

Dimensional analysis will riot tell us the value of the constant in Eq. 13.19, but it will suggest to us that for steady laminar flow in horizontal pipes a plot of pressure drop per unit length versus viscosity times average velocity divided by diameter squared might give the same straight line for all fluids, pipes, and velocities (which is experimentally verifiable). 

Powell was known for papers and books. His 1940 text deals with hydraulics in general for undergraduate students. The 1951 book is an update version, with the main chapters Hydrostatics, Fundamentals of fluid flow. Orifices, Tubes, Nozzles, Pipe flow. Flow in open channels, Effect of viscosity. Models, Properties of liquids. Dimensional analysis, and Rational basis of Nikuradse s formulas. He also co-authored the 1963 state-of-the-art paper. Powell collaborated foryears with Chesley J. Posey (1906-1991) and was also involved as secretary of the Rocl Mountain Hydraulic Laboratory, Fort Collins CO. 

Friction in pipes. For straight horizontal pipes resistance to flow arises because of viscous shear or friction at the wall. Dimensional analysis leads to the conclusion that for smooth pipes the pressure drop (scaled to make it dimensionless) is a function only of the Reynolds number. The dimensionless group containing the pressure drop is known as the friction factor ... 

Dimensional Analysis in Mass Transfer. A fluid is flowing in a vertical pipe... 

To determine the value for /ihmj we again use dimensional analysis [2]. Consider a fluid flowing inside a pipe or tube. The heat transferred to the flowing fluid is... 

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