Technical Note - Physical influences on Atkinson's friction factor

Procarione, J. A.
Organization: Society for Mining, Metallurgy & Exploration
Pages: 4
Publication Date: Jan 1, 1988
Introduction Ventilation engineers have always felt somewhat uneasy when, in the normal course of their work, they are forced to consider the flow in any mine entry as "wholly rough." Under such conditions in pipes, friction factor is no longer a function of Reynolds' number. This assumption serves to simplify the problem and overcome an evident lack of information concerning the variability of K-factors. During the summer of 1985, a unique opportunity to investigate the reaction of friction factor to changing flow rates presented itself. A Utah coal mine was temporarily directing a substantial airflow through a single entry. By regulating the quantity, it would be possible to obtain a wide range of Reynolds' numbers in an entry composed of two distinct coal thicknesses. This paper describes the conducted study. The theory covering basic fluid flows in pipes that led to the questioning of constant K-factor is presented first. Experimental procedures are then described, including the use of a view camera in determining entry area and perimeter. Finally, a discussion of the observed variations in friction factor versus Reynolds' number is given. Changes of 34% and 45% in K-factor were seen in this study for the two entry heights. The implications of ignoring the relationship in this case and a call for an expanded research effort are also presented. Theory Since a gas is a fluid, the general principles of fluid mechanics are equally applicable to airflow in a mine entry as they are to the flow of water in a pipe. The Darcy equation is, therefore, equally applicable to pressure drop calculations in both situations. This famous formula was originally derived for use in circular cross sections and may be modified for any other shape by the use of hydraulic radius (defined as the area divided by the "wetted" perimeter). The Darcy equation relates the physical parameters of the pipe (diameter, length, and resistance to flow) and fluid velocity to the energy needed to create the flow. A study by Nikuradse (1933) brought to light the suspected dependency of the friction factor term in the Darcy equation upon Reynolds' number. By performing a large number of laboratory experiments, he established some basic concepts upon which other researchers built. Chief among these contributors was Colebrook (1939). Using commercial-grade pipe, he established an empirical relationship between Darcy's friction factor and Reynolds' number that is still in use today. If Colebrook's system of equations is plotted on log-log paper (as was done by Moody, 1944), the curves have some very distinct features. Of importance to the subject at hand is the relative response of friction factor to Reynolds' number at a given relative roughness. Simply stated, a pipe can appear to be either smooth or rough, depending on the flow conditions. As the Reynolds' number increases, the friction factor decreases at a decreasing rate until it becomes nearly a constant value. At this point, the flow is said to be "wholly rough." It is upon this condition that many traditional ventilation solution methods depend. For calculating pressure drops in a mine entry, Atkinson's equation is much more popular than the Darcy equation. Both of these formulas describe the same phenomenon and so have similar forms and functions. The K-factor in the Atkinson equation is equivalent to the Darcy friction factor and can be related by simply changing units, equating the pressure drops, and canceling like terms. A generalized form of this relationship is presented below as equation 1. K = yf/28,800g = [q]f/28,800 (1) where K is Atkinson's friction factor in kg/m3 (lb - min2/ft4), Y is the specific volume in Nt/m3 (lb per cu ft), p is the fluid density in kg/m3 (slugs per cu ft), g is the acceleration due to gravity (9.8 m/s2 or 32.2 fpsz), and f is Darcy's friction factor, which is unitless. The objective behind Eq. 1 is to show the strong correlation between K and f. Since these two quantities differ only by a constant (q]/28,000), the assumed independence of K-factor and Reynolds' number is suspect in light of the known response of f. Unfortunately, the formulation does not provide information concerning the actual relationship, as entry geometry likely plays a major role in the response curve. Only an indirect indication that such a relationship might exist is provided. Previous work on this subject was performed by Thomas Falkie (1958). The study involved a laboratory model composed of sections of straight, steel duct and a range of Reynolds' numbers from 15,610 to 137,200. Results from this work show a variation in friction factor with Reynolds' number in much the same way as it was found to vary in pipes. Over the range of velocities examined, a decrease of approximately 38% in K-factor was observed. This previous experience provides additional incentives to investigate the phenomenon under field conditions. Past model studies done at the University of Utah by Ozukurt (1945), Biron (1944), and Greenhalgh (1942) present similar evidence of this variability. More recently, Kharkar's (1974) study of coal mine friction factors contains similar indications that K-factor is not constant in all situations. Taken together, the data
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