Journal of Petroleum and Gas Exploration Research (ISSN 2276-6510) Vol. 2(2) pp. 027-032, February, 2012 Available online http://www.interesjournals.org/JPGER Copyright © 2012 International Research Journals Review Exact solution of friction factor and diameter problems involving laminar flow of Bingham plastic fluids 1Prabhata K. Swamee*, 2Pushpa N. Rathie, 3Nitin Aggarwal 1JM-46 Shankar Nagar, Bhopal 462 016, India. 2Senior Research Collaborator, Department of Statistics, University of Brasilia, Brasilia 70910-900, Brazil. 3S-299, Greater Kailash-1, New Delhi 110048, India. Accepted ???? August 2011 In this paper, mathematically equivalent representations of the Buckingham-Reiner equation to compute friction factor and of the diameter equation to compute circular pipe diameter for laminar flow of Bingham plastic fluids in pipes have been given. The new forms are simple and very well- suited for accurately estimating the friction factor and diameter, because no iterative calculations are necessary. Specifically, the friction factor and diameter are expressed as computable series using Lagrange’s theorem. The equations presented in this study eliminate the need for best- fit parameters that are an integral part of the various explicit approximations proposed to date. The exact equations can also be utilized with advantage in optimization studies of pipelines. The ease, with which the new equations can be used, along with their smooth and predictable behavior, should make them the first choice of use for estimating the friction factor and diameter for laminar flow of Bingham plastic fluids in circular pipes. Key Words: Bingham fluids, diameter, darcy-weisbach equation, explicit equation, friction factor, head losses, laminar flow. INTRODUCTION There are three typical problems encountered in pipe found. These pertain to the determination of (1) friction flows, depending upon what is known and what is to be factor, (2) discharge, and (3) pipe diameter (Darby 1996). The head loss in a pipe is given by 8 fLQ 2 h f = (1) *Corresponding Author Email: [email protected]: tell. π 2 gD 5 91-181-2690931 ext 333 where f = Darcy-Weisbach friction factor, L = pipe length; Q = fluid discharge; g = gravitational acceleration; and D = pipe diameter. The continuity equation for discharge Q Notation is Following symbols have been used in this paper : π Q = D 2V (2) 4 D pipe diameter; f, friction factor; g, gravitational acceleration; where V = mean fluid velocity. Once the friction factor, f He, Hedstrom number; h, He/Re; hf, frictional head loss; L, pipe length; Q, discharge; q, nondimensional discharge; Re, is known, it becomes easier to handle the different pipe- Reynolds number; Sf, friction slope; so, σ0/ρ; V, average velocity; flow problems, for example calculating the pressure drop γ, velocity gradient; q, mass density; µ ∞, dynamic viscosity; ν∞, for evaluating pumping costs or finding the diameter of a kinematic viscosity; σ, shear stress; σ0, yield shear stress. pipe in a piping network for a given pressure drop and flow-rate. Due to the implicit nature of the friction factor equation, usually a large number of algebraic equations Subscript are generated for evaluating f when solving for fluid flow problems in a complex piping network (Brats etal., 1993). , nondimensional. * Swamee et al. 028 This task becomes cumbersome as the pipe-network Sablani et al., 2003; Danish et al., 2011; Swamee and size increases and, in such cases, numerical approaches Aggarwal, 2011. For the diameter problem, since Re and require a good initial guess of friction factor for each pipe He are unknown, it is also not possible to determine the element so that lesser computational efforts are needed. friction factor using Equation (4) by straightforward trial The same problem arises when the diameter of pipeline and error procedure for subsequently finding diameter has to be obtained that carries a discharge Q with a using Equation (1). Therefore, attempt for obtaining an given frictional loss hf. To avoid this, researchers have explicit equation for diameter has been made recently. tried to develop explicit approximations for the implicit See Swamee and Aggarwal (2011). Similarly, in the friction factor equation for flow of Newtonian fluids discharge determination problem also, as Re and He are (Swamee and Jain 1976, Haaland 1983, Serghides unknown, it is not possible to use straightforward trial 1984, Goudar, and Sonnad 2007), and Non-Newtonian and error procedure. Swamee and Aggarwal (2011) fluids (Darby 1996). found an exact equation for discharge. The overall objective of this paper is to provide exact mathematical equations for friction factor and diameter from implicit Bingham plastic fluid flow problems Buckingham-Reiner equation. The Bingham plastic model is given by the following rheological equation: Lagrange’s theorem τ = τ + ρν γ (3) o ∞ Joseph Louis Lagrange (1736-1813) in 1770 gave a where τ = shear stress; τ = yield shear stress; ν = o ∞ theorem by which solution of an implicit equation can be kinematic viscosity; ρ = mass density; and γ = velocity found in terms of an infinite series (Whittaker and gradient perpendicular to the direction of shear. Many Watson 1965; page 133). The theorem is stated as: A industrially important fluids, for e.g. various suspensions, function g(y), where y is a root of the equation slurries, drilling mud, pastes, gels, plastics, etc., exhibit non-Newtonian behavior and can be represented by the ( ) (7) Bingham plastic model. Equations. (1)-(3) are quite y = a + θφ y useful in the design of pipelines for such fluids. These where a = constant; θ = parameter, and φ = function; is materials exhibit a yield stress which must be exceeded given by before they will flow at a significant rate. Other examples ∞ θ nd n −1 () () '()()n (8) of such fluids include paint, shaving cream, and gyga= + ∑ n −1 gyyφ n =1 n! dy mayonnaise. There are also many “fluids” that may have y= a a yield stress which is not as pronounced--- blood, for where g'( y) = dg (y)/ dy ; and φ and g are differentiable example. For laminar flow of such fluids, the friction functions at a (See for a generalized result, Whittaker factor, f, is given by the Buckingham-Reiner equation and Watson 1965, page 133). (Darby and Melson 1981) 64 1 He 64 He 4 f 1 (4) Friction factor problem = + − 3 7 Re 6 Re 3 f Re Eq. (4) is written in the two parameter form as where Re = Bingham Reynolds number; and He = 4 Hedstrom number given by, 32 He 4,096 He f Re= 64 + − (9) 4Q 3 Re3 Re Re = (5) 3 () f Re πν ∞ D 2 D so Small He/Re He = 2 (6) ν ∞ Considering g(fRe) = fRe, a = 64 + (32/3)(He/Re); θ = - where s o = τo/ρ; and Q = fluid discharge. The friction 4 3 factor chart for laminar and turbulent flow conditions of (4096/3)(He/Re) and φ(fRe) = 1/( fRe) the application of Bingham plastic fluids is presented in Figure 1. See Lagrange theorem to Eq. (9) with some rearrangements Darby and Melson (1981). To solve for the friction factor, gives f as n an exact analytical solution of the Buckingham- fRe∞ 27 (4 n − 2)! Reiner equation can be obtained as it involves a fourth =1 − ∑ (10) He 256 n !(3 n− 1)! P 4n order polynomial equation in f, but due to complexity of 64 1 + n=1 the solution it is not convenient. Recently some attempts 6Re have been made to obtain friction factor using the Buckingham-Reiner equation. For example see where 029 J. Pet. Gas Explor. Res. Figure 1. Friction factor for Bingham plastic fluids 0.5 6Re 3 2 P = + 1 (11) φ fRe/He2= 192 f Re/He 2 / 3 f Re/He 2 + 16 f Re 2 /He+64 , He ()()(){ } 1 3 5 2 2 2 2 Expanding Eq. (10) and simplifying, the following series 2 256 26.66667 9.89184 4.07340 3.25977 f Re/He=++ 8 + + − + ... is obtained: hh h h h f Re 0.56988 4 0.65376 8 0.71415 12 0.75550 16 0.78545 20 =−1 − − − − ... (15 ) He P P P P P 64 1 + where h = He/Re. For fast convergence of the series, h 6Re on the right side has to be large. Eqs. (12) and (15) are (12) plotted in Figure (2). Errors involved in the Eqs. (12) and (15) are depicted in Figure 2. A perusal of Figure 2 The series on the right hand side converges fast for large reveals that the maximum error involved in the use of values of P, that is, when He/Re is small. Eqs. (12) and (15) is about 0.013%. Thus, the series solution is almost exact. The maximum error occurs at about h = 30. Thus, Eq. (12) is valid for Large He/Re He/Re≤ 30, whereas Eq. (15) is valid for He/Re≥ 30. Eq. (9) is written in the following implicit form: 3 −1 Hef Re2 64 He 1 Diameter problem = +2 − (13) Re 64He 3f Re 6 Defining 3 ν ∞ ( gSf ) Q Eq. (13) simplifies to q = 4 (16) 0.5 s o 2 3 192f Re/He gDS f 2 () (17) f Re/He= 8 + (14) D * = 2 s ()He/Re 3()f Re/He2+ 16 f Re/He+64 2 o where Sf = h f/L. Using Eqs. (1), (5), (16) and (17) one gets Applying Lagrange theorem to Eq. (14) by considering 4 πD* g(fRe 2/He) = fRe 2/He, a = 8; θ = (Re/He) 0.5 and f Re = (18) 2q Swamee et al. 030 0.03 Re f 0.02 0.01 Eq. (12) Eq. (15) 0.00 Percentage Error in Error Percentage -0.01 1 10 100 1000 He/Re Figure 2.