Utah State University DigitalCommons@USU Reports Utah Water Research Laboratory January 1974 Steady Flow Analysis of Pipe Networks: An Instructional Manual Roland W. Jeppson Follow this and additional works at: https://digitalcommons.usu.edu/water_rep Part of the Civil and Environmental Engineering Commons, and the Water Resource Management Commons Recommended Citation Jeppson, Roland W., "Steady Flow Analysis of Pipe Networks: An Instructional Manual" (1974). Reports. Paper 300. https://digitalcommons.usu.edu/water_rep/300 This Report is brought to you for free and open access by the Utah Water Research Laboratory at DigitalCommons@USU. It has been accepted for inclusion in Reports by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. STEADY FLOW ANALYSIS OF PIPE NETWORKS­ An Instructional Manual by Roland W. Jeppson Developed with support from the Quality of Rural Life Program funded by the Kellogg Foundation and Utah State University Copyright ® 1974 by Roland W. Jeppson This manual, or parts thereof, may not be reproduced in any form without permission of the author. Department of Civil and Environmental Engineering and Utah Water Research Laboratory Utah State University Logan. Utah 84322 $6.50 September 1974 TABLE OF CONTENTS Chapter Page FUNDAMENTALS OF FLUID MECHANICS Introduction . Fluid Properties Density. I Specific weight I Viscosity 1 Example Problems Dealing with Fluid Properties 2 Conservation Laws 3 In trodu cti on 3 Continuity. 3 Example Problems Applying Continuity 4 Conservation of Energy (Bernoulli Equation) 6 Example Problems Dealing with Conservation Laws 9 Momentum Principle in Fluid Mechanics 12 II FRICTIONAL HEAD LOSSES 13 Introduction . 13 Darcy-Weisbach Equation. 13 Friction Factor for Laminar Flow 13 Friction Factors for Turbulent Flows. 15 Example Problems Using the Darcy-Weisbach Equation. 16 Computer Use with Darcy-Weisbach Equation (the Newton-Raphson Method) 19 Empirical Equations. 20 Example Problems Using the Darcy-Weisbach Equation 21 Exponential Formula . 22 Example Problems in Defining Exponential Formula from the Darcy-Weisbach Equation 22 III MINOR LOSSES 25 Introduction 25 Bends, Values, and Other Fittings 25 Entrances, Exists, Contractions, and Enlargements 27 IV INCOMPRESSIBLE FLOW IN PIPE NETWORKS 29 Introduction . 29 Reducing Complexity of Pipe Networks 29 Series pipes 29 iii TABLE OF CONTENTS (Continued) Chapter Page Parallel pipes . 30 Branching system . 30 Minor losses . 30 Example Problem in Finding Equivalent Pipes . 31 Systems of Equations Describing Steady Flow in Pipe Networks 32 Flow rates as unknowns . 32 Example Problems in Writing Flow Rate Equations 34 Heads at Junctions as Unknowns . 3S Corrective flow rates around loops of network considered unknowns. 36 v LINEAR THEORY METHOD ............. 39 Introduction . 39 TransfOrming Nonlinear Energy Equations Into Unear Equations 39 Example Problems Using Unear Theory for Solution 43 Including Pumps and Reservoirs into Linear Theory Method 44 Example Problems which Include Pumps and Reservoirs 47 VI NEWTON·RAPHSON METHOD 61 Introduction . 61 Head-equation 62 Example Problems Based on the H-Equations 64 Corrective Flow Rate Equations 6S Example Problems in Solving .:lQ-Equations 68 VII HARDY CROSS METHOD 73 Introduction . 73 Mathematical Development 73 APPENDIX A: UNITS AND CONVERSION FACTORS UNIT DEFINITIONS (SI) . 77 APPENDIX B: AREAS FROM PIPE DIAMETERS. 79 APPENDIX C: DESCRIPTION OF INPUT DATA REQUIRED BY A COMPUTER PROGRAM WHICH SOLVES THE Q-EQUATIONS BY THE LINEAR THEORY METHOD. .. 81 APPENDIX D: DESCRIPTION OF INPUT REQUIRED BY A COMPUTER PROGRAM WHICH SOLVES THE .:lQ.EQUATIONS BY THE NEWTON-RAPHSON METHOD. .. 85 APPENDIX E: DESCRIPTION OF INPUT DATA REQUIRED BY PROGRAM WHICH USES THE HARDY CROSS METHOD OF SOLUTION . 87 iv CHAPTER I FUNDAMENTALS OF FLUID MECHANICS Introduction In the inter!lational system (Systeme International d'Unites) of units (abbreViated SI) which is an outgrowth This chapter provides the necessary background of the metric system, mass is measured in the unit of the information concerning the properties of fluids and the gram, gr (or kilogram, kg, which equals 1000 grams); force laws which govern their motion. Readers with some under­ is measured in Newtons, N, and length in meters, m. A standing of fluid mechanics might begin their reading with Newton is the force required to accelerate I kg at a rate of Chapter II. Those readers with better training in fluid 1 m/sec2, and equals 105 dynes which is the-force needed mechanics, equivalent to that acquired by civil, mechani­ to accelerate I gr at I cm/sec2. cal, agricultural, or aeronautical engineers, might begin their reading with Chapter IV. The units of density in the SI system are: The first section in this chapter will introduce those .1s. N-sec 2 properties of fluids which are needed to understand later m30r~ material in this manual. Along with these properties you will become acquainted with symbols which are used to The conversion of density from the ES system to the SI denote fluid properties, as well as units and dimensions system or vice versa can be determined by substituting the which are associated with these symbols. Later these equivalent of each dimension as given in Appendix A. symbols will appear in equations, and consequently as you Upon doing this read this chapter you should make the association between the symbol and what it represents. After covering I Slug/ft3 = 515.363kg/m3 basic fluid properties, with particular emphasis on liquids (or more precisely referred to as incompressible fluids; The density of water equals 1.94 slug/ft3 or 1000 kg/m3 gases are compressible fluids), the fundamental conserva­ (I gr/cm3). tion principles of mass and energy are discussed in the later portion of this first chapter with special reference to Specific weight liquid (Le., incompressible) flow in pipes and other closed conduits. The specific weight (sometimes referred to as the unit weight) is the weight of fluid per unit volume, and is Fluid Properties denoted by the Greek letter 'Y (gamma). The specific weight thus has dimensions of force per unit volume. Its Density units in the ES and SI systems respectively are: Ib N.. kg The mass per unit volume is referred to as the ft3 and m3 or m2-sec2 density of the fluid and is denoted by the Greek letter p (rho). The dimensions of density are mass per length The specific weight is related to the fluid density by cubed or M/t3. The English system of units (abbreviated the acceleration of gravity or ES) uses the slug for the unit of mass, and feet for the unit of length. The dimensions commonly used in 'Y gp . (1-1) connection with the ES system of units are force, F, length, L, and time, T. In the Systeme internationale, SI, Since g = 32.2 ft/sec2 (9.81 m/sec2), the specific weight the common dimensions are mass, M, length, L, and time, of water is T. Thus in using FLT, dimensions for the slug can be 3 obtained by relating mass to force through the gravita­ 'Y == 32.2 (1.94) = 62.4 Ib/ft (ES) tional acceleration, g, i.e. F(force) = M(mass) x g(accelera­ or tion of gravity). Thus since the units of acceleration are 'Y = 9.81 (1000) 9810 N/m3 (SI) ft/sec2, the units of density are, Viscosity . Slu g lb-sec 2 ft 3 or~ Another important fluid property is its viscosity (also referred to as dynamic or absolute viscosity). in the English system. Viscosity is the fluid resistance to flow, which reveals 1 itself as a shearing stress within a flowing fluid and The dimensions of kinematic viscosity are length squared between a flowing fluid and its container. The viscosity is per time. Common units for v in the ES and SI system given the symbol fJ. (Greek symbol mu) and is defined as respectively are: the ratio of the shearing stress T (Greek letter tau) to the 2 rate of change in velocity, v, or mathematically dv/dy. and --m This definition results in the following important equation sec sec for fluid shear. Another often used unit for kinematic viscosity in the metric system is the stoke (or centistoke = .01 stoke). The T = . (1-2) stoke equals 1 cm2/sec. The viscosity of many common fluids such as water in which dv/dy is the derivative of the velocity with depends upon temperature but not the shear stress 7 or respect to the distance y. The derivative dv/dy is called dv /dy. Such fluids are called Newtonian fluids to the velocity gradient. Equation 1-2 is valid for viscous or distinguish them from non-Newtonian fluids whose laminar flow bu t not for turbulent flow when much of the viscosity does depend upon dv / dy. Table I-I gives values apparent shear stress is due to exchange of momentum of the absolute and kinematic viscosities of water over a between adjacent layers of flow. Later, means for deter­ range of temperatures. mining whether the flow is laminar or turbulent are given. From Eq. 1-2 it can be determined that the Table 1-1. Viscosities of water. dimensions of viscosity are force multiplied by time 2 divided by length squared or FT/L . Its units in the ES Kinematic and SI systems are respectively, Temperature Viscosity, fJ. Viscosity, v 2 2 lb-sec SIU!' d N-sec Ib-sec/ft2 N-sec/m ft2/sec m /sec -- or~ an --2 or ft2 ft-sec m m-sec % °c x 10 5 x 10 3 x 105 x 106 32 C 3.746 1.793 1.931 1.794 Occasionally the viscosity is given in poises (1 poise 40 4.4 3.229 1.546 1.664 1.546 dyne sec/ cm2). One poise equal 0.1 N-sec/m2.