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Water Distribution Modeling Thomas M University of Dayton eCommons Civil and Environmental Engineering and Department of Civil and Environmental Engineering Mechanics Faculty Publications Engineering and Engineering Mechanics 2001 Water Distribution Modeling Thomas M. Walski Haestad Methods Donald V. Chase University of Dayton, [email protected] Dragan A. Savic University of Exeter Follow this and additional works at: http://ecommons.udayton.edu/cee_fac_pub Part of the Hydraulic Engineering Commons, and the Structural Engineering Commons eCommons Citation Walski, Thomas M.; Chase, Donald V.; and Savic, Dragan A., "Water Distribution Modeling" (2001). Civil and Environmental Engineering and Engineering Mechanics Faculty Publications. Paper 17. http://ecommons.udayton.edu/cee_fac_pub/17 This Book is brought to you for free and open access by the Department of Civil and Environmental Engineering and Engineering Mechanics at eCommons. It has been accepted for inclusion in Civil and Environmental Engineering and Engineering Mechanics Faculty Publications by an authorized administrator of eCommons. For more information, please contact [email protected], [email protected]. CHAPTER 2 Modeling Theory Model-based simulation is a method for mathematically approximating the behavior of real water distribution systems. To effectively utilize the capabilities of distribution system simulation software and interpret the resu lts produced, the engineer or mod­ eler must understand the mathematical principles involved. This chapter reviews the principles of hydraulics and water quality analysis that are frequently employed in water distribution network modeling software. 2.1 FLUID PROPERTIES Fluids can be categorized as either gases or liquids. The most notable differences between the two states are that liquids are far denser than gases, and gases are highly compressible compared to liquids (liquids are relatively incompressible). The most important fluid properties taken into consideration in a water di stribution simulation are specific weight, fl uid viscosity, and (to a lesser degree) compressibility. Density and Specific Weight The density of a fluid is the mass of the fluid per unit volume. The density of water is 1.94 slugs/ftl (1000 kg/m3) at standard pressure of 1 atm (1.013 bar) and standard tem­ perature of 32.0 op (0.0 oc). A change in temperature or pressure will affect the den­ sity, although the effects of minor changes are generally insignificant for water modeling purposes. The property that describes the weight of a fluid per unit volume is called speci,fic weight, and is related to density by gravitational acceleration : 20 Modeling Theory Chapter 2 y = pg (2.1) where y = fluid specific weight (M/L'ff') p = fluid density (M/U) g = gravitational acceleration constant (Lff' ) The specific weight of water, y, at standard pressure and temperature is 62.4 lb/fe 3 (9,806 N/m ). Viscosity Fluid viscosity is the property that describes the ability of a fluid to resist deformation due to shear stress. For many fluids, most notably water, viscosity is a proportionality factor relating the velocity gradient to the shear stress, as described by Newton's Law of Viscosity: dV 1 = ll­ (2.2) dy where 1 = shear stress (M!Lff') 11 = absolute (dynamic) viscosity (M/Lff) dV = time rate of strain (lff) dy The physical meaning of this equation can be illustrated by considering the two paral­ lel plates shown in Figure 2.1. The space between the plates is filled with a fluid, and the area of the plates is large enough that edge effects can be neglected. The plates are separated by a distance y, and the top plate is moving at a constant velocity V relative to the bottom plate. Liquids exhibit an attribute known as the no-slip condition, mean­ ing that they adhere to surfaces they contact. Therefore, if the magnitude of V andy are not too large, then the velocity distribution between the two plates is linear. From Newton's Second Law of Motion, for an object to move at a constant velocity, the net external force acting on the object must equal zero. Thus, the fluid must be exerting a force equal and opposite to the force F on the top plate. This force within the fluid is a result of the shear stress between the fluid and the plate. The velocity at which these forces balance is a function of the velocity gradient normal to the plate and the fluid viscosity, as described by Newton's Law of Viscosity. Thick fluids, such as syrup and molasses, have high viscosities. Thin fluids, like water and gasoli ne, have low viscosities. For most fluids, the viscosity wil l remain constant regardless of the magnitude of the shear stress that is applied to it. Returning to Figure 2.1 , as the velocity of the top plate increases, the shear stresses in the fluid will increase at the same rate. Fluids that exhibit this property conform to Newton's Law of Viscosity, and are called Newtonian fluids. Water and air are exam­ ples of Newtonian fluids. Some types of fluids, like inks and sludge, undergo changes Section 2. 1 Fluid Properties 21 in viscosity as the shear stress changes. Fluids ex hibiting this type of behavior are called pseudo-plastic fluids. Figure 2.1 Physical interpretation of Newton's Law of Vi scosit y : v /1 L_: / >--------,' dy // t ,____ ___,, y r-----w,/ / '] dV ......._, I ' I ' I ' ~--------- 11 7'~--------~~---~------~~~ " Relationships between the shear stress and the velocity gradient for typical Newto­ ni an and Non-Newtonian fluids are shown in Figure 2.2. Since most distribution sys­ tem models are intended to simulate water, many of the equations used consider Newtonian fluids only. Figure 2.2 Stress versus strain for plastics and fluids m OJ ~ (") Vl Q. 0.: ~ ~ Vl .... ro QJ .r:: Vl II l- Ideal Fluid dV/dy 22 Modeling Theory Chapter 2 Viscosity is a function of temperature, but this relationship is different for liquids and gases. In general, viscosity decreases as temperature increases for liquids, and viscos­ ity increases as temperature increases for gases. The temperature variation within water di stribution systems, however, is usuall y quite small , and thus changes in water viscosity are considered negli gible for this appli cation. Generally, water distribution system modeling software treats viscosity as a constant [assuming a temperature of 68 "F (20 "C)]. The vi scosity derived in Equation 2.2 is referred to as the absolute viscosity (or dynamic viscosity) . For hydrauli c formulas related to fluid motion, the relationship between fluid viscosity and fluid density is often expressed as a single variable. This relationship, called the kinematic viscosity, is expressed as: v = ~ (2.3) p where v = kinematic viscosity (L2ff) Just as there are shear stresses between the plate and the fluid in Figure 2. 1, there are shear stresses between the wall of a pipe and the fluid moving through the pipe. The hi gher the fluid viscosity, the greater the shear stresses that will develop within the fluid, and, consequently, the greater the friction losses along the pipe. Distribution system modeling software packages use fluid viscosity as a factor in estimating the friction losses along a pipe's length. Packages that can handle any fluid require the viscosity and density to be input by the modeler, while models that are developed only for water usuall y account for the appropriate value automaticall y. Fluid Compressibility Compressibility is a physical property of fluids that relates the volume occupied by a fixed mass of fluid to its pressure. In general, gases are much more compressible than liquids. An air compressor is a simple device that utilizes the compressibility of ai r to store energy. The compressor is essentially a pump that intermittently forces air mole­ cules into the fixed volume tank attached to it. Each time the compressor turns on, the mass of air, and therefore the pressure with in the tank, increases. Thus a relationship exists between fluid mass, volume, and pressure. This relationship can be simplified by considering a fixed mass of a fluid. Compress­ ibility is then described by defining the flu id 's bulk modulus of elasticity: dP E -Vr::- (2.4) v = lc/V 2 where E,. = bulk modulus of elasticity (M/Lff ) 2 P = pressure (M/Lff ) volume of fluid (L') V1 = All fluids are compressible to some extent. The effects of compression in a water dis­ tribution system are very small , and thus the equations used in hydrauli c simulations Section 2.1 Fluid Properties 23 Hydraulic Transients When a pump starts or stops, or a valve is opened Transients are dampened by pipe friction and the or closed, the velocity of water in the pipe effects of pipe loops that essentially cancel out the changes. However, when flow accelerates or pressure waves. Surge tanks and air chambers decelerates in a pipe, all of the water in that pipe also have dampening effects. Using slow-opening does not change velocity instantly. It takes time for valves and flywheels on pumps can minimize the water at one end of a pipe to experience the transients before they occur by reducing the effect of a force applied some distance away. acceleration or deceleration of the water. When flow decelerates, the water molecules in Fluid transients tend to be worse in long pipelines the pipe are compressed, and the pressure rises. carrying water at high velocities. The worst tran­ Conversely, when the flow accelerates, the pres­ sient effects in water systems are usually brought sure drops. These changes in pressure travel on by a sudden loss of power to a pump station. through the pipe as waves referred to as "hydrau­ Transients can also be caused by a hydrant being lic transients." When a sudden change in velocity shut off too quickly, rapid closing of an automated occurs, the resulting pressure waves can be valve such as an altitude valve, pipe failure, and strong enough to damage pipes and fittings.
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