ABSTRACT. In this study, a numerical model based on the Method of Characteristics (MOC) is developed for modeling pressure transients in viscoelastic pipes.
--
PVP-Vol. 431, Emerging Technologies for Fluids, Structures and Fluid-Structure Interaction - 2001 ASME 2001
MODELING PRESSURE TRANSIENTS IN VISCOELASTIC PIPES Warda, H.A., Kandil, H.A., Elmiligui, A.A., and Wahba, E.M. Mechanical Engineering Department, Faculty of Engineering, Alexandria University, Alexandria, 21544, Egypt ABSTRACT In this study, a numerical model based on the Method of Characteristics (MOC) is developed for modeling pressure transients in viscoelastic pipes. The model is capable of dealing with unsteady friction and viscoelastic behavior of the pipe walls. These complex phenomena cause strong distortion of the pressure waves traveling through fluid lines to that predicted by the standard MOC. A universal model, developed by the authors, for unsteady friction for both laminar and turbulent flows is used in the analysis. The viscoelastic behavior of the pipe wall is modeled through a one element KelvinVoigt-Viscoelastic Model that resulted in good agreement with the experimental data. The viscoelastic effect was shown to be the dominant damping factor of the pressure oscillations in transient flows through pipes exhibiting a viscoelastic behavior. The analysis showed also that unsteady friction has a minor effect on the damping of the pressure transient in viscoelastic pipes while it has a dominant damping effect in elastic pipes. An experimental setup was constructed to provide reliable experimental data for transient flows in PVC (viscoelastic) pipes to verify the numerical model. Eventually, the numerical model was experimentally verified to be capable of accurately and efficiently reproducing the experimentally measured pressure oscillations in viscoelastic pipes. Nomenclature A = Cross-sectional area of the flow a= Wave speed in a fluid contained within an elastic conduit D =Pipe diameter E =Young's Modulus of Elasticity for the pipe material Ej =The modulus of Elasticity of the l Kelvin-Voigt element e
=Pipe wall thickness
f = Darcy-Weisbach friction factor g = Gravitational acceleration H = Local pressure head Hb = Barometric pressure head Ho =Head of the upstream reservoir ht(t) =Friction head loss per unit length at time (t) N = Number of pipe segments P=Pressure
305
• Q = Volume flow rate R = Pipe radius RN = Reynolds Number t =Time V
= Mean velocity of the flow
V0 = Steady state mean velocity of the flow s = bistance along the pipe
z = Node elevation from a reference level Llt = Time step in method of characteristics solution A= Constraint coefficient in the wave speed formula, also used as the multiplier in the solution by the method of characteristics
v = Kinematic viscosity of the fluid p = Fluid density 't
= Dimensionless time in the unsteady friction models
SUBSCRIPTS P =Node to be calculated at time (t) R =Condition upstream at time (t-ilt) to be used in the c+ characteristic equation L =Condition downstream at time (t-Llt) to be used in the C characteristic equation
1. INTRODUCTION
This study is concerned with studying the pressure transient propagation within fluids flowing in viscoelastic pipes. The viscoelastic behavior of the pipe walls was studied thoroughly to eliminate the discrepancy between the experimental data and the results predicted numerically using the standard MOC. The MOC is a general mathematical technique that is suitable for the solution of pairs of quasi-linear hyperbolic partial differential equations. The equations defming pressure-transient propagation, within fluid piping systems belong to this family of problems. This method was first proposed by Rieman in 1860. By the early 1970s, this technique was established as the standard method for transient analysis. Studies covering the application of the MOC to transient problems such as water hammer problems developed continuously over the last 50 years. Watters [1] provided the detailed theoretical basis for estimating the wave speed in different types of pipes including thin walled pipes, thick walled pipes, circular tunnels and reinforced concrete pipes. Also, Wylie and Streeter [2] summarized the various methods of solution for the water hammer problem. They stated that the MOC is considered to be the standard numerical method by which other methods may be judged, for accuracy and efficiency, for pressure transients. Kaplan eta!. [3) showed that transients arising in long oil pipelines could be adequately simulated by the MOC which completely ta.l
