Calibration methodology for hydraulic transient solvers incorporating unsteady friction and pipe wall viscoelasticity N. Carriço*, Alexandre K. Soares**, D. Covas* *Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais, 1, 1049-001 Lisboa, Portugal **Escola de Engenharia Civil, Universidade Federal de Goiás, CEP 74605-0 Goiânia, Brasil Abstract The current paper aims at the description and application of a methodology for calibration of different types of hydraulic transient models from the classic transient solver to solvers incorporating unsteady friction models and pipe wall viscoelasticity. The proposed calibration methodology is a two-stage approach: the first stage refers to the calibration of steady state conditions and the second stage to transient state parameters. A hydraulic transient solver based on the Method of Characteristics (MOC) was developed for testing the methodology. A data collection program was carried out in an experimental facility at Instituto Superior Técnico, Lisbon (Portugal) collecting transient pressure data. Reasons for suggesting a step-wise calibration of hydraulic transient solvers, instead of the blind simultaneous calibration of all known parameters, have been discussed. Main conclusions and suggestions for future studies are discussed. Keywords: calibration, hydraulic transients, transient solver, inverse analysis, viscoelasticity, unsteady friction. INTRODUCTION When a fluid in motion is suddenly forced to change, a pressure wave occurs. This pressure wave can cause several damages in pressurized pipe systems and can result from pump failures, valves maneuvers and sudden ruptures. For that reason, this phenomenon which is called pressure surge or water hammer has been a subject of interest for most engineers involved in the design of pipe systems. The study of pressure surges has started with the works of Joukowsky (1898) and Allievi (190). The governing equations of pressurized fluid flow are derived from the conservation of mass, momentum and energy principles. In several engineering situations the energy equation can be neglected since temperature variations are insignificant. The derived equations from the mentioned principles are the so-called Navier-Stokes equations (NSE) which are a set of nonlinear, second order and partial differential equations (Freidlander and Serre, 007). Since exact mathematical solutions for the NSE equations cannot be obtained these equations need to be simplified under several assumptions. Allievi (190) developed the general theory of water hammer based on the conservation of mass and momentum principles in one-dimensional flow and showed that the convective term in the momentum equation can be neglected. Furthermore, two dimensionless parameters were introduced to characterize pipes and valve behavior (Ghidaoui et al., 005). Some refinements of the Allievi equations were made over the years resulting in the classic theory of one-dimensional hydraulic transients in pipes which was full developed in the 1960s. The classic theory of water hammer is commonly used for design, as it reasonably well describes the maximum and minimum pressure variations. This approach typically assumes linear-elastic behavior of pipe wall, steadystate friction losses, one-phase flow, completely constrained pipe from axial movement, and no lateral in/out flows from the pipe. These assumptions are not always verified in practice. Examples of these are the energy dissipation during fast transient events, the mechanical damping in plastic pipes and the two-phase flow.
Several theoretical and practical studies have been carried out on hydraulic transient, from the mathematical derivation of Allievi equations in the 19th century, to the graphical analysis of the mid-0th century, and to the current computer simulation. New approaches which takes into account one, or more, phenomena that affect hydraulic transients have been developed, namely, unsteady friction (Adamkowski. and Lewandowski, 006; Storli and Nielsen, 011; Dual et al., 01; Mitosek and Szymkiewicz, 01; Reddy et al., 01), viscoelasticity (Covas et al., 005; Soares et al., 008, 01; Duan et al., 010) and fluid structure interaction (Lavooij and Tijsseling, 1991; Tijsseling, 1996; Wiggert and Tijsseling, 001). The use of mathematical models in hydraulic transient analysis helps to better understand the behavior of a system as well as to predict extreme pressures under different operational conditions. The analysis of hydraulic transients in pressurized systems has become a common practice in engineering due to a general awareness of the economic losses and operational disturbances caused by transient events. Currently, there are new challenges for the development of more accurate and easy-to-use computational models to predict the maximum information on the behavior of a hydraulic system (Ghidaoui et al., 005). However, most of the commercial software available is based on the classic theory. The aim of the paper is to present a calibration methodology for hydraulic transient solvers incorporating unsteady friction and pipe wall viscoelasticity. This methodology will assist engineers in fitting the main parameters of hydraulic transient models to collected pressure data. The proposed methodology is a two-stage approach: the first stage refers to the calibration of steady state conditions and the second stage to transient state parameters. A Hydraulic Transient Solver (HTS) based on the Method of Characteristic (MOC) that incorporates different numerical schemes for steady state friction (i.e., 1st and nd order approximations), two formulations for unsteady friction calculation (i.e., Trikha s (1975), Vardy s (199) and Vitkovsky s (000a)) and the pipe wall non-elastic behavior (i.e., linear elastic and linear viscoelastic) has been used. The methodology was tested in a high density polyethylene (HDPE) pipe rig at Instituto Superior Técnico. A data has been collected including transient pressures and steady state flows, during transient events. The mathematical model was calibrated by following the proposed methodology using physical data collected in the laboratory. MATHEMATICAL MODEL Classic solver The continuity and momentum equations that describe one-dimensional transient flow in pressurized conduits are a set of two differential equations (Chaudhry, 1987; Almeida and Koelle, 199; Wylie and Streeter, 1993): H 1 Q hf 0 x gs t (1) H a Q 0 t gs x () where: Q = flow rate; H = piezometric head; g = gravity acceleration; S = pipe cross-sectional area; x = coordinate along the pipeline axis; t = time; a = elastic wave speed; h f = slope of the energy line. The derivation of the governing equations of one-dimensional transient flow in pressurized conduits takes into account simplifying assumptions as (i) pseudo-uniform velocity profile, (ii) linear-elastic rheological behavior of pipe wall, (iii) fluid is one-phase, homogenous and quasi-incompressible; (iv) the pipe is uniform and constrained. Equations (1) and () are valid along the pipeline and at every time, regardless of the boundary conditions. These equations are quasi-linear and hyperbolic. There are many numerical methods for solving this system of equations, being the most widely used the Method of Characteristics (MOC). Unsteady state model In water hammer classical theory, the friction term, h f, is calculated in the same manner as in the steady state regime considering a constant value of Darcy-Weisbach friction factor (steady state approximation) or a
Reynolds-dependent friction factor (quasi-steady state approximation). The application of such a simplified friction model is satisfactory only for slow transients, in which the shape of the instantaneous velocity profiles does not significantly differ from the corresponding steady state (Zidouh, 009). For rapidly varying flows or higher pulsating frequencies, these approximations are inaccurate in the description of the damping and dispersion of the pressure wave (Covas, 003). The quasi-steady model has been proven to underestimate the dampening for non-stationary flows (Adamkowski and Lewandowski, 006). In plastic pipes or in surges caused by sudden changes of the flow conditions, the governing equations need to be reformulated as this simplification is far from reality (Covas et al., 004b; 005). In order to take into account unsteady friction (UF) losses and fluid inertial effects, the head loss per unit length h f is decomposed into two terms (Covas 003): h h h (3) f fs fu where h = head loss per unit length; h = steady-state friction losses component f losses component. f s h f u = unsteady-state friction The steady or quasi-steady friction model assumes that there is no contribution due to the unsteady flow (i.e. h 0 ) and can be determined using the formula: f u h f s f QQ (4) gd S where h f s = steady-state slope of the energy line; f = Darcy-Weisbach friction factor; Q = flow-rate; g = gravity acceleration; D = pipe inner diameter; S = pipe cross-section. The flow resistance equation that describes steady state friction depends on flow regime. When flow is laminar (Reynolds number lower than 000) the Hagen-Poiseuille formulation should be used. Most of the pipe flows are turbulent. One of the difficulties of solving turbulent flow problems in pipes lies in the fact that hydraulic friction factor is a complex function of relative surface roughness and Reynolds number (Brkić, 011). Two types of friction factors are often cited in the literature. i.e., friction factors for rough pipes and for smooth pipes, respectively. In both cases the Colebrook-White can be used: 1 k.51 log 3.71 f Re D f in which f = Darcy-Weisbach friction factor; k = relative roughness; D = pipe inner diameter; Re = Reynolds number. For smooth pipes k 0 and the von Kármán-Prandtl equation is obtained. For rough pipes R e and the von Kármán is obtained. The friction factor for rough pipes is often useful for detailed pressure drop calculation. Since the Colebrook-White equation is implicit, several explicit equations have been developed (Swamee and Jain, 1976; Haaland, 1983; Serghides, 1984) to approximate Colebrook-White equation for rough pipes (Li et al., 011). The models for unsteady friction increase the damping caused by pipe friction, most of them by taking the local acceleration of the flow into account, in some sense (Rufelt, 010). The most widely used models consider extra friction losses to depend on a history of weighted accelerations during unsteady phenomena or on instantaneous flow acceleration (Adamkowski and Lewandowski, 006). Developments of the first group were initiated by Zielke (1968), who introduced an additional term representing the unsteady-friction into the momentum equation (Equation ). This term, in the form of convolution, involves the fluid accelerations from the past with a weighting function (Mitosek and Szymkiewicz, 01). Trikha (1975) proposed a less demanding version of Zielke s method, reasonably accurate for laminar flows (Equation 6). The (5)
convolution integral method was extended to turbulent flow by Vardy (199, 007) for smooth and for rough pipes. This formulation is similar to Trikha s with only two terms Y i developed for smooth turbulent flows (Equation 7). h fu h fu 16 Y gd 1 16 Y gd 1 Y Y Y 3 ni t with D Y t Y t t e m U t Ut t i i Bi t with D Y t Y t t e A U t Ut t i i 4 4 i i (6) (7) where n i and m i = Trikha s parameters which are null for steady state, and m 1 =40, n 1 =-8000, m =8.1, n =-00, m 3 =1 and n 3 =-6.4, for transient state; A i and B i = Vardy s parameters that depend on fre (Zielke, 1968) exact formulation for laminar flows is an upper bound of these weighting functions developed for smooth turbulent flows); U= average velocity in each cross-section. Within the second approach, the wall shear stress due to flow unsteadiness is proportional to flow acceleration. The proportionality coefficient has been established based on the experimental measurements (Adamkowski and Lewandowski, 006). Modifications of this model have been the subject of numerous subsequent studies (Brunone et al., 1991, 1995, 004, 011; Pezzinga, 000; Bergant et al., 001; Ghidaoui et al., 005; Brunone and Golia, 008; Storli and Nielsen, 011a, 011b). One of the formulations most known is the Vitkovsky et al. (000a) formulation which is a modification of Brunone s formulation (Brunone et al., 1991), avoiding an interpolation procedure in the numerical scheme and has good results according to experimental data. h fu k3 U U a SGN ( U) g t x where k 3 = Vitkovsky s empirical coefficient, varying between 0.03 and 0.10 (Bergant and Simpson, 1994); U= average velocity in each cross-section; SGN = sign operator. Duan et al. (010) have quantitatively shown that in plastic pipes the role of unsteady friction is relevant only in the first phases of the transients. The viscoelastic effect becomes increasingly more dominant over unsteady friction, as time progresses. Linear-viscoelastic model The rheological behavior of the pipe material determines the amplitude, the phase and the shape of the pressure wave propagation during a transient hydraulic event. In pipelines with viscoelastic (VE) mechanical behavior, in general plastic pipes like polyethylene (PE) and polyvinyl chloride (PVC), hydraulic transient events are quickly damped and suffer a time delay, as result of the retarded strain of the pipe wall (Covas et al., 005). Governing equations assume that the pipe material has a linear-elastic rheological behavior. While this assumption is valid for concrete and metal pipes, plastic pipes present a non-elastic behavior. To take into account the pipe material rheological behavior, the continuity equation must be derived again from the Reynolds Transport Theorem assuming (i) linear viscoelastic behavior of pipe material for small strains, (ii) negligible dynamic fluid-pipe interaction, assuming a constrained pipe without any axial movement, (iii) straight and uniform pipe with a constant cross-section, without lateral in/out flows; (iv) homogeneous and isotropic pipe material, and (v) constant Poisson s ratio,, of the pipe material, so that the mechanical behavior depends only on the creep function, J(t). Taking into account the relationship between pipe cross section, S, and total strain, ds S d dt dt and that the strain can be decomposed into the sum of an instantaneous-elastic strain, e, and a retarded strain, r, the continuity equation yields: H a Q a r 0 t gs x g t (8) (9)
While the third term of these equation represents the retarded strain, the instantaneous-elastic strain is included in the piezometric-head time derivative and in the elastic wave speed, a. Method of characteristics The MOC is a numerical method which converts the first order partial differential equations (PDE) into ordinary differential equations (ODE) along certain curves in the x-t plane (called characteristic lines C + and C - ), that can be solved, after being expressed in terms of finite differences. Among the main MOC advantages are the simplicity of programming and efficiency of computations, even for complex systems with numerous boundary conditions, and the accurate simulation of steep wave fronts. The result of MOC's application to Equations (1) and () is the following set of equations: C dh a dq : ahf 0 (10) dt gs dt valid along dx/dt = V±a. Generally, fluid velocity is negligible when compared to the wave speed propagation V a, leading to approximately straight characteristic lines, dx dt a. In order to take into account the viscoelastic behavior of the pipe-wall, the continuity Equation () must be replaced by the continuity Equation (9). The characteristics equations are given by: C dh a dq a r ah f : 0 dt gs dt g t These equations have two terms that cannot be directly calculated and need further numerical discretization: the slope of the energy line, h f, and retarded strain time derivative, r /t. Several mechanical models can be used combining springs and dashpots, connected in series or in parallel, to describe the behavior of simple systems, to numerically describe the viscoelastic behavior of materials. Polyethylene pipes are viscoelastic solids, being described by the generalized Kelvin-Voigt model (Covas et al., 004, 005). As such the retarded strain for each k element of the Kelvin-Voigt model can be described by: rk x, t J rk x, t k F x, t t τ τ k k,, t t t k F x t F x t t k k rk x, t Jk F x, t Jke F x, t t Jk k (1 e ) e rk x, t t (13) t, D F x t, 0 H x t H x e (14) Parameters J k and k are adjusted to the creep experimental data. Creep compliance function Jt () of the pipe material can be determinate by a creep test or by calibration of the transient model. Flow rate, Q, and piezometric head, H, for each section i and time step j are determined by the characteristic Equations (11) for all interior points and the viscoelasticity term by Equations (1-14). UNCERTAINTIES IN TRANSIENT HYDRAULICS An uncertainty can be defined as a lack of knowledge to, deterministically or numerically, describe or predict a system behavior or its characteristics. In transient hydraulics, examples of the unknown parameters are pipe roughness, wave speed, unsteady friction coefficients, creep coefficients, air cavity volumes or boundary conditions characteristics (e.g., valve maneuver or pump rated conditions). Within the hydraulics literature, the parameter identification methods have tended to focus on the estimation of pipe friction parameters and of leaks sizes (Zecchin et al., 013). Many of these have focused on customized approaches for single pipe systems with either measured (Verde et al., 006) or known boundary conditions (Wang et al., 00). One of the most widely used methods to estimate the unknown parameters is the inverse (11) (1)
transient method, also called as inverse solvers. In the recent years, there has been a significant interest in the application of the inverse transient approach for leak detection and calibration of water pipe systems (Covas and Ramos, 010; Kapelan et al., 003; Liggett and Chen, 1994; Nash and Karney, 1999; Vitkovsky et al., 000b). Even though inverse transient techniques have been widely investigated, many challenges still remain. One reason for these difficulties is that in real systems there are many kinds of uncertainties, such as pipe diameters and wave speed. The internal pipe diameter may differ from the nominal diameter that is all too commonly used in modeling. Additionally, the internal diameter often decreases over time as corrosion, tuberculation, and scaling occur (Jung and Karney, 008). According to Walski et al. (001), a 10% decrease in the pipe diameter, for example, will increase steady state headlosses by nearly 40%. Wave speed is another challenging uncertainty in a pipeline. Wave speed is a function of fluid and pipe properties (pipe diameter, thickness and material; water density, bulk coefficient, temperature, presence of air and solid; pipe restraint conditions). Some of these conditions can be accurately assessed, but many are poorly defined and uncertain (Jung and Karney, 008). For example, the accurate measurement of the air content dispersed in fluid is difficult; however, even a tiny amount of gas throughout a liquid greatly reduces the propagation velocity of a pressure wave in a pipeline (Wylie and Streeter, 1993). Inverse transient analysis attempts to estimate unknown parameters by using pressure data, collected during the occurrence of simulated pressure surges. The parameter identification is an optimization problem in which the system s behavior is simulated by a Transient Solver (TS) and the difference between observed and calculated variables is minimized by means of an optimization model - Inverse Transient Solver (ITS). An ITS is an optimization algorithm that searches for the best-fitted solution by minimizing an objective function (OF) defined by the average least-square errors between observed and calculated variables: Min OF p p 1 M 1 M T q * qp q * qp q * i q i p M i1 in which OF(p) = objective function; p = parameter-vector with N decision variables; q(p) = predicted system response vector (with M elements) for a given parameter vector p; q* = the observation-vector (with M elements), whose elements are measured heads; M = number of measurements. Observed data are typically pressure measurements. PROPOSED METHODOLOGY Brief introduction Before any pipe system model can be reliably used, the model must be firstly calibrated. Calibration involves adjusting uncertain parameters until the model results closely approximate to the observed conditions, ideally obtained from measured field data. The proposed methodology to calibrate hydraulic transient solvers is divided into two main stages: the first stage consists of the calibration of the steady state flow (Figure 1) and the second the transient state flow (Figure ). Each stage is divided into steps. This methodology is particular to a single pipe system (Tank-Pipe-Valve), but can be extended to other pipe system configurations. A coupled calibration procedure, in which steady state and transient parameters are simultaneously evaluated in a single stage, could also be defined; however, this procedure is not recommendable as some parameters have overlapping effects (e.g., pipe roughness, UF coefficients or VE coefficients) and the calibration model will tend to converge to the most sensitive parameters leading to unrealistic results. Authors experience has shown that, when dealing with real data, the ideal is to separate the effects of each parameter, for instance, to calibrate pipe roughness based on initial conditions and UF/VE coefficient based on transient flows. Steady state calibration The calibration of the steady state (Figure 1) consists of the adjustment of the calculated piezometric heads to the measured values. This calibration depends on the initial flow and, consequently, on the flow resistance equations. For example, for smooth pipes with high Reynolds numbers (R e > 4000), the von Kármán-Prandtl formulation may be use; for rough pipes and high Reynolds numbers, Colebrook-White formula may be use the; (15)
for the laminar regime, Hagen-Poiseuille equation should be use (Step 1). If the selected flow resistance equation has parameters (e.g., pipe roughness) than these have to be calibrated (in Step ). Usually, the continuous and the local head losses are incorporated in the equivalent pipe roughness. The hydraulic grade line is affected by the downstream end valve which controls the flow. This control is achieved by partially closing or opening the valve. The valve causes a local head loss described by: h valve where Q Kvalve (16) gs h valve = local head loss caused by the valve; K valve g = gravity acceleration; S = pipe cross-section. = loss coefficient of the valve; Q = flow-rate; The loss coefficient is a function of the valve opening and is, generally, experimentally determined for steady-state flows but it is assumed to be applicable for unsteady conditions. The loss coefficient of the valve can vary between zero (totally opened valve) and infinite (closed valve). The last step of calibration of the steady state calibration (Step 3) focuses on the valve opening. The calibration of this parameter can be carried out by inverting the equations to obtain a reference value or by a trial-and-error procedure. STEP 1 S Laminar regime Turbulent regime S Smooth pipe region Rough pipe Hagen-Poiseuille von Kármán- Prandtl Colebrook-White von Kármán STEP STEP 3 Adjustment of the hydraulic energy line by Figure 1. Stage I: Steady-state calibration Transient state calibration Valve maneuver The correct data input and exact characterization of the desired operation to be simulated (i.e., valve closure maneuver) is essential as it deterministically defines the results the model and their quality. The duration of the valve maneuver, the diameter and type of law (linear or non-linear) will influence the shape and values of the hydraulic grade line. The calibration of the valve maneuver requires (i) the initial time of closure based on a preliminary analysis of measurements, (ii) the total time of closure (corresponding to the inflection point of the first wave), and (iii) the maneuver (i.e., pairs of points time-opening). The valve maneuver may be linear, bilinear, or composed of linear stretches. Typically, in studies of water hammer, maneuvers are approximately linear; however, this does not
occur in most real cases in which maneuvers are complex and the method of interpolation that best fit should be chosen (Step 4). Range of wave speed values Wave speed can be estimated by (Chaudhry, 1987; Wylie and Streeter, 1993): a K 1 K D E d e where a = wave speed; K = water bulk modulus of elasticity; = water volumetric weight; E d = pipe dynamic modulus of elasticity; D = pipe inner diameter; e = pipe wall-thickness; = pipe axial constraints coefficient. For plastic pipe, the dynamic modulus of elasticity should vary between 1.5 and.0 of the static modulus of elasticity (E 0 ) provided by manufactures for PE (Covas el al., 005), and, between 1.0 and 1.5 of E 0 for PVC (Soares et al., 008). Although the dynamic modulus of elasticity leads to a higher elastic wave speed in about 10-0%, the overall wave speed calculated using the calibrated creep function is significantly lower than the one obtained by E 0, as observed by other authors (Pezzinga and Scandura, 1995; Pezzinga, 00). For non-plastic pipes, E 0 can be used. The wave speed is not only related to the bulk modulus of elasticity, K, of the fluid but also depends on the pipe properties and the physical external constraints. The pipe elastic properties are greatly influenced by the diameter, wall thickness and the most importantly the type of material. Physical constraints are related with the type and number of supports, and the ability of the conduit to move longitudinally. The bulk modulus of elasticity of a fluid depends upon the pressure and temperature. Several studies have concluded that the presence of dissolved gas tends to decrease the wave velocity significantly as they tend to come out of solution during transient low pressure peaks (Streeter, 197; Ramos, 00). Therefore, a range of wave speed values should be estimated using maximum and minimum values of dynamic modulus of elasticity, Poisson coefficient and axial constraints. Linear-elastic transient solver The use of a linear-elastic transient solver is more suitable for concrete and metal pipes, since these pipe materials have a linear-elastic rheological behavior. This solver can incorporate either a steady state friction model (Case a) or an unsteady friction model (Case b). In the former case, the wave speed is calibrated by a trial-and-error procedure or by using an optimization algorithm fitting the numerical results to the maximum and minimum observed pressures. No other parameters are calibrated. In the latter case, the wave speed is calibrated together with the unsteady friction model parameters (if any) (e.g., decay coefficient in the Vitkovsky s model) trying to fit both extreme pressures, and the pressure wave damping and phase shift (Step 6). Viscoelastic transient solver When the pipe is made of plastic (i.e., HDPE or PVC), a viscoelastic transient solver should be used. If a steady state friction loss model (Case c) is used the calibration consists of determining the number of Kelvin-Voigt elements (k), the relaxation times ( k ) and the creep parameters J k, and selection of the best fitted solution. The number of Kelvin-Voigt (K-V) elements to consider in the mathematical model depends essentially on the type of the pipe material. According to Covas (003), for HDPE pipes at least three elements of the K-V model are required to obtain a good calibration, whereas, for PVC pipes, Soares (008) states that only one element is needed. Relaxation times, k, can be estimated for each K-V element as follows: (i) the 1 st element is equal to half of the valve closure time, t c, ( 1 ); (ii) the second element is equal to 1/ period of the pressure wave, t c T, ( T ); (iii) the 3 rd element is equal to 1/3 of the simulation time, t, ( 3 t ) (Step 6). (17)
The creep coefficients J k (for a pre-set of k ) should be simultaneously calibrated with the wave speed, a (Step 7). If instead a steady state friction loss model, an unsteady model is used (Case d) than the calibration of the J k parameters for the range of celerity values should be carried out simultaneously with another calibration parameters, if any (Step 6). STEP 4 STE STEP 5 STE Case a Case b Case c Case d Steady-state Unsteady Steady-state Unsteady with parameters without parameters with parameters without parameters wave speed value of the wave speed and of UF parameters of the wave speed value wave speed value of the wave speed and of UF parameters of the wave speed value STE STEP 6 k for the range of celerity values including the UF of J k with other parameters (k 3 ) for the range of celerity values of J k for the range of celerity values including the UF STE STE STEP 7 of the best of the best Figure. Stage II: Unsteady-state calibration CASE STUDY A data collection program was carried out in an experimental facility at the Laboratory of Hydraulic and Water Resources from the Instituto Superior Técnico, Lisbon (Portugal). This experimental facility has a tank-pipevalve configuration whose piezometric line is controlled by a downstream end valve that discharges into the atmosphere. The experimental facility is composed of a closed pipe circuit in which a centrifugal pump injects water from a tank into a hydropneumatic vessel. The pipe is made of HDPE with a nominal diameter of 50 mm, a wall thickness of 3 mm and a nominal pressure of 10 bar. The pipe was installed in coil with 1 m of radius. The HDPE pipe has a total length of 199 m. The installation from the hydropneumatic vessel to the downstream valve has a total length of 03.37 m.
The downstream boundary condition of the experimental facility is an atmosphere-valve. This valve is a ball valve type and is manually operated and closed and opened as fast as possible to simulate instantaneous valve maneuvers. The inflow is controlled by another ball valve at the upstream of the tank with two-compartments. The first has a capacity of 400 liters and has inside a second one with 100 liters of capacity in which it is assembled a triangular weir with an angle of 90º. The pump sucks water from the tank to the hydropneumatic pressure and closes the circuit. The schematic configuration and plan view of the experimental facility are depicted in Figure 3. Figure 3. Schematic of the PE pipe experimental facility Two distinct data collection programs were carried out with several tests. Collected data consisted of the initial water height above the weir (to calculated flow) and transient pressures at three transducers. The transducers were installed as follows: the first (T1) at the hydropneumatic pressure vessel (x=0 m); the second (T) at a middle section of the pipe (x=101.93 m); and the third (T3) immediately upstream of the ball valve (x=03.37 m). In the first data collection program, flow measurements were obtained by measuring the level of water above the weir using an installed level scale, whilst, in the second, measurements of the water level were carried out by using an hydrometer. Every test consisted of the fast closure (almost instantaneous) of the ball valve at the downstream end of the pipeline. The tests of the first data collection program were used to calibrate the mathematical model and the tests of the second were used for model validation. RESULTS AND DISCUSSION Steady state calibration An explicit formulation of the Colebrook-White equation was used, the Zigrang and Sylvester formulation (198), as its uncertainty is lower than 0.13%. The roughness coefficient, k, is the calibration parameter. An estimation of the absolute equivalent roughness was carried out, being obtained the average value of 0.06 mm (Step ). To fit the calculated piezometric head to the measured value at the downstream end of the pipe, the valve opening was calibrated by a trial-and-error procedure (Step 3). Unsteady state calibration The calibration of the valve maneuver (Step 4) was approximated to a set of three linear lines. The static modulus of elasticity for HDPE pipes varies between 0.7 and 1.0 GPa; and considering that the dynamic modulus of elasticity is 1.5 times the static modulus of elasticity, the estimated range of wave speed is between 73 and 371 m/s (Step 5). Calibration in the next steps was carried out for a wave speed range between 70 to 380 m/s, where 10 values were adopted differing by increments of 10 m/s.
To test the methodology a linear elastic solver with unsteady friction modelling (Case b in Figure ), a viscoelastic solver with a steady state friction loss model (Case c in Figure ) and viscoelastic solver with a unsteady state friction loss model (Case d in Figure ) were used. In the latter case, the unsteady friction formulation has parameters, these have to be calibrated simultaneously with the wave speed (Step 6). Linear-elastic transient solver In the case of a linear elastic solver with unsteady friction modelling the Vitkovsky s formulation (Equation 8) has been used. In this formulation, the empirical coefficient, k 3, needs to be calibrated simultaneously with the wave speed, a. This calibration presupposes that previous Steps 1-4 have already been carried out. An Inverse Transient Solver (ITS) running the Levenberg-Maquardt optimization algorithm was used to calibrate k 3 for a fixed value of a. The procedure was repeated for each value of a within the range of expected wave speed values (i.e., from 70 to 380 m/s with a set of 10 by 10 m/s). Table 1 presents the best fitted k 3 and a values obtained for each experimental test, as well as the value of the OF. The OF is defined by the mean square error between measurement pressure at the downstream end of the pipe and calculated pressure by Equation (15), for a sample of 10 s. Table 1. Optimal values obtained for the empirical coefficient k 3 and wave speed a Flow (L/s) k 3 (-) a (m/s) OF (m ).735 0.38 90 44.6.037 0.81 300 7.4 1.136 0.310 310 9.6 0.499 0.357 30.1 The analysis of the results shows that: (i) The optimal k 3 values occur at different wave speeds a for each experimental test varying between 90 and 30 m/s, for higher and lower flow-rates, respectively. (ii) Covas (003) concluded that the calibrated empirical coefficient k 3 increases with the increase of the initial flow-rate Q: for a given creep-function, calibrated decay coefficients were 0.08, 0.030 and 0.033, respectively for Q=0.50 l/s (Re=1,600), Q=1.008 l/s (Re=5,000) and Q=1.50 l/s (Re=37,100). In the current study, this conclusion was verified as it was found that the empirical coefficient k 3 and Q vary in the same way. Calibrated decay coefficients k 3 are within the expected range of values for single-phase flows (Brunone et al., 1995; Bughazem and Anderson, 1996; 000). (iii) The OF decreases with the increase of the initial flow-rate Q, because Q has increases, higher transient pressures are and, naturally, higher are the differences between measured and calculated pressures; for each flow-rate, the OF decreases with the increase of a until it reaches a minimum value and, then, it starts increasing again. Viscoelastic transient solver The viscoelastic (VE) transient solver for describing hydraulic transients in plastic pipes was used with a steady state friction loss model (Case c in Figure ) and with unsteady friction loss model (Case d in Figure ). For the sake of simplification Vardy s (199) unsteady friction formulation without parameters was used in the latter. For both cases, the creep function coefficients J k (for a pre-set of k ) should be calibrated simultaneously with the wave speed, a. The number of Kelvin-Voigt (KV) elements used in the mathematical model depends essentially on the type of the pipe material. In the current case, three KV elements were considered for the PE pipe (Step 6). Relaxation times, k, for each K-V element considered followed Covas (003) recommendations namely 0.010 s t 0.015 s, T 0.5 s and 3 T / 30 s (Step 6). 1 c
The analysis of the initial values of the creep function, J k, introduced in the optimization model was made and it was concluded that they can take any values. However, an initial value far from the optimal solution requires a larger number of iterations and hence more computational time. Therefore, it is advisable to start the calibration of the creep function with initial values nearest to 0.1 GPa -1. Two different calibrations of the creep function were carried out: (a) in the first one, UF losses were neglected - Case (c); (b) in the second one, Vardy s formulation (199) was considered for describing UF Case (d). Table presents the results obtained for the calibration of the creep function without the unsteady state friction loss model (Steps 7-8). Table. Optimal values of the creep function neglecting the unsteady-state component of the friction losses Flow (l/s) a (m/s) 1 (s) J 1 (GPa -1 ) J (GPa -1 ) J 3 (GPa -1 ) OF (m ).735 350 0.015 0.406 0.14 0.314 0.85.037 340 0.016 0.34 0.18 0.138 0.39 1.136 370 0.011 0.413 0.16 0.680 0.18 0.499 350 0.010 0.339 0.16 0.000 0.11 Obtained results show that: (i) values of the elastic wave speed above 300 m/s do not change the creep function; (ii) for the experimental test Q = 0.499 L/s the component J 3 is null, meaning that only two elements are required to represent the creep function for low flow; (iii) the values of each element of J k decrease with the flow increase; (iv) the value of J 1 raises almost linearly with the wave speed increase; (v) the values of J and J 3 stabilize with as the wave speed increases. The calibration of the creep function considering the UF with Vardy s formulation (199) showed similar results to those presented above. An excellent agreement between numerical results and the measurements were observed showing the importance of using accurately calibrated linear viscoelastic solvers to describe transient behavior of fluids in plastic pipes, particularly for fast transient events. SUMMARY AND CONCLUSIONS This research work aimed at the establishment and application of a calibration methodology for hydraulic transient solvers incorporating unsteady friction and pipe wall linear elastic (LE) and viscoelastic (VE) behaviors. A hydraulic transient solver incorporating Vitkovsky s (000) and Vardy s (199) UF formulations and the two rheological behaviors was used to test this methodology. The model was calibrated using data collected in a laboratory facility made of PE with approximately 00 m. Reasons for suggesting a step-wise calibration of hydraulic transient solvers, instead of the blind simultaneous calibration of all known parameters, have been discussed.. Presented methodology aims to assist the user in the calibration of transient solvers and significantly simplifies the parameter fitting process. Further research can be carried out to extend this methodology to more complex pipe system with different boundary conditions and other unconventional dynamic effects (e.g., cavitation, pipe movement, dissolved air), and to overcome the practical difficulties found in the simultaneous calibration of many parameters with overlapping effects and for which the model has different sensitivities. The proposed methodology is a result of the authors experience in the calibration of numerous transient solvers using data collected in laboratory and in field conditions. Theoretically and ideally, when using an optimization algorithm all unknown parameters could be simultaneously calibrated instead of using a step-wise procedure as proposed; however, there are two main reasons for not having suggested that approach. The first reason is that when using real data, no matter how well calibrated is the hydraulic solver the objective function is never zero. First, there are always some uncertainties in the formulations used to describe considered phenomena (e.g. UF formulations, linear viscoelasticity assumption); secondly, there are other dynamic effects that are neglected in the model (e.g., small percentage of dissolved air); finally, the pressure signal (even after
node node3 node4 node5 node6 node7 node8 node9 All pipes k' Jk(0.05) Jk(0.5) Jk(1.5) Jk(5) Jk(10) Piezometric Head (m) Average dh/dxi being filtered) has always inevitable noise resulting from other electrical interferences or mechanical vibrations not described by the mathematical model. The second reason for not performing simultaneous calibration of all unknown parameters (i.e., closure time, closure maneuver, pipe wall roughness k, wave speed a, UF coefficient k 3, VE coefficients k and J k, leaks sizes and locations) is because the effect of these parameters overlaps and is not clearly distinguishable in terms of observed dissipation, delay and shape of transient pressure signal. Additionally, the inverse solver has different sensitivities to each parameter and the result is the fast convergence to the parameters with higher sensitivities and, only afterwards, to the other parameters. As a result, the final solution (i.e., the best fitted parameters) is one of many combinations of parameters that leads to minimum values of the OF (this solution can have an OF lower that the one corresponding to the true combination of parameters). Covas (003) has observed this when trying to simultaneously locate leaks and to calibrate pipe wall roughness k, UF coefficient k 3 and VE coefficients J k as presented in Figure 4. The major difference in sensitivities is between the decay coefficient ( dh/dx i ~10) and the creep coefficients ( dh/dx i ~1E9 to 1E10), whereas the other two parameters (leaks and pipe roughness) have intermediate sensitivities. This is one of the reasons why the ITS fails to accurately estimate the parameters. The second reason is that UF and VE despite being phenomena with a complete different nature, they have similar effects in the transient pressure response, which the ITS is not capable of distinguishing. (a) (b) 65 60 55 50 45 40 35 Artifitial Data Best Fitted Solution 1E+11 1E+10 1E+09 1E+08 1E+07 1E+06 100000 10000 1000 100 10 1 30 Figure 4. Artificial data vs. the best fitted solution and (b) average sensitivity of piezometric head with each parameter xi for the best solution (Unsuccessful calibration of four different parameters) REFERENCES 0 4 6 8 10 Time (s) Parameter Leaks Roughness Decay Creep Coefficient Coefficient Coefficient Adamkowski, A. and Lewandowski, M. (006). Experimental examination of unsteady friction models for transient pipe flow simulation. Journal of Fluids Engineering, 18(6):1351 1363. Allievi, L. (190). Teoria generale del moto perturbato dell acquani tubi in pressione, Ann. Soc. Ing. Arch. Ithaliana (French translation by Allievi 1904, Revue de mécanique). Almeida, A. B. and Koelle, E. (199). Fluid Transients in Pipe Networks, Computational Mechanics Publications, Elsevier Applied Science, Southampton, UK. Bergant, A. and Simpson, A. R. (1994). "Estimating Unsteady Friction in Transient Cavitating Pipe Flow." Proceedings nd International Conference on Water Pipeline Systems, Pub. BHR Group Ltd., Edinburgh, UK, 3-16. Bergant, A., Simpson, A., Vitkovsky, J., (001). Developments in unsteady pipe flow friction modelling. Journal of Hydraulic Research, IAHR, 39(3), 49 57. Brkić, D. (011). Review of explicit approximations to the Colebrook relation for flow friction. Journal of Petroleum Science and Engineering, 77(1), 34 48. Brunone, B., Golia, U. M., Greco, M., (1991). Some remarks on the momentum equation for fast transients. In: Cabrera, E., Fanelli, M. (Eds.), Proceedings of International IAHR Meeting on Hydraulic Transients and Water Column Separation. Valencia, Spain, 01 09. Brunone, B., Golia, U. M., Greco, M., (1995). Effects of two-dimensionality on pipe transients modeling. J. Hydraul. Eng., ASCE, 11(1), 906 91.
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