HYDRAULIC TRANSIENTS IN PUMPING SYSTEMS WITH HORIZONTAL PIPES

3 rd IAHR Europe Congress, Book of Proceedings, 2014, Porto -Portugal. ISBN xxx-xxxx-xx-x HYDRAULIC TRANSIENTS IN PUMPING SYSTEMS WITH HORIZONTAL PIPES JOÃO DELGADO (1), DÍDIA I.C. COVAS (2) & ANTÓNIO BETÂMIO DE ALMEIDA (3) (1) Técnico Lisboa, Lisboa, Portugal, joao.borga.delgado@ist.utl.pt (2) Técnico Lisboa, Lisboa, Portugal, didiacovas@tecnico.ulisboa.pt (3) Técnico Lisboa, Lisboa, Portugal, betamio.almeida@ist.utl.pt Abstract The current paper aims to discuss the main uncertainties associated with the hydraulic transient modelling of raising pipes systems with and without surge protection. A numerical model was developed to describe the 1-D hydraulic transients in raising pipes. The model was tested using transient pressure data collected from two laboratory hydraulic circuits. Transients were generated by the sudden stoppage of the pump. Collected data were compared with the results of the numerical modelling and used to calibrate model parameters, and a good agreement was obtained. Some tests with the hydropneumatic vessel connected lead to higher pressure surges than when there was no protection in the system. Keywords: hydraulic transients; pumping systems; hydropneumatic vessel; check valve 1. Introduction The control of hydraulic transients in pressurized pipes is a major concern for engineers and pipe system managers, for reasons related to risk, safety and efficient operation. Examples of problems caused by hydraulic transients are the occurrence of instabilities in the operation of systems due to the transient sub-atmospheric pressures and, consequently, the occurrence of cavitation, or the pipe burst due to the occurrence of overpressures. Hydraulic transients can be analysed by means of simplified formulas (e.g., Joukowsky and Michaud formulas), classic water hammer simulators (e.g., commercial models) and by using advanced simulators (nonconventional models). During a transient event caused by the sudden pump stoppage due to power failure, pressures variations can be extremely high and engineers usually design surge protection devices to control the transient-pressures. However, under some circumstances, these protection devices may worsen the pressure conditions. One of those cases is the use of hydropneumatic vessels (HPV) immediately downstream of pumping stations in pipelines with horizontal profile (Thorley, 1991). The current paper aims at the experimental and numerical analysis of the pipe flow behaviour during transient events carried out in two experimental facilities under controlled laboratory conditions. The paper includes a brief description of the developed hydraulic transient solver, the description of the experimental facilities used and the set of experimental tests carried out, as well as the comparison of experimental tests and the mathematical model results. Finally, the main uncertainties associated with these hydraulic transients modelling are discussed. 1

2. Hydraulic Transient Solver 2.1 Method of Characteristics A mathematical model for the calculation of hydraulic transients in pressurised pipes has been developed based on the classical theory of waterhammer. Equations that describe the behaviour of fluid in the pipe during transient events are based on the continuity and the momentum conservation principles independently of initial and boundary conditions. These equations are (Chaudhry, 1987; Covas, 2003; Wylie and Streeter 1993): Continuity Equation: Momentum Equation: dh dt + a2 Q gs x = 0 [1] H x + 1 dq gs dt + h f s = 0 [2] where H = piezometric head (m); a = elastic wave speed (m/s); g = gravity acceleration (m/s 2 ); S = area of the pipe cross-section (m 2 ); Q = flow rate (m 3 /s); h fs = steady-state component of friction losses (m/m); t = time coordinate (s); and x = spatial coordinate along the pipe axis (m). These equations have some assumptions inherent to the classical waterhammer theory (Covas, 2003), namely: (i) the flow is one-dimensional with a pseudo-uniform velocity profile; (ii) fluid is one-phase, homogenous and compressible and the changes in fluid density and temperature during transients are negligible; (iii) head losses during transient flow are calculated similarly to steady-state by friction formulas; (iv) the pipe material has a linear rheological elastic behaviour; (v) the dynamic fluid-pipe interaction is neglected assuming a constrained pipe without axial movement; (vi) the pipe is straight, uniform, with constant cross-section, without lateral in/out flow. The Method of Characteristics (Wylie and Streeter, 1993) allows the transformation of the previous partial differential equation into two ordinary differential equations, which can be solved by means of finite-difference schemes. A second order numerical scheme was considered for the friction resistance term. The compatibility equations form is presented below: C + : Q P = C P C ap H P [3] C : Q P = C N + C an H P [4] where C P, C a +, C N and C a = constants defined for each pipe section and time (-).The numerical solutions for each constant are presented in Table 1. Eqs. [3] and [4] describe transient flow along two straight independent lines that propagate flow and piezometric head information in the space-time domain. Table 1 Compatibility equations parameters Compatibility equations terms Additional terms C P = Q A + C a H A 1 + R A t Q A C a C ap = 1 + R A t Q A gs C a a R A = f sa 2DS C N = Q B C a H B 1 + R B t Q B C a C an = 1 + R B t Q B R B = f sb 2DS 2

2.2 Boundary conditions 2.2.1 Sudden pump stoppage For the complete mathematical representation of a pump, the relationships between discharge, Q, rotational speed, N, pumping head, H, and net torque, T, have to be specified. These are called the pump characteristics and, in most engineering situations, are not available from the manufacturer. The curves tend to have similar shapes for the same specific rotational speed. If such data are not available, then the curves must be extended by comparison with data for other specific speeds. This is an uncertain procedure, and the results of transient studies using such data must be analyses with scepticism (Wylie and Streeter, 1993). For this purpose, dimensionless parameters referring to the point of best efficiency (R=rated conditions) are used as a reference (Chaudhry, 1987): v = Q Q R ; h = H H R ; α = N N R ; β = T T R [5] where v, h, α and β = dimensionless parameters for flow rate, piezometric head, rotational speed and torque, respectively (-) and the subscript R refers to pump rated conditions. Chaudhry (1987) presents the basis for the calculations of the variables v, h, α and β, in which an iterative procedure is used. Experimental curves for dimensionless pump head, F h and dimensionless torque, F β, are presented in Figure 1. According to Marshall et al., (1965), these parameters may be plotted with angle θ S = tan 1 (α/v) and are described by: F h = h (α 2 +v 2 ) ; F β = β (α 2 +v 2 ) [6] Figure 1. Sutter characteristics curves: pump s head and; pump s torque for four specific rotational speed values (Soares et al., 2013) In this study, the suction line of the pump is considered short and the following equations are used to describe pumping head, the losses in the check valve and the decelerating torque: H P = H resu + H pump H valve [7] H valve = C V Q P Q P [8] T = WR 2π dn 2 60 dt [9] where H resu = upstream revoir level (m) H pump = pumping head in the end of each time-step (m); H valve = local headloss in the valve (m); C V = headloss coefficient in the valve (s 2 /m 5 ); and WR 2 = combined polar moment of inertia of the pump, motor, shaft and liquid inside the pump impeller (kg/m 2 ). 3

2.2.2 Valve discharging to atmosphere Considering a valve discharging to the atmosphere, the following expression may be considered: Q P H P (Z V + 2 2gS ) = C VQ P Q P [10] V where Z V = valve elevation (m) and S V = valve cross-section (m 2 ). 2.3 Hydropneumatic vessel The mathematical formulation to describe the hydropneumatic vessel (HPV) considers the following equations valid in the vessel connection with the main pipeline Eqs. [11] to [13] and inside the vessel Eqs. [14] to [17] (Figure 2). Q 1 = Q Porf + Q 2 [11] H 1 = H 2 = H P [12] ΔH Porf = C orf Q Porf Q Porf [13] H 1 = H Pair H b + Z P + ΔH Porf [14] V Pair = V air A HPV (Z P Z) [15] Figure 2 HPV scheme Z P = Z + 0.5 (Q Porf + Q orf ) t [16] A HPV H Pair V m Pair = C 0 [17] where Q Porf and Q orf = flow rate through the orifice at the end and beginning of the time-step, respectively (m 3 /s); ΔH Porf = local headloss in the HPV orifice (m); C orf = local headloss coefficient of the orifice (s 2 /m 5 ); H Pair = absolute pressure head (N/m 2 ); H b = absolut barometric pressure head (m); Z P and Z = water level in the HPV at the end and beginning of the time-step (m), respectively; V Pair and V air = volume of air inside the HPV at the end and beginning of the time-step (m 3 ), respectively; S HPV = area of the HPV cross-section (m 2 ); m = exponent of polytropic gas equation (-); and C 0 = constant of the polytropic gas equation. 3. Experimental facilities 3.1 Copper-pipe facility An experimental pipe rig was assembled in the Laboratory of Hydraulics and Water Resources, in Department of Civil Engineering and Architecture, at Instituto Superior Técnico (Figure 3a). System is composed of a coiled copper pipe with approximately 100 m of length, 20 mm of inner diameter and 1 mm of pipe-wall thickness. The system is supplied from a storage tank with 125 l of capacity by pump with nominal flow rate of 1.0 m 3 /h and nominal head of 32 m (Figure 3b). Immediately at downstream of the pump, there is a PN6 hydropneumatic vessel (HPV) with 60 l of capacity (Figure 1b). The HPV is connected or disconnected from the main pipeline by the opening or close of a ball valve, respectively. At the downstream end of the pipe, there is a ball valve with DN 3/4 that allow the generation of water hammer. Steady-state flow rates are measured by a rotameter (Figure 3c) and transient-state pressures are measured by three strain-gauge type pressure transducers located in different sections of the 4

pipe (T1 - upstream end, T2 - middle section and T3 - downstream end). Several experimental tests were carried out to collect transient data caused by the sudden stoppage of the pump. Figure 4 presents the data collected at transducer T1 for the sudden stoppage of the pump for flow rates between 100 and 500 l/h, with ad without the HPV connected to the system. (c) Figure 3. Copper-pipe facility: general view; upstream view; (c) downstream view Figure 4. Data collected at T1 in the copper-pipe facility: without HPV; with HPV Figure 4 shows that maximum pressures are extremely high when the HPV is connected to the system (Figure 4b). The reason for this is that, as the pump stops, the HPV starts supplying the pipeline at downstream as well as the pipe for upstream, between the vessel and the pump; because the check valve does not instantaneously close, allowing some reverse flow, when it actually closes, the reverse flow is quite high, inducing an extremely high upsurge. This shows that, under certain circumstances, surges protection devices can create higher pressures than when they are not installed in the system, for instance in pipelines with horizontal profile (Delgado et al., 2013). 3.2 Steel-pipe facility The system is composed of a pipeline made of steel, with a total length of 115 m, an inner diameter of 206 mm and a wall thickness of 6.3 mm. The pipeline is installed along the internal perimeter of the Laboratory (Figure 5a). The system is supplied from a storage tank through a centrifugal pump with a nominal flow rate of 30.3 l/s, a nominal elevation of 32.0 m and a installed power of 15 kw, with a ball check valve located at immediately downstream. A 1 m 3 HPV is installed at downstream the pump; this device can be connected as a side element connected through a branch or totally disconnected from the system by the opening/closing of a set of gate valves (Figure 5b). The flow can circulate in the pipeline in two directions, reason why it is called a reversible system. At the downstream end there are two ball valves with 50 mm diameter each, used to generate waterhammer and control the flow rate. 5

The facility is equipped with instrumentation for collecting steady flow data (electromagnetic flow meter ABB Processmaster FEP311DN65 and accuracy of 0.4% of measured values), transient pressure data (WIKA pressure transducers with an absolute pressure range from 0 to 25 bar and accuracy of 0.5% of full range). Figure 6 presents the data collected at transducer T1 for the sudden stoppage of the pump for flow rates between 5.0 and 20.1 l/s with and without the HPV connected. Data collected have the same features as the one collected in the copper-pipe facility. Figure 5. Steel-pipe facility: plan; side view of the reversible zone. Figure 6. Data collected at T1 in the steel-pipe facility: without HPV; with HPV 4. Model calibration and validation 4.1 Copper-pipe facility 4.1.1 Sudden pump stoppage without air vessel Pump-motor characteristics: The pump is a centrifugal pump with the following nominal parameters: Q R = 1.0 m 3 /h, pumping head H R = 32.0 m, power P R = 1.75 kw and rotational speed N R = 2900 rpm. The pump-motor inertia was estimated by Thorley and Faithfull (1992) formulation. The pump-motor inertia was estimated in I= 0.0025 kg.m 2. Wave Speed Estimation: The theoretical value of the wave speed was estimated in 1290 m/s by classic formula for a copper pipe (Young's modulus E= 177 GPa) with 20 mm inn diameter, wall thickness of 1 mm and unconstrained throughout its length. It was also estimated by the traveling time of the pressure wave between the downstream and the upstream transducer, being the obtained value 1150 m/s. This value is much lower than the theoretical value because there is gas in the liquid in a free form and cumulated in air pockets along the pipeline. Pipe roughness estimation: The pipeline has smooth walls, thus steady state friction was estimated using Nikuradse formula (Lencastre, 1983). Model calibration: The transient test carried out for the initial flow rate of 600 l/h was used for model calibration (Figure 7). The upstream level was 0.25 m and the valve elevation was set at 6

1.4 m with an opening of 7.16%. Based on the comparison of numerical results obtained and collected data, total motor/pump inertia was adjusted to 0.0025 kgm 2 and the pump power to 1.50 kw. A check valve manoeuver was calibrated (Figure 7b). Model validation: Previous values were used for model testing with Q= 300 l/h. A good fitting was obtained (Figure 8). The power was set in 2.00 kw and an opening of 4.92% was set for the downstream valve. A different check valve manoeuvre was calibrated (Figure 8b). Figure 7. Results of calibration for sudden pump stoppage without HPV for Qi = 600 l/h in the copper-pipe facility: piezometric head and flow rate, at downstream the pump. Figure 8. Results of validation for sudden pump stoppage without HPV for Qi = 300 l/h in the copper-pipe facility: piezometric head and flow rate, at downstream the pump. 4.1.2 Sudden pump stoppage with air vessel In this section, the hydraulic system is simulated but considering the connection of the HPV to the main pipeline (Figure 9). The HPV characteristics are: distance to the pump = 0.25 m; inlet orifice diameter = 20 mm; local headloss coefficient = 2; initial water level = 0.56 m; air vessel diameter = 0.35 m; initial volume of air = 30 l. Figure 9 shows that the hydraulic transient solver developed cannot reproduce the overpressures after the check valve closure, despite a significant reverse velocity occurs, at the valve closure (approximately -800 l/h which correspond to a velocity of 0.71 m/s). This is related with the HPV model that does not consider the branch pipe (it only considers an orifice between them), neglecting the fluid inertia inside the branch pipe. When the overpressure occurs, the HPV actuates quasi-instantaneously, controlling the upsurge. The system was simulated with a different validated model that considers a branch pipe (Figure 10). Maximum and minimum upsurges can be reasonably well estimated using this model. 7

Figure 9. Results of calibration for sudden pump stoppage with HPV in copper-pipe facility for Qi = 600 l/h results from the model developed: piezometric head downstream the pump; flow rate downstream the pump and check valve calibrated manoeuvre Figure 10. Results of calibration for sudden pump stoppage with HPV in copper-pipe facility for Qi = 600 l/h results from other validated model: piezometric head downstream the pump; flow rate downstream the pump and check valve calibrated manoeuvre 4.2 Steel-pipe facility 4.2.1 Sudden pump stoppage without air vessel Pump-motor characteristics: The pump have the following nominal parameters: Q R = 30.3 l/s; H R = 32.0 m; P R = 15.00 kw and N R = 2955 rpm; I = 0.075 kg.m 2 (estimated). Wave speed estimation: The theoretical value of the wave speed was estimated in 1260 m/s by classic formula for a steel-pipe (Young's modulus E= 210 GPa) with 206 mm inn diameter, wall thickness of 6.3 mm and unconstrained pipe throughout its length. It was also estimated by the traveling time of the pressure wave between the downstream and the upstream transducer, being the obtained value 950 m/s. This value is much lower than the theoretical value because there is gas in the liquid in a free form and cumulated along the pipeline. Pipe roughness estimation: This parameter was estimated based on the headlosses measured for steady-state conditions. The equivalent pipe roughness varied with the flow rate between 4 and 57 mm. Local losses are included in this parameter, reason why this parameter is not constant. Model calibration: The transient test carried out for the initial flow rate of 15.8 l/s was used for model calibration (Figure 11). For this initial flow rate the roughness was estimated in 48 mm. The upstream level was set at 0.50 m and the downstream valve elevation was set at 2.9 m with a constant opening of 25.50%. The total motor-pump inertia was adjusted to 0.075 kgm 2 and the power was adjusted to 15 kw. A check valve manoeuvre was calibrated. Model validation: Previous values were used for model testing using a different flow rate, Q= 10.0 l/s (Figure 12). The power was set in 14.5 kw and an opening of 13.38% was set for the downstream valve. A different check valve manoeuvre was calibrated. 8

Figure 11. Results of calibration for pump failure without HPV for Qi = 15.8 l/s in the steel-pipe facility: piezometric head and flow rate, at downstream the pump. Figure 12. Results of validation for pump failure without HPV for Qi = 10.0 l/s in the steel-pipe facility: piezometric head and flow rate, at downstream the pump. 4.2.2 Sudden pump stoppage with air vessel As in the copper-pipe facility the hydraulic system was simulated with the HPV connected to the system. The vessel s characteristics are: distance to the pump = 2.5 m; inlet orifice diameter = 50 mm; local headloss coefficient = 5; initial water level = 1.20 m; air vessel diameter = 0.80 m; initial volume of air = 0.5 m 3. The model cannot reproduce the pressure variations observed (Figure 13). Simulating with the other model considering a branch pipe with 1.5 m length, the maximum and minimum pressures after the check valve closure can be reasonably well estimated (Figure 14). Figure 13. Results of calibration for sudden pump stoppage with HPV in the steel-pipe facility for Qi = 15.8 l/s results from the model developed: piezometric head downstream the pump; flow rate downstream the pump and check valve calibrated manoeuvre 9

Figure 14. Results of calibration for sudden pump stoppage with air vessel in steel-pipe facility for Qi = 15.8 l/s results from other validated model: piezometric head downstream the pump; flow rate downstream the pump and check valve calibrated manoeuvre 5. Conclusions A one-dimensional solver has been developed based on the classic waterhammer theory, incorporating and three link-elements - the pump described by Suter parameters, the downstream valve described by a constant opening and a HPV described by the polytropic equation of gases. Transient tests induced by the sudden pump stoppage with and without a HPV connected have been carried out in two experimental facilities assembled at IST. Collected data has shown that, under some circumstances, the pressure surges can be higher when a HPV is connected to the system than when it is not. The vessel creates a higher deceleration of the flow, inducing higher reverse velocities through the check valve when it actually closes. The higher the velocities generated the higher the overpressure are. The analysis has shown that a good agreement between the numerical results and collected data can be obtained for the transients generated by the pump sudden stoppage, as long as a check valve manoeuver is adjusted and the total motor+pump inertia and power are calibrated based on collected transient pressures. Parameters that were also calibrated were pipe roughness or friction formula used based on steady-state data and the opening of the downstream valve that control the steady-state flow rate. The numerical model was not capable of reproducing the pressure conditions observed when the HPV was connected to the system, due to the non-consideration of the branch pipe, neglecting the fluid inertia. Therefore, as the overpressure occurs, the vessel actuated quasiinstantaneously, controlling the surge wave. Thus, the model developed in the numerical model for the HPV does not represent the reality for fast transients. This highlighted the need for the development of more complete models capable of describing these phenomena. Acknowledgments The author would like to acknowledge Fundação para a Ciência e Tecnologia (FCT) for the project PTDC/ECM/112868/2009 Friction and mechanical energy dissipation in pressurized transient flows: conceptual and experimental analysis for funding the current research in terms of experimental work and grants hold. 10

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