ANALYTICAL AND NUMERICAL SOLUTION FOR A RIGID LIQUID-COLUMN MOVING IN A PIPE WITH FLUCTUATING RESERVOIR-HEAD AND VENTING ENTRAPPED-GAS

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1 Proceedings of the ASME Pressure Vessels and Piping Division Conference PVP July 7-,, Vancouver, British Columbia, Canada PVP-9 ANAYTICA AND NUMERICA SOUTION FOR A RIGID IQUID-COUMN MOVING IN A PIPE WITH FUCTUATING RESERVOIR-HEAD AND VENTING ENTRAPPED-GAS Arris S. TIJSSEING Department of Mathematics and Computer Science Eindhoven University of Technology P.O. Box 5, 5 MB Eindhoven The Netherlands a.s.tijsseling@tue.nl Qingzhi HOU School of Computer Science and Technology; State Key aboratory of Hydraulic Engineering Tianjin University Tianjin 7 China qhou@tju.edu.cn Zafer BOZKUŞ Hydromechanics aboratory Department of Civil Engineering Middle East Technical University Ankara 8 Turkey bozkus@metu.edu.tr ABSTRACT The motion of liquid filling a pipeline is impeded when the ahead of it cannot escape freely. Trapped will lead to a significant pressure build-up in front of the liquid column, which slows down the column and eventually bounces it back. This paper is an extension of previous work by the authors in the sense that the trapped can escape through a vent. Another addition is that the driving pressure is not kept constant but fluctuating. The obtained analytical and numerical solutions are utilized in parameter variation studies that give deeper insight in the system s behavior. Key words mass oscillation, nonlinear spring, entrapped, venting, orifice, rigid column, pipe filling, analytical solution liquid flow. In particular for rapid events, caused by fast valve maneuvers and pump starting, the system pressures and temperatures may rise to unacceptably high levels. The subject has received much attention with pioneering work by C.S. Martin [a-e] and E. Cabrera [a-e] and their co-workers. Many relevant papers have appeared since, up to the latest publications [a-k]. The authors contributed to the subject by providing analytical expressions that give insight in the damped oscillation of a liquid-column/-pocket system [a]. This modest contribution is extended herein by letting the driving pressure fluctuate and by allowing the entrapped to escape through an orifice. Martin s seminal paper [a] is revisited, ly forty years after its publication, with a slightly improved model. This study is intended to form a theoretical basis for the analysis of rapid pipe filling [b], accelerating liquid slugs [c], and the start-up of liquid flow in undulating pipelines [e]. INTRODUCTION The presence of in liquid-conveying pipelines may make the system s dynamic behavior unpredictable. Entrapped pockets may store and release energy in response to unsteady C.S. MARTIN S TEST PROBEM The test problem is a reservoir-pipe-orifice system where at atmospheric pressure is trapped near the orifice. Figure is a simple sketch of the situation. Sudden opening of an Copyright by ASME

upstream valve connected to the reservoir (not shown in Fig. ) makes the liquid move towards the pocket. Depending on the size of the orifice, the liquid column will bounce back or not.

2 upstream valve connected to the reservoir (not shown in Fig. ) makes the liquid move towards the pocket. Depending on the size of the orifice, the liquid column will bounce back or not. It is to be noted that slow valve opening or large orifices will not lead to pressures higher than the reservoir pressure. The data are taken from the water-air system described in [a]. The initial length of the water column is = m and the diameter of the pipe is D =. m with corresponding flow area A. The air occupies an initial length, =.7 m with an initial absolute pressure head H, =. m; the pipe length thus is x = +, =.7 m. The absolute pressure head of the reservoir is H R, =. m, so that H R, / H, =.99. The specific constant for air is R = 87 m /(s K) and the polytropic coefficient n is either. or. herein. The skin friction coefficient for the liquid flow is f =., but valve resistance and entrance head losses are ignored by setting K v = and K e + =, where the represents reservoir head transferred to kinetic flow energy at the pipe entrance. The pipe is horizontal with θ =. The orifice has the varied diameter D or with corresponding area A or and discharge coefficient C d =. The liquid column and air pocket have initial masses of kg and.8 kg, respectively. The gravitational acceleration is g = 9.8 m/s, the mass density of the water is r = kg/m, and the initial air density is taken equal to r, =. kg/m which corresponds to an initial temperature T, = 9. K (these values were not specified in [a]). Fig. Sketch of analyzed system. FUCTUATING RESERVOIR PRESSURE HEAD One new element is that the reservoir head is allowed to fluctuate, thereby satisfying liquid volume conservation: or dhr d ARvR = AR A Av() t dt = dt = (a) A HR = HR, α( ) with α = (b) A where A R is the uniform area of the horizontal cross-section of the supply reservoir, H R is the vertical liquid level, and v R is the velocity with which this level drops or rises. The volume of the liquid in the reservoir is taken larger than the volume of the air pocket, A R H R, > A,, so that α < H R, /,. The dimensionless parameter α ( max ) / H R, defines the importance of the fluctuation. The driving pressure head decreases when the liquid column lengthens and increases when it shortens, which is expected to give a retarding effect. R When the reservoir head is kept constant, by a control device, we simply take α =. GOVERNING EQUATIONS For filling liquid from an open reservoir with fluctuating head H R () into a pipe which contains a perfect and a downstream vent (Fig. ), the governing equations for velocity v and length of the liquid column, and pressure head H and mass m of the entrapped, are ( R, + a ()) d v g H H t ( t) = d t t () g a + gsin θ f Kv vt () vt () vt () vt () D t () Ke + v ()H( t vt ()) t () d = vt () () dt d H nh v nh()d t m = + d t x t () m dt dm dt with () () ( ) m H H Cd Aor Y( H ( t)) r g A( x t ()) H if.89 H, = k + k m H C d Aor r g k k + A ( x t ()) H if >.89 H, (5) if H = H, k H k, k H k, H k H H, H Y( H ) = atm otherwise (5a) Copyright by ASME

3 Equations () and () are Eqs. (b) and (a) in Ref. [a], with the driving head H R according to Eq. (b) herein. Equations () and (5) combine Eqs. (5-8) in Ref. [a], with the auxiliary variables V = A x t () (5b) ρ = m V (5c) ρ gh() t T = R ρ (5d) P = ρ g H H (5e) atm All symbols are specified in the Nomenclature. The event starts by instantaneously opening an upstream valve that separates liquid and. The volume of the pocket V is determined by the length (t) of the liquid column, from which density r and temperature T follow. Of course, all these quantities are coupled through the governing equations (-5). The pressure difference across the orifice is P and back flow of the is not allowed herein (H > H atm ). Choking flow is assumed to occur for H >.89 H,. It is noted that for small volumes, the pocket cannot be regarded as a reservoir in the derivation of the orifice equation (5); the kinetic energy of the upstream (with velocity v > ) is then to be included in the compressible Bernoulli equation [5a-b]. The adiabatic expansion factor Y is shown in Fig. in its dependence of H as defined by Eq. (5a). With A or = (closed end) in Eq. (5), dm /dt =, so that Eq. () reduces to the confined -pocket relation: n n H = ( t ) H ( t ) () where = x t (). Y 8 H_ (m) Fig. Adiabatic expansion factor. ANAYTICA SOUTION The analytical expression for v() derived in Ref. [a] for a closed system (A or = ) and constant reservoir head (α = ) is extended herein for a fluctuating reservoir head (α ) and reads K + C v = e g K C K C e R, e d d n x ( α ) H + C + ( θ α) K + C sin e d n where C t H t (7) : =, : f C =, K := K e + K v and D v( ) =. ocal losses are absent in the limit case K =. Fully symbolic solutions can be found for special cases only [a]. Approximate symbolic solutions can be found for example by taking K = K and (x ) n = (x ) n in the integrands. NUMERICA SOUTION The governing equations (-5), with (5) substituted in (), can be casted in the autonomous form: v d y dt = f y, with y : = H (8) m The explicit Euler method has been used to solve Eq. (8) with a numerical time step t = ms for all simulations herein. RESUTS Constant reservoir-head and closed end The confined system of Martin (97) [a] is simulated to verify the numerical solutions for the case A or =, n =. and α =. Martin s results in Fig. and the current results in Fig. are consistent. Maximum values and their timings are compared in Table. Martin (unnecessarily) used instead of (t) in the first term on the right-hand side of Eq. (), which explains the small differences. Another source of discrepancy is numerical error. The maximum column length is max =.5 m, which means a growth of % compared to and this gives rise to an increase of % in maximum pressure compared to Martin s results. Table Comparison of maximum velocities and pressures. v max (m/s) t v,max (s) H air,max (m) t Hair,max (s) Martin (97) (current) (t), α = (t), α = Copyright by ASME

4 Fluctuating reservoir-head and closed end The reservoir pressure head is fluctuating in the test case A or =, n =. and α =.. Figure 5 shows that the analytical expression (7) and the numerical solution give the same result. The dimensionless parameter α ( max ) / H R, =.%. The fluctuation results in a % lower maximum pressure head, as listed in Table. Fig. air pressure head (m) Martin (97): Transient air pressure, water velocity and air volume for confined system. 8 (a) H_air H_air, H_R Fig. 5 numerical, back flow 8 Water velocity versus column length: numerical and analytical solution. air volume (m^) Fig. 8 (b) time (s) time (s) (c) 8 time (s) Current paper: (a) transient air pressure, (b) water velocity, and (c) air volume, for confined system. Constant reservoir-head and orifice end The open system of Martin (97) [a] is simulated for a maximum of s to verify the numerical solutions for the case n = k =., α =, H R, /H, =, or, and with varying A or. The calculations stop either after s or when 5% of the initial air volume is left (when =. m =.D). For small air pockets the mathematical model becomes invalid thereby leading to very high pressure heads. Figure shows the calculated column motions in terms of velocities and positions for H R, /H, =. The closed-system behavior is shown as a reference and an extended Fig. a is shown in the Appendix. The water column bounces back at least once for small orifices up to D or /D =.7. For large orifices there is a smooth liquid movement towards the pipe s end, although the build-up of air pressure finally causes a deceleration. For a fully open end the liquid column slows down mainly due to skin friction thereby satisfying the analytical solution [b, Eq. (5) for θ = ]: C C e e R, d v = gh (9) By taking = in the denominator this function can be approximated by [c, Eq. (5)]: ghr, v = e C (9a) C Copyright by ASME

5 Figure e shows solution (9) with the maximum attainable velocity v(x ) = 7. m/s (< vx ( ) = 7.7 m/s). Figure 7 shows air pressure heads as function of the monotonously decreasing air mass. Martin s results for = D in Fig. 8 and the current fv, results in Fig. 9 are consistent, keeping in mind that the variable (t) (instead of constant ) herein leads to higher air pressures (H air,max ) than in Ref. [a], see Table. Other sources of discrepancy include numerical error, and the possibly different values of T, and hence r, and to a smaller extent r. The arbitrary air-volume threshold of 5% cuts off the maximum pressures for the cases without reverse flow (v > all the time). When the water column hits the closed end, there is the potential risk of a Joukowsky pressure rise of magnitude r c v end, although the vent will relief the water hammer. 5 (c) D or /D =.7, back flow (a) D or /D =. d) D or /D =.9 5 5, back flow , back flow (b) D or /D =.5 8 e) D or /D =. 5, back flow , back flow fully open end Fig. Water velocity versus column length for different orifice sizes and H R, / H, =. 5 Copyright by ASME

6 5 5 (a) D or /D =. H_air H_air,.89 H_air, air pressure head (m) air mass (kg) 5 5 b) D or /D =.7 H_air H_air,.89 H_air, air pressure head (m) 75 5 Fig. 8 Martin (97): Effect of orifice on maximum air pressure air mass (kg).8. H_R = H_ H_R = H_ H_R = H_ air pressure head (m) c) D or /D =. H_air H_air,.89 H_air, H_air,max / H_R air mass (kg)..8.. Fig. 7 Air pressure head versus air mass for different orifice sizes and H R, / H, =. Fig. 9 D_or / D Current paper: Effect of orifice on maximum air pressure. Copyright by ASME

7 CONCUSION Previous work on entrapped air pockets has been extended with a vent. The obtained results are consistent with C.S. Martin s (97) diagrams. One new element herein is that the reservoir pressure head fluctuates in phase with the motion of the liquid column. For this situation, with a closed end, an analytical expression is found for the liquid velocity as function of column length. The work is applicable and will be extended to situations with multiple pockets and vents. ACKNOWEDGMENT The second author is grateful for the support by the National Natural Science Foundation of China (No. 5785), China Special Fund for Hydraulic Research in the Public Interest (No. 9) and Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. NOMENCATURE A = cross-sectional pipe area (m ) A or = orifice area (m ) A R = reservoir area (m ) C d = orifice discharge coefficient C, C = constants c = acoustic wave speed (m/s) D = pipe diameter (m) D or = orifice diameter (m) e = exponential function f = Darcy-Weisbach friction coefficient f = vector function g = acceleration due to gravity (m/s ) H atm = atmospheric absolute pressure head (m) H = absolute pressure head (m) H R = reservoir absolute pressure head (m) H = Heaviside step function K = total loss coefficient K e = entrance loss coefficient K v = valve loss coefficient k = adiabatic constant, ratio of specific heats = length of liquid column (m) = dummy length in integral (m) = length of pocket (m) m = mass of liquid column (kg) m = mass of pocket (kg) n = constant polytropic exponent R = specific constant (m /(s K)) T = absolute temperature (K) t = time (s) V = volume (m ) v = velocity of liquid column (m/s) v end = velocity of liquid column when hitting pipe end (m/s) v R = vertical velocity of reservoir free surface (m/s) x = axial position (m) x = pipe length (m) Y = adiabatic expansion factor y = vector of unknowns α = ratio A/A R P = pressure difference across orifice (Pa) θ = angle of downward inclination of pipe (rad) r = mass density of liquid (kg/m ) r = mass density of (kg/m ) Subscripts max = maximum value = constant initial value REFERENCES [a] Martin CS (97) Entrapped air in pipelines. BHRA Fluid Engineering, Proceedings of the Second International Conference on Pressure Surges, ondon, UK, Paper F, pp [b] Martin CS (99) Experiences with two-phase flow in fluid transients. BHR Group, Proceedings of the 7th International Conference on Pressure Surges, Harrogate, UK, pp [c] ee NH, Martin CS (999) Experimental and analytical investigation of entrapped air in a horizontal pipe. Proceedings of the rd ASME & JSME Joint Fluids Engineering Conference, Symposium S-9 Water Hammer (Editor JCP iou), San Francisco, USA, ASME - FED, Vol. 8, Paper FEDSM 99-88, pp. -8. [d] Martin CS, ee NH () Rapid expulsion of entrapped air through an orifice. BHR Group, Proceedings of the 8th International Conference on Pressure Surges, The Hague, The Netherlands, pp. 5-. [e] Martin CS, ee NH () Measurement and rigid column analysis of expulsion of entrapped air from a horizontal pipe with an exit orifice. BHR Group, Proceedings of the th International Conference on Pressure Surges, isbon, Portugal, pp [a] Abreu JM, Cabrera E, García-Serra J, Izquierdo J (99) Boundary between elastic and inelastic models in hydraulic transients analysis with entrapped air pockets. IAHR, Proceedings of the 9th Round Table on Hydraulic Transients with Water Column Separation, Valencia, Spain, pp [b] Cabrera E, Abreu JM, Pérez R, Vela A (99) Influence of liquid length variation in hydraulic transients. ASCE Journal of Hydraulic Engineering 8 (), 9-5. Discussion and Closure: (5), -. [c] Izquierdo J, Fuertes VS, Cabrera E, Iglesias P, Garcia- Serra J (999) Peak pressure evaluation in pipelines with entrapped air pockets. Proceedings of the rd ASME & 7 Copyright by ASME

8 JSME Joint Fluids Engineering Conference, Symposium S- 9 Water Hammer (Editor JCP iou), San Francisco, USA, ASME - FED, Vol. 8, Paper FEDSM 99-88, pp. -. [d] Izquierdo J, Fuertes VS, Cabrera E, Iglesias P, Garcia- Serra J (999) Pipeline start-up with entrapped air. IAHR Journal of Hydraulic Research 7 (5), [e] Fuertes VS, Arregui F, Cabrera E, Iglesias P () Experimental setup of entrapped air pockets model validation. BHR Group, Proceedings of the 8th International Conference on Pressure Surges, The Hague, The Netherlands, pp. -5. [a] Bousso S, Fuamba M () Numerical and experimental analysis of the pressurized wave front in a circular pipe. ASCE Journal of Hydraulic Engineering (), -. [b] Chosie CD, Hatcher TM, Vasconcelos JG () Experimental and numerical investigation on the motion of discrete air pockets in pressurized water flows. ASCE Journal of Hydraulic Engineering (8), art. no. 8, pp. -. [c] Zhang X., Yu B, Wang Y, Xie J, Qiu D, Sun X () Numerical study on the commissioning charge-up process of horizontal pipeline with entrapped air pockets. Advances in Mechanical Engineering, art. no. 889, pp. -. [d] Martins SC, Ramos HM, de Almeida AB (5) Conceptual analogy for modelling entrapped air action in hydraulic systems. IAHR Journal of Hydraulic Research 5 (5), [e] van Vuuren SJ (5) Effective de-aeration of pipelines and the use of captured air to mitigate dynamic pressures. BHR Group, Proceedings of the th International Conference on Pressure Surges, Dublin, Ireland, pp [f] Zhou, iu D, Karney B, Wang H, Malekpour A (5) Rapid filling of an open-ended pipeline with entrapped air. BHR Group, Proceedings of the th International Conference on Pressure Surges, Dublin, Ireland, pp [g] Malekpour A, Karney B (5) Exploring how air valves change transient responses of pipe systems during rapid filling. BHR Group, Proceedings of the th International Conference on Pressure Surges, Dublin, Ireland, pp [h] Hou Q, Wang S, Kruisbrink ACH, Tijsseling AS (5) agrangian modelling of fluid transients in pipelines with entrapped air. BHR Group, Proceedings of the th International Conference on Pressure Surges, Dublin, Ireland, pp [i] Martins NMC, Soares AK, Ramos HM, Covas DIC (5) Entrapped air pocket analysis using CFD. BHR Group, Proceedings of the th International Conference on Pressure Surges, Dublin, Ireland, pp [j] Apollonio C, Balacco G, Fontana N, Giugni M, Marini G, Ferruccio Piccinni A () Hydraulic transients caused by air expulsion. Water, 8, 5, pp. -. [k] Bergant A, Karadžić U, Tijsseling AS () Dynamic water behaviour due to one trapped air pocket in a laboratory pipeline apparatus. Proceedings of the 8th IAHR Symposium on Hydraulic Machinery and Systems, July, Grenoble, France. [a] Tijsseling AS, Hou Q, Bozkuş Z (5) Analytical expressions for liquid-column velocities in pipelines with entrapped. Proceedings of the ASME 5 Pressure Vessels and Piping Division Conference, Boston, USA, Paper PVP5-58, pp. -8. [b] Tijsseling AS, Hou Q, Bozkuş Z, aanearu J. () Improved one-dimensional models for rapid emptying and filling of pipelines. ASME Journal of Pressure Vessel Technology 8, art. no., pp. -. [c] Tijsseling AS, Hou Q, Bozkuş Z () An improved onedimensional model for liquid slugs travelling in pipelines. ASME Journal of Pressure Vessel Technology 8, art. no., pp. -8. [5a] Douglas JF, Gasiorek JM, Swaffield JA, Jack B () Fluid Mechanics, Sixth edition, Chapter 7. Pearson Education. [5b] Potter MC, Wiggert DC, Ramadan BH () Mechanics of Fluids, Fifth edition, Chapter 9. Cengage earning. APPENDIX The solution space of Eq. (8) is four-dimensional. A threedimensional trajectory is shown in Fig. a, to illustrate the nice dynamic behavior of the studied system. Fig. a Water velocity, column length and air pressure head, for D or /D =. and H R, / H, =. 8 Copyright by ASME

EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computer Science. CASA-Report May 2016

EINDHOVEN UNIVERSITY OF TECHNOOGY Department of Mathematics and Computer Science CASA-Report - May Analytical and numerical solution for a rigid liquid-column moving in a pipe with fluctuating reservoir-head