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Modeling propagation of pressure surges with the formation of an air pocket in pipelines
Computers & Fluids 32 (2003) 1179–1194 www.elsevier.com/locate/compfluid
Modeling propagation of pressure surges with the formation of an air pocket in pipelines Keh-Han Wang *, Qiang Shen, Baoxu Zhang Department of Civil and Environmental Engineering, University of Houston, Houston, TX 77204-4003, USA Received 10 May 2002; received in revised form 24 July 2002; accepted 8 October 2002
Abstract In this study, a computational model that combines the method of characteristics and the shock wave theory is developed to simulate the propagation of pressure surges in pipelines. The surge velocity and pressure change under the conditions of pressurization and depressurization are calculated. The model results show good agreement with measured data. The transient flow model is also extended to study the movement of an entrapped air pocket between interfaces. It is found the air pressure changes greatly during the early stage of formation of an air pocket. For the case of an air pocket trapped between two positive interfaces, an open surge front may be emerged from the upstream interface and eventually reverses the upstream surge to propagate upstream as a negative wave. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Hydraulic transients; Pressure surge; Method of characteristics; Entrapped air
1. Introduction Municipal sewerage systems are normally designed to operate without surcharging. However, under some conditions, such as a heavy storm or pump failure, the flow in sewers may experience a transient from free-surface flow to full pipe flow or vice versa. During the occurrence of flow transients (pressurization or depressurization), the interaction between the interfacial regions and the system boundaries, such as a drop-shaft, a manhole, or a valve, may produce a pressure wave in the system. This pressure rise may cause damage to the system.
*
Corresponding author. Tel.: +1-713-743-4277/4250; fax: +1-713-743-4260. E-mail address: [email protected] (K.-H. Wang).
0045-7930/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0045-7930(02)00103-2
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Nomenclature speed of pressure wave flow cross-sectional area cross-sectional area of air pocket top width of open-channel flow cross-section pffiffiffiffiffiffiffiffiffiffiffi propagating velocity of a disturbed surface wave, c ¼ gA=B a (in pipe flow) or c (in open-channel flow) pipe diameter gravitational constant pressure head measured above the pipe bottom air pocket pressure head air pressure energy slope time fluid velocity initial air pocket volume air pocket volume propagating velocity of an interface downstream interfacial velocity upstream interfacial velocity coordinate along the flow direction fluid depth vertical distance measured from fluid surface to the centroid of the open-channel flow cross-section Y H (in pipe flow) or y (in open-channel flow) c fluid specific weight q fluid density Dt time step Dx grid size along x direction Variable with subscript R or L variable along C or C þ characteristics at previous time step Variable with subscript p1 or p2 variable at section in front of the interface or behind the interface a A Aa B c C D g H Hb pb Sf t V V0 – Vb – w wd wu x y y
In fluid transients, typically for a near full flow, the interfacial instability may result in a complete blockage of the air flow. Consequently, the air may be trapped to form large air pockets moving in the pipelines creating air pressure surge. This is the so-called air hammer phenomenon. The trapped air pressure was found to be sufficient to pop up manhole covers. One example of this phenomenon occurred in the city of Hamilton, Ont., Canada, where a large sewer with a downstream interceptor weir and an upstream drop-pipe experienced severe pressure surges during high flows. The pressure rise was sufficient to blow off a welded manhole cover and cause basement flooding [1].
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In some cases, localized vapor cavities may form at a boundary (e.g. closed valve or dead end) or at a high point along a pipeline when the pressure drops to the vapor pressure of the liquid during fluid transients. This is commonly known as pipeline column separation. Streeter [2] and Wylie and Streeter [3] pioneered the effort in modeling piping systems with the combined water hammer and column separation effects. The effect of entrapped air as a result of column separation on transient pressures was examined by Locher and Wang [4]. They found that the inclusion of modeling entrapped air might cause the differences between analytical results and the field tests. By extending Wylie and StreeterÕs [3] approach, Bergant and Simpson [5] developed a generalized set of pipeline column separation equations to identify column separation modes for a broad range of parameters, including initial flow velocity, static head, and pipe slope. Benjamin [6] studied the properties of steady gravity flow in pipe by perfect fluid theory. His theoretical analysis covered wave breaking and energy losses. Extended to unsteady flow condition, Hamam and McCorquodale [1] investigated the hydraulic transients of flow in sewers by experiments. They analyzed factors affecting the transient pressure such as pipe size, flow rate and pipe slope. They also provided an empirical formula to predict the transient pressure. An experimental study describing the flow associated with the intrusion of an air cavity into a long horizontal duct as water draining from one end was presented by Wilkinson [7]. Wisner et al. [8] provided a discussion on the removal of air from water lines by hydraulic means. For modeling free surface to pressured transition in unsteady pipe flow, Preissmann and Cunge [9] and later Cunge and Wegner [10] introduced a ‘‘Preissmann slot’’ above the conduct to conveniently formulate the continuity equation in the event that the flow becomes pressured. Abbot [11] adopted the ‘‘Preissmann slot’’ concept to model flows in storm-sewer networks. Using the method of characteristics, Wiggert [12] developed a model to simulate transient flow in free-surface, pressured systems. Later, Song et al. [13] and Cardle [14] carried out numerical and experimental investigations of pressure surges in pipelines. An extensive summary for fluid transients study was given by Wylie and Streeter [3]. In this study, a computational model combining shock wave theory and the method of characteristics for the free-surface and full pipe flows is developed to simulate the pressurization and depressurization induced hydraulic transients in sewers. The results are compared with available experimental data for validation. The flow model is also extended to study the movement of an air pocket between interfaces. The process of the formation of the air pocket and the associate change of air pressure are investigated. 2. Basic equations The derivation of basic equations for full pipe flow is based on a one-dimensional assumption. According to the conservation of mass and NewtonÕs second law, the continuity equation and equation of motion are derived as [3] oH oH a2 oV þV þ ¼ 0; ot ox g ox g
oH oV oV þV þ þ gðSf S0 Þ ¼ 0; ox ox ot
ð1Þ ð2Þ
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where V is the fluid velocity; H , the pressure head measured above the pipe bottom; q, the density of fluid; g, the gravitational constant; S0 , the pipe slope; Sf , the energy slope and a, the speed of pressure wave. Following the similar procedure, the governing equations for the open-channel flow in a pipeline are given as oy oy c2 oV þV þ ¼ 0; ot ox g ox g
oy oV oV þ þV þ gðSf S0 Þ ¼ 0; ox ot ox
ð3Þ
ð4Þ
where y p is ffiffiffiffiffiffiffiffiffiffiffi the water depth and c represents the propagating velocity of a disturbed surface wave, i.e. c ¼ gA=B. A is defined as the flow area and B is the top width of the flow cross-section.
3. Method of characteristics The above described fundamental equations can be transformed into a set of characteristic equations describing flow in pipes and open channels when follow specified characteristic curves. The characteristic equations can be summarized in the following, dY C dV CðSf S0 Þ ¼ 0; dt g dt
ð5aÞ
dx ¼ V a; dt
ð5bÞ
where þ sign is for the C þ characteristics and sign for C characteristics. For pipe flows, Y ¼ H and C ¼ a. However, for open-channel flows Y and C represent y and c, respectively. The computational method used to solve the characteristic equations is the fixed grid method or the method of specified-time-intervals [3]. The solutions to the characteristic equations are determined at the nodal points and proceeded along the characteristic curves, the C þ and C curves. For full pipe flows, the grid system in x–t plane reflects that the C þ and C curves are straight lines since a V . The numerical procedure for solving the characteristic equations for open-channel flow is more complicated since the flow may be subcritical or supercritical. The first-order finitedifference forms of the characteristic equations ((5a) and (5b)) can be expressed as C þ : ðYp YL Þ þ CL ðVp VL Þ=g þ CL ðSf S0 ÞL Dt ¼ 0; xi xL ¼ ðVL þ CL Þ Dt; C : ðYp YR Þ CR ðVp VR Þ=g CR ðSf S0 ÞR Dt ¼ 0; xi xR ¼ ðVR CR Þ Dt:
ð6aÞ ð6bÞ ð7aÞ ð7bÞ
Here, CL ¼ CR ¼ a is applied to pipe flows. The variable with subscript R or L represents that along the C or C þ characteristics at the previous time step.
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By solving (6) and (7), we can obtain the solutions of the dependent variables Y and V at the grid point P (t ¼ tkþ1 ), provided that YR , VR , YL , and VL are already determined at the previous time step. The values of YR , VR , YL , and VL are interpolated linearly between known values at grid points i 1, i, and i þ 1 at time t ¼ tk for pipe and subcritical open-channel flows. The interpolation applied to grid points i 1, i for supercritical open-channel flows [15]. To ensure that the solution is stable, the intervals of grid lines Dt and Dx should satisfy the Courant condition, such that, for open-channel flow: Dt 6 Dx=jV cj, and for full pipe flow: Dt 6 Dx=jV aj Dx=a. The boundary conditions applied to the upstream boundary are the head provided by the upstream head tank and the C characteristics. For the downstream boundary, the outflow velocity is prescribed depending on the cases of sudden closure or open of a control valve. It is also required the application of C þ characteristics. 4. Model simulation for pressure surges 4.1. Pressurization from downstream to upstream or from upstream to downstream To test the performance of the numerical model, the phenomenon of pressure surge with a single interface is first simulated and examined. For a free-surface flow (open-channel flow) in a sewer pipe, the flow is pressurized starting from a downstream valve or pump if there is a sudden blockage caused by valve closure or failure of pump. The interface, separating the free-surface section from the pressurized section, advances upstream with a surge velocity w. The system may be divided into three sections in this transient procedure: free-surface flow region, interfacial region and pressurized flow region. For the free-surface and pressurized flow regions, the method of characteristics presented in previous section is used to calculate the values of H (or y) and V at grid points with known upstream and downstream boundary conditions. As to the interfacial region, we treat it as a shock wave or a hydraulic jump. The following equations (continuity and momentum equations) can be applied across the shock in a coordinate system moving with the interface, ðVp1 þ wÞA1 ¼ ðVp2 þ wÞA2 ;
ð8Þ
yp1 A1 ðHp2 0:5DÞA2 ¼ ðVp1 þ wÞA1 ðVp2 Vp1 Þ=g:
ð9Þ
and
The point p2 represents the section right behind the interface and the point p1 is for section in front of the interface. yp1 is defined as the vertical distance measured from the free surface to the centroid of the open-channel flow cross-section at point p1. By solving the characteristic equations of the flow regions (C1þ , C1 for the open-channel flow and C2 for the pressurized flow) and the interfacial equations (8) and (9), the unknown variables, yp1 , Vp1 , Hp2 , Vp2 , and w, can be determined to describe this transient phenomenon [14,15]. In some occasions, the inflow rate may suddenly increases and exceeds the capacity of the pipe flow somewhere upstream, the pressurization will start from upstream and develop with the interface moving towards downstream. Similar equations and solution procedure can be extended to study the case of pressurization from upstream to downstream.
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To present the model results, we select a typical case that experimental data [14] are available for comparison. The parameters used for model simulation are: pipe diameter ¼ 0.163 m, flow rate ¼ 0.00736 m3 /s, initial water depth ¼ 0.131 m, slope of pipe ¼ 0.001, and ManningÕs roughness coefficient ¼ 0.01. With a sudden closure of the downstream valve, an upstream propagating pressure surge is generated. The calculated interfacial positions by the present model are shown in Fig. 1. The experimental data are also presented in Fig. 1 for comparison. After the closure of a downstream valve, a positive surge advances upstream with nearly a constant velocity. The present model results show good agreement with the experimental measurements. Fig. 2 presents the computed and recorded pressure heads at a point of 33.53 m away from the downstream end. It appears from the experimental measurements that the pressure heads become unstable and fluctuate greatly after the arrival of the interface. Certainly, the fluctuated highpressure head is important to the design of pipelines. The model results are not shown the occurrence of the pressure fluctuations, as the effect of the air–water interaction at the interface is not included in the model equations. The phenomenon of interface instability and its effect on the propagation of pressure wave were discussed by Hamam and McCorquodale [1]. They pointed out that when the relative velocity of the air reaches the interface instability limit at which the waves become unstable, the wave height will increase rapidly until it touches the crown of the pipe. The air flow then becomes important in influencing the pressure change. Currently, there is no valid model can accurately predict the fluctuated pressures as indicated in Hamam and McCorquodale [1]. However, a two-phase flow model with the capability of handling the air–
Fig. 1. Computed and measured interfacial position.
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Fig. 2. Computed and measured pressure head at a point 33.53 m away from the downstream end.
water interaction may be adopted to improve the prediction of pressure fluctuations. A summary of two-phase flow liquid–vapor mixture equations for column separation can be found in Bergant and Simpson [5]. Here we focus on modeling the movement of liquid phase and mean pressure rise under positive surge. Generally speaking, the computed results agree reasonably well with the mean values of the recorded pressure heads. The computed pressure rise at the interface also matches the observed pressure rise at the data recording point. 4.2. Depressurization from downstream to upstream For an initially pressurized full pipe flow, depressurization may occur when the drainage capacity of pipe increases suddenly, such as the reopening of the downstream valve. The depressurization starts at the downstream and develops toward the upstream with the movement of a negative surge. The interface is defined as a negative interface. As discussed in previous section, the depressurization system may also be divided into three regions: free-surface flow region (downstream), interfacial region and pressurized flow region (upstream). The values of y (or H ) and V at the grid points of the free-surface flow region and pressurized flow region can be calculated by the method of characteristics with known upstream and downstream boundary conditions. Different from the pressurization surge, modeling the depressurization surge requires additional treatment of the leading negative wave. Anderson and Robbie [16] conducted detailed investigation
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on the behavior of free surface in upsurge (pressurization) and downsurge (depressurization). For the downsurge cases, they found out that the leading negative wave might be followed by a small positive surge (bore), depending on the surge velocity, outflow rate, and fluid depth. This phenomenon was also studied by Wilkinson [7], where he suggested that the movement of leading negative wave front could be defined as a region of energy-conserving flow. As no energy dissipation is evident upstream of the bore, it is reasonable to assume that energy is conserved in that region. Based on the experimental observations, Anderson and Robbie [16] concluded that the negative wave front is of stable at all cases tested and the fluid depth is hardly less than half of the pipe diameter. Adopting the condition of stable negative wave front and neglecting the possible energy dissipation caused by the following positive surge, the leading edge of the negative interface is modeled with the Bernoulli equation assuming the condition of a stagnation point in a coordinate system moving with the interface. Following Benjamin [6] and Wilkinson [7], the Bernoulli equation, as applied between the stagnation point (leading edge of the interface) where the pressure is atmospheric and a point further upstream where the pressure is assumed to be hydrostatic, gives ðV2 þ wÞ2 =2g þ H2 ¼ D:
ð10Þ
D is the pipe diameter. The momentum and continuity equations across the shock front are still applicable. Therefore, solving two characteristic equations together with continuity, momentum and Bernoulli equations, the solutions can be obtained to describe this negative surge with the process of depressurization.
Fig. 3. Computed and measured water depth at a point 12.2 m away from the downstream end after the arrival of a negative interface.
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Parameters used for model simulation are: diameter ¼ 0.163 m, inflow ¼ 0, initial water head ¼ 0.244 m, slope of pipe ¼ 0.001, and ManningÕs roughness coefficient ¼ 0.011. Initially, the downstream gate is closed. Water in the pipe is at rest and pressurized with upstream water head. Then, the downstream gate opens suddenly to allow water draining out of the pipe. A negative surge forms at the downstream gate. The depressurization process starts from the gate and develops with the negative interface retreating upstream. Fig. 3 presents the time variation of computed water depth at a point 12.2 m away from the downstream end after the arrival of the interface. The measured data are also marked for comparison. From Fig. 3, we note that the predicted water depths show good agreement with the measured depths when the measuring point is further away from the interface. The measurements show an initial drop of the water level, and then increase to a relatively uniform depth. The depth in the trough immediately following the interface falls slightly below the computed values. The deviation of measured and computed water depths at the early stage of flow transition is a result of the free-surface instability caused by the disturbed manual opening of a downstream valve, air entrainment and possibly the surface tension effect [14]. In addition to the water level, the predicted interfacial positions are also found to agree reasonably well with the experimental data. 5. Simulation of an entrapped air pocket between interfaces In this section, the movement and pressure change of an air pocket trapped in a sewer pipe is simulated. The principles and basic equations described in previous sections are used to develop the governing equations for modeling the movement of an air pocket. 5.1. An entrapped air pocket between positive and negative interfaces Fig. 4 shows a typical case of an air pocket trapped in sewer pipes. Initially, the downstream gate is closed while the water in the pipe is pressurized and is at rest. When the downstream gate opens suddenly, a negative surge will form and move toward upstream. After the interface moves a certain distance upstream, the downstream gate closes again to generate an upstream advancing positive surge. The air between these two upstream-propagating surges forms an air pocket. The
Fig. 4. A schematic diagram showing an entrapped air pocket between a positive and a negative surge.
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motion of the air pocket depends on the movement of the interfaces. The size and pressure of the air pocket vary with the difference of the moving velocities between these two interfaces. The air pressure, as a reaction, will also affect the movements of the interfaces. The mathematical model developed for modeling the positive and negative interfaces can be used in this air hammer study. However, the effect of air pressure, pb , needs to be included in the model formulations. In the conditions of water–air mixture, we consider the movement of a single air pocket, which separates two surge fronts. The movement of an entrapped air pocket is assumed to follow the relative motion of the upstream and downstream interfaces. According to Wylie and Streeter [3], isothermal behavior of free air moving with pressure surges is a reasonable assumption. The air pressure, pb , can be determined by means of the ideal gas law. The detailed equations can be found in Appendix A. The air pocket volume is calculated by Z t V0þ Aa ðwu wd Þ dt: ð11Þ Vb ¼– – 0
Here, – V b is the air pocket volume; – V 0 , the initial air pocket volume, Aa , the cross-sectional area of air pocket; wu , the upstream interfacial velocity, and wd , the downstream interfacial velocity. We select a typical case to illustrate the simulation. Initially, a downstream gate is closed while water is at rest in the pipe. The pipe diameter ¼ 0.163 m, slope of pipe ¼ 0.001, and ManningÕs roughness coefficient ¼ 0.011. The upstream head is 0.244 m. The initial negative surge is generated by suddenly opening the downstream gate. The gate remains open for 2.5 s, and then closes quickly for producing a positive surge. The time variations of the interfacial velocities, wu , wd , and the air pocket pressure head Hb , where Hb ¼ pb =c, are shown in Fig. 5. As indicated in Fig. 5, the
Fig. 5. Time variation of surge velocity and air pressure for an entrapped air pocket between a positive and a negative surge.
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positive surge advances upstream faster than the negative one retreating upstream after the formation of the air pocket. The air pocket pressure increases as its volume decreases. As a reaction, the pressure pushes the negative surge and slows down the following positive surge. The air pocket pressure eventually reaches an equilibrium condition. Under the equilibrium condition, the air pocket pressure becomes a constant and the negative surge moves almost identical to the positive surge. 5.2. An entrapped air pocket between two positive interfaces Another case considered in this study about the movement of an air pocket is illustrated in Fig. 6. A positive surge advances downstream and interacts with an upstream advancing positive
Fig. 6. Schematic diagrams showing an entrapped air pocket between two positive surges and the flow transition of the upstream interface with reversed propagating direction and an emerged open surge.
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surge, which is generated from the downstream end. The trapped air between these two positive interface forms an air pocket. This case is more complicated than the one discussed in previous section where the air pocket is formed between a positive and a negative surge. For the previous case, although the values of the interfacial velocities vary with the air pocket pressure, the moving directions of the interfaces remain unchanged. For the current case, the moving directions of the positive interfaces may be changed due to the pressure increase of the compressed air pocket. Actually, with the increase of the air pocket pressure, both of the interfacial velocities decrease. Eventually, the air pressure will push the upstream positive interface to propagate upstream as a negative surge. Therefore, the upstream positive surge experiences a transition from a positive surge to a negative one. This is not a sudden change in direction. During the fluid transient, a positive open surge may form and develop. The flow phenomenon for the transition of the upstream interface is shown in Fig. 6. It is assumed that the water depth in the region between the open surge and the upstream interface is uniform. The length of this open surge region increases and the corresponding water depth decreases as the fluid transition progresses. Eventually, this region merges with the downstream open-channel flow. The solution procedure for solving the flow condition around the upstream interface is described in the following. The unknowns for the flow condition are yp1 , Vp1 , yp2 , Vp2 , Hp3 , Vp3 , open surge velocity, ws , and interfacial velocity, wu . The equations adopted for determining the variables are C1þ : ðyp1 yL1 Þ þ cL1 ðVp1 VL1 Þ=g þ cL1 ðSf S0 ÞL1 Dt ¼ 0;
ð12Þ
C1 : ðyp1 yR1 Þ cR1 ðVp1 VR1 Þ=g cR1 ðSf S0 ÞR1 Dt ¼ 0;
ð13Þ
continuity equation across the open surge: ðVp1 ws ÞA1 ¼ ðVp2 ws ÞA2 ;
ð14Þ
momentum equation across the open surge: yp2 A2 yp1 A1 ¼ ðVp2 ws ÞA2 ðVp1 Vp2 Þ=g;
ð15Þ
continuity equation across the upstream positive surge: ðwu þ Vp2 ÞA2 ¼ ðwu þ Vp3 ÞA3 ;
ð16Þ
momentum equation across the upstream positive surge: ðHp3 0:5DÞA3 ðyp2 A2 þ Hb A3 Þ ¼ ðwu þ Vp3 ÞA3 ðVp2 Vp3 Þ=g;
ð17Þ
Bernoulli equation: Hp3 þ ðVp3 þ wu Þ2 =2g ¼ D þ Hb
ð18Þ
C3þ : ðHp3 HL3 Þ þ aðVp3 VL3 Þ=g þ aðSf S0 ÞL3 Dt ¼ 0:
ð19Þ
and The values of yp1 and Vp1 are directly obtained from Eq. (12) and Eq. (13). By solving Eqs. (14)– (18), we have Hp3 ¼ ½ð2A3 =A2 1:5ÞD þ yp2 A2 =A3 =ð2A3 =A2 1Þ þ Hb ;
ð20Þ
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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 g A3 A2 A2 A1 yp2 A2 yp1 A1 ¼ Vp1 VE Hp3 þ 2gðD þ Hb Hp3 Þ ; a A2 ðA2 A1 Þg
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ð21Þ
where g VE ¼ VL3 þ HL3 gðSf S0 ÞL3 Dt: a
ð22Þ
Eqs. (20) and (21), which describe the relationship between Hp3 and yp2 , can be solved to determine the values of Hp3 and yp2 . Since yp2 and A2 are functions of yp2 , an iteration is required to solve for Hp3 and yp2 . Once Hp3 and yp2 are determined, the remaining variables, Vp2 , Vp3 , wu , and ws , can be easily obtained from Eqs. (19), (17), (16) and (14). When the depth of water in the open surge region is equal to the depth of downstream freesurface flow region, the open surge disappears. At this instant, the upstream positive surge, which initially propagates toward downstream is already reversed to move upstream. The flow phenomenon is then changed into the case discussed in previous section. The solution procedure for solving the flow condition around the downstream interface is similar to that shown in previous section. Again, we select a typical case to illustrate the simulation. Initially, a positive surge advances downstream with an upstream head of 0.244 m. The water depth of the downstream free-surface flow is 0.122 m. As the upstream positive surge reaches a point about 4 m away from the downstream gate, the gate closes suddenly to generate an upstream propagating positive surge. Fig. 7 shows the results of both upstream and downstream interfacial velocities and the air pressure head under the condition of two positive surges and a trapped air pocket. It is noted that
Fig. 7. Time variation of surge velocity and air pressure for an entrapped air pocket between two positive surges.
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during the early stage of the air pocket formation, the interfacial velocities and the air pressure change greatly within a short period of time. As shown in Fig. 7, the air pressure increases rapidly when the two interfaces move towards each other. As a result, the propagating velocities of the upstream and downstream surges decrease. An open surge front is emerged from the upstream interface and eventually the upstream positive surge is pushed back and propagates as a negative wave. After the reversal of the upstream surge, the movement of the air pocket and the pressure change are similar to the case presented in previous section. The surge velocity is changed from positive to negative as indicated by solid line in Fig. 7. Comparing the results presented in Fig. 5 and Fig. 7, it is found that the variations of interfacial velocities and pressure change in the second case (air pocket with two positive surges) are much greater than those shown for the first case (air pocket with one positive and one negative surge). It is also interesting to point out that the increase of the air pressure plays an important role to influence the movement of the upstream and downstream surge waves. Other test cases also suggest that the air pressure increases with the increase of the upstream head and the pipe diameter. So far, there is no experimental data available to verify the modeling of air pocket movement. Continuing effort will be spent to obtain the experimental data for furthering the improvement of the air hammer model. 6. Conclusions In this study, a computational model is developed to simulate the hydraulic transients in sewers. The air pressure surge and the movement of an air pocket trapped in a pipeline are also studied. Movements of pressure surge are reasonably simulated. The computed results show good agreement with the experimental data. The simulation of an air pocket trapped between a positive and a negative surge shows that after the formation of the air pocket, the positive surge moves faster than the negative one and the air pressure increases as its volume decreases. Modeling of an air pocket between two positive surges shows that during the early stage of the air pocket formation, the interfacial velocities and the air pressure change greatly within a short period of time. An open surge front is emerged from the upstream interface and eventually the upstream positive surge is pushed back to propagate as a negative surge. Acknowledgement This work is sponsored by the City of Houston through Montgomery Watson. The support is greatly appreciated.
Appendix A Equations for solving air pocket between positive and negative interfaces are given below. Equations for flow conditions around the downstream interface (positive surge) are: C1þ : ðyp1 yL1 Þ þ cL1 ðVp1 VL1 Þ=g þ cL1 ðSf S0 ÞL1 Dt ¼ 0;
ðA:1Þ
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C1 : ðyp1 yR1 Þ cR1 ðVp1 VR1 Þ=g cR1 ðSf S0 ÞR1 Dt ¼ 0;
ðA:2Þ
C2 : ðHp2 HR2 Þ aðVp2 VR2 Þ=g aðSf S0 ÞR2 Dt ¼ 0;
ðA:3Þ
continuity: ðVp1 þ wd ÞA1 ¼ ðVp2 þ wd ÞA2 ;
ðA:4Þ
momentum: yp1 A1 þ Hb A2 ðHp2 0:5DÞA2 ¼ ðwd þ Vp1 ÞA1 ðVp2 Vp1 Þ=g:
ðA:5Þ
Here, the subscripts 1 and 2 represent respectively the flow conditions upstream and downstream of the interface. Equations for flow conditions around the upstream interface (negative surge) are: C1 : ðyp1 yR1 Þ cR1 ðVp1 VR1 Þ=g cR1 ðSf S0 ÞR1 Dt ¼ 0;
ðA:6Þ
C2þ : ðHp20 HL20 Þ þ aðVp20 VL20 Þ=g þ aðSf S0 ÞL20 Dt ¼ 0;
ðA:7Þ
continuity: ðwu þ Vp1 ÞA1 ¼ ðwu þ Vp20 ÞA20 ;
ðA:8Þ
momentum: yp1 A1 þ Hb A20 ðHp20 0:5DÞA20 ¼ ðwu þ Vp20 ÞA20 ðVp20 Vp1 Þ=g;
ðA:9Þ
Bernoulli: Hp20 þ ðVp20 þ wu Þ2 =2g ¼ D þ Hb :
ðA:10Þ
The subscripts 1 and 2 represent respectively the flow conditions downstream and upstream of the interface.
References [1] Hamam MA, McCorquodale JA. Transient conditions in the transition from gravity to surcharged sewer flow. 1982 National Research Council of Canada 1982. [2] Streeter VL. Transient cavitating pipe flows. J Hydraul Eng, ASCE 1983;109:1408–23. [3] Wylie EB, Streeter VL. Fluid transients in systems. Englewood Cliffs, NJ: Prentice Hall; 1993. [4] Locher FA, Wang JS. An experience with entrapped air in comparisons of analytical studies and field tests of a system with column separation. In: Proceedings of International Conference on Unsteady Flow and Fluid Transients, Balkeman, Rotterdam, 1992; pp. 327–32. [5] Bergant A, Simpson AR. Pipeline column separation flow regimes. J Hydraul Eng, ASCE 1999;125:835–48. [6] Benjamin TB. Gravity currents and related phenomena. J Fluid Mech 1967;31:209–48. [7] Wilkinson DL. Motion of air cavities in long horizontal ducts. J Fluid Mech 1982;118:109–22. [8] Wisner PE, Mohsen FN, Kouwen N. Removal of air from water lines by hydraulic means. J Hydraul Eng 1975;101:243–57. [9] Preissmann A, Cunge JA. Calcul des intumescences sur machines electroniques. In: Proceedings of 9th Congress, Dubrovnik, 1961; pp. 656–64. [10] Cunge JA, Wegner M. Integration numerique des equations dÕecoulement de Barre de St Vernant par un schema implicite de differences finies. La Houille Blanche 1964;1:33–9. [11] Abott MB. A modeling system for the design and operation of storm-sewer networks. Eng Appl Comput Hydraul, Abbott and Cunge (des) 1982;1:11–39. [12] Wiggert DC. Transient flow in free-surface pressured systems. J Hydraul Div ASCE 1992;98:11–27. [13] Song CCS, Cardle JA, Leung KS. Transient mixed-flow models for storm sewers. J Hydraul Eng 1983;109:1487– 503.
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[14] Cardle JA. An investigation of hydraulic transient in combination free surface and pressured flows. PhD Dissertation, University of Minnesota, 1984. [15] Shen Q. Hydraulic transients in pipelines and wave motion over a group of submerged horizontal plates. MS Thesis, Department of Civil and Environmental Engineering, University of Houston, 1997. [16] Anderson A, Robbie JF. The behaviour of surge tanks with horizontal expansion galleries. In: Proceedings of 5th International Conference on Pressure Surges, Hannover, Germany, 1986; pp. 261–71.
Modeling propagation of pressure surges with the formation of an air pocket in pipelines Keh-Han Wang *, Qiang Shen, Baoxu Zhang Department of Civil and Environmental Engineering, University of Houston, Houston, TX 77204-4003, USA Received 10 May 2002; received in revised form 24 July 2002; accepted 8 October 2002
Abstract In this study, a computational model that combines the method of characteristics and the shock wave theory is developed to simulate the propagation of pressure surges in pipelines. The surge velocity and pressure change under the conditions of pressurization and depressurization are calculated. The model results show good agreement with measured data. The transient flow model is also extended to study the movement of an entrapped air pocket between interfaces. It is found the air pressure changes greatly during the early stage of formation of an air pocket. For the case of an air pocket trapped between two positive interfaces, an open surge front may be emerged from the upstream interface and eventually reverses the upstream surge to propagate upstream as a negative wave. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Hydraulic transients; Pressure surge; Method of characteristics; Entrapped air
1. Introduction Municipal sewerage systems are normally designed to operate without surcharging. However, under some conditions, such as a heavy storm or pump failure, the flow in sewers may experience a transient from free-surface flow to full pipe flow or vice versa. During the occurrence of flow transients (pressurization or depressurization), the interaction between the interfacial regions and the system boundaries, such as a drop-shaft, a manhole, or a valve, may produce a pressure wave in the system. This pressure rise may cause damage to the system.
*
Corresponding author. Tel.: +1-713-743-4277/4250; fax: +1-713-743-4260. E-mail address: [email protected] (K.-H. Wang).
0045-7930/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0045-7930(02)00103-2
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Nomenclature speed of pressure wave flow cross-sectional area cross-sectional area of air pocket top width of open-channel flow cross-section pffiffiffiffiffiffiffiffiffiffiffi propagating velocity of a disturbed surface wave, c ¼ gA=B a (in pipe flow) or c (in open-channel flow) pipe diameter gravitational constant pressure head measured above the pipe bottom air pocket pressure head air pressure energy slope time fluid velocity initial air pocket volume air pocket volume propagating velocity of an interface downstream interfacial velocity upstream interfacial velocity coordinate along the flow direction fluid depth vertical distance measured from fluid surface to the centroid of the open-channel flow cross-section Y H (in pipe flow) or y (in open-channel flow) c fluid specific weight q fluid density Dt time step Dx grid size along x direction Variable with subscript R or L variable along C or C þ characteristics at previous time step Variable with subscript p1 or p2 variable at section in front of the interface or behind the interface a A Aa B c C D g H Hb pb Sf t V V0 – Vb – w wd wu x y y
In fluid transients, typically for a near full flow, the interfacial instability may result in a complete blockage of the air flow. Consequently, the air may be trapped to form large air pockets moving in the pipelines creating air pressure surge. This is the so-called air hammer phenomenon. The trapped air pressure was found to be sufficient to pop up manhole covers. One example of this phenomenon occurred in the city of Hamilton, Ont., Canada, where a large sewer with a downstream interceptor weir and an upstream drop-pipe experienced severe pressure surges during high flows. The pressure rise was sufficient to blow off a welded manhole cover and cause basement flooding [1].
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In some cases, localized vapor cavities may form at a boundary (e.g. closed valve or dead end) or at a high point along a pipeline when the pressure drops to the vapor pressure of the liquid during fluid transients. This is commonly known as pipeline column separation. Streeter [2] and Wylie and Streeter [3] pioneered the effort in modeling piping systems with the combined water hammer and column separation effects. The effect of entrapped air as a result of column separation on transient pressures was examined by Locher and Wang [4]. They found that the inclusion of modeling entrapped air might cause the differences between analytical results and the field tests. By extending Wylie and StreeterÕs [3] approach, Bergant and Simpson [5] developed a generalized set of pipeline column separation equations to identify column separation modes for a broad range of parameters, including initial flow velocity, static head, and pipe slope. Benjamin [6] studied the properties of steady gravity flow in pipe by perfect fluid theory. His theoretical analysis covered wave breaking and energy losses. Extended to unsteady flow condition, Hamam and McCorquodale [1] investigated the hydraulic transients of flow in sewers by experiments. They analyzed factors affecting the transient pressure such as pipe size, flow rate and pipe slope. They also provided an empirical formula to predict the transient pressure. An experimental study describing the flow associated with the intrusion of an air cavity into a long horizontal duct as water draining from one end was presented by Wilkinson [7]. Wisner et al. [8] provided a discussion on the removal of air from water lines by hydraulic means. For modeling free surface to pressured transition in unsteady pipe flow, Preissmann and Cunge [9] and later Cunge and Wegner [10] introduced a ‘‘Preissmann slot’’ above the conduct to conveniently formulate the continuity equation in the event that the flow becomes pressured. Abbot [11] adopted the ‘‘Preissmann slot’’ concept to model flows in storm-sewer networks. Using the method of characteristics, Wiggert [12] developed a model to simulate transient flow in free-surface, pressured systems. Later, Song et al. [13] and Cardle [14] carried out numerical and experimental investigations of pressure surges in pipelines. An extensive summary for fluid transients study was given by Wylie and Streeter [3]. In this study, a computational model combining shock wave theory and the method of characteristics for the free-surface and full pipe flows is developed to simulate the pressurization and depressurization induced hydraulic transients in sewers. The results are compared with available experimental data for validation. The flow model is also extended to study the movement of an air pocket between interfaces. The process of the formation of the air pocket and the associate change of air pressure are investigated. 2. Basic equations The derivation of basic equations for full pipe flow is based on a one-dimensional assumption. According to the conservation of mass and NewtonÕs second law, the continuity equation and equation of motion are derived as [3] oH oH a2 oV þV þ ¼ 0; ot ox g ox g
oH oV oV þV þ þ gðSf S0 Þ ¼ 0; ox ox ot
ð1Þ ð2Þ
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where V is the fluid velocity; H , the pressure head measured above the pipe bottom; q, the density of fluid; g, the gravitational constant; S0 , the pipe slope; Sf , the energy slope and a, the speed of pressure wave. Following the similar procedure, the governing equations for the open-channel flow in a pipeline are given as oy oy c2 oV þV þ ¼ 0; ot ox g ox g
oy oV oV þ þV þ gðSf S0 Þ ¼ 0; ox ot ox
ð3Þ
ð4Þ
where y p is ffiffiffiffiffiffiffiffiffiffiffi the water depth and c represents the propagating velocity of a disturbed surface wave, i.e. c ¼ gA=B. A is defined as the flow area and B is the top width of the flow cross-section.
3. Method of characteristics The above described fundamental equations can be transformed into a set of characteristic equations describing flow in pipes and open channels when follow specified characteristic curves. The characteristic equations can be summarized in the following, dY C dV CðSf S0 Þ ¼ 0; dt g dt
ð5aÞ
dx ¼ V a; dt
ð5bÞ
where þ sign is for the C þ characteristics and sign for C characteristics. For pipe flows, Y ¼ H and C ¼ a. However, for open-channel flows Y and C represent y and c, respectively. The computational method used to solve the characteristic equations is the fixed grid method or the method of specified-time-intervals [3]. The solutions to the characteristic equations are determined at the nodal points and proceeded along the characteristic curves, the C þ and C curves. For full pipe flows, the grid system in x–t plane reflects that the C þ and C curves are straight lines since a V . The numerical procedure for solving the characteristic equations for open-channel flow is more complicated since the flow may be subcritical or supercritical. The first-order finitedifference forms of the characteristic equations ((5a) and (5b)) can be expressed as C þ : ðYp YL Þ þ CL ðVp VL Þ=g þ CL ðSf S0 ÞL Dt ¼ 0; xi xL ¼ ðVL þ CL Þ Dt; C : ðYp YR Þ CR ðVp VR Þ=g CR ðSf S0 ÞR Dt ¼ 0; xi xR ¼ ðVR CR Þ Dt:
ð6aÞ ð6bÞ ð7aÞ ð7bÞ
Here, CL ¼ CR ¼ a is applied to pipe flows. The variable with subscript R or L represents that along the C or C þ characteristics at the previous time step.
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By solving (6) and (7), we can obtain the solutions of the dependent variables Y and V at the grid point P (t ¼ tkþ1 ), provided that YR , VR , YL , and VL are already determined at the previous time step. The values of YR , VR , YL , and VL are interpolated linearly between known values at grid points i 1, i, and i þ 1 at time t ¼ tk for pipe and subcritical open-channel flows. The interpolation applied to grid points i 1, i for supercritical open-channel flows [15]. To ensure that the solution is stable, the intervals of grid lines Dt and Dx should satisfy the Courant condition, such that, for open-channel flow: Dt 6 Dx=jV cj, and for full pipe flow: Dt 6 Dx=jV aj Dx=a. The boundary conditions applied to the upstream boundary are the head provided by the upstream head tank and the C characteristics. For the downstream boundary, the outflow velocity is prescribed depending on the cases of sudden closure or open of a control valve. It is also required the application of C þ characteristics. 4. Model simulation for pressure surges 4.1. Pressurization from downstream to upstream or from upstream to downstream To test the performance of the numerical model, the phenomenon of pressure surge with a single interface is first simulated and examined. For a free-surface flow (open-channel flow) in a sewer pipe, the flow is pressurized starting from a downstream valve or pump if there is a sudden blockage caused by valve closure or failure of pump. The interface, separating the free-surface section from the pressurized section, advances upstream with a surge velocity w. The system may be divided into three sections in this transient procedure: free-surface flow region, interfacial region and pressurized flow region. For the free-surface and pressurized flow regions, the method of characteristics presented in previous section is used to calculate the values of H (or y) and V at grid points with known upstream and downstream boundary conditions. As to the interfacial region, we treat it as a shock wave or a hydraulic jump. The following equations (continuity and momentum equations) can be applied across the shock in a coordinate system moving with the interface, ðVp1 þ wÞA1 ¼ ðVp2 þ wÞA2 ;
ð8Þ
yp1 A1 ðHp2 0:5DÞA2 ¼ ðVp1 þ wÞA1 ðVp2 Vp1 Þ=g:
ð9Þ
and
The point p2 represents the section right behind the interface and the point p1 is for section in front of the interface. yp1 is defined as the vertical distance measured from the free surface to the centroid of the open-channel flow cross-section at point p1. By solving the characteristic equations of the flow regions (C1þ , C1 for the open-channel flow and C2 for the pressurized flow) and the interfacial equations (8) and (9), the unknown variables, yp1 , Vp1 , Hp2 , Vp2 , and w, can be determined to describe this transient phenomenon [14,15]. In some occasions, the inflow rate may suddenly increases and exceeds the capacity of the pipe flow somewhere upstream, the pressurization will start from upstream and develop with the interface moving towards downstream. Similar equations and solution procedure can be extended to study the case of pressurization from upstream to downstream.
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To present the model results, we select a typical case that experimental data [14] are available for comparison. The parameters used for model simulation are: pipe diameter ¼ 0.163 m, flow rate ¼ 0.00736 m3 /s, initial water depth ¼ 0.131 m, slope of pipe ¼ 0.001, and ManningÕs roughness coefficient ¼ 0.01. With a sudden closure of the downstream valve, an upstream propagating pressure surge is generated. The calculated interfacial positions by the present model are shown in Fig. 1. The experimental data are also presented in Fig. 1 for comparison. After the closure of a downstream valve, a positive surge advances upstream with nearly a constant velocity. The present model results show good agreement with the experimental measurements. Fig. 2 presents the computed and recorded pressure heads at a point of 33.53 m away from the downstream end. It appears from the experimental measurements that the pressure heads become unstable and fluctuate greatly after the arrival of the interface. Certainly, the fluctuated highpressure head is important to the design of pipelines. The model results are not shown the occurrence of the pressure fluctuations, as the effect of the air–water interaction at the interface is not included in the model equations. The phenomenon of interface instability and its effect on the propagation of pressure wave were discussed by Hamam and McCorquodale [1]. They pointed out that when the relative velocity of the air reaches the interface instability limit at which the waves become unstable, the wave height will increase rapidly until it touches the crown of the pipe. The air flow then becomes important in influencing the pressure change. Currently, there is no valid model can accurately predict the fluctuated pressures as indicated in Hamam and McCorquodale [1]. However, a two-phase flow model with the capability of handling the air–
Fig. 1. Computed and measured interfacial position.
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Fig. 2. Computed and measured pressure head at a point 33.53 m away from the downstream end.
water interaction may be adopted to improve the prediction of pressure fluctuations. A summary of two-phase flow liquid–vapor mixture equations for column separation can be found in Bergant and Simpson [5]. Here we focus on modeling the movement of liquid phase and mean pressure rise under positive surge. Generally speaking, the computed results agree reasonably well with the mean values of the recorded pressure heads. The computed pressure rise at the interface also matches the observed pressure rise at the data recording point. 4.2. Depressurization from downstream to upstream For an initially pressurized full pipe flow, depressurization may occur when the drainage capacity of pipe increases suddenly, such as the reopening of the downstream valve. The depressurization starts at the downstream and develops toward the upstream with the movement of a negative surge. The interface is defined as a negative interface. As discussed in previous section, the depressurization system may also be divided into three regions: free-surface flow region (downstream), interfacial region and pressurized flow region (upstream). The values of y (or H ) and V at the grid points of the free-surface flow region and pressurized flow region can be calculated by the method of characteristics with known upstream and downstream boundary conditions. Different from the pressurization surge, modeling the depressurization surge requires additional treatment of the leading negative wave. Anderson and Robbie [16] conducted detailed investigation
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on the behavior of free surface in upsurge (pressurization) and downsurge (depressurization). For the downsurge cases, they found out that the leading negative wave might be followed by a small positive surge (bore), depending on the surge velocity, outflow rate, and fluid depth. This phenomenon was also studied by Wilkinson [7], where he suggested that the movement of leading negative wave front could be defined as a region of energy-conserving flow. As no energy dissipation is evident upstream of the bore, it is reasonable to assume that energy is conserved in that region. Based on the experimental observations, Anderson and Robbie [16] concluded that the negative wave front is of stable at all cases tested and the fluid depth is hardly less than half of the pipe diameter. Adopting the condition of stable negative wave front and neglecting the possible energy dissipation caused by the following positive surge, the leading edge of the negative interface is modeled with the Bernoulli equation assuming the condition of a stagnation point in a coordinate system moving with the interface. Following Benjamin [6] and Wilkinson [7], the Bernoulli equation, as applied between the stagnation point (leading edge of the interface) where the pressure is atmospheric and a point further upstream where the pressure is assumed to be hydrostatic, gives ðV2 þ wÞ2 =2g þ H2 ¼ D:
ð10Þ
D is the pipe diameter. The momentum and continuity equations across the shock front are still applicable. Therefore, solving two characteristic equations together with continuity, momentum and Bernoulli equations, the solutions can be obtained to describe this negative surge with the process of depressurization.
Fig. 3. Computed and measured water depth at a point 12.2 m away from the downstream end after the arrival of a negative interface.
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Parameters used for model simulation are: diameter ¼ 0.163 m, inflow ¼ 0, initial water head ¼ 0.244 m, slope of pipe ¼ 0.001, and ManningÕs roughness coefficient ¼ 0.011. Initially, the downstream gate is closed. Water in the pipe is at rest and pressurized with upstream water head. Then, the downstream gate opens suddenly to allow water draining out of the pipe. A negative surge forms at the downstream gate. The depressurization process starts from the gate and develops with the negative interface retreating upstream. Fig. 3 presents the time variation of computed water depth at a point 12.2 m away from the downstream end after the arrival of the interface. The measured data are also marked for comparison. From Fig. 3, we note that the predicted water depths show good agreement with the measured depths when the measuring point is further away from the interface. The measurements show an initial drop of the water level, and then increase to a relatively uniform depth. The depth in the trough immediately following the interface falls slightly below the computed values. The deviation of measured and computed water depths at the early stage of flow transition is a result of the free-surface instability caused by the disturbed manual opening of a downstream valve, air entrainment and possibly the surface tension effect [14]. In addition to the water level, the predicted interfacial positions are also found to agree reasonably well with the experimental data. 5. Simulation of an entrapped air pocket between interfaces In this section, the movement and pressure change of an air pocket trapped in a sewer pipe is simulated. The principles and basic equations described in previous sections are used to develop the governing equations for modeling the movement of an air pocket. 5.1. An entrapped air pocket between positive and negative interfaces Fig. 4 shows a typical case of an air pocket trapped in sewer pipes. Initially, the downstream gate is closed while the water in the pipe is pressurized and is at rest. When the downstream gate opens suddenly, a negative surge will form and move toward upstream. After the interface moves a certain distance upstream, the downstream gate closes again to generate an upstream advancing positive surge. The air between these two upstream-propagating surges forms an air pocket. The
Fig. 4. A schematic diagram showing an entrapped air pocket between a positive and a negative surge.
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motion of the air pocket depends on the movement of the interfaces. The size and pressure of the air pocket vary with the difference of the moving velocities between these two interfaces. The air pressure, as a reaction, will also affect the movements of the interfaces. The mathematical model developed for modeling the positive and negative interfaces can be used in this air hammer study. However, the effect of air pressure, pb , needs to be included in the model formulations. In the conditions of water–air mixture, we consider the movement of a single air pocket, which separates two surge fronts. The movement of an entrapped air pocket is assumed to follow the relative motion of the upstream and downstream interfaces. According to Wylie and Streeter [3], isothermal behavior of free air moving with pressure surges is a reasonable assumption. The air pressure, pb , can be determined by means of the ideal gas law. The detailed equations can be found in Appendix A. The air pocket volume is calculated by Z t V0þ Aa ðwu wd Þ dt: ð11Þ Vb ¼– – 0
Here, – V b is the air pocket volume; – V 0 , the initial air pocket volume, Aa , the cross-sectional area of air pocket; wu , the upstream interfacial velocity, and wd , the downstream interfacial velocity. We select a typical case to illustrate the simulation. Initially, a downstream gate is closed while water is at rest in the pipe. The pipe diameter ¼ 0.163 m, slope of pipe ¼ 0.001, and ManningÕs roughness coefficient ¼ 0.011. The upstream head is 0.244 m. The initial negative surge is generated by suddenly opening the downstream gate. The gate remains open for 2.5 s, and then closes quickly for producing a positive surge. The time variations of the interfacial velocities, wu , wd , and the air pocket pressure head Hb , where Hb ¼ pb =c, are shown in Fig. 5. As indicated in Fig. 5, the
Fig. 5. Time variation of surge velocity and air pressure for an entrapped air pocket between a positive and a negative surge.
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positive surge advances upstream faster than the negative one retreating upstream after the formation of the air pocket. The air pocket pressure increases as its volume decreases. As a reaction, the pressure pushes the negative surge and slows down the following positive surge. The air pocket pressure eventually reaches an equilibrium condition. Under the equilibrium condition, the air pocket pressure becomes a constant and the negative surge moves almost identical to the positive surge. 5.2. An entrapped air pocket between two positive interfaces Another case considered in this study about the movement of an air pocket is illustrated in Fig. 6. A positive surge advances downstream and interacts with an upstream advancing positive
Fig. 6. Schematic diagrams showing an entrapped air pocket between two positive surges and the flow transition of the upstream interface with reversed propagating direction and an emerged open surge.
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surge, which is generated from the downstream end. The trapped air between these two positive interface forms an air pocket. This case is more complicated than the one discussed in previous section where the air pocket is formed between a positive and a negative surge. For the previous case, although the values of the interfacial velocities vary with the air pocket pressure, the moving directions of the interfaces remain unchanged. For the current case, the moving directions of the positive interfaces may be changed due to the pressure increase of the compressed air pocket. Actually, with the increase of the air pocket pressure, both of the interfacial velocities decrease. Eventually, the air pressure will push the upstream positive interface to propagate upstream as a negative surge. Therefore, the upstream positive surge experiences a transition from a positive surge to a negative one. This is not a sudden change in direction. During the fluid transient, a positive open surge may form and develop. The flow phenomenon for the transition of the upstream interface is shown in Fig. 6. It is assumed that the water depth in the region between the open surge and the upstream interface is uniform. The length of this open surge region increases and the corresponding water depth decreases as the fluid transition progresses. Eventually, this region merges with the downstream open-channel flow. The solution procedure for solving the flow condition around the upstream interface is described in the following. The unknowns for the flow condition are yp1 , Vp1 , yp2 , Vp2 , Hp3 , Vp3 , open surge velocity, ws , and interfacial velocity, wu . The equations adopted for determining the variables are C1þ : ðyp1 yL1 Þ þ cL1 ðVp1 VL1 Þ=g þ cL1 ðSf S0 ÞL1 Dt ¼ 0;
ð12Þ
C1 : ðyp1 yR1 Þ cR1 ðVp1 VR1 Þ=g cR1 ðSf S0 ÞR1 Dt ¼ 0;
ð13Þ
continuity equation across the open surge: ðVp1 ws ÞA1 ¼ ðVp2 ws ÞA2 ;
ð14Þ
momentum equation across the open surge: yp2 A2 yp1 A1 ¼ ðVp2 ws ÞA2 ðVp1 Vp2 Þ=g;
ð15Þ
continuity equation across the upstream positive surge: ðwu þ Vp2 ÞA2 ¼ ðwu þ Vp3 ÞA3 ;
ð16Þ
momentum equation across the upstream positive surge: ðHp3 0:5DÞA3 ðyp2 A2 þ Hb A3 Þ ¼ ðwu þ Vp3 ÞA3 ðVp2 Vp3 Þ=g;
ð17Þ
Bernoulli equation: Hp3 þ ðVp3 þ wu Þ2 =2g ¼ D þ Hb
ð18Þ
C3þ : ðHp3 HL3 Þ þ aðVp3 VL3 Þ=g þ aðSf S0 ÞL3 Dt ¼ 0:
ð19Þ
and The values of yp1 and Vp1 are directly obtained from Eq. (12) and Eq. (13). By solving Eqs. (14)– (18), we have Hp3 ¼ ½ð2A3 =A2 1:5ÞD þ yp2 A2 =A3 =ð2A3 =A2 1Þ þ Hb ;
ð20Þ
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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 g A3 A2 A2 A1 yp2 A2 yp1 A1 ¼ Vp1 VE Hp3 þ 2gðD þ Hb Hp3 Þ ; a A2 ðA2 A1 Þg
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ð21Þ
where g VE ¼ VL3 þ HL3 gðSf S0 ÞL3 Dt: a
ð22Þ
Eqs. (20) and (21), which describe the relationship between Hp3 and yp2 , can be solved to determine the values of Hp3 and yp2 . Since yp2 and A2 are functions of yp2 , an iteration is required to solve for Hp3 and yp2 . Once Hp3 and yp2 are determined, the remaining variables, Vp2 , Vp3 , wu , and ws , can be easily obtained from Eqs. (19), (17), (16) and (14). When the depth of water in the open surge region is equal to the depth of downstream freesurface flow region, the open surge disappears. At this instant, the upstream positive surge, which initially propagates toward downstream is already reversed to move upstream. The flow phenomenon is then changed into the case discussed in previous section. The solution procedure for solving the flow condition around the downstream interface is similar to that shown in previous section. Again, we select a typical case to illustrate the simulation. Initially, a positive surge advances downstream with an upstream head of 0.244 m. The water depth of the downstream free-surface flow is 0.122 m. As the upstream positive surge reaches a point about 4 m away from the downstream gate, the gate closes suddenly to generate an upstream propagating positive surge. Fig. 7 shows the results of both upstream and downstream interfacial velocities and the air pressure head under the condition of two positive surges and a trapped air pocket. It is noted that
Fig. 7. Time variation of surge velocity and air pressure for an entrapped air pocket between two positive surges.
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during the early stage of the air pocket formation, the interfacial velocities and the air pressure change greatly within a short period of time. As shown in Fig. 7, the air pressure increases rapidly when the two interfaces move towards each other. As a result, the propagating velocities of the upstream and downstream surges decrease. An open surge front is emerged from the upstream interface and eventually the upstream positive surge is pushed back and propagates as a negative wave. After the reversal of the upstream surge, the movement of the air pocket and the pressure change are similar to the case presented in previous section. The surge velocity is changed from positive to negative as indicated by solid line in Fig. 7. Comparing the results presented in Fig. 5 and Fig. 7, it is found that the variations of interfacial velocities and pressure change in the second case (air pocket with two positive surges) are much greater than those shown for the first case (air pocket with one positive and one negative surge). It is also interesting to point out that the increase of the air pressure plays an important role to influence the movement of the upstream and downstream surge waves. Other test cases also suggest that the air pressure increases with the increase of the upstream head and the pipe diameter. So far, there is no experimental data available to verify the modeling of air pocket movement. Continuing effort will be spent to obtain the experimental data for furthering the improvement of the air hammer model. 6. Conclusions In this study, a computational model is developed to simulate the hydraulic transients in sewers. The air pressure surge and the movement of an air pocket trapped in a pipeline are also studied. Movements of pressure surge are reasonably simulated. The computed results show good agreement with the experimental data. The simulation of an air pocket trapped between a positive and a negative surge shows that after the formation of the air pocket, the positive surge moves faster than the negative one and the air pressure increases as its volume decreases. Modeling of an air pocket between two positive surges shows that during the early stage of the air pocket formation, the interfacial velocities and the air pressure change greatly within a short period of time. An open surge front is emerged from the upstream interface and eventually the upstream positive surge is pushed back to propagate as a negative surge. Acknowledgement This work is sponsored by the City of Houston through Montgomery Watson. The support is greatly appreciated.
Appendix A Equations for solving air pocket between positive and negative interfaces are given below. Equations for flow conditions around the downstream interface (positive surge) are: C1þ : ðyp1 yL1 Þ þ cL1 ðVp1 VL1 Þ=g þ cL1 ðSf S0 ÞL1 Dt ¼ 0;
ðA:1Þ
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C1 : ðyp1 yR1 Þ cR1 ðVp1 VR1 Þ=g cR1 ðSf S0 ÞR1 Dt ¼ 0;
ðA:2Þ
C2 : ðHp2 HR2 Þ aðVp2 VR2 Þ=g aðSf S0 ÞR2 Dt ¼ 0;
ðA:3Þ
continuity: ðVp1 þ wd ÞA1 ¼ ðVp2 þ wd ÞA2 ;
ðA:4Þ
momentum: yp1 A1 þ Hb A2 ðHp2 0:5DÞA2 ¼ ðwd þ Vp1 ÞA1 ðVp2 Vp1 Þ=g:
ðA:5Þ
Here, the subscripts 1 and 2 represent respectively the flow conditions upstream and downstream of the interface. Equations for flow conditions around the upstream interface (negative surge) are: C1 : ðyp1 yR1 Þ cR1 ðVp1 VR1 Þ=g cR1 ðSf S0 ÞR1 Dt ¼ 0;
ðA:6Þ
C2þ : ðHp20 HL20 Þ þ aðVp20 VL20 Þ=g þ aðSf S0 ÞL20 Dt ¼ 0;
ðA:7Þ
continuity: ðwu þ Vp1 ÞA1 ¼ ðwu þ Vp20 ÞA20 ;
ðA:8Þ
momentum: yp1 A1 þ Hb A20 ðHp20 0:5DÞA20 ¼ ðwu þ Vp20 ÞA20 ðVp20 Vp1 Þ=g;
ðA:9Þ
Bernoulli: Hp20 þ ðVp20 þ wu Þ2 =2g ¼ D þ Hb :
ðA:10Þ
The subscripts 1 and 2 represent respectively the flow conditions downstream and upstream of the interface.
References [1] Hamam MA, McCorquodale JA. Transient conditions in the transition from gravity to surcharged sewer flow. 1982 National Research Council of Canada 1982. [2] Streeter VL. Transient cavitating pipe flows. J Hydraul Eng, ASCE 1983;109:1408–23. [3] Wylie EB, Streeter VL. Fluid transients in systems. Englewood Cliffs, NJ: Prentice Hall; 1993. [4] Locher FA, Wang JS. An experience with entrapped air in comparisons of analytical studies and field tests of a system with column separation. In: Proceedings of International Conference on Unsteady Flow and Fluid Transients, Balkeman, Rotterdam, 1992; pp. 327–32. [5] Bergant A, Simpson AR. Pipeline column separation flow regimes. J Hydraul Eng, ASCE 1999;125:835–48. [6] Benjamin TB. Gravity currents and related phenomena. J Fluid Mech 1967;31:209–48. [7] Wilkinson DL. Motion of air cavities in long horizontal ducts. J Fluid Mech 1982;118:109–22. [8] Wisner PE, Mohsen FN, Kouwen N. Removal of air from water lines by hydraulic means. J Hydraul Eng 1975;101:243–57. [9] Preissmann A, Cunge JA. Calcul des intumescences sur machines electroniques. In: Proceedings of 9th Congress, Dubrovnik, 1961; pp. 656–64. [10] Cunge JA, Wegner M. Integration numerique des equations dÕecoulement de Barre de St Vernant par un schema implicite de differences finies. La Houille Blanche 1964;1:33–9. [11] Abott MB. A modeling system for the design and operation of storm-sewer networks. Eng Appl Comput Hydraul, Abbott and Cunge (des) 1982;1:11–39. [12] Wiggert DC. Transient flow in free-surface pressured systems. J Hydraul Div ASCE 1992;98:11–27. [13] Song CCS, Cardle JA, Leung KS. Transient mixed-flow models for storm sewers. J Hydraul Eng 1983;109:1487– 503.
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[14] Cardle JA. An investigation of hydraulic transient in combination free surface and pressured flows. PhD Dissertation, University of Minnesota, 1984. [15] Shen Q. Hydraulic transients in pipelines and wave motion over a group of submerged horizontal plates. MS Thesis, Department of Civil and Environmental Engineering, University of Houston, 1997. [16] Anderson A, Robbie JF. The behaviour of surge tanks with horizontal expansion galleries. In: Proceedings of 5th International Conference on Pressure Surges, Hannover, Germany, 1986; pp. 261–71.