Water Column Separation and Cavity Collapse for Pipelines Protected with Air Vacuum Valves: Understanding the Essential Wave Processes

Transcription

1 Water Column Separation and Cavity Collapse for ipelines rotected with Air Vacuum Valves: Understanding the Essential Wave rocesses Leila Ramezani, S.M.ASCE ; and Bryan Karney, M.ASCE 2 Downloaded from ascelibrary.org by University of Toronto on 05//7. Copyright ASCE. For personal use only; all rights reserved. Abstract: Elevated high points along a pipeline profile are the most common places where air vacuum valves (AVVs) are installed. This paper uses basic water hammer theory to semianalytically explore the effects of such AVVs. A simple frictionless reservoir-pipe-reservoir system with an exaggerated intermediate high point is considered with a sudden flow curtailment assumed upstream. Key design parameters such as the maximum air pocket volume, the duration of air pocket growth and collapse, and the maximum magnitude of the pressure spike resulting from water column rejoinder are semianalytically developed for various high point locations. The magnitude of the reduced pressure wave created by the refraction at the high point, and both its vertical and horizontal position, are demonstrated to crucially determine system performance. Numerical examples are compared with the semianalytical expressions to highlight the accuracy of the derived expressions. The effect of friction is later introduced to help reveal friction s influence on air valve performance. DOI: 0.06/(ASCE)HY American Society of Civil Engineers. Author keywords: Air vacuum valve; Column separation; Fluid transient; Water pipeline; Air pocket. Introduction Air vacuum valves (AVVs) are designed to admit air into pipelines whenever vacuum conditions occur at their location. The design goals are easily stated but less easily achieved: air valves should admit sufficient air to relieve negative pressures, while limiting any adverse influence that might be associated with this air admission. After suppressing vacuum conditions (i.e., once pressures have risen above atmospheric), AVVs should safely release the admitted air. The motivation for using AVVs for transient protection is that they are usually cheaper than alternative protection measures, such as air chambers, and they are often required in any case for line filling and draining. If vacuum conditions are to be effectively controlled, AVVs must clearly be placed and sized properly, and considerable ambiguity remains about how this should be done. If AVVs are improperly sized, they tend to induce strong secondary transient events (Lee and Leow 999; Lingireddy et al. 2004; Li et al. 2009) and can damage the valve (Li et al. 2009). Secondary transient pressures have been experimentally studied during filling scenarios in both a simple horizontal (Zhou et al. 2002a, b; Martin and Lee 202) and more complex (De Martino et al. 2008) piping system. Numerical models also addressed these secondary pressures (Lee and Leow 999; Lingireddy et al. 2004; Li et al. 2009). However, well-chosen AVVs, typically those with low outflow and high inflow capacity, can better suppress peak pressures and valve slamming while still suppressing negative pressures (Espert et al. 2008; Lee and Leow 999). Manufacturers have introduced various design modifications so that they more efficiently eliminate local surges (Zloczower 200). Yet, AVVs are often sized by considering only filling/draining scenarios with orifices large enough to exhaust/admit substantial volumes of air (AWWA 200). However, if the large valves are activated under other events such as pump failure, destructive secondary transients may occur. To effectively use AVVs for transient protection, the key physical regularities associated with their use need to be understood. Another key design factor is an AVV s location. To limit air accumulation, AVVs are most commonly sited at all local high points. This arrangement is consistent with most manufacturer recommendations and guidelines (e.g., AWWA 200) but can result in a great many AVVs being installed. However, the relative inaccessibility of some locations can discourage effective maintenance and inspection. Moreover, air valves installed for one purpose (e.g., line filling) may also be activated at other times (e.g., pump failure). AVVs selection ideally needs to consider a range of operations. Moreover, if some locations are both ineffective and inconvenient, the installation of cheaper, manually operated AVVs (i.e., pit cocks) might be preferable (Mcherson 2009). Despite research detailing the effects of AVVs on secondary transient events and experimental studies on their effect on secondary transient pressures (Bergant et al. 202; Carlos et al. 20; Lucca et al. 200; Arregui et al. 2003; Cabrera et al. 2003), ambiguity remains about what physical phenomena dominate. The objective of this paper is to understand how AVVs at particular locations influence a system s transient response. h.d. Candidate, Dept. of Civil Engineering, Univ. of Toronto, ON, Canada M5S A4 (corresponding author). leila.ramezani@mail.utoronto.ca 2 rofessor, Dept. of Civil Engineering, Univ. of Toronto, ON, Canada M5S A4. karney@ecf.utoronto.ca Note. This manuscript was submitted on November 29, 204; approved on June 27, 206; published online on September 30, 206. Discussion period open until February 28, 207; separate discussions must be submitted for individual papers. This paper is part of the Journal of Hydraulic Engineering, ASCE, ISSN Transient Response in Systems with AVVs To isolate the transient behavior of systems with AVV protection, a simple reservoir-pipe-reservoir system with an exaggerated high point is considered (Fig. ). The system is subject to a rapid depressurization associated with a sudden curtailment of the inflow (mimicking the power failure of a low-inertia pump). An upstream check valve is assumed to immediately and permanently close, preventing ASCE J. Hydraul. Eng. J. Hydraul. Eng., 207, 43(2): ---

2 Downloaded from ascelibrary.org by University of Toronto on 05//7. Copyright ASCE. For personal use only; all rights reserved. A V 0 L B AVV Fig.. ipeline having a single isolated high point subjected to a sudden depressurization caused by a sudden curtailment of the system s inflow reverse flow. When the resulting reduced pressure wave reaches the isolated high point, a large AVV with unrestricted air inflow is assumed to instantly open. This air valve action creates a reflected and refracted wave, with both waves having a magnitude (head) determined by the AVV s elevation. The idealized AVV maintains atmospheric pressure at the high point as long as air is present, and thus this boundary splits the (a) (c) (e) L 2 C H s system into upstream and downstream segments. For clarity in what follows, the terminology of upstream and downstream refers to the original (steady state) flow directions. Upstream of the AVV At the instant of its curtailment, the flow into the upstream section of pipe suddenly stops, and a reduced pressure wave is created that propagates downstream [Fig. 2(a)]. When this wave reaches the high point [Fig. 2], the whole upstream limb (A to B) (Fig. ) is under a reduced head (H ), and its velocity and momentum are lost. Yet, at the high point, the presence of the AVV prevents the pressure from dropping more than marginally below atmospheric; air is admitted, and atmospheric pressure is established and the upstream (A to B) and downstream limbs (B to C) become hydraulically separated (Fig. 3). Thus, an increased pressure (positive) wave [Fig. 2(c), ΔH 2 ] reflects upstream, and a (smaller) reduced wave [Fig. 4(a), ΔH 2 ] continues downstream. Within the upstream limb, the positive wave created at the high point [Fig. 2(c)], through a direct water hammer effect, causes a fluid velocity of V in ¼½ðgÞ=ðaÞŠΔH 2, where a is the wave speed. (d) (f) (g) Fig. 2. Transient behavior of the upstream portion of the high point with the presence of AVV: (a) 0 < t < ðl =aþ ¼ða=gAÞQ 0 ; t ¼ðL =aþ Q in ¼ 0; (c) ðl =aþ < t < 2ðL =aþ Q in ¼ ðg=aþaδh 2 ; (d) t ¼ 3ðL =aþ Q in ¼ 0; (e) 3ðL =aþ < t < 4ðL =aþ Q in ¼ ðga=aþδh 2 ; (f) t ¼ 5ðL =aþ Q in ¼ 0; (g) trend of change of discharge upstream of the high point ASCE J. Hydraul. Eng. J. Hydraul. Eng., 207, 43(2): ---

3 Check Valve ΔQ2 L,d ΔH 2 L 2,d 2 H S (a) Fig. 3. Equivalent system to (a) the upstream of the AVV (i.e., sudden valve closure); the downstream from the AVV (i.e., sudden drop in upstream reservoir water level) Downloaded from ascelibrary.org by University of Toronto on 05//7. Copyright ASCE. For personal use only; all rights reserved. This upstream moving wave increases the pressure and adjusts the fluid velocity from zero to V in. When this wave arrives upstream, the wave is reflected from the closed check valve, and another ΔH 2 wave is created that propagates downstream [Fig. 2(d)]. At a time of L =a later, where L is the length of this upstream segment, when this new high pressure wave reaches the high point [Fig. 2(d)], the pressure has increased by the second ΔH 2, and the velocity is again zero everywhere in the system s upstream limb. The pressure at the high point remains almost unchanged as long as air is present [Fig. 2(e)]. A forward flow V in is again initiated. This reduced pressure wave travels upstream, again reaching the closed check valve, and now the pressure is reduced by ΔH 2 as the velocity becomes zero. After another L =a has elapsed, this reduced pressure wave arrives at the high point, and zero velocity is established throughout the upstream system [Fig. 2(f)]. Because the effect of friction is initially neglected, the process in Figs. 2(c f) repeats as long as the air cavity is present. The overall pattern of change for discharge with time upstream of the high point is shown in Fig. 2(g). From the instant the wave first arrives at the high point, AVV behavior mimics what would have happened in a similar system experiencing a sudden increase in reservoir head (ΔH 2 ) at the high point. The high point where a large AVV is placed [Fig. 2(c)] can now be considered as a constant head reservoir with height equal to the height of reflected wave. As illustrated, this reflected wave creates a wave reversal when it encounters the closed check valve. Thereafter, this portion upstream of the high point is hydraulically equivalent to a valve closure scenario as depicted in the configuration shown in Fig. 3(a). Furthermore, the air cavity grows (slightly) when reverse flow occurs (i.e., L =a to 3L =a) and shrinks (slightly) during the intervals of forward flow (i.e., 3L =a to 5L =a). Downstream from the AVV Within the downstream limb from the high point to the reservoir (Fig., B to C), another wave ( ¼ ΔH 2 ) propagates, reducing velocity from V 0 to V ¼ V 0 ΔV (V ¼ Q =A) [Fig. 4(a)]. The unbalanced condition at the reservoir at the instant the pressure wave arrives [Fig. 4] causes the velocity to be reduced by ΔV (ΔQ =A) (corresponding to ), and a higher pressure wave is created ( ) that moves upstream [Fig. 4(c)]. The wave arrives at the high point where pressure is atmospheric [Fig. 4(d)]. Again, a reflected pressure wave ( ) develops with an associated decrease in velocity, a wave that propagates downstream and reduces the pressure and flow velocity [Fig. 4(e)]. At the instant L 2 =a later, this reduced pressure wave reaches the reservoir, further reducing the pressure and forward velocity [Fig. 4(f)]. The process described in Figs. 4(c f) is repeated until forward flow and momentum are arrested. As long as the forward flow continues, air will progressively enter the pipe through the AVV. At the instant the forward flow stops, the maximum air volume has entered the pipe. However, this momentary stoppage represents an unbalanced condition, because the head at the high point is lower than that in the reservoir. From this instant onward, reverse flow is gradually established, and the air pocket shrinks as air is forced out through the open AVV. In particular, a high pressure wave (þ ) will propagate back to the high point [Fig. 4(g)], where it will be reflected ( )ina sequence of steps or impulses. Reversed flow is increased by ΔV 0 2 over each wave trip (L 2=as). Thus, the process depicted in Figs. 4(e and f) is repeated, but this time progressively establishing a reversed flow. Each wave transit creates an unbalanced condition and thus a pressure and flow change; over each transit, the downstream pipe effectively acquires a velocity increased by ΔV [Figs. 4(g and h)]. This increase occurs every 2L 2 =as, a process that continues until all the air is removed. At that instant, the accelerated water impacts the closed AVV and a high pressure spike occurs, with a magnitude that depends directly on the velocity at the instant of cavity collapse. The overall role of the wave is to progressively reduce the velocity from its original reduced value to the negative of this value. Obviously, dissipative and resistive processes will tend to dampen this process in real systems. The transient event for the downstream portion experiences two highly significant stages: first, the air pocket at the high point grows; and then, after the water column has been reversed, shrinks until eliminated. The water columns on the upstream and downstream sides rejoin at the instant of pocket collapse. The principal task in what remains is to quantify these influences. To physically illustrate the transient event occurring in the system s downstream limb, an equivalent system can be envisioned [Fig. 3]. A reservoir is now at the high point with its water surface initially at the elevation of the system s steady hydraulic grade line. Here, the transient event is induced by suddenly dropping the elevation of the upstream reservoir to that of the high point (mimicking the behavior of the AVV at the high point). This representation allows semianalytical solutions to describe the key physical processes. A sudden drop in the water level at the upstream reservoir [Fig. 3] causes a wave to propagate downstream. Each time such a wave arrives at the reservoir or at the high point, a portion of the system s original momentum is reduced. Based on the semianalytical solution of (frictionless) water hammer, this incremental decrease equals ΔQ ¼½ðgAÞ=ðaÞŠ. This process continues in discrete steps until all forward flow first stops and then is gradually reversed to its original value. Deceleration of the forward flow and the times it takes for both flow reversal and reaching a full reversed flow depend on the driving head between the downstream and the (induced) upstream reservoirs. The larger the initial wave height, the greater will be the incremental flow reduction, and thus the faster the deceleration, and the more quickly (measured in wave trips) the final reverse flow is achieved. The design implications of this simple argument are significant: the lower the AVV s elevation, the faster the forward flow will be stopped and final reverse flow (with Q out =Q ¼ ) will be reached. ASCE J. Hydraul. Eng. J. Hydraul. Eng., 207, 43(2): ---

4 (a) Downloaded from ascelibrary.org by University of Toronto on 05//7. Copyright ASCE. For personal use only; all rights reserved. Semianalytical Formulas The equivalent systems [Figs. 3(a and b)] allow an approximate analysis to be conducted and simple semianalytical formulas for key parameters be derived. That is, the cavity grows uniformly until flow is stopped; air is discharged at the same average rate as the reverse flow until the final collapse. Consequently, and helpfully, when the cavity collapses in a frictionless system, the downstream reverse flow reaches the same value as the initially intercepted flow ( Q ). Obviously, these relationships require some modification once friction and valve resistance is included but, as is shown later, some of these effects can be readily estimated. At the instant the wave ( ) arrives at the high point, a reduced pressure wave ( ) develops and travels downstream. The residual flow depends on how much of the original wave is intercepted by the high point (c) (e) (g) Fig. 4. Transient response downstream from the AVV: (a) 0 < t < ðl 2 =aþ ΔQ ¼ðgA=aÞ Q p ¼ Q 0 ΔQ ; t ¼ðL 2 =aþ ΔQ ¼ ðga=aþ Q p ¼ Q 0 ΔQ ; (c) ðl 2 =aþ < t < 2ðL 2 =aþ ΔQ ¼ðgA=aÞ Q2 0 ¼ Q 0 ΔQ ; (d) t ¼ 2ðL 2 =aþ ΔQ ¼ðgA=aÞ Q2 0 ¼ Q 0 ΔQ ; (e) ΔQ ¼ðgA=aÞ Q3 0 ¼ Q 2 0 ΔQ ; (f) ΔQ ¼ðgA=aÞ Q3 0 ¼ Q 2 0 ΔQ ; (g) ΔQ ¼ðgA=aÞ Q n ¼ ΔQ ; (h) Q n ¼ ΔQ Q Q 0 ¼ ðþ (d) (f) (h) Q out ðtþ Q ¼ Q aδq L 2 t ΔQ ¼ ga a ð2þ where Q 0 = initial flow (m 3 =s); (m) = initial reduced pressure wave; (m) and Q p ðm 3 =sþ = wave and flow intercepted, respectively; and t = elapsed time. Time of Air Cavity Growth and Collapse: Frictionless Case As Eq. (2) indicates, the rate of change in flow at the downstream section is ΔQ=ðL 2 =aþ. Therefore, the number of wave round trips (K) required to completely stop the flow can be calculated. Obviously, the time of cavity growth can be calculated either by setting Q out ¼ 0 in Eq. (2) or by considering that as Kð2L 2 =aþ. The following relationships for K and for the time of air cavity growth (t g ) become evident: The flow reduction relates to reflections and the wave travel time in the downstream pipe K ¼ Q 0 ΔQ 2ΔQ ¼ Q 2ΔQ ð3þ ASCE J. Hydraul. Eng. J. Hydraul. Eng., 207, 43(2): ---

5 Downloaded from ascelibrary.org by University of Toronto on 05//7. Copyright ASCE. For personal use only; all rights reserved. t g ¼ L 2Q 0 ga H Assuming unrestricted inflow and outflow through the AVV in a frictionless system, the process of air cavity growth and collapse is symmetrical. Therefore, the time of air cavity collapse (t c ) is twice the time of air cavity growth t c ¼ 2L 2Q 0 ga H Eqs. (4) and (5) indicate that the time for cavity growth and collapse increase with downstream pipe length (L 2 ) and decrease with the fraction of reduced wave intercepted at the high point ( = ). Therefore, air cavity growth and collapse are strongly determined by the wave travel time in the downstream pipe and the elevation of the high point. Maximum Air Cavity Volume When the sudden curtailment of inflow occurs in Fig., the ideal AVV activates at the instant the reduced wave arrives. Considering the continuity equation at the high point and an unrestricted inflow of air from the AVV, the rate of air entering the pipe depends on the direction of flow both upstream and downstream from the high point. The possible cases of upstream and downstream flow direction and the associated sign for airflow into (positive sign) or out of (negative sign) the pipe are shown in Table. In the equivalent system upstream, over each wave round trip, flow within the upstream portion simply periodically reverses its sign. Thus, flow in the upstream limb has a negligible net effect on maximum air cavity volume. Rather, air enters continuously during the flow deceleration downstream from the high point, and thus the characteristics of the downstream limb establish the maximum air cavity volume. Of course, neglecting the effect of the upstream limb on maximum air cavity volume is only momentarily accurate at those instants when the wave at the upstream limb has completed its round trips, and more cycles would be expected in relatively frictionless systems with higher elevations of high point and shorter upstream pipes. However, in systems with friction with lower high point elevations and longer upstream pipes, such an assumption is more questionable, and the wave process in the upstream limb would be expected to fade out over time. To derive an equation for the maximum air cavity size for the system depicted in Fig., upstream and downstream sections of the AVV are separated, and the upstream section is neglected. Also, water level (i.e., the piezometric head) at the high point is considered to be constant, and the physical regularities of the equivalent system are applied. It is assumed that the AVV allows unrestricted Table. Sign for Rate of Change of Air Volume (V air =Δt) at the High oint according to the Direction of Flow in the Upstream (Q in ) and Downstream (Q out ) Sections Flow condition Q in > 0, Q out > 0 V air =Δt ¼ðQ out Q in Þ Q in < 0, Q out < 0 Q in < 0, Q out > 0 ð4þ ð5þ Q in > 0, Q out < 0 Q in < Q out + + Q in > Q out + + Q in ¼ Q out Note: + = increase in air volume; = decrease in air volume. air flow into the system, and there is no water inflow toward the high point (Q in ¼ 0). As discussed, at the instant the reduced pressure wave reaches the high point, a fraction of initial flow is intercepted by the high point (i.e., Q ). Then, at each 2L=a (i.e., round-trip time of pressure wave), a flow reduction of 2ΔQ occurs. Considering the continuity equation at the high point and the previously discussed transient process in the equivalent system [Fig. 3], the maximum air volume admitted becomes Max air ¼ XK k¼ ½Q 0 ð2k ÞΔQŠ 2L 2 a ¼ðKQ 0 K 2 ΔQÞ 2L 2 a ¼½KðKþÞΔQŠ 2L 2 a ð6aþ Substituting K [Eq. (2)] and ΔQ [Eq. (3)] in the preceding formula, Eq. (6a) results in Max air ¼ L 2Q 0 H aq0 a 2gA H þ ð6bþ in which Max air = maximum air cavity volume (m 3 ). Effect of Downstream ipe Length on Max air Based on Eq. (6b), the maximum air cavity volume is primarily a function of the wave travel time in the downstream pipe length, its cross-sectional area, initial pipe flow, and the fraction of original wave intercepted by the high point. Assuming a given pipe diameter, wave speed, and initial pipe flow, the effect of pipe length and intercepted wave height on the amount of air volume can now be better understood. According to Eq. (6b), the air volume has a direct relationship to the pipe length, as each wave round trip takes longer when the downstream pipe is longer, allowing more air into the pipe. However, the rate of change in air volume with pipe length is constant, and therefore, the ratio of air volume to pipe volume is independent of the pipe length. Cavity Collapse: Frictionless Case After reverse flow occurs and as water accelerates toward the high point, air is discharged through the AVV. Assuming no air compression, air discharge occurs at the same rate as the water flow behind it and the AVV closes once all the air is removed. At this instant of cavity collapse, a water hammer pressure spike occurs, and the resulting high pressure wave travels both upstream of and downstream from the AVV location. The magnitude of the resulting pressure spike is half the value arising from the Joukowsky equation [Eq. (7)], because the wave propagates both upstream and downstream (Wylie and Streeter 993) ΔH max ¼ a 2g ðv up V ds Þ where ΔH max = pressure spike as a result of a column rejoinder event (m); V up = flow upstream of the AVV (m=s); and V ds = flow downstream from the AVV (m=s). Eq. (7) shows the magnitude of the pressure rise depends on the amount of both upstream and downstream flow at the time of air cavity collapse. Turning back to the initial problem (Fig. ), the downstream velocity (i.e., Q =A) at the time of air cavity collapse ð7þ ASCE J. Hydraul. Eng. J. Hydraul. Eng., 207, 43(2): ---

6 (a) (c) (d) Downloaded from ascelibrary.org by University of Toronto on 05//7. Copyright ASCE. For personal use only; all rights reserved. can be estimated from Eq. (3). The worst case scenario for the largest peak pressure occurs when the instant of air cavity collapse coincides when the upstream water column has stopped. Under such conditions, the resulting pressure spike may intensify when superimposed with the wave traveling within the upstream pipe section [ΔH 2 in Figs. 2(d and f)]. Thus, transient pressures at the high point can likely vary by ΔH 2 depending on the exact time of air cavity collapse ΔH max;min ¼ 2K ΔH 2 For lower high point elevations (i.e., higher ), because reverse flow at the high point (Q ) is smaller, less severe pressure spikes occur. These pressure spikes have a reciprocal relationship with the magnitude of intercepted wave height and a direct relationship with reverse flow velocity. Furthermore, the resulting pressure spikes are more severe when superimposed onto a source transient wave (ΔH 2 ) upstream of the AVV. t n ¼ n 2L a (e) Fig. 5. Transient behavior upstream of the high point with the presence of AVV (considering frictional head loss): (a) 0 < t < ðl =aþ ¼ða=gÞðQ 0 =AÞ; t ¼ðL =aþ Q ¼ 0; (c) ðl =aþ < t < ð2l =aþ Q ¼ ðga=aþδh 2 ; (d) t ¼ 3ðL =aþ Q ¼ 0; (e) 3ðL =aþ < t < 4ðL =aþ Q 2 ¼ðgA=aÞ½ΔH 2 h f ðq ÞŠ; (f) t ¼ 5ðL =aþ Q ¼ 0 ð8þ Effect of Friction Considering frictional effects introduces more realism but generally requires a numerical computation of the transient event. Because the current purpose is to introduce a clear physical understanding of the phenomenon, a semianalytical estimate is developed by applying the Joukowski formulation in combination with friction losses. Introducing friction causes the transient wave to attenuate with each wave trip as illustrated by Figs. 5(a f) and, therefore, results in attenuating the upstream discharge. The phenomenon described in [Figs. 5(c) 2(f)] continues until the downstream and upstream columns rejoin. Here, the process is the same as before, except that the transient wave attenuates resulting in the reduction of upstream discharge with time, so that the effect of the upstream limb gradually becomes inconsequential. Step Function Adjusted to Consider Friction The following step function represents the transient event with friction upstream of the AVV. Obviously, a frictionless system is a particular case of such a step function 8 >< Q n ¼ Q in ¼ ga a ðδh 2Þ for n ¼ 0 ; n ¼ 0; ; 2; ::: >: Q n ¼ ga ΔH a 2 Xn i h f ðq i Þ Cos½ðn þ ÞπŠ for n i¼0 (f) As described in Figs. 6(a h), the process of discharge reduction is different in the downstream limb in that friction attenuates the transient wave with each wave trip. This causes the forward flow to take longer to stop, and therefore, allows more air to enter through the AVV. The process in Figs. 6(a f) continues until forward flow stops, and then backward flow begins [Fig. 6(h)]. The formula describing this event can be shown as a step function with variable ΔQ 0. Discharge at each wave trip and the amount of air volume entering the pipe can be computed by the following procedure. Accordingly, the time of air cavity growth can be calculated by considering the time at which the maximum air cavity enters the pipe, and the time of air cavity collapse is the time at which all the air discharges out of the pipe ASCE J. Hydraul. Eng. J. Hydraul. Eng., 207, 43(2): ---

7 (a) (c) (d) Downloaded from ascelibrary.org by University of Toronto on 05//7. Copyright ASCE. For personal use only; all rights reserved. t n ¼ L a þ n L 2 a Q n ¼ Q Xn i¼ ΔQ 0 i n ¼ ; 2; 3; 4; ::: ΔQ 0 n ¼ ga a ½ h f ðq 0 Þþh f ðq n ÞŠ if Q n > 0; h f ðq n Þ¼f L 2 Q 2 n D 2gA 2 ΔQn 0 ¼ ga a ½ h f ðq 0 Þ h f ðq n ÞŠ if Q n 0 n air ¼ 2 L X 2 n at Max n air ðq a i Þ i ¼ 0; 2; 4; ::: i¼0 at n air ¼ 0 t n ¼ t g t n ¼ t c where t n = time (s); Q n = discharge (m 3 =s); ΔQ 0 = incremental change in discharge (m 3 =s); h f ðq n Þ = head loss associated with Q n ; and V air = air volume (m 3 ). Comparing Transient Behavior in Frictional Systems Fig. 7 depicts transient behavior upstream of and downstream from the AVV in both frictionless and real (frictional) systems by applying the previously defined step functions. As expected, upstream discharge attenuates with time. In the downstream limb, the discharge increment is smaller for systems having friction. This results in a smaller final reverse flow at the instant of cavity collapse. The reduced discharge upstream and the smaller final reverse flow downstream account for the less severe pressure spikes when friction effects are included. Indeed, friction will usually result in reduced pressure rises. (e) (g) Fig. 6. Transient responses downstream from the AVV (considering friction): (a) 0 < t < ðl 2 =aþ ΔQ 0 ¼ðgA=aÞH p Q 0 ¼ Q 0 ΔQ 0 ; t ¼ðL 2 =aþ ΔQ 0 ¼ðgA=aÞH p Q 0 ¼ Q 0 ΔQ 0 ; (c) ðl 2=aÞ < t < ð2l 2 =aþ ΔQ 0 2 ¼ðgA=aÞΔH 0 2 Q 0 2 ¼ Q 0 ΔQ 0 2 ; (d) t ¼ ð2l 2 =aþ ΔQ 0 2 ¼ðgA=aÞΔH 0 2 Q 0 2 ¼ Q 0 ΔQ 0 2 ; (e) ΔQ 0 3 ¼ðgA=aÞ½ΔH 0 2 þ h f ðq 0 2 ÞŠ Q 0 3 ¼ Q 0 2 ΔQ 0 3 ; (f) ΔQ 0 3 ¼ðgA=aÞ½ΔH 0 2 þ h f ðq 0 2 ÞŠ Q 0 3 ¼ Q 0 2 ΔQ 0 3 ; (g) ΔQ 0 6 ¼ðgA=aÞΔH 0 3 Q 6 ¼ ΔQ 0 6 ; (h) ΔQ 0 6 ¼ðgA=aÞΔH 0 3 Q 6 ¼ ΔQ 0 6 (f) (h) Effect of Downstream ipe Length on t g and t c Considering Eq. (2) and the step function for frictional case, the effect of the downstream pipe length (L 2 ) on the trend of variation in discharge, time of air cavity growth and collapse, and the final reverse flow at the time of collapse is examined. Figs. 7(c and d) show such a trend for two ratios of intercepted wave height ( = ¼ 0.2 and 0.5) at the high point and three fractions of downstream pipe length (L 2 ) to total pipe length (L). As shown, t g and t c increase with downstream pipe length. Because friction is directly proportional to pipe length, for each = ratio when downstream length increases, friction increases. Also, ΔQ 0 decreases resulting in larger Q after each round trip of the wave. Both larger pipe length and larger Q account for the increased frictional losses. Therefore, the number of wave round trips for flow stoppage increases. Consequently, for longer downstream pipes lengths, the differences between t g and t c is greater for frictional systems. However, for shorter downstream pipes lengths, because friction loss is reduced, the magnitude of ΔQ and ΔQ 0 becomes more similar, and the difference in t g and t c between the two systems less pronounced. Also, when = decreases, the total number of round trips to reach to final reverse flow increases. Because the discharge is higher with each wave trip, so is friction. Therefore, frictional effects are more pronounced with lower = ratios because of the direct relationship of head loss to both Q and wave travel time. That is, = decreases, ΔQ 0 decreases, Q is greater for each wave trip, and L 2 is also more, therefore, the effect of friction increases. However, the difference between systems with and without friction is considerably less for higher ratios of =. Because the number of round trips is lower and friction has less opportunity to affect the system [Fig. 7(d)]. ASCE J. Hydraul. Eng. J. Hydraul. Eng., 207, 43(2): ---

8 Q in /Q 0 (a) Qin/Q0 frictionless Qin/Q0 with friction t/(l 2 /a) Q out /Q Qout/Q (frictionless) Qout/Q with friction H/DH (frictionless) H/DH with friction t/(l 2 /a) H/ Downloaded from ascelibrary.org by University of Toronto on 05//7. Copyright ASCE. For personal use only; all rights reserved. Q out /Q (c) Effect of Intercepted Wave Height on t g and t c For a given flow curtailment event, the intercepted wave height at the high point depends on the high point s vertical location. Obviously, the acceleration or deceleration rate in the downstream pipe has a direct relationship with the magnitude of intercepted wave height at the high point. Consequently, the time of cavity growth and collapse and the maximum air cavity volume all have a reciprocal relationship with this key parameter. By application of Eqs. (2) and (4) and the step function, the difference in the trends of change in discharge downstream from the AVV at different elevations of high points are shown and compared in frictional and frictionless systems (Fig. 8). Obviously, the greater the intercepted wave height, the lower the intercepted discharge (Q =Q 0 ) by the high point. Also, the air cavity s time of growth and collapse decreases with intercepted wave height. For smaller ratios of =, the differences between t g and t c for systems with and without friction is greater. However, as the ratio of = increases, the total number of round trips required to achieve final reverse flow decreases and frictional effects are less pronounced. Hence, the trend of discharge is similar for both systems. However, there are situations in which the final reverse flow experiences higher values than Q. This is mainly attributable to the high ΔQ 0 s at higher ratios of =. Eventually, during the reverse flow, there are situations when the reverse flow has approximately approached Q but there is still a small amount of remaining air in the system ( = ¼ 0.5). In such conditions, the transient resumes adding a high ΔQ 0 to the previous discharge, and therefore, the final reverse flow at the time of air cavity collapse gets a higher value than expected. Obviously, this results in higher pressure spikes at the time of cavity collapse. Semianalytical Formulas with Friction 0 - L2/L=0.5, f=0.07 L2/L=0.7, f=0.07 L2/L=, f=0.07 L2/L=0.5, f=0 L2/L=0.7, f=0 L2/L=, f= t/(l 2 /a) Semianalytical formulas for the parameters under study for frictional systems are developed in this paper. This is based on the Q out /Q (d) L2/L=0.5, f=0.07 L2/L=0.7, f=0.07 L2/L=, f=0.07 L2/L=0.5, f=0 L2/L=0.7, f=0 L2/L=, f= t/(l 2 /a) Fig. 7. Discharge variation and secondary transient pressures by application of the step function (H p = ¼ 0.): (a) upstream of AVV; downstream from AVV; effect of downstream pipe length on discharge pattern downstream of AVV; (c) = ¼ 0.2; (d) = ¼ 0.5 physical reasoning of the transient behavior in real systems with AVVs. Such approximations allow comparisons of systems with an AVV and with different geometries of the high point (i.e., horizontal and vertical distances to the associated boundaries). They can provide preliminary guidance for evaluating real systems using AVVs. When friction is introduced into the analysis, the upstream discharge reduces with a variable rate. To analytically approximate the flow with time, a constant rate of flow reduction is considered in Eq. (9). It is assumed that the rate of change in discharge is equal to the rate of change in discharge at the first round trip (N ¼ ), and therefore, a decay function can describe the upstream discharge attenuation Q out /Q t/(l 2 /a) H/DH=0.05 f=0 H/DH=0. f=0 H/DH=0.5 f=0 H/DH=0.2 f=0 H/DH=0.3 f=0 H/DH=0.5 f=0 H/DH=0.05 f=0.07 H/DH=0. f=0.07 H/DH=0.5 f=0.07 H/DH=0.2 f=0.07 H/DH=0.3 f=0.07 H/DH=0.5 f=0.07 Fig. 8. Comparing effect of intercepted wave height at the high point on trend of discharge at the downstream pipe in systems with and without friction ASCE J. Hydraul. Eng. J. Hydraul. Eng., 207, 43(2): ---

9 Downloaded from ascelibrary.org by University of Toronto on 05//7. Copyright ASCE. For personal use only; all rights reserved. ΔH 2 ¼ ; Q in ¼ ga a ΔH 2; Q N¼ ¼ ga a H N¼ ¼ ga a ½ h f ðq in ÞŠ; r ¼ Q N¼ Q in Q in r ¼ fl g 2Da 2 ðδh 2Þ in which Q in = initial backward flow when the wave arrives at the high point; and N = number of round trips (2L =a) since the initial wave arrived and initial reverse flow occurs in the upstream section. To estimate the upstream flow at each time step, a decay function is introduced. Substituting the approximate rate of discharge reduction in the following decay function, the approximate analytical formula for the upstream discharge is expressed as follows: Q Nð2L =aþ ¼ Q in e Nr Q N ¼ exp N fl g Q in 2Da 2 ðδh 2Þ Cos½ðN þ ÞπŠ for N ¼ 0; ; 2; 3; ::: ð0þ in which Q N = upstream discharge at the Nth round trip after initial reverse flow has occurred at the high point (m 3 =s). Friction also dissipates the upstream head with each wave trip. To estimate the reduced upstream head, Eq. (0) is written in terms of head at the high point, resulting in the following approximate analytical equation: H N ¼ exp N fl g ΔH 2 2Da 2 ðδh 2Þ Cos½ðN þ ÞπŠ for N ¼ 0; ; 2; 3; ::: ðþ where H N = upstream head at the Nth round trip after initial flow reversal (m). To estimate the upstream head and discharge at the time of air cavity collapse, it is worthwhile to estimate N C, which is the number of round trips (2L =a) required before the air cavity collapses N C ¼ t c 2L =a ¼ al 2Q 0 H 2L ga þ þ h fðq 0 Þ h fðq 0 Þ þ ð9þ ð2þ The upstream head at the time of cavity collapse is required to estimate the associated pressure spike. By substituting N C in Eq. (), the upstream head at the time of cavity collapse is computed as follows: fl H NC ¼ ΔH 2 exp N g C 2Da 2 ðδh 2Þ Cos½ðN C þ ÞπŠ ð3þ As previously discussed, the rate of discharge increments in transient events in the downstream limb with friction is a more complex function of time. Therefore, two assumptions are made: mean ΔQ 0 s are assumed for both forward and backward flows. Also, it is assumed that final backward flow reaches Q at the time of cavity collapse regardless of the magnitude of the intercepted wave height. Then, the same concept as previously discussed in frictionless systems is considered and the subsequent formulas are derived. Friction reduces the magnitude of the wave height and the associated discharge increment over each wave trip. To roughly estimate this effect, a mean value is assumed for the incremental rate of flow reduction in Eq. (4). This assumption may slightly underestimate the final reverse flow in the downstream limb at the time of cavity collapse and the resulting secondary transient pressure. However, this simplification is reasonable considering this paper s goal of exploring the trend of change in secondary transient pressure as a function of the system configuration; the focus is not on the maximum pressures per se ΔQ Max ¼ ga a ½ h f ðq 0 Þþh f ðq ÞŠ ΔQ Min ¼ ga a ½ h f ðq 0 Þþh f ðq ¼ 0ÞŠ ΔQ Mean ¼ ΔQ ¼ ga a ½ h f ðq 0 Þþ0.5h f ðq ÞŠ ð4þ The mean value for ΔQ is substituted in Eq. () resulting in the following formula: Q out ðtþ ¼ aδq t ð5þ Q Q L 2 Time of Air Cavity Growth and Collapse: Considering Friction As discussed previously, t g occurs when Q out ðtþ ¼0. Substituting this value in Eq. (5) and writing it in a nondimensional form will result in t g L 2 =a ¼ aq 0 ga þ h fðq 0 Þ H ð6þ Friction reduces wave heights and the associated discharge increment at each wave trip. To estimate these effects, a mean value is assumed for the incremental rate of flow increase ΔQ Max ¼ ga a ½ h f ðq 0 Þ h f ðq ¼ 0ÞŠ ΔQ Min ¼ ga a ½ h f ðq 0 Þ h f ðq ÞŠ ΔQ Mean ¼ ΔQ 2 ¼ ga a ½ h f ðq 0 Þ 0.5h f ðq ÞŠ The mean value for ΔQ 2 is substituted in Eq. () resulting in the following formula: Q out ðtþ ¼ aδq2 ðt t Q Q L g Þ ð7þ 2 Time of collapse, t c, occurs when Q out ðtþ ¼Q. Substituting this value in Eq. (7) and writing it in a nondimensional form produces t c L 2 =a ¼ t g L 2 =a þ aq 0 ga h fðq 0 Þ þ H ð8þ Eqs. (6) and (8), by expressing the air cavity growth and collapse time, permit systems with and without friction to be compared ASCE J. Hydraul. Eng. J. Hydraul. Eng., 207, 43(2): ---

10 Downloaded from ascelibrary.org by University of Toronto on 05//7. Copyright ASCE. For personal use only; all rights reserved. t/(l 2 /a) (a) (Fig. 9). The effect of friction is clearly greater with lower intercepted wave heights. Although the difference in t g is quite small, the difference in t c becomes larger as frictional effects increase owing to the consequentially lower rate of discharge increments in the reverse flow. Maximum Air Cavity Volume: Considering Friction Maximum air cavity volume occurs as soon as forward flow stops. In other words, the area under Eq. (5) is defined as the maximum air volume entering the pipe, leading to air max ¼ ipe þ h fðq 0 Þ H 2 ð9þ For a given pipe diameter, initial discharge and wave speed and by applying Eqs. (6b) and (8), the trend of change in the maximum air cavity size with intercepted wave height is compared in systems with and without friction (Fig. 2). There is a reciprocal relationship between the relative air volume and intercepted wave height at the high point. According to the previous discussion, a smaller intercepted wave at the high point implies a smaller fraction of flow is intercepted (i.e., Q is larger, and ΔQ is smaller). Hence, it takes longer for the downstream column to reverse and longer for the air cavity to grow. Moreover, the effect of friction means the difference in maximum air volume between the two systems is higher at lower intercepted wave heights. This is a result of the lower rate of discharge increments in the backward flow from the effect of friction and higher friction losses because of the higher values of discharge at each wave trip. In summary, when the high point is farther upstream (i.e., a longer downstream pipe) and its elevation is higher (smaller intercepted wave height), it takes longer for the forward flow to subside. Therefore, the system will experience reduced pressure for a longer duration and there will be more opportunity for the air cavity to grow in size. In contrast, a high point located closer to the downstream reservoir and lower in height, provides a reduced opportunity for the air cavity to grow, resulting in smaller air exchanges and cavities. Column Rejoinder Event with Friction tg/(l2/a) f=0 tg/(l2/a) f=0.07 tc/(l2/a) f=0 tc/(l2/a) f=0.07 hf(q0)/h Q/Q Hp/ Considering unrestricted air flow at the AVV, a semianalytical formula is developed for maximum and minimum transient pressure at cavity collapse. It is assumed that discharge in the downstream limb Q /Q 0 Or h f (Q 0 )/ V air /V pipe Vair/Vipe f=0 Vair/Vpipe f= H p / Fig. 9. (a) Comparing time of growth and collapse of air cavity with intercepted wave height in systems with friction and frictionless systems; effect of friction on trend of maximum air volume with relative to wave height hf(q0)/h achieves Q. Also, the worst-case scenario is when Q in in the upstream section is zero. Therefore, ΔH max;min can be computed as ΔH max;min ¼ 2K 0 ½ h f ðq 0 Þ 0.5h f ðq ÞŠ H Nc ð20þ Using this, the effect of friction on pressure spikes at the time of cavity collapse is shown in Fig. 0 for L 2 =L ¼ 0. Fig. 0 shows that the effect of friction is higher at lower intercepted wave heights. Hence, it takes more time for the reverse flow to reach Q. During this time, the upstream discharge has more time to attenuate. Therefore, at lower intercepted wave heights, ΔH max is lower, and ΔH min is higher. With regard to the ratio of h f ðq 0 Þ=, maximum H max occurs at the knee in the h f ðq 0 Þ= curve. Because friction loss is directly proportional to pipe length, this knee alters as the length of the downstream pipe is increased. Therefore, depending on the distance of the high point to the downstream boundary, maximum H max occurs at a specific ratio of =. For instance, maximum H max for L 2 =L ¼ 0 occurs when = ¼ 0.5. Such a trend is shown in Fig.. Comparing Semianalytical to Numerical Solutions with Friction A numerical solution and program was developed to verify the analytical discussion. This model uses the method of characteristics ΔH max,min / Q/Q / Q /Q 0 Or h f (Q 0 )/ DHcmax/DH DHcmin/DH 2K'H'/DH Q/Q0 hf(q0)/h Fig. 0. Effect of friction on pressure spikes at AVV location at the instance of cavity collapse (L 2 =L ¼ 0) Q /Q 0 Or h f (Q 0 )/ ASCE J. Hydraul. Eng. J. Hydraul. Eng., 207, 43(2): ---

11 Downloaded from ascelibrary.org by University of Toronto on 05//7. Copyright ASCE. For personal use only; all rights reserved. Maximum ΔH max / or / H/DH Maximum DHmax/DH L 2 /L Fig.. Trend of change of local maximum secondary transient pressure with downstream pipe length (MOC) to solve the one-dimensional water hammer governing equations. These equations and the AVV boundary condition are presented in classic texts such as Wylie and Streeter (993). The results obtained for semianalytical formulas and numerical analyses for different intercepted wave heights in systems with and without friction are compared in Figs. 2(a d). As shown, time of growth and collapse of air cavity and maximum air cavity volume are in good agreement with the numerical results. The small discrepancy between the air volumes clearly justifies ignoring the tg/(l/a) (a) V max air /V ipe /(L/a) effect of the portion of the system upstream of the AVVs in the semianalytical solution. Also, for more elevated high points, maximum pressure spikes are in good agreement with the numerical results. However, for lower AVVelevations, the differences between numerical and semianalytical data increase. This is likely because assuming the flow downstream from the AVV changes linearly rather than as a step function. Assuming a linear change underestimates the downstream flow and the associated secondary transient pressure. The semianalytical approach clearly approximates t g and V air and assumes that final reverse flow achieves Q ; but numerically there are situations when the reverse flow has almost approached Q, but a small amount of air is retained ( = ¼ 0.5). In such conditions, the numerical analysis resumes adding another ΔQ 0 to the previous discharge, and therefore, the final reverse flow at the time of air cavity collapse achieves a higher value than Q. Obviously, this results in higher values of t c =ðl=aþ and H max =. This effect is more pronounced at high values of ΔQ 0 at higher ratios of =. For example, for = ¼ 0., the relative difference between numerical and analytical solution for t g =ðl=aþ, t c =ðl=aþ, V air =V ipe, and H max = are 5.9, 2.9, 2.6, and 4.9%, respectively; whereas for = ¼ 0.5, the relative difference between numerical and analytical solution for t g =ðl=aþ, t c =ðl=aþ, V air =V ipe, and H max = are 55.5, 55.5, 75, and 24.4%, respectively. Another possible error occurs by assuming a mean frictional head loss, but significantly, both numerical and semianalytical results continually display similar trends tg/(l/a) (Analytical) f = tg/(l/a) (Numerical) f = 0 40 tc/(l/a) (Analytical) f = 0 tg/(l/a) (Analytical) f = tc/(l/a) (Numerical) f = 0 6 tg/(l/a) (Numerical) f = tc/(l/a) (Analytical) f = 0.07 tc/(l/a) (Numerical) f = / / Vair/Vpipe (Analytical) f = 0 VAir /Vipe (Numerical) f = 0 Vair/Vpipe (Analytical) f = 0.07 VAir /Vipe (Numerical) f = 0.07 tc/(l/a) ΔH max / DHmax/DH (Analytical) f = 0 Hmax/DH (Numerical) f = 0 DHmax/DH (Analytical) f = 0.07 Hmax/DH (Numerical) f = (c) /ΔH H (d) / Fig. 2. Comparing semianalytical and numerical results in systems with and without friction ASCE J. Hydraul. Eng. J. Hydraul. Eng., 207, 43(2): ---

12 Downloaded from ascelibrary.org by University of Toronto on 05//7. Copyright ASCE. For personal use only; all rights reserved. Limitations and Suggestions for Future Work The analysis of the AVV boundary condition conventionally assumes that the water level at an open AVV is equal to the high point elevation. However, this level must clearly drop while flow is supplied downstream; this reality somewhat alters the hydraulics because H p, the wave travel time, and consequently, the deceleration and acceleration rates, change. Moreover, this work currently assumes a simple undulating pipeline containing an AVV at the sole high point; more complex pipelines, particularly those containing interacting AVVs or other surge protection devices, will need special consideration as air processes at different part of the system interact. erhaps most significantly, the current work naïvely assumes that air flow is unrestricted so that the air pressure at any operational AVV is nearly atmospheric. Future work should consider restricted inflow and outflow adjustments in the height of the AVV location to estimate such influences and to allow the semianalytical formulas to be more readily compared with published data (e.g., Lingireddy et al. 2004). Conclusion Transient behavior in an AVV-protected pipeline is physically explored through a variety of semianalytical formulas. The downstream limb is shown to play the dominate role in determining the air volume entering the system and establishing the magnitude of any pressure spike occurring at the instant of air cavity collapse. Key parameters, including the maximum air volume, time of growth and collapse of the air cavity, and the water velocity as well as the maximum and minimum pressure spikes at the instance of air cavity collapse, are estimated for both frictionless and frictional systems. For the frictional case, the effect of the upstream limb on the secondary transient pressures is estimated. Both semianalytical and numerical results show that maximum air volume is directly proportional to the wave travel time in the downstream pipe, but that there is a reciprocal relationship with the reduced wave height intercepted at the high point. Thus, as the high point elevation increases, more time is required to stop the forward flow, allowing more air to be admitted. Thus, air valves with larger inlet diameters are often preferable as the AVV elevation increases, particularly when it is farther from the downstream delivery point. The rejoining of the water columns is analytically and numerically discussed through the preliminary (though obviously naïve) assumption of unrestricted air flow for both the inflow and outflow processes. Results show that the maximum pressure spike (H max )at the instant of cavity collapse increases with the elevation of the high point. This is because the final reverse flow associated with higher elevations is larger. Consequently, water velocity at the time of impact is higher, thereby, creating more severe pressure spikes. However, in systems with friction, the maximum H max occurs at an intercepted wave height at which a turning point occurs in the relative friction curve. Therefore, depending on the length of the downstream pipe and the magnitude of friction, H max occurs at a specific intercepted wave height. In other words, there is a local maximum for H max. Such a finding emphasizes that the importance of a careful selection of outlet diameter of air valves depends on the elevation of the high point. Although simplifying assumptions are invoked, the semianalytical formulas derived provide helpful insight. To confirm the semianalytical results, numerical examples are presented and shown to be in consistent with tentative conclusions arising from the semianalytical approach. Overall, the results confirm an obvious but crucial reality: if air valves are to be used effectively, it is not enough that they be present, they must also be carefully sized and located. Notation The following symbols are used in this paper: A = cross sectional area of the pipe (m 2 ); a = wave velocity (m=s); D = pipe diameter (m); f = friction factor; g = acceleration owing to gravity (m=s 2 ); H = pressure head (m); H N = head at the Nth round trip after initial reverse flow at the high point (m); H NC = upstream head at the moment of cavity collapse (N C ) (m); = reflected wave height; h f ðq n Þ = head loss associated with Q n (m) (various subscripts are used as required); hf = head loss at the upstream limb of the high point (m); h f ðq in Þ = head loss associated with Q in at the upstream limb of the high point (m); K; K 0 = number of wave round trips in the downstream limb before flow reversal in frictionless systems (K) and systems with friction (K 0 ); L = total pipe length (L þ L 2 ) (m), L for upstream and L 2 for downstream length; MaxV n air = maximum air cavity volume at the nth wave trip (m 3 ); N, N C = number of round trips (2L 2 =a) until flow reversal (N) and cavity collapse (N C ) upstream; n = number of wave trips; Q 0 = initial steady state discharge; QðtÞ = discharge as a function of time (m 3 =s); Q i = discharge at the ith wave trip (m 3 =s); Q in (t) = discharge at the limb upstream of the high point as a function of time t (m 3 =s); Q in = initial backward flow when the wave arrives at the high point (with friction) (m 3 =s); Q N = discharge at the Nth round trip after initial flow reversal at the high point (m 3 =s); Q n = discharge at the end of each nth wave trip (m 3 =s); Q Nð2L=aÞ = discharge at time Nð2L =aþðm 3 =sþ; Q N¼ = discharge when N ¼ (m 3 =s); Q out (t) = discharge in the limb downstream from the high point at time t (m 3 =s); Q = reduced downstream discharge associated with wave at the high point (m 3 =s); r = approximate rate of discharge reduction in the upstream limb, friction included; t = time (s); t c and t g = time of air cavity collapse and growth (s); t n = time at the end of each nth wave trip (s); V 0 = initial flow velocity (m=s); V air = air cavity volume (m 3 )(V Max air = maximum air cavity volume); V n air = air cavity volume at nth wave trip (m3 ); V = reduced velocity downstream from the high point at the instant of wave arrival (m=s); V pipe = pipe volume (m 3 ); ASCE J. Hydraul. Eng. J. Hydraul. Eng., 207, 43(2): ---