Identifying Head Accumulation due to Transient Wave Superposition in Pipelines

First, let us define the hydraulic pipeline impedance (in $\mathrm{s}/{\mathrm{m}}^2$), which is $B=a/(gA)$ (Wylie and Streeter 1993), where $a$ is the wave speed in $\mathrm{m}/\mathrm{s}$, $A$ is the cross-sectional area in ${\mathrm{m}}^2$, and $g$ is gravity in $\mathrm{m}/{\mathrm{s}}^2$. The system that is considered to understand the head accumulation in a pipe is shown in Fig. 1(a). It consists of two pipes in a series connected to a reservoir at the upstream end and with a dead end at the downstream end. The upstream pipe has a diameter $D_1$, length $L_1$, wave speed $a_1$, and hydraulic impedance $B_1$, while the downstream pipe characteristics are $D_2,L_2,a_2$, and $B_2$. The transient event is generated by the closure of a side discharge valve at an interior point of Pipe 2 marked as $G$ in Fig. 1(a). This point is located at a distance $L_G$ from the dead end. $H_0$ and $Q_G$ represent the initial head and flow in the pipeline (which is also the flow through the side discharge valve).

As shown in Fig. 1(a), the length of Pipe 1 is greater than the length of Pipe 2 and its diameter is smaller. This cross-sectional area configuration $(D_1<D_2)$ is chosen to guarantee that the hydraulic impedance of Pipe 1 is larger than the hydraulic impedance of Pipe 2 $(B_1>B_2)$. However, this same condition can be present with the same cross-sectional area in both pipes and a larger wave speed in Pipe 1 $(a_1>a_2)$, as shown in Fig. 1(b).

This relation between the hydraulic impedances $(B_1>B_2)$ is necessary for the development of the head accumulation as explained in the subsequent sections. Fig. 1 summarizes the different systems that will be used in this paper to explain the head accumulation phenomenon.

Wave reflection and transmission are now studied in Pipe 2. A transient event is generated by closing the side discharge valve at $G$, assuming zero friction in the system. The analysis is carried out using the method of characteristics (MOC). This process is illustrated in Fig. 3 and divided into six different stages when the transient generation point (Point $G$ in Fig. 3) is located close to the dead end.

After the valve is fully closed in a single step at $t_1=0$, referring to Fig. 3 (Stage 1), a transient wave is generated with head $H_A$. The head and flow conditions in the pipeline after the closure of the side discharge valve can be determined using the MOC according to Fig. 2, where there are three unknowns at $t_1:H_A,Q_{A_u}$, and $Q_{A_d}$. Using the two compatibility equations available and the continuity equation at the junction [Fig. (2)], the magnitude of the head is shown in Eq. (1) and the magnitude of the flow is shown in Eq. (2). From Eq. (1), the incident head rise in the pipe at the side discharge valve $\mathrm{\Delta }{H}_i$ can be calculated using Eq. (3). According to Eq. (2), the flow in the pipeline after the side discharge valve closure is not zero; it is half of the flow that was initially going through the side discharge valve and it is propagated in both directions of the pipeline. However, when the transient wave reaches the dead end and reflects, the discharge becomes zero

The incident transient wave $H_A$ propagates in both directions along the pipeline, reaching first the dead end at the downstream end of the pipeline if the transient generation point ($G$) is closer to this boundary condition. The reflection at the dead end occurs at $t_2=L_G/a_2$, inducing a head shown in Fig. 3 (Stage 2) as $H_B$, where the magnitude may be determined from the MOC [following the same approach presented in Fig. (2)] and is shown in Table 1 in terms of the incident head rise at the side discharge valve $\mathrm{\Delta }{H}_i$.

The propagation of the transient wave in the upstream direction reaches the junction between Pipe 2 and Pipe 1 at time $t_3$, while the transient wave with magnitude $H_B$ is transmitted in the upstream direction from the dead end (Stage 3). Time $t_3$ also depends on the wave speed $a_2$ and the distance between the generation point and the junction $(L_2-L_G)$. To determine the magnitude of the head $H_C$, three equations can be used, including Eq. (1), and the two compatibility equations from the MOC [shown in Fig. (3)] that apply at the series pipe junction arewhere $H_C$ and $Q_C$ = head and flow reflected once the transient wave has reached the series junction and $C_p$, $B_p$, $C_m$, and $B_m$ = definitions taken from Wylie and Streeter (1993). By solving Eqs. (1), (4), and (5) for $H_C$ and $Q_C$, Eqs. (6) and (7) are obtained. Expressions for $H_C$ and $Q_C$ can be obtained from Eqs. (4) and (5), but to express the head and the flow at the series pipe junction just in terms of the incident head rise $H_A$, Eq. (1) is required. An equivalent form of Eqs. (6) and (7) was presented by Wylie (1983) in a study where three compatibility equations were solved to find the reflection and transmission of an incident wave after reaching a pipe junction. However, for the case analyzed in this paper, the incident wave corresponds to $H_A$ as a result of the closure of a side discharge valve in contrast to Wylie (1983), where an inline valve is closedwhere $\mathrm{\Delta }{H}_i=B_2(Q_G/2)$. Analysis of Eq. (6) shows that the magnitude of the head at the junction depends on the relation between the pipes’ hydraulic impedances $B_1$ and $B_2$. If the impedance of Pipe 1 is smaller than the impedance of Pipe 2, $H_C$ will be smaller than the incident transient wave. However, if the impedance $B_1$ is larger (either because the wave speed is larger and/or the area is smaller as in the case being analyzed in this section), the head in the junction will also be larger than the incident transient head, as reported by Wylie (1983).

If the generation point ($G$) is located closer to the pipe junction than to the dead end, the transient wave reflection at the interface ($H_C$) would occur before the reflection at the dead end ($H_B$), but the rest of the head accumulation process would develop in the same way. The expressions presented in this section are valid as long as the transient event is generated at any interior point of Pipe 2.

Subsequently, at instant $t_4$, the reflection from the dead end ($H_B$) traveling upstream and the reflection from the junction ($H_C$) propagating downstream meet at an interior point of Pipe 2 (Stage 4). The superposition of these two waves induces a head increase when compared with the initial transient head rise at the side discharge valve. The magnitude of the head ($H_D$) and the flow in the pipe at that time is shown in Table 1 and was developed again from the compatibility equations using the MOC. This transient wave is again propagated in both directions toward the pipe junction and toward the dead end.

Depending on the position of the generation point along the pipeline, the transient wave with head $H_D$ may reach either the junction first (as shown in Fig. 3, Stage 5) or the dead end first, but both of those reflections will induce a new head accumulation with magnitudes $H_E$ and $H_F$, respectively, at times $t_5$ and $t_6$. As can be seen in Table 1, the magnitudes $H_E$ and $H_F$ are the same, leaving the pipe at the end of the reflection cycle with head

Finally, as shown in Fig. 3, the transient waves $H_E$ and $H_F$ approach each other to meet at the generation point at the end of the head accumulation cycle (Stage 6). The total time ($t_T$) of one head accumulation cycle is defined as

Once these transient waves meet, a new cycle of head accumulation will start with an initial head defined by Eq. (8). The evolution of several head accumulation cycles will be analyzed in the next section.

If the transient generation point was at the end of the pipe instead of at any interior point of the pipe, a head accumulation phenomenon would also have occurred, but in a simpler way. When comparing this process with Fig. 3, only a reflection at the junction and at the dead end would occur. However, when the transient wave propagation analysis is developed for this situation, the final head accumulation can also be computed with Eq. (8) and the time from Eq. (9).

By analyzing a single pipeline, with the transient source $G$ at the end, an understanding of the transient wave propagation is simpler to achieve than the one shown in Fig. 3 and could have been used to define the repetitive head accumulation; however, in a real system, it is more common either to find transient events induced along the pipe as a result of user demands, fire tests, or valve operation, rather than at the end of the pipeline.

For this first approach, Pipe 1 in Fig. 1(a) is considerably longer than Pipe 2, meaning that there would not be an interaction of the transient wave that returns from the upstream reservoir with the multiple reflections between the dead end and junction in Pipe 2. In this sense, the maximum head accumulation computed for this system is only a potential maximum. The real head accumulation is analyzed in the next section. A more general equation for calculating the head in Pipe 2 at the end of any head accumulation cycle is presented in Eq. (10) aswhere $n$ represents a cycle that starts with a value of one and can take any positive integer value. This equation expresses the head in Pipe 2 at the end of each cycle. There are intermediate head values (e.g., visible in Fig. 4) that are not explicit in this definition. These values are presented in Table 1 for the first cycle and can be computed for subsequent cycles by multiplying the following term by $\mathrm{\Delta }{H}_i$:

A better way to express the head accumulation is by computing it relative to the initial steady head in terms of the incident transient wave head increase ($\mathrm{\Delta }{H}_i$) and the characteristics of the system. This relative increase after the $n$-th head accumulation cycle iswhere $\mathrm{\Delta }{H}_N=H_N-H_0$ and $H_N$ = head at the end of $N$ accumulation cycles. In contrast to the situation analyzed by Wylie (1983), where wave focusing was studied for several consecutive changes in the cross-sectional area, in the present paper $n$ corresponds to each head accumulation cycle due to multiple reflections from one junction. Eq. (12) is only valid when $B_1\ne B_2$ given that if the two impedances are equal, no extra reflections would occur. If there is no head reduction mechanism for this accumulation process, as analyzed in this first case, $N$ will take a large value (it could become infinite) and Eq. (12) can be simplified to

This equation determines the maximum potential head accumulation in Pipe 2. The criteria that can be used to detect if a head accumulation will take place are: (1) if the hydraulic impedance of Pipe 1 ($B_1$) is larger than the impedance of Pipe 2 ($B_2$). This occurs if the cross-sectional area of Pipe 1 is smaller ($A_1<A_2$) or if the wave speed in this pipe is larger $(a_1>a_2)$; and (2) if Pipe 1 is long enough to allow the development of a complete accumulation (which is the case analyzed in this section). On the other hand, Eqs. (13) and (3) can be combined to obtain

This equation shows that the potential maximum head accumulation in Pipe 2 can also be computed as twice the incident head rise if a transient event had been generated along Pipe 1 (rather than in Pipe 2). Therefore, the larger the impedance of Pipe 1, the larger the maximum potential head accumulation in Pipe 2.

To illustrate this finding, Figs. 4(a and b) show the normalized head rise at the side discharge valve [calculated by normalizing the head rise at the transient generation point at any time with the initial head rise $\mathrm{\Delta }{H}_i$ from Eq. (3)] from a numerical simulation using the MOC for two different systems. Example A is the system described in Fig. 1(a) and Example B is a system in which the diameter of both pipes is the same, but the pipe material is different and is shown in Fig. 1(b). Pipe 1 is assumed to be a metallic pipe and Pipe 2 is a plastic pipeline. The main parameters of both examples are summarized in Table 2.

As can be seen in Table 2, the steady-state conditions for both examples are assumed to be the same (head and flow); however, the characteristics of Pipe 2 (in which the transient event is generated) are different. Therefore, the initial head rises are also different. Fig. 4 presents the normalized head rise at the side discharge valve for both cases, showing that head accumulates to almost 22 times the initial transient head rise in Example A, while in Example B, head builds up to 8 times the incident head rise.

Using the information in Table 2 and Eq. (13), it is possible to obtain the exact values for the maximum head accumulation in the systems. In Example A, the impedance of Pipe 1 is $\mathrm{4,169}\mathrm{s}/{\mathrm{m}}^2$, the impedance of Pipe 2 is $383.6\mathrm{s}/{\mathrm{m}}^2$, and the initial head rise is 7.46 m; therefore, using Eq. (13), the maximum normalized head rise is 21.73 [the same value as in Fig. 4(a)]. In Example B, the impedance of Pipe 1 is $383.6\mathrm{s}/{\mathrm{m}}^2$, the impedance of Pipe 2 is $95.17\mathrm{s}/{\mathrm{m}}^2$, and the initial head rise is 1.85 m, resulting in a maximum normalized head accumulation of 8.07 [this value is also evident in Fig. 4(b)]. The same results can be obtained using Eq. (14), showing that the maximum potential head accumulation in Pipe 2 is a function of the hydraulic impedance of Pipe 1 and not of the characteristics of the pipe in which the transient event is generated.

Fig. 4 shows some of the intermediate head accumulations described in Table 1. For instance, in Example A, the head at the beginning of the second cycle of accumulation [obtained by using Eq. (8) and modified equations of Table 1] is 63.54 m, which corresponds to a relative head rise of 4.49 times. This value is visible in Fig. 4(a); however, two extra reflections are visible in the figure before the relative head rise reaches 4.49. Those reflections correspond to $H_B$ and $H_D$ in Table 1. Not all the reflections described in Table 1 are visible in Fig. 4 because, as shown in Fig. 3, some of these reflections never reach the transient generation point.

According to Eq. (13), the maximum head accumulation only depends on the impedance of Pipe 1 (the one causing the repetitive transient wave reflections) and not on the characteristics of the pipe in which the transient event is induced. However, this is only true when the pipe causing the head accumulation is significantly long. In a system that receives reflections from an upstream reservoir or another source in a relatively short time, the maximum head accumulation can be considerably smaller than the one described by Eq. (13). In the next section, a more realistic system, in which the length of Pipe 1 is not excessively long, is presented to determine the real maximum head accumulation.

A potential maximum head accumulation in a system with a long Pipe 1 was described in the preceding section. To obtain Eq. (13), the limit in Eq. (12) when $N$ tends to infinity was computed. However, this limit is only valid when no reflections from the upstream reservoir (on any other element in a different system) interact with the pipe in which the transient event is generated. In this section, Pipe 1 is assumed to have a finite and shorter length [Fig. 1(c)] and an estimate of the number of head accumulation cycles that will occur ($n$) is required to determine the maximum head accumulation.

This number of cycles is related to the time that it will take the first transient wave to return from the upstream reservoir. This time can be computed using Eq. (15) and then compared to: (1) the time that it takes for each head accumulation cycle to develop, as described by Eq. (9); and (2) the time it takes to reach the maximum head accumulation computed with Eq. (16)where $t_R$ corresponds to the time of arrival of the first transient wave from the upstream reservoir. $L_1$, $L_2$, $L_G$, $a_1$, and $a_2$ are defined in Fig. 1(c), $t_{NT}$ is the time required to reach the maximum head accumulation, and $N_t$ represents the number of cycles when the head accumulation is considered to be maximum. The maximum head accumulation would only be reached when $N_t$ is infinite; however, in a real system, from a certain point the head accumulation could be considered the maximum given the fact that with each cycle, the extra head accumulation decreases (behavior that is visible in Fig. 4). Considering this behavior, $N_t$ can be computed aswhere $ln$ = natural logarithm. The term $\epsilon$ corresponds to a threshold value that defines when the maximum head accumulation is considered to have been reached and is based on the relative difference between the potential maximum head accumulation [given by Eq. (13)] and the real head accumulation after $N_t$ cycles. Upon applying Eq. (17), the term $\epsilon$ can be used in the same way as a tolerance: a smaller threshold will result in a larger number of cycles and a maximum head accumulation that is closer to the potential maximum head accumulation.

Using the previous equations, the head accumulation in a real pipeline can be computed. If $t_R$ is greater than $t_{NT}$, the maximum potential head accumulation will be reached and the maximum head in the pipe can be approximately computed using Eq. (13) or (14). If $t_{NT}$ is greater than $t_R$, the first transient wave coming back from the upstream reservoir will interact with Pipe 2 to reduce the head before reaching the potential maximum head accumulation. In this case, the number of head accumulation cycles can be determined using Eq. (18) and the head accumulation can be calculated using Eq. (12)

In order to illustrate how these equations represent a general case, a numerical model using the MOC was developed to obtain the head variation for two systems with the same characteristics as Example A and B (as previously described in Fig. 4) except for the length of Pipe 1, which was defined as 100 m. These new cases are now referred to as Example C and Example D and their characteristics are shown in Figs. 1(c and d). The results of these numerical simulations are shown in Figs. 5(a and b).

Fig. 5(a) shows that the maximum head accumulation is not reached [when compared to Fig. 4(a)]. Therefore, using Eqs. (15)–(17), it is found that the required time to obtain the maximum head accumulation (with an $\epsilon$ value assumed to be =1%) is $t_{NT}=1.22\mathrm{s}$ and the reflection from the upstream reservoir returns to the junction at Pipe 2 at $t_R=0.18\mathrm{s}$. With this comparison, it is possible to determine that four complete head accumulation cycles will take place in Pipe 2 and the maximum expected accumulated head will be 114.7 m, which corresponds to a relative head rise of 11.35. When comparing this value with the results presented in Fig. 5(a), it is evident that the maximum relative head obtained with the MOC is larger (12.25 corresponding to a head of 121.4 m). This difference is due to the fact that in the equations presented in this section, a detailed analysis of the interaction between the reflections from the upstream reservoir and the transient wave that had been accumulating is ignored; depending on the stage of the cycle described in Fig. 3 during which the reflection arrives, an extra head accumulation can develop. However, this extra head accumulation is small when compared to the rest of the head accumulation phenomenon.

For Example D, $t_{NT}=1.89\mathrm{s}$ and $t_R=0.28\mathrm{s}$, which results in only two cycles of head accumulation and an expected maximum relative head of 5.14 $(\mathrm{\Delta }{H}_i=9.50\mathrm{m})$ or a maximum head of 39.50 m. Based on Fig. 5(b), the maximum relative head is 4.41 (or $\mathrm{\Delta }{H}_i=8.15\mathrm{m}$ or a maximum head of 38.16 m). In this case, the extra interaction between the accumulated transient wave and the reflected transient wave from the upstream reservoir causes a reduction in head (diminishment mode) instead of an increase in head (magnification mode), as shown in Example C. These two examples support the statement of Starczewska et al. (2014) about the possibility of having both destructive or constructive interactions between transient waves depending on the characteristics of a water pipeline system.

Experiments in the laboratory of the University of Adelaide have been conducted to validate the existence of head accumulation in a pipeline after the generation of a transient event. The test pipeline was a 50.83 m copper pipe with nominal diameters of 25.4 and 76.2 mm, internal diameters of $D_1=22.14\mathrm{mm}$ and $D_2=72.94\mathrm{mm}$, and a wall thickness $e_0=1.63\mathrm{mm}$. The length of the segment of the pipeline with an internal diameter of 22.14 mm was 13.23 m. The upstream end of the pipeline was connected to a larger water main that served as a reservoir to the system and the downstream end was a closed in-line valve (creating a dead end). The element that would induce the head accumulation is represented by a segment of the pipe with a smaller diameter.

A transient wave was generated in the downstream segment of the pipeline by sharply closing a side-discharge solenoid valve located 12.87 m upstream of the closed in-line valve. The valve closure time was estimated as 3 ms. Head variation responses were measured at the side-discharge valve with a sampling rate of 10 kHz. The pressure transducer was a Druck PDCR 810 with an absolute pressure range of 0–15 bars. The pipeline system is illustrated in Fig. 8.

Several tests were conducted to verify the repeatability of the experiments using the same pipeline configuration. The head trace obtained for Test 1 is shown in Fig. 9(a). The transient response of the system shows signal attenuation due to the dissipation of the head in the pipeline. However, in the first $4L_2/a\mathrm{s}$, the head rises in comparison with the value associated with the initial generated transient wave.

Fig. 9(b) presents an enlarged view of Fig. 9(a) for the first $4L_2/a\mathrm{s}$ for three different tests. In this figure, it is evident that the existence of head accumulation occurs in the pipeline, considering that the initial head rise was 2.16 m for Test 1. This figure shows that the results between different tests are similar with a reflection in the middle of the first plateau that will be discussed later in the paper.

The preceding head variation traces were used to observe and analyze the head accumulation in the pipeline; however, a second pressure transducer (located 12.2 m upstream of the first pressure transducer) was installed to determine the wave speed of the downstream section of the pipe. An enlarged plot for the initial head rise for Test 1 (Fig. 10) shows that the transient wave front of the incident transient wave is not a vertical step, but is a curve with a total rise time of 3 ms. This transient wave front travels along the pipe, reaching the second pressure transducer with a certain delay. Using these two traces, the wave speed in the downstream segment of the pipe of diameter 72.94 mm was estimated to be $\mathrm{1,060}\mathrm{m}/\mathrm{s}$, which corresponds to the results presented by Wang (2002).

On the other hand, to estimate the wave speed of the upstream segment of the test pipeline, the formula proposed by Wylie and Streeter (1993) was used, considering a parameter $c_1$ of 1.0 and obtaining a wave speed of $\mathrm{1,198}\mathrm{m}/\mathrm{s}$. The steady-state head was estimated to be $H_o=61.43\mathrm{m}$ by calculating the average of the head values within a short time interval before the valve closure. In the same way, the incident transient wave can be estimated to be $H_A=63.59\mathrm{m}$ by averaging the head between 35.527 and 35.533 s. As a result, the magnitude of the incident transient wave is determined to be $\mathrm{\Delta }{H}_i=2.16\mathrm{m}$. Finally, the impedance of the upstream segment of the pipeline was calculated to be $\mathrm{317,200}\mathrm{s}/{\mathrm{m}}^2$ and the impedance of the downstream segment was $\mathrm{25,860}\mathrm{s}/{\mathrm{m}}^2$.

Using the information related to the initial conditions of the system and the initial generated transient, an analysis of the potential and real head accumulation was conducted. Table 4 summarizes the relevant characteristics of the pipeline to use the set of equations proposed in this paper. Considering the dimensions of the pipeline, a maximum head accumulation was not expected, mainly due to the length of the upstream segment. However, the verification was made using Eqs. (15) and (16). The time of arrival of the first transient wave from the upstream reservoir ($t_R$), assuming a vertical step wave, is 0.057 s. On the other hand, with the information in Table 4 and a threshold of $\epsilon =1\%$, Eq. (17) states that to reach the maximum potential head accumulation in the pipeline, 29 accumulation cycles are required, which would take 2.78 s. Therefore, a maximum head accumulation will not occur under these conditions.

Fig. 8 shows that the upstream segment responsible for the head accumulation is shorter than the pipe in which the transient is generated. Thus, a dramatic accumulation was unlikely. Using Eq. (18), only one head accumulation cycle is likely to develop in the system, resulting in a head accumulation of 7.99 m as calculated with Eq. (12). This result implies that the head in the pipe by the end of the head accumulation phenomena would be 69.41 m.

For the conducted laboratory experiment, the set of equations proposed successfully predicted the maximum head with an error of 1.9% (considering that the maximum head obtained in the laboratory was 68.13 m). Differences in the results are associated with two features of the experimental setup: valve maneuverability and extra transient wave reflections. First, even with the use of a solenoid valve, the closure of this element is instantaneous, thus it generates a transient wave front that is not vertical (as shown in Fig. 10) compared with the instantaneous closure considered to obtain the proposed equations.

Second, the presence of additional features in the pipeline might have affected the reflections of the incident transient wave and therefore, disturbed the head accumulation process. In Fig. 10, a drop in head is visible after the arrival of the first transient wave. This head drop corresponds to the presence of a flow meter that has different characteristics (cross-sectional area and wave speed) from the pipeline and might have altered the head accumulation in the rest of the experiment. Regarding possible sources of error in the equations, neglecting the effect of the friction (including unsteady friction) in the system can influence the predicted maximum head accumulation. An analysis of the experimental setup can be conducted using the preceding non-dimensional equations, but is not shown in the paper for the sake of brevity.