Influence of connection stub parameters and valve closure time on transient measurement accuracy of a pressure transducer

In the last decade, various water hammer or transient-based non-invasive pipe condition assessment methods have been developed. These include the inverse transient analysis method (Liggett & Chen 1994; Vítkovský et al. 2000, 2007; Kapelan et al. 2003; Covas & Ramos 2010; Stephens et al. 2013), the transient-damping method (Wang et al. 2002, 2005), the frequency response method (Mpesha et al. 2001; Ferrante & Brunone 2003; Covas et al. 2005; Lee et al. 2005, 2008, 2013; Sattar & Chaudhry 2008; Duan et al. 2011, 2013, 2014), and the time domain reflectometry (Brunone 1999; Lee et al. 2007; Gong et al. 2013). For all these methods, a transient is injected into the pipe system and the pressure variation at one or more points is measured using high speed pressure transducers. These measured pressure traces are then analyzed in either the time domain or the frequency domain. As a result, the transient pressure measurement accuracy is a primary prerequisite for the success of transient-based methods.

Often, when measuring a transient pressure in a pipeline, it is not possible to insert the pressure transducer such that the diaphragm of the transducer is flush with the inside of the pipe. Thus, the pressure transducer is usually installed at the end of a connection stub attached to the outside diameter of the pipeline. An example of a connection stub is shown in Figure 1 (from point B to point C). For steady state flow conditions, because there is no flow in the connection stub, the pressure as measured by a pressure transducer at the end of the stub (point C in Figure 1) is the same as that of the point B in Figure 1 (assuming a negligible difference in elevation). In contrast, under transient conditions, two questions now arise, including: (1) does the connection stub have an influence on the transient in the main pipe? and (2) does the pressure variation as measured at the end of the stub (point C in Figure 1) provide an accurate representation of the pressure variation in the pipeline (point B in Figure 1)?

This paper presents a series of numerical simulation results for an end-valve closure transient for the system in Figure 1 by using a method of characteristics (MOC) water hammer analysis. The influence of the dimensions of the stub (length L₃ and diameter D₃) and the closure time of the swing check valve, t_c, on the transient in the main pipe and the pressure measurement accuracy are investigated.

The classical one-dimensional water hammer model comprises the unsteady pipe flow continuity equation and the unsteady momentum equation (Wylie & Streeter 1993). These equations are solved using a standard MOC formulation.

Figure 2 shows the two characteristic grids for both the main pipe and the stub in Figure 1. The first grid applies to the main pipeline where the general section is designated as i. The transient flow and head in the main pipeline are designated as and ⁠. The second grid is for the stub and the general section is designated as j. The transient flow and head in the stub are designated as and ⁠.

where

The same equations as Equations (1)–(4) (except with the subscript j) would apply to modeling the stub.

The boundary condition of the junction between the main pipe and the stub (see point B in Figures 1 and 2) can be expressed as:

The parameters of the system defined in Figure 1 are summarized in Table 1, in which the friction factors for main pipe and stub are calculated by using Chézy formula and Manning formula, and the roughness coefficient is 0.012. It is assumed that the following holds for the system in Figure 1: L₃ < L₂ < L₁.

In order to investigate the influence of the presence of a pressure transducer connection stub on the transient in the main pipe, the end-valve closure transient in a system is first simulated without a pressure transducer connection stub. The results will be used as the baseline for the comparison analysis as presented in the following sections.

In Figure 1, consider the system without the stub, when the end valve at point A is closed at a time of t = 0.0 s, a positive water hammer wave is generated and will travel back and forth in the system with wave return time of 2(L₁/a₁ + L₂/a₂) or 0.2 s. The wave will be reflected only at point D and point A.

The simulation results of the transient pressure at point B under the condition of the absence of a connection stub (subscripted as ‘WS’, without stub) for different cases are presented in Table 2, in which t₀ represents the time for a water hammer wave to travel from point A to point B, t₀ = L₂/a₂; t₁ represents the time for a water hammer wave to travel from point B to point D and back to point B, t₁ = 2L₁/a₁; H_WS-Bmax represents the maximum head at point B during the period from the time of t = 0.0 s to a time of t = (t_c + t₀ + t₁); t_WS-Bmax represents the time of occurrence of H_WS-Bmax. Various swing check valve closure times as shown in Column 1 of Table 2 are simulated.

For the frictionless case, the head at point B reaches its maximum value at the time of t = t_c + t₀. If t_c is less than t₁ (the wave return time from point B to the upstream reservoir at point D and back to point B), the value of H_WS-Bmax is not affected by the wave reflection from point D and can be calculated by the Joukowky formula. In addition, it does not change with the change of t_c. However, if t_c is larger than t₁, a reflected wave will return from point D to point B before the velocity at point B reduces to zero. Thus, H_WS-Bmax will be less than that for the case of t_c being larger than t₁. In addition, the maximum head decreases with an increase of t_c.

For the frictional case, the value of H_WS-Bmax decreases with an increase of the valve closure time t_c for all cases. If t_c is less than t₁, the maximum head H_WS-Bmax will occur at the time of t = t₀ + t₁ rather than t = t_c + t₀ because of the effect of line pack. However, if t_c is larger than t₁, the maximum head will occur at time of t = t_c + t₀ because of the influence of the reflected wave from point D. Note that the initial velocity for the frictionless case (V₀ = 1.98 m/s) is less than the initial velocity for the frictional case (V₀ = 1.82 m/s). These velocities were calculated based on the fully open valve loss coefficient of K_o = 10.0 as shown in Table 1.

In Figure 1, when the valve at the end of the pipe is closed at a time of t = 0.0 s, a positive water hammer wave, W₀, is generated and travels from point A in the upstream direction. At a time of t₀ = L₂/a₂ s, the water hammer wave reaches point B and splits into three parts at the junction B, i.e. waves W₁, W₂ and W₃. Wave W₁ will continue traveling upstream from point B towards the constant head reservoir at point D and is reflected back towards point B where it arrives after t₁ = 2L₁/a₁ s. Meanwhile, wave W₂ is reflected back towards point A and will return to point B again after t₂ = 2L₂/a₂ seconds after being reflected from the downstream end valve. Finally, wave W₃ will repeatedly travel in the connection stub back and forth between point B and point C with a return time of t₃ = 2L₃/a₃ seconds. The waves traveling back and forth in the stub will interact with the waves traveling in the main pipe and it will be difficult to separate the effect of each wave. Superposition of these various water hammer waves in the main pipe and stub can only be determined using a MOC model.

In order to simplify the analysis and analyze the influence of a connection stub on the transient in the main pipe, the end valve is assumed to close instantaneously (the valve closure time will be varied later on in the paper) and the analysis time is limited to the period from t = 0.0 s to a time of t = (t_c + t₀ + t₂) s for this system to ensure there is no influence from waves W₁ and W₂. Moreover, in this section, in order to highlight the influence of the ratio of the stub diameter to the diameter of the main pipe (D₃/D), the system is assumed to be frictionless in both the main pipe and the connection stub. The friction will be considered when discussing the influence of the stub length and the valve closure time in the following sections.

The simulation results for influence of D₃/D on transient head at point B are shown in Table 3, in which t₄ represents the time of the valve closure time plus the time for a water hammer wave to travel from the valve to the junction B, t₄ = t_c + t₀; t₅ represents the time of the valve closure time plus the time for a water hammer wave to travel from the valve to the junction B and then up the stub to point C and back to the junction B, t₅ = t_c + t₀ + t_3..

The heads at point B for two cases, (for the case of without the presence of a connection stub) and (for the case of with the presence of a stub), are compared at a time of t = t₄ and t = t₅. A variable, Err_B-B(ws), representing the percentage difference between and ⁠, is defined as follows: ⁠.

In Table 3, at a time of t = t_4, the head at point B is reduced as a result of the presence of the stub (see Column 8 in Table 3). The reduction of head increases markedly as the size of the stub increases to be the same order as the diameter of the main pipe (in Column 6, a change occurs down the column from a value of 212.01 m to 144.69 m for both a 100 mm and 1.0 m stub length as the stub diameter changes from 15 mm to 1.0 m). Because the system is assumed to be frictionless, the length of the stub has no influence on the head at point B at the time of t = t_4. These head values compare with 212.03 m for the case of no stub.

Now consider the condition where the wave has traveled along the stub, reflected from point C (which is a dead end), and returned back to point B at a time of t = t₅. The head at point B is now greater than the head for the case of without the stub (see Column 9 in Table 3). In addition, the length of the stub has no influence on the head at point B at the time of t = t₅. _.In Column 7, a change occurs from a value of 212.05 m to 234.48 m for both a 100 mm and 1.0 m stub length as the stub diameter changes from 15 mm to 1.0 m. Thus, at a time of t₅, the head in the connector stub is larger than the head in the main pipe. The larger the stub diameter ratio value of D₃/D, the more energy the connector stub has, and the more influence it has on the head of point B.

The results of Columns 8 and 9 in Table 3 are plotted in Figure 3.

From the above analysis, it has been found that the larger the value of D₃/D, the larger the influence of the connector stub on the transient in the main pipe. So in order to decrease this influence, the value of stub diameter D₃ should be as small as practicable. Since D₃ = 15 mm is typically the smallest size that one would expect in usual engineering applications, this stub diameter value is now selected for the following analysis.

In order to analyze the influence of the length of the connection stub, L₃, on the transient pressure measurement accuracy, the transient for the case of a stub diameter of D₃ = 15 mm is simulated for two different conditions. The first condition is for a valve closure time of less than 2L₃/a₃, while the other condition is for a valve closure time of larger than 2L₃/a₃.

Consider the case where a stub diameter, D₃, is 15 mm, the closure time of the end valve, t_c, is 0.0 s, and the shortest stub length, L₃, is 0.02 m. Thus, t_c is less than 2L₃/a₃ = 0.00004 s (a₃ = 1,000 m/s, as shown in Table 1).

The simulation results for the case assuming zero friction and for a frictional condition with the stub length L₃ varying from 0.02 m to 1.0 m are shown in Table 4, in which H_Cmax represents the maximum head at point C for the situation of the presence of a stub during the period from a time of t = 0.0 to the time of t₅ = (t_c + t₀ + t₃) for this system; H_WS-Bmax represents the maximum head at point B under the condition of no stub; t_Cmax represents the time of occurrence of H_Cmax; Err_C-B(ws) represents the percentage difference between and ⁠, and is defined as follows: ⁠.

For the frictionless case, H_Cmax and Err_C-B(ws) for all the cases of different stubs lengths in Table 4 are the same. Thus, they are not influenced by the stub length L₃. The water hammer wave that travels along the stub from point B to point C reflects from the dead end at point C and causes a very significant increase in head (⁠ reaches 414.02 m – compared to a value of 212.03 m at point B for the case with no stub present). The time at which the maximum head at point C occurs, t_Cmax, is given in Column 5 in Table 4.

For frictional case, H_Cmax and Err_C-B(ws) are the same for the cases of stubs lengths of 0.02 m, 0.05 m and 0.1 m. While both H_Cmax and Err_C-B(ws) decrease with an increase of the length of the stub L₃, when the length of the stub is longer than 0.1 m. This is because the larger the stub length L₃, then the larger the friction loss in the stub. And when the length of the stub is less than 0.1 m, the influence of the friction loss on transient pressure in the stub is almost the same. For example, compare the three cases where the stub length L₃ is 0.02 m, 0.1 m and 1.0 m, respectively. The maximum head at point C at the end of the connection stub, H_Cmax is 381.75 m, 381.75 m and 381.02 m, respectively. In contrast, for the case without a connection stub, the head at point B, H_WS-Bmax, is 196.03 m. Thus, the percentage differences of head of Err_C-B(ws) are 94.74%, 94.74% and 94.37%, respectively, for these three lengths. The time of t_Cmax is given in Column 5 in Table 4.

For the above three cases, H_Cmax is very much larger than H_WS-Bmax, and the percentage difference Err_C-B(ws) is correspondingly large, which means the measurement accuracy is very low.

Consider the case in which the stub diameter, D₃, is 15 mm, the closure time of the end valve, t_c, is 0.003 s, and the shortest stub length, L₃, is 0.02 m. A small valve closure time, 0.003 s, has been selected for illustrative purposes. Thus, t_c is larger than 2L₃/a₃ (0.00004 s) and less than both 2L₂/a₂ (0.02 s) and 2L₁/a₁ (0.18 s) to ensure the value of H_Cmax is not affected by waves W₁ and W₂.

The simulation results for the case assuming zero friction and for a frictional condition with stub lengths L₃ from 0.02 m to 1.0 m are shown in Table 5 and Figure 4.

For all the cases in Table 5, the valve closure time t_c is 0.003 s (3 milliseconds), which is larger than the stub water hammer wave return time of 2L₃/a₃ (for the maximum length stub of 1.0 m in Table 5, the maximum value of 2L₃/a₃ is 0.002 s). The maximum head at the end of the connection stub (point C) of H_Cmax will increase with an increase of stub length L₃, as depicted in Table 5, and the percentage difference Err_C-B(ws) will also increase with an increase of stub length L_3, as depicted in Figure 4. Thus, the measurement accuracy of a transducer at point C to indicate the true pressure head at point B gets worse as the stub gets longer. Moreover, the smaller the connection stub length of L₃, then the smaller the stub water hammer wave return time of 2L₃/a₃, thus there will be more positive and negative water hammer waves reflecting back and forth and experienced at point C before the velocity of point B reaches 0.0 m/s. As a result, the maximum head at the transducer location of H_Cmax and the percentage difference of Err_C-B(ws) both decrease with a decrease of stub length L₃.

For the frictionless case in Table 5 and Figure 4, H_Cmax is 225.64 m, 245.73 m and 278.14 m, which overestimates the head by 6.42%, 15.89% and 31.18%, for a stub length of 0.02 m, 0.05 m and 0.1 m, respectively. While for a stub length of 1.0 m, the value is 409.23 m which is an overestimate of head of 93.01% and considerably less accurate. The time at which the maximum head at point C occurs, t_Cmax, is given in Column 5 in Table 5.

For the frictional case in Table 5 and Figure 4, H_Cmax is 208.81 m, 228.07 m and 258.97 m, which is an overestimate of pressure head of 6.52%, 16.34% and 32.11%, for a stub length of 0.02 m, 0.05 m and 0.1 m, respectively. While for a stub length of 1.0 m, the value is 377.04 m, which again is an overestimate of pressure head of 92.34%. Thus, the influence of friction in this case is small, which is also clearly shown in Figure 4. The time of t_Cmax is given in Column 5 in Table 5. Often the stub length L₃ is less than 1.0 m, so 2L₃/a₃ is less than 0.002 s and, as a result, the valve closure time t_c is usually larger than the stub return time of 2L₃/a_3. Thus, according to analysis above, the length of the connection stub L₃ should be as small as possible to improve the measurement accuracy. Now the effect of varying the closure time of the end valve will be considered. In the following analysis, L₃ is selected as 0.02 m, 0.05 m and 0.1 m, respectively.

Simulation results for the cases of D₃ = 15 mm, L₃ = 0.02 m, 0.05 m and 0.1 m with different valve closure times t_c are shown in Table 6 and Figure 5.

For the same reasons as described above, the larger the valve closure time t_c, the more times a wave will travel back and forth between point B and point C before the velocity of point B reaches 0.0 m/s. As a result, both H_Cmax and Err_C-B(ws) decrease with an increase of the valve closure time t_c. This is clearly shown in Table 6 and Figure 5. For example, for the case of stub length of 0.1 m, when the valve closure time t_c = 1.0 s (which is 5,000 times the stub water hammer wave return time of 2L₃/a₃ = 0.0002 s), then H_Cmax is 124.28 m and very close to H_WS-Bmax value of 124.17 m or an overestimate of 0.111 m or a Err_C-B(ws) value of 0.09% which exhibits satisfactory measurement accuracy. It also can be found that under the same value of t_c/(2L₃/a₃), the values of Err_C-B(ws) are almost the same for different stub lengths; while under the same value of t_c, the value of Err_C-B(ws) increases with the increase of stub length. Thus, as long as the valve closure time exceeds the value of 500 times the stub water hammer wave return time of 2L₃/a₃, the pressure transducer at point C will measure the pressure variation in the main pipeline at point B with an acceptable accuracy (the percentage difference between maximum heads at point C and point B is no more than 0.94%). In the case of a swing check valve slam where the time of closure is smaller than the value of 500 times 2L₃/a₃, one must be careful about using a transducer at the end of a connection stub (even if it is very short and has a small diameter) as the pressure head measured will differ (perhaps exceeding say 0.94% of percentage difference) from the actual pressure in the main pipeline.

This paper investigates the influence of connection stub parameters and valve closure time on transient measurement accuracy of a pressure transducer in a reservoir-pipe-valve-reservoir system by using the MOC analysis.

Firstly, the influence of the ratio of the stub diameter to the main pipe diameter on the transient in the main pipe is analyzed under an assumption of zero friction. It is observed that the larger the diameter of the stub, the larger its influence on the transient in the main pipe.

Then the influence of connection stub length on the difference between transient head at the measurement point (the junction of stub and main pipe) and the end point of stub (the installing point of a transducer) is investigated for both frictionless and frictional cases. The results show that this difference increases with an increase of the length of the stub when the closure time of the end valve is greater than the time for the water hammer wave to travel back and forth between the measurement point and the end point of stub, which means a decrease of the measurement accuracy with an increase of the stub length. In contrast, this difference decreases with an increase of the length of the stub only when the closure time of the end valve is less than the water hammer wave return time as defined immediately above.

Finally, this paper investigates the influence of the valve closure time on the difference between the transient pressure at the measurement point and the end point of stub. The results show that this difference decreases with an increase of the closure time of the end valve, which means an improvement of the measurement accuracy with an increase of the closure time of the end valve.

In practice, the closure time of the end valve is normally longer than the stub water hammer wave return time. As a result, the diameter and length of connection stub for pressure-transducer should both be selected to be as small as possible. Another aspect that should be considered is the presence of air in the connection stub. If there is air trapped in the transducer connection, the transducer measurements can be completely wrong. So the transducer connection stub should not be installed in the upper side of the pipe to avoid the accumulation of air bubbles.

This work was partly supported by the National Natural Science Foundation of China (grant 51409197).