Abstract

Consider that a transient pressure measurement occurs at the end of the connection stub attached to a pipeline. The question arises as to whether the pressure being recorded at the end of the stub is an accurate representation of pressure in the pipeline. In this study, the influence of three parameters, including pressure transducer connection stub length, stub diameter and valve closure time, on the measurement accuracy of transient pressure is investigated through numerical simulation on a reservoir-pipe-valve-reservoir system. The results show that the larger the diameter of stub, the larger its influence both on the transient in the pipe and on measurement error; the measurement accuracy increases with an increase of the length of the stub only when the closure time of the end valve is less than the time for the water hammer wave to travel back and forth between the measurement point and the end point of stub. In contrast, when the closure time of the end valve is greater than the water hammer wave return time, the measurement accuracy will decrease with an increase of the stub length; the measurement accuracy is improved as the closure time of the end valve increases. As a result, in practice, diameter and length of connection stub for the pressure-transducer should both be selected to be as small as possible.

INTRODUCTION

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)?

Figure 1

Reservoir-pipe-valve-reservoir system with a pressure transducer connection stub.

Figure 1

Reservoir-pipe-valve-reservoir system with a pressure transducer connection stub.

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, tc, on the transient in the main pipe and the pressure measurement accuracy are investigated.

MATHEMATICAL MODELS

One-dimensional water hammer modeling and the MOC solution

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 .

Figure 2

The x-t plane for computation of the main pipe and stub.

Figure 2

The x-t plane for computation of the main pipe and stub.

Along the C+ characteristic line , the compatibility equation is:  
formula
(1)
or  
formula
(2)
where H represents the piezometric head; Q represents the flow; a is the wave speed; A is the cross sectional area of pipe; D is the diameter of pipe; t represents time; x represents the spatial coordinate along pipeline; g is the gravitational acceleration; and f represents the Darcy-Weisbach pipe friction factor; ; ; ,; is the reach length; and is time step.
Along the C characteristic line , the compatibility equation is:  
formula
(3)
or  
formula
(4)

where

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

Boundary condition of the end point of a stub

The end point of the stub (point C in Figures 1 and 2) can be considered as a dead end of the pipe. Thus, the flow at the end of the stub is . Consider the C+ characteristic line from point E to point F in Figure 2. To compute the head at the stub end (point C), the C+ compatibility equation is used:  
formula
(5)

Boundary condition of the swing check valve and the downstream reservoir

The downstream end valve in the main pipeline is modeled using the dimensionless equation as given by Wylie and Streeter (1993) (in Example 3-1 on page 46) as:  
formula
(6)
The C+ compatibility equation to the swing check valve at point A (see Figure 1) is given by:  
formula
(7)
The energy equation written across the valve from the upstream side at point A to the downstream reservoir (see Figure 1) is:  
formula
(8)
where is the dimensionless valve opening; is the initial valve opening; is the final valve opening; tc is the closure time of the valve; EM is the exponent that defines the valve curve; is the valve loss coefficient when the valve is fully open (the general fitting or minor head loss equation is , where K is the loss coefficient); AV is the cross sectional area of valve; and EL2 is the downstream reservoir elevation.

Boundary condition of the junction between the main pipe and 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 C+ and C compatibility equations in the main pipe are:  
formula
(9)
 
formula
(10)
The C compatibility equation in the stub towards the junction at point B is:  
formula
(11)
Other required equations are as follows, including the continuity of flow at the junction:  
formula
(12)
 
formula
(13)
where and are the flows upstream and downstream, respectively, of point B in main pipe; and is the flow in the stub at the junction.

Boundary condition of upstream constant head reservoir

The boundary condition at point D (see Figure 1) is a standard constant head condition. Thus, the head at point D is . Along the C characteristic line, the boundary condition at point D can be expressed as:  
formula
(14)
where EL1 is the upstream reservoir elevation.

System configuration

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.

Table 1

System parameters for Figure 1 

SymbolParameterValue
L (m) Total length of main pipe 100 
L1 (m) Main pipe length from measurement point to upstream reservoir 90 
L2 (m) Main pipe length from measurement point to end valve 10 
L3 (m) Length of stub Varies 
EL1 (m) Upstream reservoir elevation 10.0 
EL2 (m) Downstream reservoir elevation 8.0 
D1 = D2 = D (mm) Main pipe diameter 1,000 
D3 (m) Stub diameter Varies 
f1 = f2 Friction factor for main pipe 0.018 
f3 Friction factor for stub (for D3 = 15 mm) 0.073 
hf (m) Head loss in main pipe for frictional case 0.3 
a1=a2 = a3 (m/s) Wave speeds in main pipe and stub 1,000 
V0 (m/s) Initial velocity for frictionless system 1.98 
V0-f (m/s) Initial velocity for frictional system 1.82 
 Initial dimensionless valve position 1.0 
 Final dimensionless valve position 0.0 
EM Exponent that defines the valve curve 1.0 
K0 Valve loss coefficient when the valve is fully open 10.0 
Δx (m) Reach length in MOC calculations 0.1 
Δt (s) Time step in MOC calculations 0.00001 
tc (s) Time of closure of valve Varies 
SymbolParameterValue
L (m) Total length of main pipe 100 
L1 (m) Main pipe length from measurement point to upstream reservoir 90 
L2 (m) Main pipe length from measurement point to end valve 10 
L3 (m) Length of stub Varies 
EL1 (m) Upstream reservoir elevation 10.0 
EL2 (m) Downstream reservoir elevation 8.0 
D1 = D2 = D (mm) Main pipe diameter 1,000 
D3 (m) Stub diameter Varies 
f1 = f2 Friction factor for main pipe 0.018 
f3 Friction factor for stub (for D3 = 15 mm) 0.073 
hf (m) Head loss in main pipe for frictional case 0.3 
a1=a2 = a3 (m/s) Wave speeds in main pipe and stub 1,000 
V0 (m/s) Initial velocity for frictionless system 1.98 
V0-f (m/s) Initial velocity for frictional system 1.82 
 Initial dimensionless valve position 1.0 
 Final dimensionless valve position 0.0 
EM Exponent that defines the valve curve 1.0 
K0 Valve loss coefficient when the valve is fully open 10.0 
Δx (m) Reach length in MOC calculations 0.1 
Δt (s) Time step in MOC calculations 0.00001 
tc (s) Time of closure of valve Varies 

RESULTS AND DISCUSSION

Examination of the system behavior assuming the absence of a stub

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; HWS-Bmax represents the maximum head at point B during the period from the time of t = 0.0 s to a time of t = (tc + t0 + t1); tWS-Bmax represents the time of occurrence of HWS-Bmax. Various swing check valve closure times as shown in Column 1 of Table 2 are simulated.

Table 2

Simulation results of the transient pressure at point B in Figure 1 (without stub)

tc (s)Frictionless (V0 = 1.98 m/s)
With friction (f1 = f2 = 0.018, V0 = 1.82 m/s)
HWS-B at t = (tc + t0) (m)HWS-Bmax (m)tWS-Bmax (s)Formula for calculation of tWS-BmaxHWS-B at t = (tc + t0) (m)HWS-Bmax (m)tWS-Bmax (s)Formula for calculation of tWS-Bmax
Col. (1)Col. (2)Col. (3)Col. (4)Col. (5)Col. (6)Col. (7)Col. (8)Col. (9)
0.0 212.03 212.03 0.01 tc + t0 195.76 196.03 0.19 t0 + t1 
0.0005 212.03 212.03 0.0105 tc + t0 195.76 196.03 0.19 t0 + t1 
0.001 212.03 212.03 0.011 tc + t0 195.76 196.03 0.19 t0 + t1 
0.003 212.03 212.03 0.013 tc + t0 195.76 196.03 0.19 t0 + t1 
0.005 212.03 212.03 0.015 tc + t0 195.76 196.03 0.19 t0 + t1 
0.01 212.03 212.03 0.02 tc + t0 195.76 196.02 0.19 t0 + t1 
0.05 212.03 212.03 0.06 tc + t0 195.77 195.97 0.19 t0 + t1 
0.1 212.03 212.03 0.11 tc + t0 195.78 195.91 0.19 t0 + t1 
0.2 211.57 211.57 0.21 tc + t0 195.42 195.42 0.21 tc + t0 
0.5 189.04 189.04 0.51 tc + t0 176.26 176.26 0.51 tc + t0 
1.0 129.76 129.76 1.01 tc + t0 124.17 124.17 1.01 tc + t0 
tc (s)Frictionless (V0 = 1.98 m/s)
With friction (f1 = f2 = 0.018, V0 = 1.82 m/s)
HWS-B at t = (tc + t0) (m)HWS-Bmax (m)tWS-Bmax (s)Formula for calculation of tWS-BmaxHWS-B at t = (tc + t0) (m)HWS-Bmax (m)tWS-Bmax (s)Formula for calculation of tWS-Bmax
Col. (1)Col. (2)Col. (3)Col. (4)Col. (5)Col. (6)Col. (7)Col. (8)Col. (9)
0.0 212.03 212.03 0.01 tc + t0 195.76 196.03 0.19 t0 + t1 
0.0005 212.03 212.03 0.0105 tc + t0 195.76 196.03 0.19 t0 + t1 
0.001 212.03 212.03 0.011 tc + t0 195.76 196.03 0.19 t0 + t1 
0.003 212.03 212.03 0.013 tc + t0 195.76 196.03 0.19 t0 + t1 
0.005 212.03 212.03 0.015 tc + t0 195.76 196.03 0.19 t0 + t1 
0.01 212.03 212.03 0.02 tc + t0 195.76 196.02 0.19 t0 + t1 
0.05 212.03 212.03 0.06 tc + t0 195.77 195.97 0.19 t0 + t1 
0.1 212.03 212.03 0.11 tc + t0 195.78 195.91 0.19 t0 + t1 
0.2 211.57 211.57 0.21 tc + t0 195.42 195.42 0.21 tc + t0 
0.5 189.04 189.04 0.51 tc + t0 176.26 176.26 0.51 tc + t0 
1.0 129.76 129.76 1.01 tc + t0 124.17 124.17 1.01 tc + t0 

For the frictionless case, the head at point B reaches its maximum value at the time of t = tc + t0. If tc 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 HWS-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 tc. However, if tc 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, HWS-Bmax will be less than that for the case of tc being larger than t1. In addition, the maximum head decreases with an increase of tc.

For the frictional case, the value of HWS-Bmax decreases with an increase of the valve closure time tc for all cases. If tc is less than t1, the maximum head HWS-Bmax will occur at the time of t = t0 + t1 rather than t = tc + t0 because of the effect of line pack. However, if tc is larger than t1, the maximum head will occur at time of t = tc + 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 Ko = 10.0 as shown in Table 1.

Influence of the ratio of the stub diameter to the main pipe diameter (D3/D) on the transient in the main pipe

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 = (tc + 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 = tc + 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 = tc + t0 + t3..

Table 3

Influence of D3/D on transient head of point B (f = 0.0, tc = 0.0)

L3 (m)D3 (mm)D3/DWithout connection stub (m)
With connection stub (m)
ErrB-B(WS)(%)
At t = t4At t = t5At t = t4At t = t5At t = t4At t = t5
Col. (1)Col. (2)Col. (3)Col. (4)Col. (5)Col. (6)Col. (7)Col. (8)Col. (9)
0.1 15 0.015 212.03 212.03 212.01 212.05 −0.01 0.01 
20 0.02 212.03 212.03 211.99 212.07 −0.02 0.02 
50 0.05 212.03 212.03 211.78 212.28 −0.12 0.12 
100 0.1 212.03 212.03 211.03 213.03 −0.47 0.47 
200 0.2 212.03 212.03 208.07 215.84 −1.87 1.80 
500 0.5 212.03 212.03 189.58 229.49 −10.59 8.23 
1,000 212.03 212.03 144.69 234.48 −31.76 10.59 
1.0 15 0.015 212.03 212.03 212.01 212.05 −0.01 0.01 
20 0.02 212.03 212.03 211.99 212.07 −0.02 0.02 
50 0.05 212.03 212.03 211.78 212.28 −0.12 0.12 
100 0.1 212.03 212.03 211.03 213.03 −0.47 0.47 
200 0.2 212.03 212.03 208.07 215.84 −1.87 1.80 
500 0.5 212.03 212.03 189.58 229.49 −10.59 8.23 
1,000 212.03 212.03 144.69 234.48 −31.76 10.59 
L3 (m)D3 (mm)D3/DWithout connection stub (m)
With connection stub (m)
ErrB-B(WS)(%)
At t = t4At t = t5At t = t4At t = t5At t = t4At t = t5
Col. (1)Col. (2)Col. (3)Col. (4)Col. (5)Col. (6)Col. (7)Col. (8)Col. (9)
0.1 15 0.015 212.03 212.03 212.01 212.05 −0.01 0.01 
20 0.02 212.03 212.03 211.99 212.07 −0.02 0.02 
50 0.05 212.03 212.03 211.78 212.28 −0.12 0.12 
100 0.1 212.03 212.03 211.03 213.03 −0.47 0.47 
200 0.2 212.03 212.03 208.07 215.84 −1.87 1.80 
500 0.5 212.03 212.03 189.58 229.49 −10.59 8.23 
1,000 212.03 212.03 144.69 234.48 −31.76 10.59 
1.0 15 0.015 212.03 212.03 212.01 212.05 −0.01 0.01 
20 0.02 212.03 212.03 211.99 212.07 −0.02 0.02 
50 0.05 212.03 212.03 211.78 212.28 −0.12 0.12 
100 0.1 212.03 212.03 211.03 213.03 −0.47 0.47 
200 0.2 212.03 212.03 208.07 215.84 −1.87 1.80 
500 0.5 212.03 212.03 189.58 229.49 −10.59 8.23 
1,000 212.03 212.03 144.69 234.48 −31.76 10.59 

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, ErrB-B(ws), representing the percentage difference between and , is defined as follows: .

In Table 3, at a time of t = t4, 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 = t4. 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.

Figure 3

Influence of the value D3/D on the head at point B relative to a system without a connection stub (f = 0, tc = 0).

Figure 3

Influence of the value D3/D on the head at point B relative to a system without a connection stub (f = 0, tc = 0).

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.

Influence of connection stub length on transient pressure measurement accuracy

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.

Case of a valve closure time less than 2 L3 /a3 seconds

Consider the case where a stub diameter, D3, is 15 mm, the closure time of the end valve, tc, is 0.0 s, and the shortest stub length, L3, is 0.02 m. Thus, tc 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 HCmax 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 = (tc + t0 + t3) for this system; HWS-Bmax represents the maximum head at point B under the condition of no stub; tCmax represents the time of occurrence of HCmax; ErrC-B(ws) represents the percentage difference between and , and is defined as follows: .

Table 4

Simulation results for the case of a stub diameter D3 = 15 mm with various stub lengths (L3) and an instantaneous valve closure of tc = 0.0 s

fWithout connection stub HWS-Bmax (m)With connection stub
ErrC-B(ws) (%)
L3 (m)HCmax (m)tCmax (s)Formula for tCmax
Col. (1)Col. (2)Col. (3)Col. (4)Col. (5)Col. (6)Col. (7)
Frictionless
f1 = f2 = f3 = 0 
212.03 0.02 414.02 0.01002 tc + L2/a2 + L3/a3 95.26 
0.05 414.02 0.01005 tc + L2/a2 + L3/a3 95.26 
0.10 414.02 0.01010 tc + L2/a2 + L3/a3 95.26 
0.20 414.02 0.01020 tc + L2/a2 + L3/a3 95.26 
0.30 414.02 0.01030 tc + L2/a2 + L3/a3 95.26 
0.40 414.02 0.01040 tc + L2/a2 + L3/a3 95.26 
0.50 414.02 0.01050 tc + L2/a2 + L3/a3 95.26 
0.60 414.02 0.01060 tc + L2/a2 + L3/a3 95.26 
0.70 414.02 0.01070 tc + L2/a2 + L3/a3 95.26 
0.80 414.02 0.01080 tc + L2/a2 + L3/a3 95.26 
0.90 414.02 0.01190 tc + L2/a2 + L3/a3 95.26 
1.00 414.02 0.01100 tc + L2/a2 + L3/a3 95.26 
With friction
f1 = 0.018
f2 = 0.018
f3 = 0.073 
196.03 0.02 381.75 0.01006 tc + L2/a2 + 3L3/a3 94.74 
0.05 381.75 0.01015 tc + L2/a2 + 3L3/a3 94.74 
0.10 381.75 0.01030 tc + L2/a2 + 3L3/a3 94.74 
0.20 381.67 0.01060 tc + L2/a2 + 3L3/a3 94.70 
0.30 381.60 0.01090 tc + L2/a2 + 3L3/a3 94.66 
0.40 381.50 0.01120 tc + L2/a2 + 3L3/a3 94.61 
0.50 381.42 0.01150 tc + L2/a2 + 3L3/a3 94.57 
0.60 381.34 0.01180 tc + L2/a2 + 3L3/a3 94.53 
0.70 381.26 0.01210 tc + L2/a2 + 3L3/a3 94.49 
0.80 381.18 0.01240 tc + L2/a2 + 3L3/a3 94.45 
0.90 381.10 0.01270 tc + L2/a2 + 3L3/a3 94.41 
1.00 381.02 0.01300 tc + L2/a2 + 3L3/a3 94.37 
fWithout connection stub HWS-Bmax (m)With connection stub
ErrC-B(ws) (%)
L3 (m)HCmax (m)tCmax (s)Formula for tCmax
Col. (1)Col. (2)Col. (3)Col. (4)Col. (5)Col. (6)Col. (7)
Frictionless
f1 = f2 = f3 = 0 
212.03 0.02 414.02 0.01002 tc + L2/a2 + L3/a3 95.26 
0.05 414.02 0.01005 tc + L2/a2 + L3/a3 95.26 
0.10 414.02 0.01010 tc + L2/a2 + L3/a3 95.26 
0.20 414.02 0.01020 tc + L2/a2 + L3/a3 95.26 
0.30 414.02 0.01030 tc + L2/a2 + L3/a3 95.26 
0.40 414.02 0.01040 tc + L2/a2 + L3/a3 95.26 
0.50 414.02 0.01050 tc + L2/a2 + L3/a3 95.26 
0.60 414.02 0.01060 tc + L2/a2 + L3/a3 95.26 
0.70 414.02 0.01070 tc + L2/a2 + L3/a3 95.26 
0.80 414.02 0.01080 tc + L2/a2 + L3/a3 95.26 
0.90 414.02 0.01190 tc + L2/a2 + L3/a3 95.26 
1.00 414.02 0.01100 tc + L2/a2 + L3/a3 95.26 
With friction
f1 = 0.018
f2 = 0.018
f3 = 0.073 
196.03 0.02 381.75 0.01006 tc + L2/a2 + 3L3/a3 94.74 
0.05 381.75 0.01015 tc + L2/a2 + 3L3/a3 94.74 
0.10 381.75 0.01030 tc + L2/a2 + 3L3/a3 94.74 
0.20 381.67 0.01060 tc + L2/a2 + 3L3/a3 94.70 
0.30 381.60 0.01090 tc + L2/a2 + 3L3/a3 94.66 
0.40 381.50 0.01120 tc + L2/a2 + 3L3/a3 94.61 
0.50 381.42 0.01150 tc + L2/a2 + 3L3/a3 94.57 
0.60 381.34 0.01180 tc + L2/a2 + 3L3/a3 94.53 
0.70 381.26 0.01210 tc + L2/a2 + 3L3/a3 94.49 
0.80 381.18 0.01240 tc + L2/a2 + 3L3/a3 94.45 
0.90 381.10 0.01270 tc + L2/a2 + 3L3/a3 94.41 
1.00 381.02 0.01300 tc + L2/a2 + 3L3/a3 94.37 

For the frictionless case, HCmax and ErrC-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, tCmax, is given in Column 5 in Table 4.

For frictional case, HCmax and ErrC-B(ws) are the same for the cases of stubs lengths of 0.02 m, 0.05 m and 0.1 m. While both HCmax and ErrC-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, HCmax 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, HWS-Bmax, is 196.03 m. Thus, the percentage differences of head of ErrC-B(ws) are 94.74%, 94.74% and 94.37%, respectively, for these three lengths. The time of tCmax is given in Column 5 in Table 4.

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

Case of a valve closure time larger than 2 L3/a3 s

Consider the case in which the stub diameter, D3, is 15 mm, the closure time of the end valve, tc, 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, tc 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 HCmax 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.

Table 5

Simulation results for the case of a stub diameter of D3 = 15 mm for various stub lengths (L3) and a valve closure time of tc = 0.003 s

fWithout stub HWS-Bmax (m)With stub
ErrC-B(ws) (%)
L3 (m)HCmax (m)tCmax (s)Formula for tCmax
Col. (1)Col. (2)Col. (3)Col. (4)Col. (5)Col. (6)Col. (7)
Frictionless
f1 = f2 = f3 = 0 
212.03 0.02 225.64 0.01302 tc + L2/a2 + L3/a3 6.42 
0.05 245.73 0.01305 tc + L2/a2 + L3/a3 15.89 
0.10 278.14 0.01310 tc + L2/a2 + L3/a3 31.18 
0.20 330.01 0.01320 tc + L2/a2 + L3/a3 55.64 
0.30 362.21 0.01330 tc + L2/a2 + L3/a3 70.83 
0.40 379.93 0.01340 tc + L2/a2 + L3/a3 79.19 
0.50 391.51 0.01350 tc + L2/a2 + L3/a3 84.65 
0.60 397.47 0.01360 tc + L2/a2 + L3/a3 87.46 
0.70 401.41 0.01370 tc + L2/a2 + L3/a3 89.32 
0.80 404.60 0.01380 tc + L2/a2 + L3/a3 90.82 
0.90 407.28 0.01390 tc + L2/a2 + L3/a3 92.09 
1.00 409.23 0.01400 tc + L2/a2 + L3/a3 93.01 
With friction
f1 = 0.018
f2 = 0.018
f3 = 0.073 
196.03 0.02 208.81 0.01302 tc + L2/a2 + L3/a3 6.52 
0.05 228.07 0.01305 tc + L2/a2 + L3/a3 16.34 
0.10 258.97 0.01310 tc + L2/a2 + L3/a3 32.11 
0.20 307.44 0.01320 tc + L2/a2 + L3/a3 56.83 
0.30 336.52 0.01330 tc + L2/a2 + L3/a3 71.67 
0.40 352.16 0.01340 tc + L2/a2 + L3/a3 79.65 
0.50 362.18 0.01350 tc + L2/a2 + L3/a3 84.76 
0.60 367.28 0.01360 tc + L2/a2 + L3/a3 87.36 
0.70 370.60 0.01370 tc + L2/a2 + L3/a3 89.05 
0.80 373.26 0.01380 tc + L2/a2 + L3/a3 90.41 
0.90 375.45 0.01390 tc + L2/a2 + L3/a3 91.53 
1.00 377.04 0.01400 tc + L2/a2 + L3/a3 92.34 
fWithout stub HWS-Bmax (m)With stub
ErrC-B(ws) (%)
L3 (m)HCmax (m)tCmax (s)Formula for tCmax
Col. (1)Col. (2)Col. (3)Col. (4)Col. (5)Col. (6)Col. (7)
Frictionless
f1 = f2 = f3 = 0 
212.03 0.02 225.64 0.01302 tc + L2/a2 + L3/a3 6.42 
0.05 245.73 0.01305 tc + L2/a2 + L3/a3 15.89 
0.10 278.14 0.01310 tc + L2/a2 + L3/a3 31.18 
0.20 330.01 0.01320 tc + L2/a2 + L3/a3 55.64 
0.30 362.21 0.01330 tc + L2/a2 + L3/a3 70.83 
0.40 379.93 0.01340 tc + L2/a2 + L3/a3 79.19 
0.50 391.51 0.01350 tc + L2/a2 + L3/a3 84.65 
0.60 397.47 0.01360 tc + L2/a2 + L3/a3 87.46 
0.70 401.41 0.01370 tc + L2/a2 + L3/a3 89.32 
0.80 404.60 0.01380 tc + L2/a2 + L3/a3 90.82 
0.90 407.28 0.01390 tc + L2/a2 + L3/a3 92.09 
1.00 409.23 0.01400 tc + L2/a2 + L3/a3 93.01 
With friction
f1 = 0.018
f2 = 0.018
f3 = 0.073 
196.03 0.02 208.81 0.01302 tc + L2/a2 + L3/a3 6.52 
0.05 228.07 0.01305 tc + L2/a2 + L3/a3 16.34 
0.10 258.97 0.01310 tc + L2/a2 + L3/a3 32.11 
0.20 307.44 0.01320 tc + L2/a2 + L3/a3 56.83 
0.30 336.52 0.01330 tc + L2/a2 + L3/a3 71.67 
0.40 352.16 0.01340 tc + L2/a2 + L3/a3 79.65 
0.50 362.18 0.01350 tc + L2/a2 + L3/a3 84.76 
0.60 367.28 0.01360 tc + L2/a2 + L3/a3 87.36 
0.70 370.60 0.01370 tc + L2/a2 + L3/a3 89.05 
0.80 373.26 0.01380 tc + L2/a2 + L3/a3 90.41 
0.90 375.45 0.01390 tc + L2/a2 + L3/a3 91.53 
1.00 377.04 0.01400 tc + L2/a2 + L3/a3 92.34 
Figure 4

Variation of relative difference between maximum heads at point C and point B with L3 (tc = 0.003 s, D3 = 15 mm).

Figure 4

Variation of relative difference between maximum heads at point C and point B with L3 (tc = 0.003 s, D3 = 15 mm).

For all the cases in Table 5, the valve closure time tc 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 HCmax will increase with an increase of stub length L3, as depicted in Table 5, and the percentage difference ErrC-B(ws) will also increase with an increase of stub length L3, 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 HCmax and the percentage difference of ErrC-B(ws) both decrease with a decrease of stub length L3.

For the frictionless case in Table 5 and Figure 4, HCmax 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, tCmax, is given in Column 5 in Table 5.

For the frictional case in Table 5 and Figure 4, HCmax 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 tCmax 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 tc is usually larger than the stub return time of 2L3/a3. 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.

Influence of the valve closure time on the difference of the head between point B and point C

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 tc are shown in Table 6 and Figure 5.

Table 6

Simulation results for the cases with different tc (D3 = 15 mm, friction included)

L3 (m)tc (s)2L3/a3 (s)tc/(2L3/a3)HWS-Bmax (m)HCmax (m)ErrC-B(ws) (%)
Col. (1)Col. (2)Col. (3)Col. (4)Col. (5)Col. (6)Col. (7)
0.02 0.0 0.00004 196.03 381.75 94.74 
0.0001 2.5 196.03 367.59 87.52 
0.0002 196.03 336.60 71.71 
0.001 25 196.03 234.53 19.64 
0.002 50 196.03 215.24 9.80 
0.01 250 196.03 199.67 1.86 
0.02 500 196.01 197.72 0.87 
0.04 1,000 195.98 196.75 0.39 
0.1 2,500 195.91 196.17 0.13 
0.2 5,000 195.42 195.62 0.10 
0.05 0.0 0.0001 196.03 381.75 94.74 
0.00025 2.5 196.03 367.59 87.52 
0.0005 196.03 336.60 71.71 
0.0025 25 196.03 234.53 19.64 
0.005 50 196.03 215.24 9.80 
0.025 250 196.00 199.67 1.87 
0.05 500 195.98 197.72 0.89 
0.1 1,000 195.91 196.76 0.43 
0.25 2,500 193.38 193.77 0.20 
0.5 5,000 176.26 176.44 0.10 
0.1 0.0 0.0002 196.03 381.75 94.74 
0.0005 2.5 196.03 367.59 87.52 
0.001 196.03 336.60 71.71 
0.005 25 196.03 234.53 19.64 
0.01 50 196.03 215.25 9.80 
0.05 250 195.98 199.68 1.89 
0.1 500 195.91 197.75 0.94 
0.2 1,000 195.42 196.40 0.50 
0.5 2,500 176.26 176.61 0.20 
5,000 124.17 124.28 0.09 
L3 (m)tc (s)2L3/a3 (s)tc/(2L3/a3)HWS-Bmax (m)HCmax (m)ErrC-B(ws) (%)
Col. (1)Col. (2)Col. (3)Col. (4)Col. (5)Col. (6)Col. (7)
0.02 0.0 0.00004 196.03 381.75 94.74 
0.0001 2.5 196.03 367.59 87.52 
0.0002 196.03 336.60 71.71 
0.001 25 196.03 234.53 19.64 
0.002 50 196.03 215.24 9.80 
0.01 250 196.03 199.67 1.86 
0.02 500 196.01 197.72 0.87 
0.04 1,000 195.98 196.75 0.39 
0.1 2,500 195.91 196.17 0.13 
0.2 5,000 195.42 195.62 0.10 
0.05 0.0 0.0001 196.03 381.75 94.74 
0.00025 2.5 196.03 367.59 87.52 
0.0005 196.03 336.60 71.71 
0.0025 25 196.03 234.53 19.64 
0.005 50 196.03 215.24 9.80 
0.025 250 196.00 199.67 1.87 
0.05 500 195.98 197.72 0.89 
0.1 1,000 195.91 196.76 0.43 
0.25 2,500 193.38 193.77 0.20 
0.5 5,000 176.26 176.44 0.10 
0.1 0.0 0.0002 196.03 381.75 94.74 
0.0005 2.5 196.03 367.59 87.52 
0.001 196.03 336.60 71.71 
0.005 25 196.03 234.53 19.64 
0.01 50 196.03 215.25 9.80 
0.05 250 195.98 199.68 1.89 
0.1 500 195.91 197.75 0.94 
0.2 1,000 195.42 196.40 0.50 
0.5 2,500 176.26 176.61 0.20 
5,000 124.17 124.28 0.09 
Figure 5

Variation of relative difference between maximum heads at point C and point B with tc /(2L3/a3) (D3 = 15 mm).

Figure 5

Variation of relative difference between maximum heads at point C and point B with tc /(2L3/a3) (D3 = 15 mm).

For the same reasons as described above, the larger the valve closure time tc, 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 HCmax and ErrC-B(ws) decrease with an increase of the valve closure time tc. 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 tc = 1.0 s (which is 5,000 times the stub water hammer wave return time of 2L3/a3 = 0.0002 s), then HCmax is 124.28 m and very close to HWS-Bmax value of 124.17 m or an overestimate of 0.111 m or a ErrC-B(ws) value of 0.09% which exhibits satisfactory measurement accuracy. It also can be found that under the same value of tc/(2L3/a3), the values of ErrC-B(ws) are almost the same for different stub lengths; while under the same value of tc, the value of ErrC-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.

CONCLUSIONS

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.

ACKNOWLEDGEMENTS

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

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