Analytical study on water hammer pressure in pressurized conduits with a throttled surge chamber for slow closure

Yong-liang ZHANG*, Ming-fei MIAO, Ji-ming MA

State Key Laboratory of Hydroscience and Hydraulic Engineering, Department of Hydraulic Engineering,Tsinghua University, Beijing 100084, P. R. China

1 Introduction

Since the provision of a surge chamber (also referred to as a surge shaft or surge tank)in a pressurized pipe system can transform rapid flow change generated by closing or opening a valve/turbine into mass oscillation in the chamber, and lead to the reduction of water hammer pressure, the hydraulic characteristics of this arrangement have been extensively studied experimentally and theoretically (Jaeger 1977; Chaudhry 1987; Zhang and Liu 1992). These studies cover various types of surge chambers, including simple, throttled (orifice), differential,one-way, and air cushion (closed)surge chambers. Much effort so far has been devoted to the behavior of water hammer generated at the valve or turbine using analytical methods for pipe systems with surge chambers. A comprehensive review of analytical studies on pressure transmission in pipe systems has been conducted (Almeida and Koelle 1992).

Earlier analytical studies (Allievi 1913)used transient flow theory to derive formulas for water hammer in a simple pipe of constant diameter and wall thickness with a very large reservoir located upstream and a valve positioned downstream. These studies have been extended to piping systems with throttled surge chambers or other types of surge chambers,such as differential or air cushion chambers, and transmission pressure through the chamber in the case of rapid or instantaneous valve closure has been investigated analytically (Jaeger 1933; Zienkiewicz and Hawkins 1954; Shima and Hino 1960; Mosonyi and Seth 1975; Wang and Ma 1986; Zhang and Liu 1992; Ma 1996). These analytical formulas provide a good design/analysis tool for engineers and researchers, particularly in the hydropower industry. All of these studies are based on the assumption that

where Tsis the valve closure time; L1and L2are the length between the water surface and the tunnel-penstock-chamber junction and the penstock length, respectively; and a1and a2are the wave speeds in the chamber and the penstock, respectively. However, the valve/turbine actually does not close instantaneously, and its normal closing time TS(≈10 s)is at least ten times longer than 2L1a1(＜ 1 s). Therefore, the aforementioned assumption is not valid and the analytical formula is not applicable to actual hydroelectric pipe systems. This is evident from graphical (Escande 1949; Zienkiewicz and Hawkins 1954; Shima and Hino 1960),numerical (Peng and Yang 1986; Prenner and Drobir 1997)and experimental (Shima and Hino 1960; Bernhart 1975; Peng and Yang 1986; Wang and Yang 1989; Prenner and Drobir 1997)investigation of transmission pressure in many hydropower plants with slow valve/turbine closure. To the authors’ knowledge, no analytical formula has been reported for transmission pressures for the case of slow closure of the turbine/valve.

This paper presents analytical formulas of maximum water hammer pressures at the downstream end of the tunnel and the valve, for a hydraulic pressurized pipe system with a throttled surge chamber subjected to linear and slow valve closure. In the system, a throttled surge chamber is located at the junction between a tunnel and a penstock, and a valve is positioned at the downstream end of the penstock. The analytical results are then compared with numerical ones obtained using the method of characteristics to demonstrate the validity of the formulas.

2 Mathematical model

2.1 Water hammer equations

Consider a hydraulic system consisting of a diversion tunnel, a surge chamber, and a penstock, as shown in Fig. 1. The fluid is described by the piezometric head H (x, t)and cross-sectional average velocity V (x, t), where x is the spatial coordinate along the pipeline and t is the temporal coordinate. In this study, the friction loss was assumed to be small and was therefore neglected. The equations for water hammer are

where a is the wave speed and g is the gravitational acceleration. The general integrals of the two simultaneous partial differential equations above are

where φ is a function that can be interpreted as a wave moving in the +x direction, and ψ is a function that can be interpreted as a wave moving in the -x direction; H0is the initial piezometric head; and V0is the initial cross-sectional average velocity in the tunnel. The exact forms of functions φ and ψ depend on particular boundary conditions.

Fig. 1 Sketch of pressurized pipe system with throttled surge chamber

2.2 Tunnel-surge chamber-penstock joint equation

The general assumptions made in the course of the analysis of the junction of the tunnel,the chamber, and the penstock (Fig. 2)are as follows:

(1)the continuity equation is valid for the system;

(2)at any instant there are identical pressure heads at points B0, B1, B2, and B3,which denote the entrance to the surge chamber just above the orifice, the entrance to the surge chamber just below the orifice, the entrance to the penstock, and the entrance to the tunnel, respectively;

(3)the velocity is uniformly distributed across each conduit at the junction;

(4)incident and reflected pressure waves from the junction are plane-fronted;

(5)at any instant the difference of pressure on each side of the orifice is equal to the water head loss corresponding to the flow under steady conditions;

(6)the inertia force of the water column and the friction in the surge chamber are small and can be neglected;

Fig. 2 Tunnel-surge tank-penstock joint

(7)the upstream tunnel is long enough that, upon the full closure of the valve, the reflected wave ψ3does not arrive at the junction.

The conditions at the junction can be expressed as

For steady flow, they can be expressed as

where H0is the initial pressure head at the entrance to the surge chamber just above the orifice; Q0is the initial flow rate in the tunnel; H0(t)is the pressure head at the entrance to the surge chamber just above the orifice at time t; H1(t)and Q1( t)are the pressure head and the flow rate at the entrance to the surge chamber just below the orifice at time t, respectively;H2(t)and Q2(t)are the pressure head and the flow rate at the entrance to the penstock,respectively; H3(t)and Q3(t)are the pressure head and the flow rate at the downstream end of the tunnel, respectively; z is the increment of water level in the surge chamber; Asis the cross-sectional area of the surge chamber; and k is the coefficient defined as follows by Zienkiewicz and Hawkins (1954):

where C is the contraction coefficient of the orifice and f0is the area of the orifice.

In this study we considered linear valve closure and introduced a set of dimensionless quantities of relative water hammers at the valve and the entrance of the penstock, defined as follows:

where f3is the cross-sectional area of the tunnel. It is indicated in Eq. (11)that mass oscillation Z*will exceed HA(0)ξ*and H2(0 )for a certain range of L3, so that ξ*andmight not be the largest in this condition. The second peak of the water hammer wave in Fig. 4 is larger than the first peak, and the value of H2(t)at timeis larger than that at time Ts+ (n - 1)θ2. This is due to the fact that the increment of the water level in the surge chamber exceeds the decrement of transmission pressure. Fig. 5 shows the variation of the increment of the water level subtracted from the pressure head at the entrance of the penstock with time. It can be seen from Fig. 5 that the value H2(t)- H2(0)- z increases significantly with time, reaches its maximum at time Ts+θ2, and then oscillates. The peak value of H2(t)- H2(0)- z decreases slowly after time Ts+θ2. This shows that the maximum transmission water hammer is reached at time Ts+θ2.

Fig. 3 Pressure head at valve

Fig. 4 Pressure head at bottom of surge chamber

Fig. 5 Transmitted pressure minus increment of water level in chamber

The period of mass oscillation in the surge chamber (100 s to 500 s)is much larger than the time taken for valve closure Ts(10 s), so it is in the time interval0≤ t ≤ (Ts+ θ2), with Q1( t)≥ 0, as shown in Fig. 6. Therefore, at t=Ts+θ 2 the absolute value sign in Eq. (5)can be removed.

Fig. 6 Flow rate into surge chamber

Provided that the diversion tunnel is long enough, the reflected pressure wave ψ3has not yet arrived at the junction when the valve has just fully closed. This yields

Substituting Eq. (12)into Eqs. (3)and (4), the following equations are obtained:

where u3=gf3a3; a3is the wave speed in the tunnel.

Combining Eqs. (13)and (14)yields

It is indicated from Eq. (15)that Q3(t)decreases with increasing H3(t), and that there is a linear relation between Q3(t)and H3(t).

Transforming Eqs. (1)and (2)into four ordinary differential ones using the method of characteristics, C+equations (the positive direction of the x-axis is A pointed to B)are obtained:

Integrating Eq. (16)along a C+line from A to P (Fig. 7)yields

where u2=gf2a2.

Since the penstock is usually short (＜ 700 m),the water head loss in the penstock, Hwm(0), is small compared with the pressure head, and can be neglected. In this study, HA(0)was approximately equal to H0, which was expressed as

Fig. 7 Characteristic line of penstock

Combining Eqs. (6), (15), and (18)yields

The front of pressure wave φ2arrives at B at t=θ2. Meanwhile,Q1( t)and z begin to increase from 0. According to the continuity equation for the surge chamber,

Combining Eqs. (19)and (21)yields

Substituting Eqs. (19)and (22)into Eq. (5), the following equation is obtained:

2.3 Interlocking equations of water hammer in penstock

2.3.1 Improved interlocking equations for conduit with surge chamber

In previous studies, the pressure head H2(t)at the upstream boundary was assumed to be constant in the course of deriving interlocking equations of water hammer. This is reasonable for a simple surge chamber located upstream with an infinitely large area. However,such interlocking equations are not applicable to hydraulic pipe problems with a throttled surge chamber, as H2(t)varies significantly with time upon closure of the valve. Modified interlocking equations should be derived for pressurized conduits with a throttled surge chamber located upstream.

Eq. (3)at the upstream end of the penstock (point B2)can be written as

Substituting t=ti-θ 2 into Eq. (24), the following equation is obtained:

which can be written in the general form

or in the form

where VAand HAare the flow velocity and pressure head at the downstream end of the penstock, respectively.

Substituting Eq. (26)into Eqs (27)and (28)yields

After manipulations, the following equations can be obtained:

When pressure head at the entrance to the surge chamber just below the orifice is constant,i.e., hp( t)≡0, Eq. (33)becomes identical to the interlocking equations derived by Allievi(1913). It can be seen that Allievi’s interlocking equations are a special case of Eq. (33).

2.3.2 Interlocking equations in condition of final water hammer

For the system with a valve located at the downstream end of the penstock, the Bernoulli equation is assumed to be valid (Wylie and Streeter 1993):

wherev( t)= VA(t)VA(0), and τ( t)is the relative valve opening.

Substituting this into Eq. (34)yields the following equation:

The hydraulic system mentioned before (Figs. 4 and 5), in which θ = 1.4 s and Ts= 13.0 s, is considered, and some distinguishing features of the functions hpand ξ, which will be used for deriving equations in the next section, are found:

(1)The curve ofξ( t)turns at time 0, θ, Ts, and Ts+nθ (n = 1, 2, 3…), and the curve of hp(t)turns at timeTs+ (n + 1 2)θ, (n = 1, 2, 3…).

(2)Provided that closure of the valve is continuous and differentiable with respect to time,ξ( t)and hp(t)are continuous and differentiable with respect to time in any time intervals between any two successive points mentioned above.

Subtraction of the two equations above yieds the following equation:

For linear valve closure,

Magnitude analysis of the right-side term in Eq. (37)yields the following equations:

In hydro-electric power plants, θ Ts≈110. It can be seen that R2is two orders of magnitude smaller than R1, and can be neglected. Therefore, Eq. (37)can be rewritten as

Using the Taylor expansions of ξnand ξn-1at time (n-1 2)θ to the first order and then adding them to each other leads to the following equation:

Eq. (43)is valid when nθ≈Ts. It can be viewed as the fitting curve of ξ and hpin the adjacent time interval of Ts. Eq. (43)is approximately valid at Ts, yielding

Eq. (44)indicates the relation between hp(Ts)and ξ*. Because hp(t)is a smooth function in a time interval 0～ (Ts+θ2), Taylor expansion of hp(Ts)at time Ts+θ2 to a first order term yields

It can be clearly seen that T2is one order of magnitude smaller than T1, and can be neglected.In θ 2～(Ts+θ 2), as mentioned before, the flow rate into the surge chamber can be approximated by a linear function with time, viz.Q1( t)∝ (t -θ 2), so we can obtain

The derivative of parabola y=x2at x=x0is 2x0. The slope of the secant line of y=x2at x=x0is x0. The derivative of hp(t)is twice as large as the secant slope of hp(t). Magnitude analysis of hp(t)yields the following equations:

We set ?=θ Ts, which is about 1/10 in practical hydraulic engineering projects. Combining Eqs. (44)and (50)yields the following equation:

2.4 Analytical formulas of and ξ*

The two variables ξ*andcan be solved from Eqs. (23)and (51). Eq. (51)can be rewritten as

Since ξ*is much smaller than 1, the second order term can be ignored. Thus, Eq. (53)can be rewritten as

Such an approximation was used by Alleivi (1913):

It is shown from Eq. (56)that there is an approximately linear proportion betweenand k,provided that Q0remains stable. Combining Eqs. (51)and (54)yields a linear function of ξ*and

Dimensionless terms are defined as follows:

Substituting them into Eq. (57), the following equation in dimensionless form is obtained:

The reasonable solution of Eq. (58)is

Eqs. (59)and (60)are the analytical formulas ofand ξ*, which are derived from the classic water hammer equations using the approximate method dealing with the term

3 Results and discussion

The conditions for using Eqs. (59)and (60)are (1)that the upstream tunnel is long enough that at time Ts+θ2 the front of ψ3has not yet arrived at the junction, i.e., L3≥a3Ts;(2)that the type of water hammer is final water hammer, i.e., τ0μ＞1, and Tsis much larger than θ (the order of magnitude of Tsis close to that of 10θ)to ensure that indirect water hammer occurs; and (3)that the opening of the valve/turbine decreases linearly. In available experiments, the fast valve and needle valve are adopted, neither of which closes linearly. Also,in many experiments, Ts＜θ, and direct water hammer is generated.

Zienkiewicz and Hawkins (1954)used the Schnyder-Bergeron graphical method to calculate transmission pressure, achieving good agreement between theoretical and experimental results. The graphical method has been replaced by numerical methods that have better accuracy and efficiency, so the results have also verified the numerical methods. Peng and Yang (1986)computed the transient pressure and found that the numerical result was in good agreement with the experiment. This verified that the basic assumptions are reasonable.Prenner and Drobir (1997)conducted an experiment using four different throttle-type orifices to study the pressure wave transmission through the surge chamber, and also made a numerical calculation using the method of characteristics (MOC), which showed good agreement with the experiment. All of these studies show that the MOC leads to good agreement with the experimental data. In this study, results obtained using the described analytical formula were compared with the numerical results obtained using the MOC to examine the validity of the approximate equation.

Numerical solution of one-dimensional fluid transient flow in pipe systems has been developed for half a century. The MOC, which has desirable accuracy, simplicity, and numerical efficiency, is very popular. The characteristics method of one-dimensional fluid transient flow and the boundary treatment technique can be found in some standard reference books (Wylie and Streeter. 1993; Chaudhry 1987). The time step in this study was 10-2s. The results tend to be convergent as the number of grid cells increases. A relative numerical error of less than 10-6was adopted. The downstream end condition was treated as a valve. The formula of the head loss coefficient used here was Eq. (9). In this study, the contraction coefficient C = 0.700. We set w=f0f3. Other parameters were held constant;and ξ*increased with decreasing w. The validity of the present analytical formulas for a hydraulic piping system with a surge chamber was examined across a range from 0.1 to 1.0, which covers the scope of practical situations. The physical and geometric parameters of the system were as follows: the area of the surge chamber was 450.0 m2, and the area of the orifice was in the range of 11.309 7 m2to 113.097 3 m2; the turbine was simplified as a valve and its closure time was 10 s; the water levels of the upstream reservoir and downstream river were 1 658.0 m and 1 314.6 m, respectively; and other parameters associated with a penstock and a tunnel are given in Table 1.

Table 1 Geometric parameters of tunnel and penstock

Table 2 Analytical and numerical results of and

Table 2 Analytical and numerical results of and

1/10 109.95 110.43 0.44 137.61 140.05 1.74 2/10 43.33 43.48 0.33 75.17 76.68 1.98 3/10 24.15 24.22 0.30 57.14 58.52 2.36 4/10 16.64 16.68 0.27 50.07 51.41 2.61 5/10 13.03 13.07 0.25 46.68 48.00 2.76 6/10 11.06 11.09 0.23 44.82 46.14 2.86 7/10 9.88 9.90 0.22 43.71 45.02 2.92 8/10 9.12 9.14 0.21 42.99 44.31 2.96 9/10 8.61 8.63 0.20 42.51 43.82 2.99 10/10 8.25 8.26 0.19 42.17 43.48 3.01

At time t=0, the flow rate in the penstock is Q0and the head loss due to friction in the penstock is Hwm(0). At time t=Ts+θ 2, the flow velocity in the penstock is reduced to nearly zero, so the friction is reduced to nearly zero. The decrease of flow velocity in the tunnel is relatively small. Head recovery due to friction decrease can be estimated by Hwm(0)H0for ξ*, and is negligible for. The modified formula for ξ*is

Table 3 shows that the modified formula of ξ*, Eq. (61), is in better agreement with the numerical results than Eq. (60). The relative error increases with the decrease of w. This is due to the fact that the value of ξ*increases with the decrease of w. The residual error of approximate treatment of Eq. (43)also increases. The maximum relative error of all ten cases was less than 1%.

Table 3 Theoretical value of H0ξ* calculated by Eq. (61)and its relative error

In this section, ξ*is modified by adding a penstock friction term. Analytical results of ξ*andare compared with numerical results obtained using the MOC, showing that they are in a good agreement.

4 Conclusions

In this study, water hammer and transmitted pressure in a hydro-electric power plant with a long diversion tunnel and a throttled surge chamber were examined. Two equations, namely a tunnel-surge chamber-penstock joint equation and a water hammer interlocking equation for a penstock with a surge chamber located upstream were derived, in which the maximum water hammer pressure ξ*at the valve and the maximum transmitted pressurewere two unknown variables. The analytical formulas ofand ξ*were deduced by solving these two equations. Taking friction in the penstock into account, the analytical formula of ξ*was improved in accuracy.

The results obtained using the proposed analytical formulas are in good agreement with the numerical results obtained using the method of characteristics for various sizes of the orifice.

Under the assumption that the reflected wave from the inlet of the division tunnel does not arrive at the bottom of the surge tank at time Ts+θ2, the proposed formulas are valid for hydro-electric power plants with long diversion tunnels. For short tunnels, the reflected wave from the inlet of the tunnel has to be taken into consideration. For such a case, an analytical study will be quite complicated, but deserves further exploration.

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