water hammer

C H A P T E R

WATER

10

HAMMER

10.1 Introduction In this chapter we are concerned with the effects of rapid valve closure in pipes connected to wave reflection points (e.g., reservoirs, pumps, and turbines and rapid starting and stopping of turbomachines). These turbomachines are connected in turn via conduits to wave reflection points. The pressure energy generated by these actions may destroy or severely damage parts of the system. The energy is of two kinds: the kinetic energy of the moving liquid and the elastic energy stored in the liquid and pipes. Both forms are converted to pressure energy, and the rapidity of the conversion is of the utmost importance in terms of the ensuing damage that may result. Such energy must be dissipated in a controlled, nondamaging way. We consider first the case of instantaneous valve closure at the end of a horizontal pipeline with a flowing liquid, as shown in the sequence of events illustrated in Figure 10-1. The valve is at the fight-hand side of the pipeline with a reservoir at the left with a head H, which supplies the necessary potential energy for the flow. The sequence of events after valve closure is as follows: 1. A wave of positive pressure is generated at the valve and travels upstream (to the left of the valve) with the velocity of sound as in Figure 10-1 (1). During this time, behind the wave the pipe is expanded elastically in the redial direction. 2. When the wave reaches the reservoir, shown in Figure 10-1 (2), the wave pressure falls to the reservoir pressure. The reservoir acts as a reflecting surface. 3. A negative pressure wave now travels downstream with the velocity of sound, and behind the wave the pipe contracts. The fluid velocity behind the wave is negative, and in front of the wave it is zero. 4. When the wave reaches the fight-hand side, the valve, it is reflected upstream. 5. The velocity behind the wave traveling toward the reservoir is zero. 6. The negative wave reaches the reservoir, and the pressure rises to reservoir level. 7. A positive reflected wave travels toward the valve. 8. The reflected wave reaches the valve, and one cycle is completed.

281

282

Incompressible Flow Turbomachines

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M

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Figure 10-1 Water hammer-generated wave. 1. Wave propagation upstream immediately after valve closure; 2. wave reaches the reservoir, pipe fully expanded; 3. reflection at reservoir with a negative velocity in the fluid; 4. refection of negative wave at the valve; 5. propagation of negative wave upstream; 6. negative wave at reservoir; 7. negative wave reflected at reservoir; 8. reflected wave reaches the valve, one cycle completed.

Figure 10-2 shows the magnitude of wave velocities in pipes made of steel and cast iron as a function of pipe size.

10.2 Equations Describing Wave Generation and

Propagation The equations describing the relationship of pressure, liquid velocity, and wave velocity are: ( 1/oQ)(M2V/Mx 2) -- (M2V/Mt 2)

(10.1)

( 1/pQ)(M2p/Mx 2) -- (M2p/Mt 2)

(10.2)

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Hammer

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1200 .......

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m

r o >

800

600

40( I 0

50

100

150

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l

200

,

250

I

300

Dlb

Figure 10-2 Wave velocity as a function of the ratio (pipe diameter/pipe thickness), D/b, for steel and cast iron pipes.

where: p = liquid density V -- velocity Q = (1/k + D/bE) k = bulk compressibility modulus D = pipe diameter b = pipe wall thickness E = modulus of elasticity of pipe wall material The wave velocity, a, is related to p and Q by means of the equation: a = 1/(pQ)~

(10.3)

Equations (10.1) and (10.2) may be manipulated (Rich, 1945) to give: (a2)(M2p/Mx 2) -- (M2p/Mt 2) - ( M p / M x ) = p(MV/Mt)

(10.4) (~0.5)

284


1.2

0.8

0.4

i

-0.4

-0.8

-1.2 0

2L/a

4L/a

6L/a

8L/a

lOL/a

12L/a

14L/a

16L/a

18L/a

20L/a

Figure 10-3 Pressureofwave in conduit as a function of time. The time intervals are given as one-cycle intervals. The maximum and minimum values of pressure = +paV 0 and -paV O.

The boundary conditions are: at x = 0; p = P0 and at x - L; V = 0. Equations (10.4) and (10.5) apply to frictionless flow. Rich (1945) has presented solutions of these equations in terms of series summations of two wave components. Plots of pressure and velocity as a function of time solutions of Equations (10.4) and (10.5) are presented in Figures 10-3 and 10-4. The time intervals are given as one-cycle intervals--that is, the time taken for the wave to travel from the valve to the reservoir and back to the valve. Since friction has not been included in these solutions, the waves are not attenuated. Figures 10-5 and 10-6 indicate the effects of friction on the attenuation of the waves. In this case, the friction factor has been greatly exaggerated to show the effect. For all practical purposes, in most analyses the magnitude of friction would be such that it would have no appreciable effect on the water hammer pressure and velocities and might be safely ignored. The plot is similar to Figure 10-4 except that the effects of friction are included.

10.2.1 Valve Opening or Closure Position as a Function of Time The position of the end valve in terms of its effective area of opening has a marked effect on wave reflection. The effective area of opening also depends on the type of valve. Figure 10-7 shows the effective area change in terms of valve type and degree of opening. Types C and D are very close in terms of their characteristics, and very little error would be incurred by assuming the same curve for each valve.

Water

Hammer

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-0.8

-1.2

0 tea

3L/a

5L/a

7L/a

9L/a

11L/a

13L/a

15L/a

17L/a

19L/a

21L/a

Figure 10-4 Velocity in conduit as a function of time. The time intervals are given as one-cycle intervals, beginning at L/a. The velocity = velocity at reservoir, x = O.

1.2

0.8

i

j I

0.4

,

__...-,,

-0.4

-0.8

-1.2

0

2L/a

4L/a

6L/a

8L/a

lOL/a

12L/a

14L/a

16L/a

18L/a

20L/a

Figure | O-S Pressure of wave in conduit as a function of time. The plot is similar to Figure 10-3 except that the effects of friction are included.

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1.2 0.8

0.4

-0.4

-0.8

-1.2

0 L/a

5k/a

3L/a

7k/a

9k/a

11k/a

13L/a

15L/a

17L/a

19L/a

21L/a

Figure 10-6 Velocity in conduit as a function of time. 100

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=

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20

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0.8

0.6

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Angular opening or vertical stroke Figure 10-7 Characteristics of different valves in terms of effective area of flow and closure position. A-Disk gate valve; B-ring follower gate valve; C-plug valve; D-butterfly valve

287

Water Hammer

10.3 Graphical Solution Undoubtedly the pioneer in the theory of water hammer is Allievi (1925) who developed charts for the series equations, describing the flow. Unfortunately the method is somewhat cumbersome to use. Angus (1935) developed a more rapid graphical method. The uses of Allievi charts and other graphical methods have since been superseded by computer solutions, but because the graphical method is a useful illustration of the solution of the differential equations, it will be described in some detail here. The basic equations are rewritten in the form of head rather than pressure. The convention used by Angus is that x = 0 at the valve and x = L at the reservoir. The basic equations have the form: -(0H/0x) = (1/g)(0V/0t)

(10.6)

- ( 0 V / 0 x ) = (g/a2)(0H/0t)

(10.7)

The general solution of Equations (10.6) and (10.7) is: H - H0 = F(t - x/a) + f(t + x/a)

(10.8)

V0 - V = (g/a)F(t - x/a) - f(t + x/a)

(10.9)

and

The high value of wave velocity compared to cycle time means that intervals of valve gate closure intervals may be executed at cycle time intervals of (2L/a), (4L/a), (6L/a), (8L/a), and so on. Therefore, for successive intervals, Equations (10.8) and (10.9) may be written as: H1 = Ho + F I : H 2 = Ho + F2 - F I : H 3 = Ho + F3 - F2: . . .

V1

-- V0

-

(g/a)Fl:

V2

=

V0

-

(g/a)(F1 + F2): V3 = Vo - (g/a)(F2 + F3): . . .

(10.10)

(10.11)

F is a function of the gate setting of the valve; for practical purposes, it may be assumed to be linearly variable. Eliminating F from Equations (10.10) and (10.11), we obtain: H1 - Ho = (a/g)(Vo - V1)

(10.12)

HI - H2 - 2Ho = (a/g)(V1 - V2)

(10.13)

H2 - H3 - 2Ho = (a/g)(V2 - V3)

(10.14)

Hn - Hn-1 - 2H0 = (a/g)(Vn-1

-- Vn)

(10.15)

~~


Equations (10.12) through (10.15) give the relationship between pressure and velocity at the start of each successive interval (2L/a). The function F is the sum of all the positive pressures at time t at position x. Similarly, the function f is the sum of all the negative pressures at time t at position x. Successively adding and subtracting Equations (10.8) and (10.9):

H - Ho = ( - a / g ) ( V o - V) + 2F(t - x/a)

(10.16)

H - H0 = ( + a / g ) ( V 0 - V) + 2f(t + x/a)

(10.17)

and

Figure 10-8 presents the notation to be used for direct and reflected wave transmission. Considering two sections of the pipe A and B, Equation (10.16) may be written for these sections as

HBt

-

-

HB0 = (--a/g)(VBo -- VBt) -k- 2F(t - x/a)

HAtl -- HA0 -- (--a/g)(VA0 -- VAtl) -k- 2F(tl - x l / a )

(10.18) (10.19)

Since (t - tl) = (x - Xl)/a

.'.

F(t - x/a) = F(tl - x l / a )

(10.20)

The pipe is of uniform diameter; therefore, VA0 = VB0. We assume HA0 = HBO. Subtracting Equation (10.19) from Equation (10.18) gives:

HBt -- HAtl -- (-+-a/g)(VBt

-- VAtl)

(10.21)

The same reasoning for the reflected wave yields:

HBtl -- HB0 = (+a/g)(VB0 - VBtl) q- 2f(tl + x l / a )

(10.22)

HAt -- HA0 -- (+a/g)(VA0 -- VAt) -+- 2f(t + x/a)

(10.23)

Subtracting Equation ( 10.21 ) from Equation (10.22) gives:

HAt -- HBtl = (--a/g)(VAt -- VBtl)

(10.24)

289

Water Hammer

Direct wave B

A

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x

Reflected wave C

B

A

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