42. Water hammer in pipes

Water hammer is the name given to a sudden change of pressure that develops inside pipes when a valve is closed rapidly.

The theoretical and experimental investigation of water-hammer phenomena were first carried out by N. E. Joukowski. The expla­nations set forth here are based on the basic conclusions of his funda­mental work, Water Hammer^ published in 1899.

Water hammer is a very rapid process in which elastic deforma­tions of both liquid and pipe take place.

Consider a liquid flowing through a pipe with a velocity u_Oi and let a valve be closed instantaneously (Fig. 108a). The velocity of the liquid particles next to the valve becomes zero and their kinetic energy turns into work done in deforming the pipe walls and the fluid. As a result, the pipe walls suffer an expansion and the fluid a compression corresponding to the pressure increase Ap_Wh-* Other particles collide with the halted particles and also lose their velocity. The result is that the section n-n travels to the right with a velocity a; the transient region in which the pressure changes by kp_wh is called the shock, or pressure, wave.

When the shock wave reaches the reservoir, the liquid in the pipe is at rest and in compression and the pipe walls are in tension; the pressure increase Ap_wh extends throughout the whole pipe (Fig. 1086).

This state, however, is unstable. Due to the pressure difference kpwhi the liquid particles start to flow towards the reservoir, the motion beginning at the cross-section adjoining the reservoir. The section n-n travels back to the valve with the velocity a, leaving in its wake the unloaded pressure p₀ (Fig. 108 c).

The liquid and the walls are assumed to be absolutely elastic, therefore they revert to their initial state corresponding to the pres­sure p₀. The work done in the deformation all turns back into kine­tic energy and the liquid obtains its initial velocity v₀, but in the opposite direction.

Now the liquid tends to move away from the valve (Fig. 108c?) and a negative shock wave — Ap_wh develops which travels from the valve to the reservoir with the velocity a, leaving in its wake the pipe walls contracted and the liquid expanded, due to the reduced pressure —kp_wh (Fig. 108e). The kinetic energy of the liquid once again turns into the work done in deformation, but with the opposite sign.

*Iα this case liquid compressibility cannot be neglected, as is customary, in solving hydraulics problems, because the low compressibility of liquids isthe very cause for the development of the high water-hammer pressure &p_wfl.

The state of the pipe when the negative shock wave reaches the reservoir is shown in Fig. 108/. As in Fig. 1086, the state is unsta­ble, and Fig. 108g shows the levelling out of the pressure in the pipe and in the reservoir and the accompanying development of the velocity u₀.

It is evident that as soon as the shock wave —Ap_wh is reflected from the reservoir and reaches the valve the same conditions thai were produced by the closure of the valve reappear and the cycle is repeated.

Joukowski in his experiments registered 12 complete cycles, with Apwh gradually degrading due to friction and dissipation of energy in the reservoir.

The history of a water hammer is illustrated diagrammatically in Fig. 109.

The upper diagram shows the change in the pressure Ap_wh that would be recorded by an indicator installed at the valve at A (the closure is assumed to be instantaneous).

The shock wave reaches B, in the middle of the pipe, after a time interval , and the pressure remains during the timeit takesthe shock wave to run from B to the reservoir and back; the pressure at B then reverts to p₀ (i. e., Ap_wh = 0) and remains such till the negative shock wave comes from the valve after a time interval .

The value of the water-hammer pressure Ap_wh is found from a force equation in which the kinetic energy of the liquid turns into the work done in the deformation of the pipe walls and the liquid. Tho kinetic energy of a liquid in a pipe of radius R is

The work done in the deformation constitutes half the product of the force by the extension. Expressing the work done in expanding the pipe walls as the work done by the pressure forces in a displace­ment Ai? (Fig. 110), we obtain

From Hooke's law,

where o = normal stress in the pipe wall, which is related to the pressure Ap_wh and the wall thickness 6 by the well-known equation

Taking the expression for ΔR from Eq. (10.7) and the expression for a from (10.8) we obtain the following expression for the work done in the deformation of the pipe walls:

The work done in compressing a volume W of liquid can be rep­resented as the work done by the pressure forces in the displacement Δl (Fig. 110):

By analogy with Hooke's law for linear extension, the relative reduction of the liquid volume is connected with the pressureby the relationship

where K volume modulus of the liquid (see Sec. 4.)

If W is the volume of liquid in a pipe, the expression for the work done in the compression of the liquid is

The equation of the acting forces takes the form

Solving with respect to Ap_wh gives Joukowski's formula:

The quantity

has the dimension of velocity. Its physical meaning can be interpreted by assuming the pipe to have absolutely rigid walls, i.e.,. Then the last expression turns into, which is the velocity with which sound propagates ("sonic"or "acoustic" velocity) in a homogeneous elastic medium of density ρ and volume modulus K.

Acoustic velocity in water is 1,435 m/sec, in gasoline, 1,116 m/sec, and in oil, 1,400 m/sec.

If, as is the case here,E =£ go, the quantity

represents acoustic velocity in a liquid filling a pipe with elastic walls.

It follows from the very nature of acoustic vibrations that the velocity of sound is the speed with which rarefactions or compres­sions of infinitely small amplitude propagate through a medium. When the density change is relatively small Hooke's law is valid, and it was used in developing Eq. (10.9). We come, thus, to the con­clusion that the quantity a is the velocity of the shock wave exam­ined in Fig. 108. Consequently, Joukowski's formula can be written in the form

This formula is valid for "instantaneous" closure of a valve, i. q., when the time of closure

At T_c > T_o water hammer is "incomplete" as the shock wave makes its round trip from the reservoir before the valve is com­pletely closed. The pressure rise Ap'_Wh is smaller, and it is commonly assumed that

Equation (10.10) and the expression for T_o yield instead of the foregoing

Thus, unlike Δp_wh, the quantity Δp_wh depends on the length of the pipe and does not depend on the velocity a.

It is evident from all that has been said that water-hammer effects can be reduced by increasing the closure time T_c. If this is not feasi­ble for some reason or other, air chambers or surge tanks are in­stalled in pipelines. They absorb the water-hammer shock, which is equivalent to increasing the time of valve closure.

Example. A receiver in an aircraft hydraulic system is cut out by means of an electromagnetic valve. The time of complete closure is r_c = 0.02 sec.

Determine the pressure increase at the valve at the moment of cut-out under the following conditions (Fig. Ill):

Length of pipe from valve to accumulator which absorbs shock wave / =» = 4 m; pipe diameter d = 12 mm; thickness of pipe wall 6 = 1 mm; material, steel (E = 2.2 X 10^e kg/cm²); volume modulus of AMIMO fluid # = 13,300 kg/cm², density q = 90 kg sec²/m⁴; pipe flow velocity v_% = 4.5 m/sec.

Solution, Determine the speed of propagation of a shock wave through pipe filled with AMIMO fluid:

or

Total water-hammer pressure if the closure were instantaneous would be

In this case, however, water hammer is less pronounced because the return trip of the shock wave takes

i. e., less than the time of the complete closure T_c. The pressure increase at the valve is only