Study on Wave calculation of an air cushion surge chamber

Xiaohong ZhangXi’an University of Technology, Institute of

Water Resources and Hydro-Electric Engineering, Xi’an,Shaanxi

Abstract.

To constant flow, the varying rule between gaseous volume and pressure meets Boyle Gas Law. But, in transient process of water diversion system, kinetic characteristic of the gas in air cushion surge chamber has certain complexity when there is great fluctuation in surge chamber.

Therefore, when computing surge of air cushion surge chamber, firstly, it is necessary that gaseous volume and pressure meet actual rule and important to establish correct mathematic model. Secondly, initial height and pressure of the chamber should be confirmed correctly.

Then the computing method of water level surge of air cushion surge chamber is acquired.

The basic equations and boundary conditions of the surge chamber were established in this article. Parameters selection, structure design and surge calculation of the air cushion surge chamber were conducted.

The effect of the initial air height and Polytrophic Exponent n in the chamber on the fluctuations of pressure and changes of water level is discussed.

The results are quite important to fluid transient study of the air cushion surge chamber and may promote its use in water power plant.

1.Basic equations of the transient flow in Pressure pipelines

1.1 motion equation and continuity equation

0sin

02

2

Vx

HV

x

V

g

a

t

H

D

VVf

t

V

x

VV

x

Hg

1.2 Characteristic equation and its differential form

C

adt

dxD

VfV

dt

dV

dt

dH

a

g0

2

C

adt

dxD

VfV

dt

dV

dt

dH

a

g0

2

Differential form of Characteristic equations:

pimpi

pippi

BQCHC

BQCHC

:

:

1111

1111

iiiim

iiiip

QRQBQHC

QRQBQHC

2.Surge chamber boundary conditions

Boundary conditions of air cushion surge chamber are showed in Figure 1.

Fig. 1..Boundary conditions of Air Cushion Surge Chamber

equation:

NSppNSp QBCHC ,111,1: 1,2221,2: pMp QBCHC

ppNSp HHH 1,2,1 1,2,1 PcNSp QQQ

)(2 010 cc

c

QQF

tZZ

2

2

21

2,1

,1 2)1(

2 c

caNSP

NSp gF

Qk

g

PZ

g

P

gf

QH

nLLPP )/( 1010

HP1,NS,、 QP1,NS, QP2,1,、 Z,、 QC,、P, and L are seven unknown parameters in t moment when calculating water hammer pressure and surge chamber fluctuations jointly. The air pressure change, water level fluctuation and other physical parameters can be determined by solving the boundary equations of the chamber jointly.

3 Examples and results analysis

A hydropower station has 100 meters high roller compacted concrete arch dam. The total installed capacity is 3×2.4 million watt. The entire power generation system consists of water intake tower, diversion tunnel, surge chamber, pressure pipeline fork tubes and branch pipe. The hydropower station design discharge is 93m3/s, design head is 90m, maximum flood level is 512.5m, dead water level is 485m,

pressure diversion tunnel is 2715.213m long (to centerline of surge chamber), tunnel centerline elevation near surge chamber is 463.524 meters, and tunnel diameter after lined is 6.0m. Considering the terrain and environmental protection requirements, air-cushion type surge chamber is designed. After the chamber is buried pressure pipe with 5.5 meters inner diameter and 134.281 meters long main pipeline. The main pipe divided into three 3.0m inner diameter branches before enter the main power house and the distance between them is about 102 meters.

3.1 Results analysis of air-cushion surge chamber calculation

3.2.1 Influence of initial heights to the pressure and surge of air cushion surge chamber

In order to compare and show the relationship between initial height and surge of air chamber , L0 is chose as 10 m (L0/24=0.42), 12m (L0/24=0.50) and 16 meters(L0/24=0.67) respectively, and partial results are showed in Table 1, Figure 2, and Figure 3.

Fig. 2 ． Fluctuation Course of Surge in air cushion Surge Chamber (Lo=10m, n=1.4)

484

485

486

487

488

489

490

491

492

493

0 200 400 600 800 1000 1200

t/ s

Zp/m

Fig. 3 ． Fluctuation Course of Pressure in air cushion Surge Chamber (Lo=10m, n=1.4)

0

10

20

30

40

50

60

70

80

0 200 400 600 800 1000 1200

t/ s

P/m

Table 1. Surge Water Level When Three Sets Discard all Loads at the same time in Air Cushion

Surge Chamber

Based on calculation, It was found that under the same calculation conditions, same surface area of surge chamber, and same polytrophic exponent, with different initial height or a different initial pressure of air chamber (volume), the highest surge (maximum pressure chamber) comes in quite close time with different value, and has some rules.

It shows that the lower the initial air chamber height Lo the higher the maximum and minimum swell height of air chamber, and vice versa. Therefore increasing the initial pressure can reduce the height of the air chamber and the total amount of excavation. However, increase of the initial pressure is restricted by the minimum water level of air chamber. Air is not allowed to enter the diversion tunnel.

In air cushion surge chamber the existence of "air cushion" buffer the effects of water level fluctuations. Its water level fluctuation differs to the conventional normal open chamber. When the hydropower station load change, water level fluctuations of the conventional open surge chamber likes sine wave attenuation curve, but the water level fluctuations of air cushion surge chamber have the following characteristics: water level fluctuation waveform has steep wave crest and mild trough. It can be clearly seen from fig.3

that the amplitude of water level fluctuation of air cushion surge chamber is smaller than that of the conventional open surge chamber. While along with continuing fluctuation time, the attenuation speed is slower than that of conventional open surge chamber.

3.2.2 The impact of polytrophic exponent n on the pressure and surge of air cushion surge chamber

The indoor air cushion surge chamber characteristics ranges from constant temperature and adiabatic process in the transition process. In order to illustrate the impact of different states of the transition process, a different combination of n value is calculated.

Fig. 4． Fluctuation Course of the maximum pressure of air cushion surge chamber with in

dex nLo=10m

I ni t i al hei ght of ai r chamber

60

62

64

66

68

70

0 0. 5 1 1. 5

Pol ytropi c exponent(n)

Maxi

mum

pres

sure

of

air

cham

ber(

m)

58

60

62

64

66

68

0 0. 5 1 1. 5

Pol ytropi c exponent(n)

Ma

ximu

m pr

essu

re o

fai

r ch

ambe

r(m)

Lo=12mI ni t i al hei ght of ai r chamber

56

58

60

62

64

66

0 0. 5 1 1. 5

Pol yt ropi c exponent (n)

Maximum pressure of

air chamber(m) Lo=16m

I ni t i al hei ght of ai r chamber

5 Conclusion

Through a real example this article is about more in-depth study on air cushion surge chamber fluctuations in the course of several main parameters and the relationship between them, shows the range of initial height L0 of air cushion surge chamber (initial air surge chamber pressure P0), and polytrophic exponent n. That has great important influence to the application of air cushion surge chamber in China.

Thank you