1

The Calculation of the Transient Response of Small Orifices in the Lateral Pipe of a T-Junction.

Reinhard K. Prenner1

Abstract

This paper concerns the calculation of the hydraulic resistance of small circular orifice plates at the connection of the pressure tunnel to the surge tank during pressure wave transmission. It presents the results of some recent investigations in which the extent of transmission and reflection of pressure waves in such construction elements were investigated. At present, there is an unsteady difference between quasi-steady calculations and the real physical behavior of a pressure wave passage in such smaller orifice plates. In order to ascertain these unsteady influences of these orifices on the transmission of pressure waves, a simple experiment with a T-junction in a 12 m long pipe model was conducted in which 4 symmetrical circular-orifice plates with butt edges (orifice area/pipe cross-section area ratios of 1:16, 1:32, 1:64 and 1:100) were used under various steady basic flows with single pressure waves.

The results of the experimental investigations are depicted, after elimination of additional physical influences such as unsteady fluid friction and pseudo-pressure transmission, in the form of diagrams. Furthermore, a suitable mathematical approach for computation of the transmisson behavior is developed. The suitability of application for this unsteady throttle headloss in the usual unsteady calculation is tested by a standard characteristic method (MOC) using control calculations of the experiments. Influences of basic flow conditions of the transmission- and reflection behavior is also described and evaluated. Keyword: fluid transients, waterhammer, pressure surges, pressure wave transmission, pressure duct system, throttled

surge tanks, throttle elements, throttle orifice plates.

1 Introduction

The unsteady dynamic behavior of throttle elements could be of great importance to its influences on the pressure surges in pressure duct systems, e.g. supply or control systems for water, oil, gas and chemical fluids in industry and aerospace. It is particularly necessary for the design of surge tanks of high-head plants to know the behavior of extreme throttle elements in the conduits during the expected pressure surge development induced by control and regulation processes. Understanding the unsteady hydraulic resistance behavior of the restriction orifice is of great importance for the dynamic calculation according to the elastic transient fluid theory, for it influences the pressure oscillation in the pressure duct system. Under certain circumstances of the non-linear throttle headloss, higher-frequency pressure waves can pass through the

1 Associate Professor, Institute of Hydraulic Engineering, University of Technology Vienna, A-1040 Vienna, Karlsplatz 13, Austria

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pressure tunnel and dangerous oscillation of surge pressure can arise, the stresses of which may cause massive damage to the concrete tunnel lining (Figure1). The resulting repair works and stoppage of the power plant can result in considerable financial losses.

Figure 1: Pressure duct system of a high-head powerplant, maximum pressure surges stimulated

the 2nd harmonic oscillation of the headrace tunnel.

Unsteady influences are also important if very steep pressure waves, caused by failures of pumps with small rotating masses hit on a partially opened valve. In this case, the throttle characteristic of quick closing valves or non-return valves can also be changed due to this unsteady transmission behavior. For this reason, there are not only scientific but also economic motivations to investigate the transmission and reflection behavior of these small throttle elements accurately and to verify them mathematically. 2 Previous Studies - Theoretical Approach

Many studies (e.g. Jaeger, 1933; Zienkiewicz et al., 1954; Seth, 1973; Bernhart, 1977) have developed methods of calculating water hammer behavior at orifice plates of throttled surge tanks based on Allievi's (1903) equations. In all studies known to the authors, the calculations with steady headloss coefficients were developed only for the transmitted pressure wave in the pressure tunnel and included additional influences such as closure time and the wave reflection from the free surge tank water level. None of the studies isolated the pressure wave transmission and distribution in the junction with a throttle in the lateral pipe. Investigations of the transmission and reflection behavior of pressure waves on the throttle element itself have only been published for straight pipelines and shock tubes. As it is difficult to obtain the measurements, only a few such studies have been published, i.e., by Contractor, 1965; Trengrouse et al., 1966; Funk et al., 1972; and Berger, 1978; in which the behavior of pressure wave transmission and

powerhouse

pressure shaft or penstock

reservoir

pressure according to elastic calculations (elastic theory)

maximum

cracks caused by pressure surges

pressure according to mass oscillations calculations

(unelastic theory)

minimum

headrace tunnel

upper surge chamber

lower surge chamber

throttle element

3

reflection at several throttle elements are investigated. The results of these studies can only be related to bifurcated systems with orifice plates in a qualitative sense, because interaction arises when pressure waves are distributed among the three pipes. Lately, because the experimental requirements for the investigations are very difficult and complicated, the studies of Prenner, 1997; are worthy of note. This paper represents one of the current research priorities of the Institute of Hydraulic Engineering at the University of Technology, Vienna.

For an unthrottled pipe branch, the transmission approach according to Jaeger, 1933; was applied to constant pipe cross-sections of A1=A2=A3 and constant pressure wave speeds of a1=a2=a3. The following simplified equation for the transmission coefficients for the branches yields (1).

66.032

3s

3A

2A

1A

2A2

2s ===

++

⋅= ( 1 )

Ai cross section areas of pipes, s 2 transmission factors into the main branch, s3 transmission factors into the side branch

Headlosses in the pipe junction were generally taken into consideration by a headloss coefficient for steady flows of ξ orifice, which is related to the kinetic energy (2).

g2

2pipe

v2

1orifice

Apipe

A

g2

2pipe

v

orificeorifice,fh

⋅⋅

−

⋅µ=

⋅⋅ζ= ( 2 )

g gravitational acceleration, h piezometric head, h f,orifice orifice head loss, v velocity, ξorificeorifice headloss coefficient,

µ coefficient of contraction (A contraction / Aorifice)

v20

pipe 2 (tunnel)

pipe 1 (penstock)

v

v1

v2

v3 h20

h30

v30

throttle elements

transmitted wave (tunnel) H2 transmitted wave (surge tank) H3

reflected wave (penstock)

( )30orifice,f

vh

( )3orifice,f

vh

Hinc.wave

h10

r . Hinc.wave

s3 . Hinc.wave s2 . Hinc.wave

pipe 3 (surge tank)

incident wave (penstock) H1

Figure 2. Theoretical pressure distribution in a T-junction after a shock wave passage under steady flow conditions

4

The reflection factor r (3) and the transmission factors s2 (4) and s3 (5) in a throttled pipe branch with the above assumptions, are given by

hg23va

r∆⋅⋅

⋅−= ( 3 )

tunnels

1H

2H

r12

s ==+= ( 4 )

ktansurges

1H

3H

r23

s ==⋅−= ( 5 )

r reflection factor in the approach pipe ∆h change in pressure head ai wave propagation velocities H local pressure head at a node i as subscript denotes number of pipe branches

The solution for v3 in equation (6) is positive, when the pressure wave passes in flow direction (v30 is positive) through the throttle element (Figure 2). For a pressure wave passage against the steady flow direction (v30 is negative) through the throttle, v3 becomes negative, until hJou (∆h) is reached and exceeded. The headloss coefficients ξ inflow or ξoutflow are used corresponding to the flow direction in the surge tank pipe 3.

orificeorifice

302

30

2

orificeorifice

3

hg2

va3v

a3a3v

ξ∆

⋅⋅±ξ

⋅⋅++

ξ

⋅±

ξ⋅

= µ ( 6 )

ai wave propagation velocities hjou Joukowsky head variation a⋅∆v/g ∆v change in velocity

For the calculation of the unsteady pipe flow of the experiment, the momentum (7) and the continuity (8) equations were used, based on the assumptions of the one-dimensional fluid flow theory. These equations were solved numerically, under inclusion of boundary conditions with an explicit characteristic method (Wylie and Streeter, 1993).

∂∂

∂∂

∂∂

hx

1g

vt

vvx

I f 0+ ⋅ + ⋅

+ = ( 7 )

Avx

g Aa

ht

vhx

Ax

vg Aa

v⋅ +⋅

⋅ + ⋅

+ ⋅ +⋅

⋅ ⋅ =∂∂

∂∂

∂∂

∂∂

α2 2 0sin ( 8 )

a wave propagation velocity, If pipe friction gradient, t time, as a subscript denotes partial differentiation,

x distance along pipe axis, as a subscript denotes partial differentiation, α slope of pipe axis

Calculations using a steady quadratic resistance law of the experimental investigations indicate that unsteady differences are of significance for relatively small orifice area/pipe area cross-section ratios of 1:32 (ξorifice>2000) or less (Figure 5).

5

The influence of additional inertia forces of the orifice flow, the delayed scaling-off of the boundary layer, cavitation and unsteady friction effects in the orifice flow were taken into consideration. To minimize these unsteady differences for practical use in numerical simulation, a suitable unsteady headloss approach (9) was developed. ( 9 )

w(t0-t) weighting function for the prehistory of time steps, t 0 number of time steps for the prehistory,

ξ trans,orifice unsteady headloss coefficient

This assumption includes the prehistory of the incident pressure wave (Arlt, 1983) with a transient headloss coefficient to consider the instantaneous strong contraction of streamlines during the starting process of the incident pressure wave. This physical effect takes place especially at the beginning when the wave hits on the orifice plate and the flow takes the same direction as the wave. 3 Experimental Investigations

The size of the experiment was essentially determined by the amount of room available in the laboratory (Prenner, 1997). A 100-mm-diameter steel pipe system was rigidly fixed to a foundation plate with a special steel construction to prevent shifting of the T-section during pressure wave passage (Figure 3).

Figure 3: General layout of experimental arrangement

dt)tt(wtv

h 0

1t

0

orifice,transorifice,trans ⋅−⋅∂∂

⋅ξ= ∫

700

flowmeter flowmeter

upstream inflo

inflow

outflow outflow Ch2

Ch1

Ch3 Ch6

Ch5

fixpoints

throttle element (brass) (4 sizes)

flow direction of the incident single pressure

wave

750 3700 mm

50 downstream

M1 - M4 cross sections for measuring

3150 mm

pressure wave initiator rubber damped piston

3000 mm

steel pipe DN 100, PN16

inside diameter = 100 mm, pipe wall thickness = 3.6 mm

M2 M1 M3

Ch1 - Ch6 channels for pressure gauges

700

inflow

outflow

Ch4

3150 mm

M4

6

The single pressure waves were produced by the impact of a pendulum mallet on a rubber damped piston and introduced into the lower half of the penstock pipe. The pipe lengths after the measuring points were chosen so that no reflection waves from the ends of the pipe branches would influence the behaviour of the incident and transmitted single pressure waves. The pressure waves were recorded by high-resolution inductive pressure gauges (0-100 meter) and transferred by triggering via an A/D converter to a PC measuring program. Measurement of the flow discharges was carried out via two magnetic inductive flowmeters. In the experiment, four orifice plates with square edges were investigated under various steady basic flow conditions (Figure 4). The limited length of the pipe model only allowed the production of very steep pressure waves.

Figure 4: Program of experimental investigations, basic flow conditions with relative main streamflow discharges, experimental data of steady headloss coefficents ξorifice

The experimental part of this work deals with a wide range of single pressure waves (with pressure heights from 10 m to 70 m) during transmission through the throttle element. Because of the constant geometry, different gradients of the incident pressure wave resulted. These fundamental investigations on a T-junction model, with constant orifice plates and different orifice-to-pipe cross-section ratios (m=1:16, 1:32, 1:64 and 1:100), required high accuracy both in the measuring technique and in the execution. Particular care was necessary to record these exactly for reproduction.

For the interpretation of results, it was necessary to eliminate additional physical influences such as unsteady fluid friction and pseudo pressure transmission (induced by deformation of orifice plates and unavoidable elastic fixation in the foundation plate) in order to arrive at the pressure wave transmission

Aorifice Apipe

1/16, ξinflow = 620 ξoutflow = 640 1/32, ξinflow = 2400 ξoutflow = 2500 1/64, ξinflow = 6200 ξoutflow = 8500 1/100, ξinflow = 17000 ξoutflow = 19500

=

case 3

0

case 6

case 2

case 8 case 5

0

0

case 4

case 7

case 1

0 0

0

0,5

0,5 0,5

0,5

0,5 0,5 1

1

1

1

1 1 1

1

1

1

1

Aorifice

10 mm

100 mm

7

factor itself. This was corrected by control calculations based on measurements at the unthrottled and the fully throttled branches, which deviated from the transmitted values by a maximum of 4%. A comparison of the transmission factors of the investigated small orifice plates can be seen in Figure 5. These diagrams make it possible to estimate the efficiency of a throttle element regarding the reflection and transmission behavior in a duct system. The diagram also shows the unsteady differences between the experimental data and the data computed using the quadratic law of resistance.

Figure 5: Comparison of transmission factors in a T-junction, without basic flow (v = 0) orifice plate with butt edges

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0

0 . 1 4 0

0 . 1 6 00 . 1 8 0

0 . 2 0 0

0 . 2 2 0

0 . 2 4 00 . 2 6 0

0 . 2 8 0

0 . 3 0 0

0 . 3 2 0

0 . 3 4 00 . 3 6 0

0 . 3 8 0

0 . 4 0 0

0 . 4 2 0

0 . 4 4 0

0 . 4 6 00 . 4 8 0

0 . 5 0 0

0 . 5 2 0

0 . 5 4 0

0 . 5 6 00 . 5 8 0

0 . 6 0 0

0 . 6 2 0

0 . 6 4 00 . 6 6 0

0 . 6 8 0

0 . 7 0 0

0 . 7 2 0

0 . 7 4 00 . 7 6 0

0 . 7 8 0

0 . 8 0 0

0 . 8 2 0

0 . 8 4 0

0 . 8 6 00 . 8 8 0

steady thrott le headloss

ca lculat ion

1 / 6 4 o r i f i c e

1 / 1 6 o r i f i c e

1 / 3 2 o r i f i c e

1 / 1 0 0 o r i f i c e

H1

H3

s

surg

e ta

nk =

Def in i t ion

s k e t c h :

H1

H2

pressure wave

A pipe

A orifice

H 1= H inc.wave

p r e s s u r e h e a d o f t h e i n c i d e n t s i n g l e p r e s s u r e w a v e H inc.wave [ m ]

stu

nn

el =

Api

pe

H 2

H 3

exper imenta l data

a v e r a g e i n c r e a s e g r a d i e n t o f t h e i n c i d e n t p r e s s u r e w a v e [ m / s ]

0 3 0 0 0 02 5 0 0 02 0 0 0 01 5 0 0 02 5 0 0 1 0 0 0 05 0 0 0

8

4 Comparison of Measured Test Results and Results of Calculations

Results are not entirely satisfactory for throttle area ratios of 1:16 to 1:32 (Figure 6). But in principle, it is also possible to simulate the transmission process for these orifice cross-section ratios of m=1:16 to m=1:32 relatively well with a quasi-steady loss assumption (ξ throttle ≤2350) depending on the flow direction. In case of superpositioned basic flows, these phenomena decrease if the wave and the flow direction is the same, or increase if the wave direction is against the flow direction. The calculations of the experiments show that the physical behavior of the investigated throttles with a steady head loss coefficient ξ throttle >2000 cannot be reproduced accurately with the steady quadratic resistance law.

Figure 6: Computed (MOC) measurement for single pressure wave with quasi-steady headloss

assumption (basic flow - case 3), throttle ratio m=1:16

For ratios smaller than 1:50 (Figure 7), it is expedient to take into consideration the influence of the sudden flow contraction induced by the high local acceleration in the vicinity of the orifice at the beginning of mass flow. The inclusion of these phenomena in the one-dimensional characteristic method was carried out with a transient throttle headloss assumption (Figure 8) as described in 2.

-5

0

5

10

15

20

25

30

35

40

45

50

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4

0.00

5

0.00

6

0.00

7

0.00

8

0.00

9

0.01

0

0.01

1

time [sec]

H [

m]

H1 (measurement) incident wave

HJ (measurement) junctionHJ (computation)

H2 (measurement) tunnelH2 (computation)

H3 (measurement) surge tankH3 (computation)

ξinflow = 620 ξoutflow = 640 ssurge tank (measurement)=0,557; s tunnel (measure.) =0,732

ssurge tank (computation) =0,558; s tunnel (comput.) =0,720

incident wave

reflected

wave

transmitted wave

(surge tank)

transmitted wave (tunnel)

Re = 42 500

case 3

0,50 m/s

incident wave

∆H/∆t ≈20000 m/s

9

Figure 7: Computed (MOC) measurement for single pressure wave with a quasi-steady headloss

assumption (without basic flow - case 1), throttle ratio m=1:100 Figure 8: Computed (MOC) measurement for a single pressure wave with a transient headloss

assumption (equ. 9), (basic flow - case 7), throttle ratio m=1:64

5 Conclusion

The transmission factors of the investigated orifice plates indicate that the plates hinder the transmission of pressure waves more or less depending on the throttle ratio. Measuring results deviate only marginally (maximum 7%), depending on the pressure wave gradient and the steady flow direction, from the generally

-505

1015202530354045505560

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4

0.00

5

0.00

6

0.00

7

0.00

8

0.00

9

0.01

0

0.01

1

time [sec]

H [m

]

H1 (measurement) incident waveHJ (measurement) junctionHJ (computation)H2 (measurement) tunnelH2 (computation)H3 (measurement) surge tankH3 (computation)

ξinflow = 17000 ξoutflow = 19500 ssurge tank (measurement)=0,331; stunnel (measure.) =0,838

ssurge tank (computation) =0,379; stunnel (comput.) =0,811

incident wave

reflected wave

transmitted wave (tunnel)

unsteady difference

transmitted wave (surge tank)

-505

10152025303540

45505560

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4

0.00

5

0.00

6

0.00

7

0.00

8

0.00

9

0.01

0

0.01

1

time [sec]

H [

m]

H1 (measurement) incident waveHJ (measurement) junction

HJ (computation)

H2 (measurement) tunnelH2 (computation)H3 (measurement) surge tankH3 (computation)

ξ inflow = 6200 ξoutflow = 8500 s surge tank (measurement)=0,516; s tunnel (measure.) =0,765

s surge tank (computation) =0,500; s tunnel (comput.) =0,750

incident wave

reflected

wave

transmitted wave (tunnel)

transmitted wave (surge tank)

case 1

0 m/s

incident wave

∆H/∆t ≈25000 m/s case 1

0 m/s

incident wave

∆H/∆t ≈25000 m/s

0,15

Re = 6 400 and 12 800

case 7 0,075

0,075 m/s

incident wave

∆H/∆t ≈25000 m/s

10

accepted steady throttle headloss assumption. In case of superpositioned basic flows, unsteady phenomena decrease if the the wave and the flow direction is the same, or increase if the wave direction is against the flow direction. The calculations also show that the commonly used steady headloss for throttle ratios greater than 1:32 are moderately suitable. Additional approaches to describe the unsteady resistance behavior of the orifice plate are necessary only for orifice area/pipe cross-section area ratios 1:64 and smaller. The unsteady differences during a steep pressure wave passage are caused by the phenomenon of a instantaneous tail back of the flow in front of the throttle plate which also induced a deceleration in the recovering of pressure of the transmitted wave. For the mathematical simulation of very narrow orifice plates a relatively simple unsteady headloss aproach was developed to get more accurate results.

References

Allievi, L.: „Teoria generale del moto perturbato dell’acqua in pressione“. Annali della Societa degli Ingegneri ed Architetti Italiana, Milano, 1903.

Arlt, H.: ″Experimentelle Untersuchungen über das instattionäre, turbulente Reibungsverhalten bei aufgeprägten Druckimpulsen in einer Rohrleitung mit Kreisquerschnitt″.Institut für Wasserbau und Wasserwirtschaft, Technische Universität Berlin, Mitteilung Nr. 102, 1983.

Berger, H.: “Druckstossverhalten an Blenden bei Berücksichtigung der Anlaufströmung”. Institut für Wasserbau und Wasserwirtschaft, Technische Universität Berlin, Mitteilung Nr. 89, 1978.

Bernhart, H.H.: „Einfluß der Schließzeit auf die DruckstoßtransmiVersuchsanstalt für Wasserbau und Kulturtechnik, Theodor - Rehbock - Flußbaulaboratorium, Universität Fridericiana Karlsruhe, Mitteilung Heft Nr. 164, 1977.

Contractor, D.N.: ″The reflection of Waterhammer Pressure Waves from Minor Losses″. Transactions of ASME, Journal of Basic Engineering, June 1965, pp. 445-452.

Funk, J.E., Wood, D.J. and Chao, S.P.: ″The Transient Response of Orifices and very Short Lines″. Transactions of the ASME, Journal of Basic Engineering, June 1972, pp. 483-491.

Jaeger, CH.: „Théorie générale du coup bélier“. Dunod, Paris, 1933.

Prenner, R.K.: ″Das Widerstandsverhalten von Kreisblenden in Druckstoßsystemen″. Dissertation, Technische Universität Wien, Fakultät für Bauingenieurwesen, November 1997.

Seth, H.B.S.: „Pressure wave transmission at an orifice surge tank“. Water Power, August 1973, pp. 305-308.

Trengrouse, G.H., Imrie, B.W. and Male D.H.: ″Comparison of Unsteady Flow Discharge Coefficients for Sharp-Edged Orifices with Steady Flow Values″. Journal of Mechanical Engineering Science, Vol.8 No.3 1966, pp. 322-329.

Wylie, E.B. and Streeter, V.L.: ″Fluid Transients in Systems″.Prentice-Hall, New Jersey, 1993.

Zienkiewicz, O.C. and Hawkins, P.: „Transmission of Water-Hammer Pressures Through Surge Tanks“. Proceedings of the Institution of Mechanical Engineers, 1954, Vol. 168, Nr. 23. pp. 629-642.