THE ANALYSIS OF THE EFFECT EXERTED BY THE DIFFERENTIAL PRESSURE TRANSDUCER AND THE

IMPULSE PIPING ON THE ACCURACY OF THE GIBSON METHOD DISCHARGE MEASUREMENT

Authors: Adam ADAMKOWSKI, Waldemar JANICKIPresenter: Janusz Steller

The Szewalski Institute of Fluid-Flow Machinery of the Polish Academy of Sciences

Fiszera Street 14, 80-952 Gdansk, Poland

Introduced in 1923 by Norman R. Gibson.

Used mainly for determining the flow rate in water turbine penstocks.

Recommended by the IEC (60041, 62006 Draft) and ASME standardson water turbine, pump and pump-turbine performance tests

The measurement accuracy not worse than +/-(1-2)% and not differing from that of other basic methods

Till ninetieth of the XXth century used predominantly in the USA and Canada.

At present, ever more attractive all over the world.

The tools needed for practical use of the method were developed in Poland by the team of IF-FM, Gdansk, in 1996-97.

Basic information about THE GIBSON METHOD

OBJECTIVE OF THIS STUDY

Determining the influence exerted on the Gibson method accuracy by the dynamic parameters of a differential transducer and the size of the measurement system impulse tubes.

METHODOLOGY

Simulation tests by means of validated computational models of the differential pressure transducer and the impulse tube.

Schedule

1. Introduction

2. Gibson method principle

3. Dynamic model of a differential pressure transducer and an impulse pipe

4. Computational software

5. Experimental validation of the calculation method

6. The impulse pipe length and the differential pressure transducer time constant impact on the flow rate measurement results

7. Conclusion

Measuring pressure difference

p1-2=p2+gz2–p1-gz1

1

2

z1

z2

GIBSON METHOD PRINCIPLE

Gibson method utilizes the effect of water

hammer phenomenon in a pipeline when

water flow is stopped using a cut-off device.

The flow rate is determined by integrating the

measured pressure difference change caused

by the water hammer (inertia effect).

dt

dQ

A

LPgzpgzp f

2211

p1, p2 – mean static pressures;z1, z2 – hydrometric section

weight center elevations;

Pf – pressure drop caused by friction losses; – water density

Discharge value in the initial conditions:

kt

t

kf QdttPtpL

AQ

0

)]()([0

p = p2 + gz2 - p1 - gz1 – static pressure difference ,

Qk – discharge under final conditions,

(t0, tk) – time interval

2-2

1-1

manifold

z1z2

pressure

difference measurement

p = p2 – p1

THEORETICAL BASIS

Equation of motion

GIBSON METHOD VERSIONS

• Version II makes use of separate measurements of pressure changes in two hydrometric cross sections of a pipeline

• Version III is based on measurement of pressure changes in one hydrometric cross section of a pipeline and relating these changes to pressure in an open reservoir, to which the pipeline is directly connected

p

p1 p2

p1p2

• Version I is based on direct measurement of pressure difference between two hydrometric cross sections of a pipeline using a pressure differential transducer

Exemplary application

Dychow Hydropower Plant

Exemplary application

Dychow Hydropower Plant

DYNAMIC MODEL OF A DIFFERENTIAL PRESSURE TRANSDUCER

Equation of the first order inertial component:

)()( tkxtydt

dyTc

Operational transmittance:

1)(

)()(

csT

k

sX

sYsG

Notation

Tc – time constant, k – factor of proportionality (gain), x – input signal,y – output signal, t – time, s – Laplace transform variable,X – Laplace transform of input signal, Y – Laplace transform of the output signal.

Y – electric output

signal

X - input signal

(pressure

difference) 1sT

k

c

Diagram of a differential pressure transducer

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

ste

p f

un

cti

on

resp

on

se h

(t)

[-]

time t [s]

measurement

simulation

Tc = 3.25 s

Tc = 0.85 s

Tc = 0.25 s

Comparison of the realistic and simulated response

of the Rosemount 1151 smart transducer to a stepwise pressure variation

The equations of continuity and motion:

01

2

x

V

t

p

a

032

sin1

2

V

Dg

t

V

x

p

x – length coordinate along the pipe axis,

t – time,

V – averaged liquid velocity,

p – static pressure,

D – internal pipe diameter,

DYNAMIC MODEL OF AN IMPULSE PIPE

after Laplace transformation

0),(1),(

2

sxps

ax

sxV

0),(),(1

sxVsK

x

sxph

– liquid density,

g – acceleration of gravity,

a – pressure wave speed,

– dynamic liquid viscosity,

– angle of pipe inclination.

NOTATION

a

sKsr h )(

r

sKz h

c

)(

p(L,s) p(0,s)

a

1

+

+

f

a a

LKh

e 2

sa

L

e

+

–

f

+

+

a

LKh

e 2

sa

L

e

+

+ +

–

f

s

Kh

2

s

Kh

2

V(0,s) V(L,s)

+

–

f

BLOCK DIAGRAM OF AN IMPULSE PIPE MODEL

),(),(),0(

),0( sLVesLpz

e

z

spsV rL

c

rL

c

),(),0(),0(),( sLVzsVezspesLp crL

crL

Relationships between the input and output quantities in the Laplace domain

COMPUTATIONAL SOFTWARE

Purpose:Modelling of the pressure transducer and impulse piping system

with due consideration of their dynamic properties

Programming environment:Matlab–Simulink

Computational method:Runge-Kutta algorithm of the fourth order

Additional features:The effect exerted by the membrane deformability due to pressure variations

on its both sides on the dynamic performance of the connecting piping /

transducer system was taken into account.

+

–

f

Pipe 1 p1

Pipe 2 p2

1

1

sTc

pout

Differential pressure transducer

pin

p1 – pressure in penstock hydrometric section 1-1,

p2 – pressure in penstock hydrometric section 2-2,

Δpin – pressure difference,

Δpout – output signal of the differential pressure transducer.

FLOW CHART OF THE DEVELOPED COMPUTATIONAL CODE

Validation principle:

Comparison between the realistic and simulated deviation

of discharge measurement results

by means of independent and differential pressure transducers

EXPERIMENTAL VALIDATION OF A DIFFERENTIAL TRANSDUCER SIMULATION METHOD

Dychow Hydropower Plant

-1.5

-1

-0.5

0

60 70 80 90 100 110 120

q

[%]

Qa [m3/s]

measurement

calculationTc = 0.2 s

-1.5

-1

-0.5

0

100 105 110 115 120 125

q

[%]

Qa [m3/s]

measurement

calculationTc = 0.85 s

-1.5

-1

-0.5

0

70 80 90 100 110 120

q

[%]

Qa [m3/s]

measurement

calculationTc = 3.25 s

Simulated and realistic influence of a differential pressure transducer time constant

on the deviation between flow measurement results

EXPERIMENTAL VALIDATION OF A DIFFERENTIAL TRANSDUCER SIMULATION METHOD

The influence of a pressure transducer time constant on the flow rate value determined by means of the Gibson method - numerical simulation

Δq – relative discharge value deviationfrom that calculated in case of a zero time constant

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0 0.5 1 1.5 2 2.5 3 3.5

rela

tiv

e d

ev

iati

on

q

[%]

time constant Tc [s]

for the tested course of water flow stopping

for twice accelerated course of water flow stopping

for twice enlarged course of water flow stopping

for twicely shrinked time axis

for twicely extended time axis

THE IMPULSE PIPE LENGTH IMPACT ON THE RESULTS OF FLOW RATE MEASUREMENT

BY MEANS OF THE GIBSON METHOD

-0.04

-0.02

0

0.02

0.04

1 1.2 1.4 1.6 1.8 2

q

[%]

relative length of pressure tube L/Lo [-]

Lo = 17.4 m

Δq – relative flow rate deviation from the value calculated for the reference pipe length L

CONCLUSION

1. A numerical method for analysing the effect of a differential pressure transducer and the impulse piping characteristics on the discharge measurement using the classic version of the Gibson method has been developed and verified experimentally.

2. Exemplary calculations show an influence of the transducer time constant on the accuracy of discharge measurement using the Gibson method. This effect depends on the flow cut-off rate.

3. The calculations performed do not provide any evidence of a significant impact of impulse pipe length on the flow rate measurement results despite observed changes in the pressure variation pattern.

Thank you for your attention!