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 JANICKI Presenter: Janusz Steller
The Szewalski Institute of Fluid-Flow Machinery of the Polish Academy of Sciences Fiszera Street 14, 80-952 Gdansk, Poland
Basic information about THE GIBSON METHOD 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 standards on 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.
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. 2. 3. 4. 5. 6. 7.
Introduction Gibson method principle Dynamic model of a differential pressure transducer and an impulse pipe Computational software Experimental validation of the calculation method The impulse pipe length and the differential pressure transducer time constant impact on the flow rate measurement results Conclusion
GIBSON METHOD PRINCIPLE Measuring pressure difference p1-2=p2+gz2–p1-gz1
z1
1 z2
2
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).
THEORETICAL BASIS Equation of motion
p1 gz1 p2 gz2 Pf p1, p2 z1, z2 Pf
L dQ
pressure difference measurement p = p2 – p1
A dt
– mean static pressures; – hydrometric section weight center elevations; – pressure drop caused by friction losses; – water density
2-2 1-1
Discharge value in the initial conditions: tk
A Q0 [p(t ) Pf (t )]dt Qk L t0 p = p2 + gz2 - p1 - gz1 – static pressure difference , Qk – discharge under final conditions, (t0, tk) – time interval
z1
z2
manifold
GIBSON METHOD VERSIONS • Version I is based on direct measurement of pressure difference between two hydrometric cross sections of a pipeline using a pressure differential transducer p
• Version II makes use of separate measurements of pressure changes in two hydrometric cross sections of a pipeline p1
p2
• 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 p1
p2
Exemplary application Dychow Hydropower Plant
Exemplary application Dychow Hydropower Plant
DYNAMIC MODEL OF A DIFFERENTIAL PRESSURE TRANSDUCER Equation of the first order inertial component: Tc
dy y(t ) kx(t ) dt
Operational transmittance: G(s)
Y ( s) k X (s) sTc 1
Notation Tc k x y t s X Y
– time constant, – factor of proportionality (gain), – input signal, – output signal, – time, – Laplace transform variable, – Laplace transform of input signal, – Laplace transform of the output signal.
Diagram of a differential pressure transducer X - input signal (pressure difference)
k Tc s 1
Y – electric output signal
Comparison of the realistic and simulated response of the Rosemount 1151 smart transducer to a stepwise pressure variation 1 measurement
step function response h(t) [-]
simulation
0.8 Tc = 3.25 s
0.6
0.4
Tc = 0.85 s
0.2 Tc = 0.25 s
0 0
1
2
3 time t [s]
4
5
DYNAMIC MODEL OF AN IMPULSE PIPE The equations of continuity and motion: after Laplace transformation
1 p V 0 2 a t x
1 p V 32 g sin V 0 2 x t D
V ( x, s) 1 2 sp( x, s) 0 x a
1 p( x, s) Kh sV ( x, s) 0 x
NOTATION x t V p D
– length coordinate along the pipe axis, – time, – averaged liquid velocity, – static pressure, – internal pipe diameter,
– liquid density, g – acceleration of gravity, a – pressure wave speed, – dynamic liquid viscosity, – angle of pipe inclination.
BLOCK DIAGRAM OF AN IMPULSE PIPE MODEL p(0,s)
V(0,s)
1 + a
+ – f
Kh 2s
Kh L 2a
e
+ f
L s a
e
+ +
L s a
e
K L h 2a
e
+ – f
Kh 2s
– f
p(L, s) erL p(0, s) zcerLV (0, s) zcV (L, s)
+
V(L,s) +
Relationships between the input and output quantities in the Laplace domain
p(0, s) erL V (0, s) p(L, s) erLV (L, s) zc zc
p(L,s)
+
a
r
zc
s( K h s) a
(Kh s) r
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.
FLOW CHART OF THE DEVELOPED COMPUTATIONAL CODE p1
p2
Pipe 1
Pipe 2
+
pin
1 Tc s 1
pout
– f Differential pressure transducer
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.
EXPERIMENTAL VALIDATION OF A DIFFERENTIAL TRANSDUCER SIMULATION METHOD
Dychow Hydropower Plant 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 q [%]
0 -0.5
measurement
-1
Tc = 0.2 s
calculation
-1.5 60
70
80
90
100 Qa
110
120
[m3/s]
q [%]
0 -0.5
measurement
Tc = 0.85 s
calculation
-1 -1.5 100
105
110
115 Qa
q [%]
0
120
measurement
Tc = 3.25 s
calculation
-0.5
125
[m3/s]
-1 -1.5
70
80
90
100 Qa
110
120
[m3/s]
Simulated and realistic influence of a differential pressure transducer time constant on the deviation between flow measurement results
The influence of a pressure transducer time constant on the flow rate value determined by means of the Gibson method - numerical simulation
0 for twice enlarged course for twicely extended time of water flow stopping
relative deviation q [%]
-0.2
-0.4
-0.6
for the tested course of water flow stopping for twice accelerated course
-0.8
for twicely shrinked time axis of water flow stopping
-1
-1.2 0
0.5
1
1.5 2 time constant Tc [s]
2.5
Δq – relative discharge value deviation from that calculated in case of a zero time constant
3
3.5
axis
THE IMPULSE PIPE LENGTH IMPACT ON THE RESULTS OF FLOW RATE MEASUREMENT BY MEANS OF THE GIBSON METHOD 0.04
q [%]
0.02 0 -0.02
Lo = 17.4 m
-0.04 1
1.2
1.4 1.6 1.8 relative length of pressure tube L/Lo [-]
Δq – relative flow rate deviation from the value calculated for the reference pipe length L
2
CONCLUSION 1.
2. 3.
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. 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. 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.
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