Experimental verification of the kinetic differential pressure method for flow measurements

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  • This article was downloaded by: [Linkping University Library]On: 19 August 2014, At: 18:57Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK

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    Experimental verification of the kinetic differentialpressure method for flow measurementsAyaka Kashimaa, Pedro J. Leeb, Mohamed S. Ghidaouic & Mark Davidsonda PhD Student, Department of Civil and Natural Resources Engineering, University ofCanterbury, Private Bag 4800, Christchurch 8140, New Zealandb Senior Lecturer, Department of Civil and Natural Resources Engineering, University ofCanterbury, Private Bag 4800, Christchurch 8140, New Zealand Email:c (IAHR Member), Professor, Department of Civil and Environmental Engineering, HongKong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Email:d (IAHR Member), Associate Professor, Department of Civil and Natural ResourcesEngineering, University of Canterbury, Private Bag 4800, Christchurch 8140, New ZealandEmail:Published online: 01 Oct 2013.

    To cite this article: Ayaka Kashima, Pedro J. Lee, Mohamed S. Ghidaoui & Mark Davidson (2013) Experimental verificationof the kinetic differential pressure method for flow measurements, Journal of Hydraulic Research, 51:6, 634-644, DOI:10.1080/00221686.2013.818583

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  • Journal of Hydraulic Research Vol. 51, No. 6 (2013), pp. 634644http://dx.doi.org/10.1080/00221686.2013.818583 2013 International Association for Hydro-Environment Engineering and Research

    Research paper

    Experimental verification of the kinetic differential pressure method for flowmeasurementsAYAKA KASHIMA, PhD Student, Department of Civil and Natural Resources Engineering, University of Canterbury, Private Bag4800, Christchurch 8140, New ZealandEmail: ayaka.kashima@pg.canterbury.ac.nz (author for correspondence)

    PEDRO J. LEE, Senior Lecturer, Department of Civil and Natural Resources Engineering, University of Canterbury, Private Bag4800, Christchurch 8140, New ZealandEmail: pedro.lee@canterbury.ac.nz

    MOHAMED S. GHIDAOUI (IAHR Member), Professor, Department of Civil and Environmental Engineering, Hong KongUniversity of Science and Technology, Clear Water Bay, Kowloon, Hong KongEmail: ghidaoui@ust.hk

    MARK DAVIDSON (IAHR Member), Associate Professor, Department of Civil and Natural Resources Engineering, University ofCanterbury, Private Bag 4800, Christchurch 8140, New ZealandEmail: mark.davisdon@canterbury.ac.nz

    ABSTRACTThe accuracy of the kinetic differential pressure (KDP) method for measuring unsteady flow in pressurized fluid conduits is investigated in this paper.The method is based on the linearized one-dimensional governing equations for pipe transient flow and utilizes two pressure measurements to estimatethe unsteady flow. Experiments conducted on the pipeline aparatus in the hydraulics laboratory at the University of Canterbury are used to test andvalidate the operation of the technique. The validation is conducted over a range of flow Reynolds numbers and flow pertubation sizes. The resultsshow that the method can accurately capture flow pulses in a system within the range of Reynolds number considered in this study. It was also foundthat the KDP flow measurement method is sensitive to the accuracy of the input parameters and numerical distortions were introduced into the flowresponse when the input parameters were incorrect. A robust method for correcting these distortions is required before this method can be applied ingeneral flow measurement applications.

    Keywords: Flow measurements; prediction accuracy; transfer matrix; transient flow; unsteady flow

    1 Introduction

    The need to measure rapidly changing flows occurs in variouscommercial applications, for example, in the control of pipelinesystems with piston pumps or fast switching valves, the monitor-ing of pharmaceutical processes with fast-batch filling systemsand for maintaining the optimum fuel to air ratio in internal com-bustion engines (Clark and Cheesewright et al. 2006, Unsal et al.2006, Brereton et al. 2008, Manhartsgruber 2008). In all theseexamples, the rapidly changing flow occurs in the form of asharp flow pulse superimposed onto a system base flow. Theability to measure rapid flow changes is also important for theresearch efforts into unsteady friction, pipewall viscoelasticity,

    fluidstructure interaction, aswell as the application of fluid tran-sient signals for pipe defect detection (Rachid and Mattos 1998,Pezzinga 2000, 2009, Wang et al. 2002, 2005, Lee et al. 2005,YongLiang and Vairavamoorthy 2005, Soares et al. 2008, Fer-rante et al. 2011). While pressure transducers are available forthe accuratemeasurement of rapid pressure changes in a pipeline,a non-intrusive device capable of measuring flow at the transienttime scale is not commercially available and experimental studiesin these areas have been largely limited to pressuremeasurementsalone. The ability to non-intrusivelymeasure discharge in rapidlychanging flowswill increase the amount of information availablefrom each experiment, which will have a significant impact onresearch in the field.

    Revision received 3 July 2012; accepted 19 June 2013/Open for discussion until 30 June 2014.

    ISSN 0022-1686 print/ISSN 1814-2079 onlinehttp://www.tandfonline.com

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  • Journal of Hydraulic Research Vol. 51, No. 6 (2013) Experimental verification of the kinetic differential pressure method 635

    Flow pulses from a piston pump, for example, can be directlymeasured from the displacement of the piston pump but usingthis approach the flow can only be determined at the pis-ton location and at the time of signal generation. Also thechanges imposed on the flow pulses as it travels through thesystem as well as subsequent system reflections cannot becaptured.

    Magnetic flow meters are known to give accurate flowmeasurements (Doebelin 1990, Clark and Cheesewright 2006,Catania and Ferrari 2009). The operational principle of mag-netic flow meters is based on Faradays law of induction whichstates that the voltage induced as a conductive fluid passes atright angles through a magnetic field proportional to the velocityof that fluid. Magnetic flow meters consist of magnetic coils togenerate a magnetic field. As a conductive fluid flows throughthe magnetic field, a voltage is induced which is measured bymeter electrodes. Magnetic flow meters can achieve a responsetime of around 30ms and do not create any obstructions to theflow. However, they need to be inserted at the point of flowmea-surement within the pipeline system and the size of the meterincreases with the pipe size. Furthermore, since Faradays lawof induction only applies to conductive fluids, working fluidsfor magnetic flow meters are limited by their conductivities.Some commercial magnetic flow meters can deal with fluidswith conductivities as low as 0.5S/cm, however, gasoline hasa conductivity of 108 S/cm and cannot be measured by mag-netic flow meters (Doebelin 1990, Catania and Ferrari 2009).The pipeline material must also be non-conductive and metallicpipes require a non-conductive rubber liner installed for thesemeters to operate accurately.

    A clamp-on ultrasonic flow meter is the most non-intrusiveflow meter. Pressure disturbances in the fluid propagate at avelocity that is the sum of the fluid flow velocity and the pres-sure wave speed (Doebelin 1990), with the disturbance travellingfaster in the direction of the flow. Ultrasonic flow meters consistof a pair of sending and receiving transducers externally attachedto the pipeline which measure the time required by an ultra-sonic wave to travel between the transducers. The time requiredfor the wave to travel in one direction will be different to thetime required to travel in the opposite direction because of thefluid flow velocity and this time difference is used to determineflow rate (Catania and Ferrari 2009). Commercial clamp-on flowmeters can be applied to a range of pipe materials and are mostaccurate when the ratio of the pipe diameter to wall thickness isgreater than 10. Despite their non-intrusive property, clamp-onultrasonic flowmeters generally has a response time of hundredsof milliseconds, which is well above the relevant transient flowtime scales.

    An ideal flow meter should be small in size, and causes mini-mum disruption to the existing flow and has a fast response time.Unfortunately, no commercial flow meter is capable of captur-ing rapidly changing unsteady flow without placing significantrestrictions on the pipe material, fluid content and cost. Alter-natively, a number of attractive techniques exist in the research

    literature that utilizes the behaviour of unsteady flows within thepipeline to determine the flow rate.

    The kinetic differential pressure (KDP) method is one exam-ple of a pressure-based flow measurement method that exists inthe literature. Pressure transducers are well-developed deviceswith high response rates. Modern pressure sensors are physi-cally small and can be flush mounted into the pipe wall causingminimal disturbance to the flow. Flow rates can be determinedfrom themeasuredpressure at the same sampling frequency as theoriginal pressure signals using relationships between theflowandpressure in the pipe. There are a number of pressureflow rela-tionships that have been applied in the literature. The Joukowskyequation (Eq. 1) describes the pressure change in response to achange in flow in a pipe and is given as

    H = agA

    Q (1)

    where H and Q are changes in head and flow rate, respec-tively, a is transient wave propagation speed, A is cross-sectionalarea of the pipe andg is acceleration due to gravity. The sign in theequation takes into account whether the measurement positionis located upstream or downstream of the source of the inducedflow change. The positive sign is used when the measurementpoint is downstream of the flow change and the negative sign isused when themeasurement point is located upstream of the flowchange (Chaudhry 1979, Wylie and Streeter 1993). In typicaltransient signals where the trace consists of a myriad of bound-ary and system reflections, the sign of the Joukowsky equationfor different parts of the signal will change depending on thesource of the reflection and its application becomes too complexfor practical purposes.

    Catania and Ferrari (2009) proposed a relationship betweenthe unsteady pipe flow and pressure head by determining thetotal unbalanced force acting on a control volume. After takingfluid compressibility into account, this net force is equated to theacceleration of the fluid within the control volume and the mea-surement of the pressure difference across the control volume inreal time allows the calculation of the flow change from an initialvalue. The calculation of the flow rate in this method assumessmall spatial variations in head and flow and is most valid forsmall spacings between the pressure transducers.

    Yokota et al. (1992) proposed a method for estimating theunsteady flow rate which theoretically has no limitation on themeasurement spacing. The technique is based on the work donebyDSouza andOldenburger (1964) andChaudhry (1979)wherethe dynamic relationship between pressure and flow rate in theLaplace and frequency domains are derived. The importance ofthe knowledge of system parameters such as the system wavespeed and fluid viscosity when using the method in Yokota et al.(1992) was pointed out byManhartsgruber (2008) who placed anadditional pressure sensor for identification of these parameters.Washio et al. (1996) extended this method and called it theKDP method to allow additional flexibility in the choice of the

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  • 636 A. Kashima et al. Journal of Hydraulic Research Vol. 51, No. 6 (2013)

    flow prediction point. Using the KDP method the unsteady flowrate can be determined at a point where no pressure measurementis taken. This property is useful when there is not enough space atthe point of interest or where the point of interest is in a sensitivelocation where any flow disturbance, even minimum ones fromflushed faced pressure sensors, are not permitted. Furthermore,unlike the method developed in Catania and Ferrari (2009), themodel in Washio et al. (1996) gives flow rates at a point in thepipeline instead of a spatial average flow across the control vol-ume. There are few restrictions on the placements and the spacingbetween pressure sensors, meaning this method can utilize pres-sure sensors that already exists within the system, reducing theeffort for implementation.

    While the KDP method has good potential, a detailed investi-gation of the accuracy of the method has not be conducted in theliterature. Two possible sources of error can be identified in theapplication of the KDP method in reality (Kashima et al. 2012).One is the error from the linear approximation of the governingequation at the core of the technique and the other is the errorwhich represents the inability of the one-dimensional model forreplicating real transient behaviour. In Kashima et al. (2012), itwas found that the error from the linear approximation was in theorder of 1% for non-resonant flow conditions. This paper con-tinues the verification of the KDP method by investigating theerror when the technique is applied in a real pipeline system.

    2 Derivation of the KDP flow measurement method

    The one-dimensional governing equations for unsteady pipeflows are given in Eqs. (2) and (3) and these equations assumeelastic pipe behaviour, weakly compressible fluids and flowvelocities that are significantly smaller than the pressure wavespeed such that the advective terms may be ignored.

    1gA

    Qt

    + Hx

    + hf = 0 (2)

    a2

    gAQx

    + Ht

    = 0 (3)

    where x is the distance along the pipeline, Q the instantaneousdischarge, t time, hf the friction loss term, D the pipe diameter,andH the instantaneous piezometric head at the centreline of thepipe above the specified datum. The instantaneous flow and headin Eqs. (2) and (3) can be assumed to be composed of two parts

    Q = Q0 + qv (4)H = H0 + h (5)

    where Q0 is the time-averaged mean discharge, q the time-varying discharge about the mean state, H0 the time-averagedmean pressure head, and h the time-varying pressure head aboutthe mean state. Equations (2) and (3) can be linearly approxi-mated and combined to give the transfer matrix equation which

    Figure 1 Pipe section which is further divided into two subsectionsa and b

    relates the head and flow conditions on either side of a pipesubsections. For example, the transfer matrix relationship acrosssections a and b in Fig. 1 can be given as (Chaudhry 1979)

    (h2()q2()

    )=

    [fa11 fa12fa21 fa22

    ] (h1()q1()

    )(6a)

    (h3()q3()

    )=

    [fb11 fb12fb21 fb22

    ] (h2()q2()

    )(6b)

    where h and q are fluctuating head and flow responses at aparticular frequency , and the subscript denotes the loca-tion of these responses in Fig. 1. The entries of the matricesare given as fs11 = fs22 = cosh(ls), fs12 = Zc sinh(ls) andfs21 = sinh(ls)/Zc, where l is the length of the pipe section, = (2/a2 + jgAR/a2), j = 1, Zc = a2/jgA asgiven in Chaudhry (1979) and the subscript s is an indicatorfunction which takes on values of a or b. The overall resistanceterm R appears in the propagation operator, , which takes intoaccount both steady and unsteady friction effects

    R = RS + RUR = RS + RU (7)

    whereRS is the resistance term for steady friction = 32/(gAD2)for a laminar flow and RS = fQ/(gDA2) for a turbulent flow, the kinematic viscosity of the fluid, f the steady-state frictionfactor, and RU the resistance term for unsteady friction.

    The unsteady resistance term,RU , is given as (Vtkovsk et al.2003)

    RU = 4jgA 0

    e(jD2/4)W ( ) d (8)

    where W is weighting function and is a dummy variable.For laminar flow, the weighting function,W , is computed usingthe weighting function model presented in Zielke (1968). Theunsteady friction model proposed in Vardy and Brown (1996) isemployed for smooth pipe turbulent flow. Details of these mod-els in the frequency domain are given in Vtkovsk et al. (2003).Note that the resistance termmaybeneglected provided the trans-ducer spacing is small. The matrices of Eqs. (6a) and (6b) arecombined in the KDP method to produce an explicit expressionfor the flow pertubation at a point (point 3) as a function of twomeasured pressure fluctuations in the pipeline (points 1 and 2).The use of two transfer matrices in the KDP method providesflexibility in the flow prediction point as it allows the point tobe located away from the points of pressure measurement. The

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  • Journal of Hydraulic Research Vol. 51, No. 6 (2013) Experimental verification of the kinetic differential pressure method 637

    equation for the flow perturbation at point 2 is derived fromEq. (6b) as

    q2 = fa21h1 + fa22(h2 fa11h1

    fa12

    )(9)

    and the flow rate at point 3 can be calculated by combiningEqs. (6b) and (9) resulting in

    q3 = fb21h2 + fb22[fa21h1 + fa22

    (h2 fa11h1

    fa12

    )](10)

    Equation (10) shows that the time-varying flow component at apoint can be calculated from two pressure measurements, h1 andh2, spaced la apart in the pipeline.

    There are a number of requirements for implemention of theKDP method. First, users must know the conditions of the pipesections a and b (Fig. 1) prior to flow prediction, as well asthe system parameters such as the system wave speed, distancesbetween pressure transducers and the pipe diameter. In the subse-quenct experimental verification, only the first pulse of the entiretransient trace is used. However, the method also works in thepresence of boundary reflections provided all the pressure trans-ducers for the method detect the same reflections. This secondrequirement is owing to the assumption that, when going fromthe time to the frequency domain, the transferred signal in timerepeats itself indefinitely. Hence, if a part of signal was missingfrom one of the measured traces, the missing part would not beproperly acknowledged by the method.

    3 Laboratory apparatus

    The experimental verification of the KDP method was carriedout on the pipeline system in the hydraulics laboratory at theUniversity of Canterbury. The schematic of the pipeline systemis shown in Fig. 2 and consists of a stainless steel pipeline of41.6m length and a diameter of 22.25mm. The pipe is boundedby pressurized tanks that are part filled with water and wherethe pressures within each tank are maintained through the injec-tion of compressed air. The pressure in the tanks is adjusted tocreate laminar and turbulent flow conditions. The inline valve atthe downstream end of the system is closed to establish a staticsteady-state condition.

    Controlled flow pertubations for the validation of the KDPmethod are introduced using two hydraulic devices: an electron-ically controlled solenoid valve and a manually operated sidedischarge valve which are located 8.5m from the downstreamreservoir. Example pressure traces created by the solenoid valveand the side dicharge valve are presented in Fig. 3a and 3b,respectively. The time t on the x-axis is non-dimensionalized bythe period of the pipeline system, t = t/T (T = 2L/a or 4L/awhere L is the pipe length). The pressure head on the y-axis isdivided by the head at the generator to give h. The solenoid valvehas a flow diameter of 1.6mm and the side discharge valve has aflow diameter of 8mm. The introduced discharge perturbationsare in the form of sharp pulses which contain a wide spectrum offrequencies for the rigorous testing of KDPmethod. This type ofperturbation is encountered inmany real life situations, includingindustrial batch filling processes as well as internal combustionengines. The pulse from the solenoid valve is created by rapidlyopening the valve followed immediately by a sharp closure, witha pulse duration in time of 8ms. The solenoid valve is usedunder static and laminar flow conditions. In turbulent flow, adischarge pulse is created from the manual operation of a sidedischarge valve which is placed close to a reservoir boundary,in this case the downstream boundary. The side discharge valveis initially open and then rapidly shut, creating a high-pressurewave that propagates away from the valve in both directions.When the wave front moving downstream implinges upon thereservoir boundary it is reflected as a pressure restoring wavewhich moves upstream, following the high-pressure wave. Thesum total of these two waves is a pressure pulse. The duration ofthe pulse created in this way is 24ms. While the pulse createdby the side discharge valve is slower than that from the solenoidvalve, it has a larger magnitude and is suitable in cases of largersystem base flow.

    Pressure traces for theKDPmethod aremeasured using piezo-electric transducers located 20.8 and 26.9m from the upstreamreservoir and the data is collected at the sampling frequency of10 kHz. The pressure sensors are accurate to 1% of the measuredpressure. The distance between the transducers and the time dif-ference between the two measured pressure signals were used todetermine the system wave speed. The experimental verificationof the KDP method is conducted with and without a steady baseflow. The flow Reynolds number ranges from 325.6 to 53374.9.

    Figure 2 The schematic of the laboratory pipeline system

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  • 638 A. Kashima et al. Journal of Hydraulic Research Vol. 51, No. 6 (2013)

    Figure 3 (a) Pressure signal generated by the solenoid valve and (b) pressure signal generated by the side discharge valve

    The accuracy of the KDP method is given as the absolute dif-ference between a measured flow response and a predicted flowresponse by the KDP equations. The measured flow response isobtained from the pressure reponse measured at the generatorthrough the Joukowsky equation. The system wave speed insidethe equation is very sensitive to the air content of fluid and dif-ficult to obtain its true value. The error in the wave speed leadsto the error in the height of the flow pulse. In order to minimizethe error, the pulse height is adjusted according to the measureddischarge volume out of the generator. In the case of the solenoidvalve, a transparent tube is connected to the valve outlet. A risein the water height inside the vertical tube and the tube diameterprovide the volume of fluid released by the valve operation. Therise was measured by a vernier scale and the maximum standarddeviation of themeasurementswas 2.0%.With the side dischargevalve, the volume is given by themeasuredmass of the dischargeout of the valve. The standard deviation of themassmeasurementwas 2.3%.

    The accuracy of the KDP is quantified by three norms. Thedifference in the area under the flow pulse signal in the time pro-vides the error in the volumetric measurement of the discharge,EVolume. The error in capturing the maximum flow pulse ampli-tude is given the symbol, EMax. Finally, the difference in thespectral content of the predicted response describes the errorin the shape of the flow pulse profile, EProfile and it is given asa root mean sum of the error across all the frequency compo-nents of the signal. All three errors are given relative to the trueresponse.

    4 Accuracy of the KDP method for measuring rapid flowchanges

    Previous studies on theKDPmethod included only unsteady fric-tion for laminar flow (Washio et al. 1996) and the technique wasonly studied under laminar flow condition. This study providesthe experimental investigation of the KDP method under static,laminar and turbulent flow conditions with the flow Reynoldsnumbers ranging up to the smooth pipe turbulent zone.

    4.1 Static steady-state condition

    Under the static steady-state condition, four different pulse sizeswere used: size 1 = 1.4 105 m3/s, size 2 = 1.5 105 m3/s,size 3 = 1.9 105 m3/s and size 4 = 2.4 105 m3/s. Theseflow rates are average flow rates out of the solenoid valve whichare estimated from the measured discharge volume and the pulseduration. The choice of the pulse size was governed by the limi-tations of the solenoid valve. Errors are summarized in Table 1.The results show that the average error across all pulse sizes is inthe order of 0.1%. The KDPmethod captured the maximum flowrate well but the error in the pulse profile was relatively large forall pulse sizes. The largest error in these tests was 2.0% for thebiggest pulse size and the errors were found to generally increasewith the pulse magnitude.

    The predicted responses for each pulse size are shown inFig. 4. They are compared with the true responses indicated bythe grey line. The time t on the x-axis is non-dimensionalizedby the period of the pipeline system, t = t/T (T = 4L/a, whereL is the pipe length). The flow rate on the y-axis is divided bythe respective average flow rate to give q.

    4.2 Laminar and turbulent flow conditions

    Five flow scenarios with different Reynolds numbers were testedfor each flow regime. The pulse size in the tests range from 1.85to 43.9% of the steady-state flow. The flow prediction from theKDP method is compared with the true flow pulse in Fig. 5 andthe errors in theKDPflowprediction are shown in Tables 2 and 3.

    Table 1 Summary of percentage errors in the flow predictions ofdifferent flow pulse magnitudes

    Pulse size EVolume EMax EProfile

    Size 1 3.78 103 3.37 103 8.14 103Size 2 8.10 103 1.40 103 8.20 103Size 3 3.63 103 2.59 103 7.25 103Size 4 4.10 103 1.70 103 2.00 102

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  • Journal of Hydraulic Research Vol. 51, No. 6 (2013) Experimental verification of the kinetic differential pressure method 639

    Figure 4 Predicted flow response (black solid line) compared with the reference flow response (grey line) in the time domain. (a) Time domaingraph of size 1 flow pulse, (b) time domain graph of size 2 flow pulse, (c) time domain graph of size 3 flow pulse and (d) time domain graph of size4 flow pulse

    Over all, the errors are larger than the static case and againthe KDP prediction is shown to have the greatest error in theprediction of the shape of the flow profile, with a maximumerror of 6.41% under turbulent flow condition. The errors inthe prediction was found to generally increase with Re withthe average error in the order of 0.1% for the laminar flow and1% for the turbulent flow. The current set of results has shownthat the KDP method can measure rapid changes in flow withacceptable accuracy for the range of Reynolds number consid-ered in this study. It is also worth mentioning that the accuracy ofthe method degrades with the transducer spacing as was shownnumerically in Kashima et al. (2012). The 6-m spacing usedin the experiments was the maximum spacing possible in thelaboratory.

    5 Effect of input parameter error on the accuracy of theKDP method

    The operation of the KDP method requires the estimation of anumber of system and flow parameters and these are used as theinputs to the model. In real pipelines, the input parameters willoften contain errors and their effect on the accuracy of themethodforms an important consideration in this study.

    The sensitivity of the KDP method to the accuracy of theinput parameters was tested in a numerical pipeline system.The pipeline is 2000m long and it is bounded by constant head

    reservoirs giving a head difference across the pipeline of 30m.The pipe diameter is 0.3m and a flow perturbation source islocated at the middle of the system. The pressure response ismeasured at points located 700 and 900m from the upstreamreservoir. The wave speed of the system is 1000m/s. The flowpertubation is a flow pulse of a magnitude of 1.0% of thesteady base flow. The base flow has the Reynolds number of7.3 105. Transient responses of the system are produced by afinely discretized method of characteristics model which dividesthe pipe into 1000 reaches. This numerical pipeline system ismuch larger than the laboratory system used in the previoussection, thereby showing the applicability of the method on largesystems.

    To investigate the sensitivity of theKDPmethod, Eq. (6a) wasrearranged to give the following equation for the flow responseat point 2 (Fig. 1):

    q2 = h1csc h(mla)/Zc h2 coth(mla)/Zc (11)

    System parameters involved in this calculation are the pipe diam-eter (D), the pipe friction factor (f ), the base flow (Q), the systemwave speed (a) and the transducer spacing (la). The subscript ais dropped from here on for simplicity. The sensitivity of theKDP method was studied by differentiating Eq. (9) with respectto each input parameters and the resultant equations are given inappendix. Values obtained from Eqs. (A1)(A5) indicate the sig-nificance of each input parameter. For a valid assessment, these

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  • 640 A. Kashima et al. Journal of Hydraulic Research Vol. 51, No. 6 (2013)

    Figure 5 Predicted flow response by KDP method in various base flow condition

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  • Journal of Hydraulic Research Vol. 51, No. 6 (2013) Experimental verification of the kinetic differential pressure method 641

    Table 2 Summary of average errors in the flow predictions in variouslaminar flows

    Test case EVolume EMax EProfile

    Case (a) (R = 327.9) 7.80 103 2.10 103 1.61 102Case (b) (R = 593.4) 6.90 103 3.10 103 1.14 102Case (c) (R = 845.2) 5.70 103 6.90 103 1.48 102Case (d) (R = 1164.5) 1.04 102 3.20 103 1.67 102Case (e) (R = 1641.2) 1.31 102 1.07 102 2.11 102

    Table 3 Summary of average errors in the flow predictions in variousturbulent flows

    Test case EVolume EMax EProfile

    Case (f) (R = 28933.8) 1.87 102 2.04 102 4.27 102Case (g) (R = 36309.8) 1.32 102 1.10 103 5.31 102Case (h) (R = 42624.7) 8.40 103 9.10 103 3.92 102Case (i) (R = 48248.6) 3.83 102 1.77 102 6.41 102Case (j) (R = 53374.9) 1.87 102 1.93 102 5.48 102

    Table 4 Summary of sensitivity analysis

    Input parameter EVolume

    Pipe diameter, D 1.55 104Friction factor, f 5.62 106Base flow, Q 5.62 106Transducer spacing, l 6.68 102System wave speed, a 6.67 102

    values were normalized by the value of the input parameters andthe results are summarized in Table 4.

    The results show that the influence of the transducer spac-ing and the system wave speed was in the order of 24 timeslarger than other input parameters and the robustness of the KDPequations against estimation errors in the pipe friction factor andthe base flow through the system are evident. Although it is notshown in the paper, the sensitivity of Eq. (11) was also testednumerically and it reached the same conclusion as the aboveanalytical study.

    Predicted responses in the cases of the incorrect pipe diameterand transducer spacing are presented in Fig. 6a and 6b, respec-tively. The predicted response with correct input parameters isrepresented by the grey line and the broken line indicates the pre-diction with the incorrect input parameter. In each graph, the tipof the flow pulse is enlarged and presented in a separate windowfor easier comparison.

    Figure 6 clearly shows that errors in the pipe diameter and thetransducer spacing have a different impact on the predicted trace.The error in the pipe diameter led to a change in the character-istics of the pulse. It was found that the incorrect friction factorand the base flow also affect the pulse in a similar way. On theother hand, the error in the transducer spacing caused minimalchange to the pulse characteristics, but instead, it gave rise tonumerical contamination which repeats regularly for the rest of

    Figure 6 (a) Predicted flow response with incorrect pipe diameterand (b) predicted flow response with incorrect transducer spacing. Thepredicted flow response with incorrectly assumed input parameter ispresented by the broken line while the grey line shows the predictedresponse with correctly assumed input parameters

    the response. The same phenomenon was observed in the casewith the incorrect wave speed and further study found that thenoise profile was identical for the same ratio of the transducerspacing to the wave speed.

    The periodic nature of the noise points to the hyperbolicsine and cosine functions of the transfer matrix as the sourceof the contamination. The transfer matrix method simulates thetransient behaviour by first decomposing the signal to a set offrequency components and then transfers them a given distancealong the pipe and at a givenwave speed in the formof hyperbolicsine and cosine waves. The system wave speed and the length ofthe pipe is therefore central to the transfer matrix model and areused to characterize the fundamental behaviour of the pipe seg-ment of interest. The mismatch between the true and estimatedsystem characteristics is a result of the discrepancy between theobserved travel time for a signal to go from one pressure trans-ducer to another in the measured pressure traces and the traveltime calculated from the input parameters, and it manifests itselfas numerical contamination in the predicted flow response.

    The relationship between the error in the travel timet(= LT/a, where LT is the transducer spacing) and the noiseprofile was investigated through two test scenarios. In the firstscenario, noise profiles from two different travel timeswere com-pared.The twodimensionless travel timest = t/T were0.05

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  • 642 A. Kashima et al. Journal of Hydraulic Research Vol. 51, No. 6 (2013)

    Figure 7 (a) Recurrence period of noise. The broken line representsthe noise with the dimensionless travel time of 0.05 and the solid linerepresents the noise with the dimensionless travel time of 0.075. Theminor x-axis grid having a width of 0.01 of dimensionless time wasadded to give a better idea of time scale, (b) noise profileswhen the traveltime is estimated 0.5% (solid grey line), 1.0% (broken black line), 1.5%(broken grey line), 3.0% (dotted black line), 5.0% (solid black line), and10.0% (dotted grey line) greater than the correct value

    and 0.075. In both cases, the travel time was estimated 1.0%greater than the correct value. The second scenario examined theeffect of the varying error in the travel time on the noise profile.In this test, the correct dimensionless travel time was 0.05 and itwas assumed incorrect by 0.5% (solid grey line), 1.0% (brokenblack line), 1.5% (broken grey line), 3.0% (dotted black line),5.0% (solid black line) and 10.0% (dotted grey line) greater thanthe correct value.

    Results from the first scenario are presented in Fig. 7a. Thebroken and solid lines indicate the case with the dimensionlesstravel time of 0.05 and 0.075, respectively. It is observed that,with the dimensionless travel time of 0.05, the noise repeats witha dominant recurrence period tR of 0.0505 which coincides withthe estimated dimensionless travel time.When the dimensionlesstravel time was 0.075, the recurrence period was 0.07575 whichagain agrees with the estimated dimensionless travel time of thepipe segment. The influence of the error in the travel time on thenoise profile is illustated in Fig. 6b which shows that the noisemagnitude increases proportinally to the error in the travel time.

    The results from the two sets of studies support the idea that thenoise is related to disagreement in the estimated and actual traveltimes of the signal.

    The error in the input parameters degrades the quality offlow predictions. The sensitivity analysis showed that the pipediameter, transducer spacing and the system wave speed are keyparameters for accurate flow predictions. In reality, however, thepipe diameter can be measured most accurately among otherparameters and its measurement error is expected to be minimal.The quality of the predicted response therefore hinges on theaccuracy of the estimated characteristic time of the pipe segmentbounded by the pressure transducers. The impact of the noisebecomes more problematic when dealing with continuous sig-nals where the contamination will be superposed on the predictflow trace. Depending on the degree of error in the travel time,the correct transient trace might not be clearly visible. In thispaper, the verification of the KDP method was carried out usingdiscrete pulse signals, commonly seen in the batch processingapplications or fuel injection lines. For these signals, the partsof the flow predictions that is related to the real response can beclearly identified as they do not overlap with the part of the signalaffected by the numerical contamination. The noise can thereforebe removed from the response and does not affect the accuracyof the results. However, it is important to note that this distor-tion of the predicted flow response would need to be addressedbefore the technique can be applied in more complex and longerduration flow measurement situations.

    6 Conclusions

    The KDP method for measuring the unsteady flow has advan-tages in terms of the high sampling rate, minimal disturbanceand low space requirement over other techniques. The use ofpressure transducers means that the existing transducers in thesystemmay be used for flowmeasurements which reduces initialcosts. The unique advantage of the method is its flexibility in theflow measurement point.

    The authors pointed out two sources of error of thismethod: the numerical error due to linear approximation of theone-dimensional governing equation and themodelling error rep-resenting the inability of the one-dimensional model to replicatethe real transient behaviour. The present paper looks into themodelling error using a real pipeline system.

    Experimental verification of the method was carried out usingthe pipeline system at theUniversity ofCanterbury. The accuracyof the KDP method was quantified in terms of three parameters:the accuracy in predicting the discharge volume, the maximumflow and the shape of the flow pulse. The study showed that thethe average error was in the order of 0.1% for the static steady-state and the laminar flow conditions, and 1% for the turbulentflow condition. These results indicate that the KDP method wasable to predict the flow response with acceptable accuracy. Thegreatest error occurred in the shape of the flow profile with amaximum error of 6.41%.

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  • Journal of Hydraulic Research Vol. 51, No. 6 (2013) Experimental verification of the kinetic differential pressure method 643

    The sensitivity of the KDP method was studied. It was foundthat the errors in the pipe diameter, friction factor or the base flowlead to a change in the flow pulse characterics while the errorsin the transducer spacing or the system wave speed contami-nates the predicted response with periodically repeating noise.This numerical noise is a reflection of disagreement between thetheoretical and actual travel time of the pipe segment boundedby the transducers. The noise amplitude and its recurrence fre-quency are a function of the degree of error in the travel time.The effect of noise is not severe when dealing with discrete pulsesignal and the current study showed the validity of themethod forthis signal type. For continuous signal for long-period flowmea-surements, the input parameters, especially the travel time of apipe sectionmust carefully bemeasured to avoid large numericalcontamination.

    Appendix

    Equations used for the sensitivity analysis are

    q2D

    = lZ2c

    ZcD

    [h1csc h(l) h2 coth(l)]

    + lZc

    12

    D[h1 coth(l)csc h(l)+ h2csc h2(l)]

    (A1)

    q2f

    = lZ2c

    Zcf

    [h1csc h(l) h2 coth(l)]

    + lZc

    12

    f[h1 coth(l)csc h(l)+ h2csc h2(l)]

    (A2)

    q2Q

    = lZ2c

    ZcQ

    [h1csc h(l) h2 coth(l)]

    + lZc

    12

    Q[h1 coth(l)csc h(l)+ h2csc h2(l)]

    (A3)

    q2a

    = lZc

    12

    a[h1 coth(l)csc h(l)+ h2csc h2(l)]

    (A4)

    q2l

    = Zc

    [h1 coth(l)csc h(l)+ h2csc h2(l)] (A5)

    where

    = 2 (A6)

    a= 2

    a3

    (2 + jfQ

    D

    )(A7)

    D= jfQ

    a2D2(A8)

    f= jQ

    a2D(A9)

    Q= jf

    a2D(A10)

    ZcD

    = 8jgD3

    12jfQjgD4

    (A11)

    Zcf

    = 4jQjgD3

    (A12)

    ZcQ

    = 4jfjgD3

    (A13)

    Notation

    A = cross-sectional pipe area (m2)a = transient wave propagation speed (m/s)D = pipe diameter (m)EVolume = error in the volumetric measurement of the

    discharge ()EMax = error in capturing the maximum flow rate ()EProfile = error in the shape of the flow pulse profile ()f = friction factor ()g = acceleration due to gravity (m/s2)H = instantaneous pressure head at pipe axis

    above specified datum (m)H0 = mean pressure head (m)H = change in pressure head (m)h, h = fluctuating pressure head (m)j = (1)1/2 ()L = pipe length (m)LT = distance between two transducers (m)l = length of arbitrary pipe section (m) = kinematic viscosity (m2/s)Q = instantaneous discharge (m3/s)Q0 = time-averaged discharge (m3/s)Q = change in discharge (m3/s)q, q = time-varying discharge (m3/s)R = Reynolds number (-)RS = resistance term for steady friction ()RU = resistance term for unsteady friction ()T = period of pipeline system = 2L/a for a

    reservoirreservoir system, 4L/a for areservoir-closed valve system (s)

    t = time (s) = dummy variable for weighting function ()tR = recurrence period (s)W = weighting function ()x = distance (m)Zc = characteristic impedance () = propagation constant () = angular frequency (rad/s)

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    IntroductionDerivation of the KDP flow measurement methodLaboratory apparatusAccuracy of the KDP method for measuring rapid flow changesStatic steady-state conditionLaminar and turbulent flow conditions

    Effect of input parameter error on the accuracy of the KDP methodConclusionsNotation

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