White Paper Quantifying And Specifying The Dynamic Repone Of Flowmeter DP Flow ABSTRACT The dynamic repone characteritic of flowmeter are often incompletely or incorrectly pecified. Thi i often the reult of inadequate dynamic teting. Thi paper decribe a robut method for determining the dynamic repone characteritic of flowmeter. The reult of frequency repone tet on a number of flowmeter technologie are preented. A concie way of pecifying the dynamic performance of flowmeter i preented. INTRODUCTION A common method for electing a flowmeter i to compare the performance pecification of the device under conideration. The performance parameter mot often ued in thee comparion i the uncertainty in the meaurement of the flow rate. It i generally pecified by flowmeter manufacturer and i relatively eay to verify in the laboratory under controlled teady tate condition. However, becaue real world flow application are not teady tate, a more meaningful comparion of candidate flowmeter can be made if their repone to flow tranient i included a one of the metric in the evaluation. Such a comparion i often complicated by incomplete and ometime incorrect pecification of the dynamic characteritic of flowmeter. Thi paper will decribe a method for quantifying the dynamic repone of flowmeter a well a a concie way of pecifying their dynamic performance. STEP RESPONSE TESTS The tep repone tet i often ued to determine the dynamic repone characteritic of intrument. In conducting a tep repone tet the input to the device under tet i abruptly changed while the output i monitored. During uch tet it i highly deirable to have a fat reponding, independent device to monitor and provide a true a repreentation of the input a poible. The repone of many intrument to tep input look imilar to that hown in Figure 1 on page 6. For an increaing tep input the output of the device aymptotically approache it final value in a decaying exponential fahion. Thi behavior i characteritic of device whoe time repone can be decribed mathematically by a firt order differential equation. They are called firt order device or firt order lag. A ingle parameter, the time contant, τ, i ufficient to completely decribe the repone of a firt order lag to tranient input. The time contant i the time required for the output of the device to reach approximately 63% of it final value. When the reult from a tep repone tet look imilar to thoe hown in Figure 1 on page 6 it i natural to aume that the behavior can be decribed by one firt order differential equation. That thi i not alway the cae i illutrated in Figure 2 on page 6. Here a tep input and the repone of a device decribed by a pair of firt order lag are hown. Determining the 63% point of the output repone reult in a time contant of approximately 0.67 econd. Uing thi time contant the time repone of thi ingle lag approximation i alo hown in Figure 2 on page 6. The ingle lag approximation i een to be a poor etimator of the time repone of the real device. The underetimation of the order of a flowmeter and reulting incorrect pecification of the dynamic performance ha ramification in flow control application [1, 2]. In addition to underetimating the order of the device, the tep repone tet ha three other eriou hortcoming for teting flowmeter. Firt, pump and valve cannot change intantaneouly. The typical input become a ramp input rather than a tep input. Thi complicate the interpretation of the data and make it even more important to have a fat reponding monitoring device to properly characterize the ramp input. Second, ince many flowmeter make ue of digital circuitry, the www.roemount.com
DP Flow White Paper flowmeter output will exhibit dead time. The timing of the tep input relative to the ampling of the digital circuitry will affect the apparent dead time. Repeated tet need to be performed with the larget meaured value of dead time interpreted a the flowmeter dead time. Finally, tep tet often conit of large amplitude, unidirectional input. Such input are unrepreentative of real world input and make it difficult to identify any aymmetrical or non-linear behavior of the device under tet. Frequency Repone Tet The evaluation of tep repone tet reult in the time domain, while it may eem familiar and intuitive, i not the bet way to quantify the dynamic repone characteritic of flowmeter. A far more reliable method for determining the dynamic repone characteritic of any device i to perform a frequency repone tet. In uch a tet a inuoidally varying input i applied to the device and the output i compared to the input. Figure 3 on page 6 how that the two parameter of interet are the gain and the phae hift. The gain i a meaure of how much the output of the device i amplified or attenuated compared to the input. The phae hift i a meaure of the time lead or lag a the device output follow the input. The gain and phae hift value vary a a function of the frequency of the input. Conequently, it i more convenient to look at the data in the frequency domain. To make the leap from the time domain to the frequency domain require the ue of ome mathematical tool ued in automatic control theory [3,4]. One of thee tool i the Laplace tranform of a function of time, f(t), defined a: F = 0 t ft e t d F ft t = Laplace tranform of f(t) = time function = Laplace tranform variable = time One advantage of uing the Laplace tranform i that it tranform differential equation into algebraic equation making them much eaier to manipulate and olve. Another convenient mathematical concept i the tranfer function, defined a the ratio of the Laplace tranform of the output divided by the Laplace tranform of the input. Uing the Laplace tranform notation the tranfer function of a firt order lag i: TF = Thi tranfer function expreion i characteritic of all firt order device. It i only τ that change from device to device. A final ueful mathematical tool i the Bode plot. The Bode plot i a two-part graph that plot the gain v. frequency and the phae hift v. frequency. For both graph the frequency i plotted on a logarithmic cale. The gain i generally converted to decibel uing the following equation: For all reult preented here the phae hift will be in degree and the frequency will be in Hz. A powerful feature of tranfer function i that when the output of one element erve a the input to another the tranfer function of the two element are multiplied. Becaue of the logarithmic nature of the Bode plot, the gain and phae hift value of the two element are additive. An example of a Bode plot for a device whoe tranfer function i given by two firt order lag and dead time i hown in Figure 4 on page 6 and Figure 5 on page 6. The tranfer function i given by: ------------- 1 1 + TF G db = 20 log10g numerical TF compoite G numerical G db = tranfer function = time contant (typically in econd) = Laplace tranform variable = numerical value of gain = gain in decibel. d e S = ------------------------------------------- 1 + 1 1 + 2 TF compoite d 12 = compoite tranfer function = dead time (econd) = time contant of the two firt order lag (econd) = Laplace tranform variable. 2
White Paper DP Flow It hould be noted that the dead time make no contribution to the gain. It only contribute to phae hift, a fact that ha important ramification in control application [1, 2]. In Figure 4 on page 6 the gain contribution of the two firt order lag add to produce the compoite gain curve. Similarly, in Figure 5 on page 6 the phae hift from the two firt order lag and the dead time add to produce the compoite phae hift curve. Experimental Tet Setup The experimental apparatu ued to conduct frequency repone tet on flowmeter i hown in Figure 6 on page 6. It conited of a recirculating water flow loop with 3-inch pipe. The mechanical part of a 3-inch turbine meter were ued a the reference meter. The pule from the turbine meter were ent to a fat Frequency Voltage converter to produce an analog ignal repreentative of the flow rate. Two identical 3-inch globe-tyle valve with poitioner were intalled downtream of the flowmeter. They were the fatet available indutrial control valve. One wa held contant at 50% open. The poition of the other valve wa modulated to provide the inuoidally varying flow rate. A HP 3563A control ytem analyzer wa ued to generate the inuoidal ignal, to proce the data and to produce the Bode plot data. The analyzer wept the frequency from 0.01 Hz to 10 Hz. The input to the valve wa 12 ma ± 0.5 ma (50% of travel ± 3%). At the low frequency end the nominal flow rate and variation were approximately 96 gpm ± 6 gpm. The control ytem analyzer performed a fat Fourier tranform on the input from the flowmeter under tet and the fat turbine meter/f-v converter to generate the gain and phae hift value. The attenuation of the valve travel caue the inuoidal change in flow rate to top at approximately 2 Hz. The HP3563A continue to weep through the frequency range and perform the FFT in an attempt to generate phae hift and gain data. The reult i that both the gain and phae hift curve become very noiy for frequencie above approximately 2 Hz. The experimental tet etup i decribed in more detail elewhere [5]. Frequency Repone Tet Method and Reult The flowmeter technologie on which the frequency repone tet were run included differential preure/orifice, vortex, electromagnetic and corioli. The tet involved eleven different model of differential preure tranmitter from five manufacturer, ix different model of vortex meter from four manufacturer and three different model of magmeter from two manufacturer. A ingle corioli meter wa alo teted. The ame orifice meter run (β = 0.67) wa ued for all differential preure/orifice meter tet. It wa potulated that ome of the flowmeter technologie would have part of their dynamic repone performance determined by element whoe characteritic would not change a the flowmeter damping wa changed and part that would vary with the damping. An example of thi i the differential preure tranmitter. It wa anticipated that the oil-filled enor would have fixed dynamic. It wa further potulated that the filtering due to the uer-adjutable damping would reult in variable dynamic performance. Conequently, the differential preure/orifice flowmeter were modeled mathematically a two firt order lag and dead time. Since the uer-adjutable damping value wa known there were two parameter to be determined the fixed time contant and the dead time. Similar mathematical model were potulated for the other flowmeter technologie. Multiple frequency repone tet were conducted on each device with the uer-adjutable damping et to different value for each tet. Mathematical model baed on the potulated tranfer function were plotted with the experimental data. The value of the time contant and dead time were adjuted until all of the mathematical model fit the experimental data. The equation ued for the mathematical model were: 1 Gain 1tOrder = 20log10 ------------------------------------ ; 1 + 2 f 2 PhaeShift 1tOrder = Tan 1 2 f (5), (6) Gain DeadTime = 0 PhaeShift DeadTime = d 2 f ; (7), (8) 3
DP Flow White Paper Gain f d = gain (decibel) = frequency (Hz) = time contant and dead time The gain value are in decibel and the phae hift value are in degree. An example of the how the mathematical model fit the experimental data i hown in Figure 6 on page 6 and Figure 7 on page 7. A wa previouly noted, the attenuation of the valve travel at frequencie greater than approximately 2 Hz wa the caue of the noie hown in Figure 6 on page 6 and Figure 7 on page 7(identified a the Sytem Limit ). Thi phenomenon wa oberved in all tet. It wa an artifact of the tet ytem and hould not be conidered repreentative of the behavior of any of thee device. The mathematical model are een to fit the experimental data very well and revealed that the dead time wa 0.050 ec and the fixed time contant wa 0.030 ec. The uer-adjutable damping wa et at the value lited in the legend of Figure 6 on page 6 and Figure 7 on page 7. The tranfer function parameter for all flowmeter teted are ummarized in Table 1 through 4. The nomenclature for deignating the meter in the table i DP, V, M and C for differential preure/orifice, vortex, electromagnetic and corioli, repectively. Thi i followed by a numerical indicator to deignate the manufacturer and then by a letter to differentiate meter from a given manufacturer. For example DP1B indicate the econd differential preure/orifice meter from manufacturer #1. Table 1 on page 8 how the reult for the differential preure/orifice meter tet. Table 2 on page 8 how the reult for the vortex meter, Table 3 on page 9 how the reult for the magmeter and Table 4 on page 9 how the reult for the corioli meter. It hould be noted that for ome of the flowmeter there were dicrepancie between the etting of the uer-adjutable damping value and the value ued in the mathematical model. Thi i probably related to the oftware implementation in thoe particular device. The tet reult how that for all differential preure/orifice meter the dynamic repone i characterized by two firt order lag, one with a fixed time contant, and dead time. Four of the ix vortex meter were characterized by a ingle firt order lag with a time contant that varied with the uer-adjutable damping and dead time. For the other two vortex meter, both from the ame manufacturer, an additional firt order lag with a fixed time contant wa required to fit the experimental data. The magmeter were all characterized by two firt order lag, one having a fixed time contant with the other varying with the uer-adjutable damping and dead time. Since only a ingle corioli meter wa teted it i difficult to generalize the reult. However, the phyic of thee device demand a different type of model to fit the experimental data. The corioli meter teted wa characterized by a critically damped econd order lag with a fixed natural frequency and damping ratio, a variable time contant firt order lag and dead time. The time contant of the firt order lag and the dead time were both found to vary with the uer-adjutable damping. Thee reult can be ummarized in general tranfer function notation uing the following expreion: For differential preure/orifice meter, vortex meter and magmeter: TF = For the corioli meter: TF = d e -------------------------------------------------------- 1 + fixed 1 + adj d e (12) ------------------------------------------------------------------------------------------- n 2 + 2 n + 1 1 + adj TF d fixed adj n (13) = tranfer function = Laplace tranform variable = dead time = fixed time contant = adjutable damping time contant = undamped natural frequency (rad/ec) = damping ratio. 4
White Paper DP Flow SPECIFYING DYNAMIC PERFORMANCE The typical way of pecifying the dynamic performance of flowmeter i to tate the dead time and a range of value for the uer-adjutable damping time contant. The implication of thi are that the dead time i contant and that a ingle firt order lag will adequately decribe the dynamic performance. For mot of the device teted the aumption of a contant dead time i a good one. The econd aumption, that the dynamic performance can be quantified with a ingle firt order lag, will often reult in a mitatement of the dynamic performance. The tet reult how that an additional firt order lag aociated with the mechanical and/or electrical deign of the device contribute to the dynamic performance of many flowmeter. At minimum damping etting thi additional lag i actually the main contributor to the dynamic performance. A more robut and decriptive way of pecifying the dynamic performance i to ue the tranfer function. The tranfer function provide a way of unambiguouly pecifying the dynamic performance of flowmeter. It reveal all of the term that contribute to the dynamic performance of the device. It make clear ditinction between element common to all flowmeter (i.e., uer-adjutable damping) and deign-dependent element (i.e., enor and/or electronic deign characteritic). Thi allow uer to do a complete and comprehenive comparion of flowmeter performance. CONCLUSIONS The difficultie aociated with uing tep repone tet to quantify the dynamic repone characteritic of flowmeter have been addreed. Thee include difficultie in providing a true tep input, difficultie in quantifying the dead time and the tendency to underetimate the dynamic performance by pecifying it with a ingle time contant. The frequency repone tet method ha been hown to be uperior to the tep repone tet. It provide input to the flowmeter that are more repreentative of actual flow application. Furthermore, it provide for eaier quantification of the flowmeter dead time. In addition, the frequency repone tet method make it much eaier to completely quantify the dynamic repone characteritic of flowmeter. The tet reult how that within a given technology the mechanical and electrical deign of the device play a large role in determining the dynamic repone characteritic. The uer-adjutable damping reult in a firt order lag for all flowmeter teted. For differential preure tranmitter the enor mechanical deign and electronic ignal proceing deign contribute to both an additional firt order lag time contant and the dead time of the device. For magmeter and vortex meter an additional firt order lag i affected primarily by the electronic ignal proceing deign. The phyic of Corioli meter lead to a different mathematical model than other flowmeter. In general, the fatet flowmeter were the differential preure/orifice meter, although there wa much variation in repone within thi technology. One of the magmeter had comparable performance to the fatet differential preure/orifice meter. The remaining magmeter, vortex meter and the corioli meter were ignificantly lower in their repone characteritic. It i alo clear that, regardle of the flowmeter technology employed, etting the damping to the minimum value reult in fater repone. Finally, the reult of the frequency repone tet how that the commonly ued ingle time contant method of pecifying dynamic performance i inadequate and inaccurate. A conitent method of pecifying dynamic performance i required to allow uer to comprehenively compare one flowmeter to another within a given type of technology and to make the ame kind of comparion of flowmeter of differing technologie. The tranfer function i preented a a mean of pecifying dynamic performance that would eliminate any ambiguity in the pecification of dynamic repone characteritic and allow the uer to make a more informed deciion when electing flowmeter. 5
DP Flow White Paper REFERENCES 1. Wiklund, David and Peluo, Marco, Flowmeter Dynamic Repone Characteritic, Part 2: Effect in Variou Flow Application, Fifth International Sympoium on Fluid Flow Meaurement, 2002. 2. Wiklund, David and Peluo, Marco, Reducing Proce Variability By Uing Fater Reponding Flowmeter in Flow Control, ISA 2002. 3. Ogata, Katuhiko, Modern Control Engineering, Prentice-Hall, 1970. 4. Lloyd, Sheldon and Anderon, Gerald, Indutrial Proce Control, Fiher Control, 1971. 5. Wiklund, David and Peluo, Marco, Flowmeter Dynamic Repone Characteritic, Part 1: Quantifying Dynamic Repone, Fifth International Sympoium on Fluid Flow Meaurement, 2002. FIGURE 1. Typical repone of a device to a tep input FIGURE 4. Gain part of Bode plot demontrating the addition of gain contribution FIGURE 2. Step repone of dual lag device and ingle lag approximation FIGURE 5. Phae hift part of Bode plot demontrating the addition of phae hift contribution FIGURE 3. Time repone howing attenuation and phae hift of output relative to input. FIGURE 6. Experimental tet etup. 6
White Paper DP Flow FIGURE 7. Gain for a differential preure/orifice meter, experimental and mathematical model data. FIGURE 8. Phae hift for a differential preure/orifice meter, experimental and mathematical model data. 7
DP Flow White Paper Table 1. Tranfer Function Parameter for Differential Preure Tranmitter T d Meter DP2B Meter DP5A Meter DP3A T d Dead Time 0.070 0.00 0.031 0.150 0.00 0.0. 0.300 0.00 0.227 0.20 0.20 0.25 0.25 0.25 0.227 T fixed 0.40 0.40 T fixed 0.50 0.50 T fixed 0.50 0.199 0.010 0.80 0.80 0.200 1.00 1.00 0.199 1.00 0.318 1.60 1.60 2.00 2.00 2.00 0.796 Meter DP2A Meter DP1A Meter DP2C T d T d Dead Time Uer 0.070 0.00 0.005 0.170 0.00 0.02 0.400 0.112 0.079 0.20 0.20 T fixed 0.16 0.16 0.224 0.159 T fixed 0.40 0.40 0.306 0.32 0.32 T fixed 0.448 0.398 0.050 0.80 0.80 0.48 0.48 0.010 0.896 0.796 1.60 1.60 1.00 1.00 1.792 1.768 Meter DP2D Meter DP1B Meter DP3B T d Dead Time Uer Dead Time 0.100 0.00 0.020 0.200 0.00 0.122 0.400 0.00 0.079 0.10 0.10 0.16 0.318 0.50 0.159 T fixed 0.20 0.20 T fixed 0.32 0.455 T fixed 1.00 0.637 0.080 0.40 0.40 0.306 0.48 0.637 0.032 2.00 1.592 0.80 0.80 1.00 1.224 4.00 3.386 1.60 1.60 Meter DP4B Meter DP4A T d T d 0.125 0.10 0.06 0.270 0.20 0.20 0.50 0.50 0.50 0.50 T fixed 1.00 1.00 T fixed 1.00 1.00 0.100 2.00 2.00 0.250 2.00 2.00 Table 2. Tranfer Function Parameter for Vortex Meter Meter V1A Meter V4A Meter V2A T d T d Dead Time Uer 0.170 0.20 0.15 0.300 0.00 0.15 0.630 0.00 0.01 T fixed 0.50 0.42 T fixed 0.50 0.20 T fixed 2.00 1.60 NA 1.00 0.90 NA 1.00 0.40 0.700 4.00 3.50 2.00 1.90 2.00 0.90 T d Meter V3A Meter V3B Meter V2B T d Dead Time Uer 0.250 0.20 0.20 0.400 0.20 0.20 0.630 0.00 0.01 T fixed 0.50 0.50 T fixed 0.50 0.50 T fixed 2.00 1.60 NA 1.00 1.00 0.700 1.00 1.00 0.200 4.00 3.50 2.00 2.00 0.700 2.00 2.00 8.00 7.00 8
White Paper DP Flow Table 3. Tranfer Function Parameter for Magmeter Meter M1B Table 4. Tranfer Function Parameter for Corioli Meter Meter M1A Dead Time Uer Dead Time Uer 0.070 0.01 0.05 0.220 0.01 0.05 T fixed 0.25 0.25 T fixed 0.25 0.25 0.05 0.50 0.50 0.05 0.50 0.50 1.00 1.00 1.00 1.00 2.00 2.00 2.00 2.00 Meter M2A Dead Time Uer 0.200 0.20 0.053 T fixed 0.50 0.265 0.279 1.00 0.723 2.00 1.768 Meter C1A Uer Dead Time Undamped Natural Freq, ωn(rad/ec) Ratio, ζ 0.00 0.030 2.39 1.00 0.001 0.40 0.050 2.39 1.00 0.400 0.80 0.200 2.39 1.00 0.800 1.60 0.400 2.39 1.00 1.600 9
DP Flow White Paper The Emeron logo i a trade mark and ervice mark of Emeron Electric Co. Roemount and the Roemount logotype are regitered trademark of Roemount Inc. All other mark are the property of their repective owner. Standard Term and Condition of Sale can be found at www.roemount.com/term_of_ale Emeron Proce Management Roemount Flow 7070 Wincheter Circle Boulder, Colorado USA 80301 Tel (USA) 1 800 522 6277 Tel (International) +1 303 527 5200 Fax +1 303 530 8549 Emeron Proce Management Blegitrae 23 P.O. Box 1046 CH 6341 Baar Switzerland Tel +41 (0) 41 768 6111 Fax +41 (0) 41 768 6300 Emeron FZE P.O. Box 17033 Jebel Ali Free Zone Dubai UAE Tel +971 4 811 8100 Fax +971 4 886 5465 Emeron Proce Management Aia Pacific Pte Ltd 1 Pandan Crecent Singapore 128461 Tel +65 6777 8211 Fax +65 6777 0947 Service Support Hotline : +65 6770 8711 Email : Enquirie@AP.EmeronProce.com 00840-0100-XXXX Rev XX, 9/12