A two-thermocouple probe technique for estimating thermocouple time constants inflows with combustion: In situ parameter identification of a first-order lag systemM. Tagawa, T. Shimoji, and Y. Ohta

Citation: Review of Scientific Instruments 69, 3370 (1998); doi: 10.1063/1.1149103 View online: http://dx.doi.org/10.1063/1.1149103 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/69/9?ver=pdfcov Published by the AIP Publishing

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:177.20.130.9 On: Tue, 10 Dec 2013 18:23:53

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REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 69, NUMBER 9 SEPTEMBER 1998

This artwith various physical phenomena, the severe measurementenvironment of turbulent combustion has long prevented ac-curate measurement of fluctuating temperature. Among cur-rent techniques available for such fluctuating temperaturemeasurement, a fine-wire resistance thermometer of0.6 3 mm in diameter ~usually called cold wire! iswidely used because of its fast response to fluid temperaturefluctuation.2 The typical cut-off frequency reaches the orderof 103 Hz without any response compensation. However, thewire is mechanically very weak and is not durable enough towithstand the high temperature of combustion. A fine-wirethermocouple of 20 50 mm in diameter, on the other hand,is generally superior to a cold wire in durability, whereas thisis a slow thermometer whose cut-off frequency remains1 10 Hz. Thus, appropriate compensation for this slow re-sponse is indispensable for accurate measurement of tem-perature fluctuations. The thermocouple is a standard ther-mometer for combustion experiments, and has frequentlybeen expected to serve as a reference thermometer in variouslaser-based techniques for temperature measurement.3 Forthe above reasons, we need to establish an accurate compen-sation method for the thermocouple response.

mocouple time constant is almost always measured/calibrated by internal or external heating of the thermocouplewire.4,6,7 The time constant thus obtained is effective only inthe calibration condition, so it would be next to impossible toperform such a dynamic calibration of the thermocouple tocover all experimental conditions. We have an easier alter-native than the dynamic calibration, which may predict thetime-constant value with the aid of a correlation equation ofthe heat transfer coefficient of a fine wire.4 However, there isa serious problem in its application to a combustion experi-ment, since every available correlation equation is effectiveonly for flows without combustion. Thus, we should not relyon this alternative method, unless we are confident of theuniversality of the correlation equation applied.

A two-thermocouple probe, which is composed of twofine thermocouples of unequal diameters, is a unique tool formeasuring temperature fluctuations, and the dynamic calibra-tion of the thermocouple is not needed.812 If a good data-reduction scheme for the two-thermocouple probe techniqueis available, we will be free of the very demanding task ofdynamic calibration of the thermocouple response, i.e., mea-surement of a time constant. The two-thermocouple probetechnique provides in situ measurement of the time constant,and the spatial and temporal variations in the time-constanta!Electronic mail: tagawa@megw.mech.nitech.ac.jpA two-thermocouple probe techniqueconstants in flows with combustion: Iof a first-order lag system

M. Tagawa,a) T. Shimoji, and Y. OhtaDepartment of Mechanical Engineering, Nagoya Institute o~Received 16 March 1998; accepted for publication 1

A two-thermocouple probe, composed of two fine-wnovel technique for estimating thermocouple time conthermocouple response. This technique is most suitaturbulent combustion. In the present study, the reliaappraised in a turbulent wake of a heated cylinderthermometer ~cold wire! of fast response is simultaneoA quantitative and detailed comparison between thethermocouple ones shows that a previous estimationsmaller than appropriate values, unless the noise in thespatial resolution of the two-thermocouple probeimproved so as to maximize the correlation coefficienoutputs. The improved scheme offers better compepresent approach is generally applicable to in situsystem. 1998 American Institute of Physics. @S00

I. INTRODUCTION

To clarify heat transport processes in turbulent combus-tion, we need a highly reliable technique for measuring ve-locity and temperature simultaneously,1 which should bebased on accurate measurement of temperature fluctuationsof up to 102 104 Hz normally existing in turbulent flames.Although this is not a very rapid fluctuation when compared3370034-6748/98/69(9)/3370/9/$15.00icle is copyrighted as indicated in the article. Reuse of AIP content is subje

177.20.130.9 On: Tue, 1or estimating thermocouple timesitu parameter identification

Technology, Nagoya 466-8555, Japan

June 1998!

e thermocouples of unequal diameters, is aants without any dynamic calibration of thee for measuring fluctuating temperatures inlity and applicability of this technique areithout combustion!. A fine-wire resistancesly used to provide a reference temperature.ld-wire measurement and the compensated

scheme gives thermocouple time constantshermocouple signals is negligible and/or the

sufficiently high. The scheme has beenetween the two compensated-thermocouple

sation of the thermocouple response. Thearameter identification of a first-order lag4-6748~98!03109-8#

The frequency response of a fine-wire thermocouple canbe described as a first-order lag system. In turbulent combus-tion fields, velocity, temperature, density and gas composi-tion will vary spatially and temporally, and so does the timeconstant of the thermocouple response.4,5 Hence, for accuratemeasurement of fluctuating temperature, we need a correcttime constant together with a reliable and robust compensa-tion method for the thermocouple response. So far, the ther-0 1998 American Institute of Physicsct to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:0 Dec 2013 18:23:53

This artvalue can naturally be reflected in the compensation of thethermocouple response.12 In short, the two-thermocoupleprobe technique makes no distinction, hitherto existing, be-tween the dynamic calibration of the thermocouple and thedata collection.

In general, if a measurement system can be approxi-mated by a linear dynamic system of appropriate order, thesystem is fully described by a set of system parameters.These parameters can usually be determined from a calibra-tion experiment, in which a known input such as a sinu-soidal or a step-wise signal is given to the system, and thesystem output is then measured. On the contrary, we areproposing a different approach which enables us to deter-mine the system parameter ~the thermocouple time constant!from the unknown input ~fluid temperature!. In short, akind of in situ parameter identification of the system is real-ized and is applicable not only to the two-thermocoupleprobe technique but also to a general first-order lag system.

In the present study, the reliability and applicability ofthe two-thermocouple probe technique are evaluated using afine-wire resistance thermometer of fast response ~cold wire!.So far, a few studies1315 investigating the response charac-teristics of a thermocouple are available, where a fine coldwire was used as a reference thermometer which can providea true temperature. It should be noted here that the mainsubject of the studies was to show how accurately the tem-perature fluctuation can be reproduced by compensating thethermocouple response with a given time constant. In thisstudy, on the other hand, we are aiming at establishing areliable scheme for estimating/measuring the thermocoupletime constant, and are dealing with the cold-wire measure-ment from a different point of view. First, we appraise aprevious scheme12 for the two-thermocouple probe techniqueand then improve the scheme to give it wider applicabilityand higher reliability.

II. METHOD FOR ESTIMATING TIME CONSTANT ANDITS IMPROVEMENT

Two kinds of schemes have so far been proposed forestimating the thermocouple time constants using the two-thermocouple probe technique: one for determining the timeconstants from a cross spectrum of two temperature signals8and the other for obtaining the time constants by assumingthe ratio of two time constants is a known constant.911 Re-cently, we have proposed an alternative scheme for estimat-ing the time constants,12 in which there is no need for so-phisticated data reduction and introduction of any a prioriassumption. In its practical applications, however, we cometo realize that the proposed scheme is sensitive to thechanges in the signal-to-noise ratio of thermocouple signalsand in the spatial resolution of the two-thermocouple probe.As a result, the time-constant values tend to be underesti-mated. In the following, we outline briefly the previousscheme and then propose an improved version for the two-thermocouple probe technique.

The fluid temperatures sensed by the two thermocouples,

Rev. Sci. Instrum., Vol. 69, No. 9, September 1998Tg1 and Tg2 , can be given by the following first-order lagsystems:12

icle is copyrighted as indicated in the article. Reuse of AIP content is subje177.20.130.9 On: Tue, 1Tg15T11t1G1Tg25T21t2G2J , ~1!

where T , t and G are a hot-junction temperature ~raw/uncompensated temperature!, a time constant and the deriva-tive of T , respectively. The subscripts 1 and 2 denotethe thermocouples of d1 and d2 in diameter, respectively.

As for the two adjacent thermocouples of the two-thermocouple probe, the relation Tg1.Tg2 holds. Thus, theunknown time constants t1 and t2 can be determined fromthe minimization of the difference between Tg1 and Tg2 :

e5~Tg22Tg1!2. ~2!

If t1 and t2 are kept constant through time averaging in Eq.~2!, we obtain time constants that will minimize e:12

t15@G22G1DT212G1G2G2DT21#/D ,

~3!t25@G1G2G1DT212G12G2DT21#/D ,

where DT215T22T1 and D5G12G222(G1G2)2. The

physical meaning of the derivation of Eq. ~3! is clear. How-ever, as mentioned above, this scheme tends to underesti-mate the time constants, unless the noise in the thermocouplesignals is below a negligible level and/or the measurementvolume formed by the two thermocouples is sufficientlysmall compared with a characteristic length scale of a turbu-lent flow measured. The reason for this underestimation maybe explained as follows: compensation of the thermocoupleresponse amplifies not only a temperature signal itself butalso noises and/or high-frequency signal components of nocorrelation between the two thermocouples; therefore, Eq.~3! will give time constants smaller than correct ones tominimize the difference e as a whole.

To diminish the above effect of the noise and the signalcomponents of no correlation, we introduce a new criterioninstead of e for estimating the thermocouple time constants.The criterion is based on a similarity between the twocompensated temperatures. The similarity can be quantifiedby a cross-correlation coefficient between the fluidtemperature fluctuations measured by the two compensatedthermocouples. We can therefore obtain the highest similar-ity between Tg1 and Tg2 by maximizing the correlation co-efficient R:

R5Tg18 Tg28

ATg18 2ATg28 2, ~4!

where the prime denotes a fluctuating component. By ex-pressing Tg and G as Tg5T g1Tg8 and G5G 1G8 and usingEq. ~1!, we obtain the relations Tg18 5Tg12T g15T181t1G18and Tg28 5Tg22T g25T281t2G28 . Now, we can rewrite Eq.~4! using the following set of equations:

Tg1825a1112b11t11g11t1

2,

Tg2825a2212b22t21g22t2

2, ~5!

3371Tagawa, Shimoji, and OhtaTg18 Tg28 5a121b12t11b21t21g12t1t2 ,ct to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:0 Dec 2013 18:23:53

This artwhere every element of the coefficients a, b and g is definedby

a115T182, a225T28

2, a125T18T28,

b115G18T18, b125G18T28, b215G28T18, b225G28T28,~6!

g115G182, g225G282, g125G18G28.

The time constants to maximize R will satisfy the followingconditions:

]R]t1

50,

~7!]R]t2

50.

The substitution of Eqs. ~4! and ~5! into Eq. ~7! yields thequadratic equations for t1 and t2 :

t121

^a ,d&1^b ,c&^b ,d& t11

^a ,c&

^b ,d& 50,

t221

^a ,d&1^c ,b&^c ,d& t21

^a ,b&^c ,d& 50,

~8!

where the bracket ^, & denotes a mathematical operator de-fined as ^p ,q&[p1q22p2q1 , and the elements are given by

a15a11b122a12b11 , a25a22b212a12b22 ,

b15b11b122a12g11 , b25a22g122b12b22 ,~9!

c15a11g122b11b21 , c25b21b222a12g22 ,

d15b11g122b21g11 , d25b22g122b12g22 .

Rewriting Eq. ~8! as t121B1t11C150 and t2

21B2t21C250 and solving these equations on condition that t1.0 andt2.0, we finally obtain

t15~2B11AB1224C1!/2,~10!

t25~2B21AB2224C2!/2.Compensation of the thermocouple response using the timeconstants thus obtained will give a maximum correlation co-efficient between the two thermocouples. It is noted here,however, that Eq. ~10! becomes mathematically trivial whena fluid temperature Tg fluctuates in an exactly sinusoidalform of a single frequency. This is because the correlationcoefficient R becomes maximum (R51) whenever no phasedifference exists between the two compensated temperaturesTg1 and Tg2 irrespective of their amplitudes. Fortunately,there is no such case where real turbulent temperature field iscomposed strictly of a single sinusoidal fluctuation, andtherefore this defect in Eq. ~10! causes hardly any problem inthe application of the present scheme.

Before a practical application of the two-thermocoupleprobe technique, we need to examine whether the frequencyresponse of the thermocouples can be represented by thefirst-order lag system @Eq. ~1!#. Equation ~1! holds if a ther-

3372 Rev. Sci. Instrum., Vol. 69, No. 9, September 1998mal inertia term of the local heat balance equation of a finewire is balanced with a convective heat transfer term. This

icle is copyrighted as indicated in the article. Reuse of AIP content is subje177.20.130.9 On: Tue, 1means that all other terms, i.e., conductive ~axial and radial!and radiative heat transfer, internal heating due to electriccurrent, and surface reaction ~catalytic heating!, should benegligible. As for a fine-wire thermocouple, we may neglectthe conductive heat transfer in the radial direction16 and theinternal heating. In the present experiment, the thermal ra-diation and the surface reaction contribute little to the heattransfer, and thus the assessment of the axial-direction heatconduction becomes a key issue. It is widely recognized thatone can neglect the axial-direction heat conduction whenl/d.200 (l is wire length; d is wire diameter!17 or l/lc.10 @lc is cold length5(awt)1/2; aw is thermal diffusivity ofwire; t is time constant#6 for a fine-wire thermocouple, andwhen l/d.1000 for a cold wire.18 The difference in the re-quirement between a thermocouple and a cold wire resultsfrom the principles of operation of these thermometers. Acold wire will sense the temperature spatially averaged overthe entire wire length, while a fine-wire thermocouple candetect the temperature at the hot junction ~normally locatedat the midpoint between the supports!. Petit et al.6 and Du-pont et al.19 have shown that the first-order lag system canrepresent the frequency response of a fine-wire thermocouplewhich meets the above requirement. We therefore make thetwo-thermocouple probe along the above guideline.

The present scheme is applicable to various temperaturesensors as far as their frequency responses are expressed inEq. ~1!. If this is not the case, when for example, a second-or a higher-order system applies well2022 rather than thefirst-order system, we may not obtain reliable results. Thus,we have to confirm prior to the application that the first-orderlag system is an appropriate model for a sensor used.

From now on, we term the estimation scheme of Eq. ~3!an emin method and that of Eq. ~10! an Rmax method, respec-tively, and test these methods.

III. EXPERIMENTAL APPARATUS AND PROCEDURE

The experimental apparatus and the coordinate systemare shown in Fig. 1. The origin of the coordinate axes islocated in the center of a wind-tunnel exit ~some details ofthe wind tunnel are described in a previous article12!. An airflow behind a heated cylinder, i.e., the wake, forms turbulentvelocity and thermal fields. The cylinder is a quartzglass

FIG. 1. Experimental apparatus and coordinate system.

Tagawa, Shimoji, and Ohtapipe 11 mm in diameter and 230 mm in length, which con-tains an electric heater of power consumption 400 W. In the

ct to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:0 Dec 2013 18:23:53

This artpresent experiment, the cylinder is placed at the position 25mm above the wind-tunnel exit. Measurement is performedat x585 mm and z50 mm over the region of y>0 mm. Themean velocity in the x direction, U , is 4 m/s at the wind-tunnel exit (z50 mm). Free stream turbulence is artificiallygenerated using a perforated plate ~hole size: f5 mm; stag-gered arrangement with 7 mm pitches!, which is installed ata position 100 mm upstream from the exit.

The details of a measurement probe are depicted in Fig.2. The two-thermocouple probe is composed of two K-type~chromelalumel! thermocouples 25 and 51 mm in diameter~bare wire: OMEGA SP!, respectively. A hot junction ismade by butt welding making the bead size as close to thewire diameter as possible. The thermocouple wires should beset along the z direction to minimize the effect of heat con-duction along their axes ~the wire axes are normal to theairflow!. As for the two thermocouples used in the presentexperiment, l/d5320 and l/lc.13 for the 25 mm thermo-couple, and 216 and 11, respectively, for the 51 mm one.Hence, we would be safe in concluding that the thermo-couples do not suffer from the axial-direction heat conduc-tion, and their frequency responses can be well representedby the first-order lag system. In addition, the effectivelengths of the thermocouple wires are probably greater thanthe spans between the supports, since the chromel and thealumel wires are epoxy bonded to the tips of the steel sup-ports ~Fig. 2! with little thermal conduction ~there is no elec-tric conduction!. The end of each wire ~cut 20 mm in lengthfrom the tip! is connected to a thicker wire of the samematerial which leads to a cold junction. The cold junctionsare immersed in an ice bath.

For an accurate estimate of the thermocouple time con-stants using the two-thermocouple probe, it is crucial tomake the normal distance between them as small as possible.However, if the wire separation is too small to be free fromfluid dynamical interference, the temperature measurementwill be smeared by turbulence introduced to an original flowfield. A proper wire separation depends primarily on the mi-croscale of a turbulent flow, i.e., the Kolmogorov scale.23,24Since no data are available on the Kolmogorov scale, wedetermine a proper wire separation by trial and error to makefluid-dynamical interference between the wires negligible.The main objective of the present study is to develop a ro-

FIG. 2. Details of two-thermocouple probe with cold wire.

Rev. Sci. Instrum., Vol. 69, No. 9, September 1998bust scheme for estimating the thermocouple time constants,which will be insensitive to instrumentation noises and a

icle is copyrighted as indicated in the article. Reuse of AIP content is subje177.20.130.9 On: Tue, 1decline in the spatial resolution of the probe. We need there-fore to test the time-constant estimation scheme under severeexperimental conditions. Although the present two-thermocouple probe may not have enough spatial resolution,we can take advantage of this situation to disclose the differ-ence between the emin and the Rmax methods.

As shown in Fig. 2, the probe has a fine-wire resistancethermometer ~cold wire! between the two thermocouples toprovide a reference temperature for the two-thermocoupleprobe technique. The cold wire is a fine tungsten wire 3.1mm in diameter ~Japan Tungsten Co., VWW1E! and thesensing part is 1.5 mm in length. Both ends of the cold wireare copper plated to become 30 mm in diameter and 1.4 mmin length, and are soldered on steel needles ~prongs! for sup-port. Since the time constant of the cold wire remains 0.30.4 ms under the present experimental condition, its fre-quency response is far faster than that of the thermocouples.In the cold-wire measurement, however, the crucial problemis that the frequency response of the cold wire can deterioratein a low-frequency region due to heat conduction along itsaxis.18,25 Hence, we need to take into account not only theordinary response delay to high-frequency temperature fluc-tuations but also the deterioration in response at the low-frequency region ~see the Appendix!. After compensating thecold-wire response in the manner shown in the Appendix, wemay obtain a reference ~true! fluid temperature Tg .

Here, we have some reason to use a somewhat long coldwire (l51.5 mm). In the cold-wire techniques,2528 the twocontradictory properties of the cold wire should be taken intoaccount:29 one is the attenuation of the low-frequency com-ponents of temperature fluctuations due to the axial-directionheat conduction, and the other the attenuation of high-frequency ones due to eddy averaging,30 resulting from thefinite spatial resolution of the cold wire.31 We can compen-sate the attenuation due to the heat conduction as shown inthe Appendix. On the one hand, the eddy-averaging effectcan be estimated by the Wyngaards analysis.31 From theanalysis for the cold wire used (l51.5 mm), we find that theroot-mean-square ~rms! value of temperature fluctuationswill diminish by 4% at most. If we apply the same analysisto a much shorter cold wire, l50.8 mm as an example, wehave 1.6% decrement in the rms value. This shows that theuse of a shorter cold wire improves slightly the accuracy inthe rms measurements. However, the attenuation due to theaxial-direction heat conduction becomes more pronouncedwhen l50.8 mm, and the gain in the low-frequency regiondecreases by about 15% from that shown in Fig. 8. Conse-quently, the use of a short cold wire will need large compen-sation for reproducing fluid ~true! temperature. In addition, ashort cold wire is likely to cause fluid-dynamical interferencebetween the cold-wire stubs and the thermocouple junctions.We thus use the somewhat long cold wire of l51.5 mm togive a sufficient gain to the low-frequency componentswhich make a key contribution to the power spectrum den-sity of fluid temperature fluctuations.

The electromotive forces ~emfs! of the thermocouplesare amplified by a factor of 1000 with high-precision instru-

3373Tagawa, Shimoji, and Ohtamentation amplifiers. An electric current driving the coldwire is 0.27 mA, and internal heating of the wire due to this

ct to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:0 Dec 2013 18:23:53

This artcurrent is negligible.18 The voltage across the wire is ampli-fied by a factor of 500. These outputs are digitized by a12-bit analog to digital ~A/D! converter ~Canopus ADXM! ata sampling frequency of 5 kHz with a range 05 V, thenstored and processed on a personal computer ~CPU: IntelDX4, 100 MHz!. The number of samples is 30 000 for eachwire. The noise in the instrumentation-amplifier output andthe quantized errors due to the A/D conversion, when evalu-ated in terms of temperature, are 0.03 and 0.03 K for thethermocouples, and 0.15 and 0.20 K for the cold wire, re-spectively.

IV. RESULTS AND DISCUSSIONA. Preliminary appraisal of emin and Rmax methods incombustion flow

To test the performance of the emin @Eq. ~3!# and theRmax @Eq. ~10!# methods, we applied these schemes to a com-bustion flow case,12 where two kinds of two-thermocoupleprobes were used depending on the flow conditions: one iscomposed of 40 and 100 mm R-type thermocouples and theother of 25 and 60 mm thermocouples. Table I shows a com-parison of the time constants estimated by the emin and Rmaxmethods. These two different schemes give virtually thesame results. The previous experiment12 was carefully ar-ranged and the measurement conditions were controlled tobe suitable for the appraisal of the two-thermocouple probetechnique. Consequently, these two schemes may workequally well in the previous combustion experiment.

B. Mean characteristics of velocity and thermal fieldsin the wake of a heated cylinder

To outline the velocity and thermal fields of the wake,we have measured the mean and rms velocities using a hot-wire probe and a mean temperature with a K-type thermo-couple probe. In the present experiment, the mean flow ve-locity is U .4 m/s at the wind-tunnel exit. We haveperformed the velocity measurement under the isothermalcondition in order to utilize the hot-wire technique. This willbe justified because the temperature can be regarded as apassive scalar in this experiment. Figure 3 shows the profilesof these measurements. The mean velocity profile takes aminimum at y50 mm where the cylinder is located, andapproaches the free stream velocity in the y.20 mm region.The rms velocity has a gentle profile ranging from 0.4 to 0.8m/s in the y direction. From spectral analysis of the velocity

TABLE I. Comparison of the time constants estimated by the emin and Rmaxmethods using previous combustion measurements by Tagawa and Ohta.a

Scheme

Case A (U 51.4 m/s) Case B (U 55.1 m/s)

t1(d1 :40 mm!

t2(d2 :100 mm!

t1(d1 :25 mm!

t2(d2 :60 mm!

emin method 98.4 ms 31.4 ms 8.3 ms 31.9 msRmax method 101.5 ms 34.9 ms 9.9 ms 34.0 ms

aSee Ref. 12.

3374 Rev. Sci. Instrum., Vol. 69, No. 9, September 1998fluctuations, it is found that the power spectrum densitiesbecome maximum around 2060 Hz, and the frequency

icle is copyrighted as indicated in the article. Reuse of AIP content is subje177.20.130.9 On: Tue, 1components higher than 1 kHz indicate little contribution tothe power spectra. The mean temperature profile becomessymmetrical at y50 mm naturally and is similar to a Gauss-ian distribution. The maximum difference in the mean tem-perature profile is at most 22 K. The temperature gradientbecomes steepest at y.15 mm, where the rms temperaturepeaks ~Fig. 6!.

C. Efficient calculation of derivative of hot-junctiontemperature

The emin and the Rmax methods need the accurate deriva-tive values of the hot-junction temperature ~raw/uncompensated temperature!. Thus, a polynomial curve-fitting method32 should be useful. Although the method isintrinsically suitable for smoothing noisy data, it also pro-vides efficient computation of the derivative G .32,33 Here, weuse the second-order polynomial for the curve fitting. Theconcrete procedure follows below.

Now, let Dt and yn1i be a sampling time interval of theA/D conversion and discrete temperature data at the time(n1i)Dt , respectively, where i denotes the offset from thetime nDt . Then, a curve-fitted value yn1i for the data yn1iwill be expressed as a function of i:

yn1i5ani21bni1cn , ~11!

where the coefficients an , bn and cn can be determinedthrough the minimization of the following mean squarevalue:

5 (i52m

m

~yn1i2yn1i!2, ~12!

where m is a length of the data window. Differentiation ofEq. ~11! on i leads to dyn1i /diu i505bn . Thus, the first-order derivative at the time n , Gn , is given by Gn5bn /Dt , and we derive the following formula for bn :

bn53

m~m11 !~2m11 ! (i52mm

iyn1i . ~13!

As seen from Eq. ~13!, we can compute bn very efficiently inthe same manner as the moving-average method. Since thepolynomial curve-fitting method works as a kind of low-pass

FIG. 3. Mean characteristics of the wake of a heated cylinder.

Tagawa, Shimoji, and Ohtafilter, high-frequency fluctuations will be attenuated by in-creasing the window length m in Eq. ~12!. Because of this,

ct to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:0 Dec 2013 18:23:53

This artwe need to find what is the most adequate value for m .Figure 4 may give guidelines on this matter, where the varia-tions of the correlation coefficient Rm and the time constantt1 ~both obtained by the Rmax method! are indicated as afunction of m . As seen in Fig. 4, Rm increases gradually withincreasing m . This is because both the noise and the high-frequency temperature fluctuations, which show no correla-tion between the two thermocouples, are increasingly dimin-ished as m becomes large. For the further large m value,however, attenuation of the high-frequency components be-comes too strong due to the low-pass filtering effect of Eq.~13! to reproduce the fluid temperature, and will cause agentle decrease in the Rm value ~somewhat difficult to dis-tinguish from the decrease in Fig. 4!. The time constant ofthe 25 mm thermocouple, t1 , also indicates only a slightvariation over a wide range of m . The same applies to the 51mm thermocouple case. On the basis of the results, we havedetermined to use m515 in the stage of estimating the timeconstants to maximize Rm , because both thermocouples arethen participating. When compensating each thermocoupleresponse, on the other hand, we have used m53 so as toreproduce high-frequency components of temperature fluc-tuation within a permissible level of noise.

D. Estimation of time constants in a nonisothermalturbulent wake

In this section, we estimate the thermocouple time con-stants t1 and t2 using the emin @Eq. ~3!# and the Rmax @Eq.~10!# methods with the aid of the above scheme for calculat-ing G . The results are shown in Fig. 5. The upper part of thefigure shows the distributions of t1 (d1525 mm) and thelower one those of t2 (d2551 mm). When the time con-stants estimated by the emin method ~n! are compared withthose by the Rmax method ~s!, we find that the emin methodgives consistently smaller values. The ratio ranges from 0.6to 0.8. Needless to say, the use of a thermocouple instead ofa cold wire would be inappropriate for measuring ordinarytemperature with small amplitude fluctuation. Hence, thepresent measurement may suffer from a much lower signal-to-noise ratio than the previous combustion case in Sec.IV A. In addition, the cold wire placed between the two ther-mocouples ~Fig. 2! makes the measurement volume of the

FIG. 4. Effect of window length m on time constant t and correlationcoefficient Rm which are obtained by the Rmax method.

Rev. Sci. Instrum., Vol. 69, No. 9, September 1998probe, i.e., size of the sensing part, larger than that of anoriginal two-thermocouple probe. These two factors may

icle is copyrighted as indicated in the article. Reuse of AIP content is subje177.20.130.9 On: Tue, 1cause the discrepancy in the results between the emin and theRmax methods. It is noted here that this discrepancy is notalways linearly reflected in the measurements of turbulencequantities.

Apart from the above estimation, we can obtain areference/true time constant by minimizing the time-averaged difference between the cold-wire measurement Tgand the compensated-thermocouple one Tg1 or Tg2 , which ismathematically expressed as (Tg2Tg1)2 or (Tg2Tg2)2. Inshort, this procedure will fit the compensated-thermocoupletemperature Tg1 or Tg2 to the fluid temperature Tg . Theresult is included in Fig. 5 ~l!. A comparison between thereference time constant thus estimated and those obtainedusing the two-thermocouple probe technique shows that theRmax method gives better results than the emin method. FromTable I and Fig. 5, we may conclude that the Rmax method isrobust and has higher reliability than the emin method.

As stated in Sec. I, if the heat transfer coefficient of afine wire is given, we can predict a time constant of the finewire. We have performed this prediction using the well-known CollisWilliams law34 for the heat transfer coeffi-cient together with the mean velocity and temperature distri-butions shown in Fig. 3. The physical properties of theK-type ~chromelalumel! thermocouple wires were replacedby those of nickel, since both chromel and alumel consistmainly of nickel ~the nickel content is more than 90%!. Asfor a flow without combustion, as in the present experiment,it is widely accepted that the above approach can predict thetime constant of a fine wire fairly accurately. The time-constant values thus predicted are also included in Fig. 5. Acomparison between this prediction and the Rmax result for

FIG. 5. Thermocouple time constants.

3375Tagawa, Shimoji, and Ohtathe 51 mm thermocouple shows that both are in good agree-ment for all but the regions around y50 mm and the

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This artthermal-boundary-layer edge where the temperature fluctua-tions become small ~Fig. 6!. As for the 25 mm thermocouplecase, on the other hand, the discrepancy between them be-comes a bit large into the y.10 mm region. In the presentstage, we cannot unequivocally state the reason for the dis-crepancy, but it is legitimate to ascribe it to limited applica-bility of the CollisWilliams law. In addition, as for the 25mm thermocouple, we can hardly make a sufficiently smallbead size relative to the wire diameter, and this may lead tothe larger discrepancy compared with the 51 mm thermo-couple case.

From the results of Fig. 5, we use the Rmax method forfurther compilation of the data.

E. Compensation of thermocouple response usingthe Rmax method

Figure 6 shows the rms values of the uncompensatedtemperatures T1 and T2 ~s,n! and those of the compensatedones Tg1 and Tg2 ~d,m!. The compensated rms values be-come 34 times as large as the uncompensated ones in the25 mm thermocouple case, and 56 times as large as those inthe 51 mm thermocouple case. In reality, a thermocouple isnot suitable for measurement of ordinary temperature fluc-tuations with small amplitude. In this sense, the present ex-periment should be regarded as a rigorous test for thecompensated-thermocouple technique. It is clear from Fig. 6that the compensated rms temperatures are in good agree-ment with the cold-wire measurement ~h!, and therefore wemay expect that the Rmax method will work well even underdemanding experimental conditions.

The good agreement of the compensated rms tempera-tures does not always guarantee correct reproduction of in-stantaneous temperature signals. Thus, we compare the in-stantaneous signal traces of the compensated thermocoupleswith that of the cold wire at y514 mm as an example. Theresult is shown in Fig. 7. The cold wire provides a referencesignal. The uncompensated signals are presented in the upperpart of Fig. 7, which cannot trace the reference signal at all.This means that the uncompensated thermocouple does notallow us to obtain turbulence characteristics such as prob-ability density function or power spectrum density. The com-

FIG. 6. rms temperature (Rmax method!.

3376 Rev. Sci. Instrum., Vol. 69, No. 9, September 1998pensated signals shown in the lower part of Fig. 7, on theother hand, can reproduce the instantaneous fluid tempera-

icle is copyrighted as indicated in the article. Reuse of AIP content is subje177.20.130.9 On: Tue, 1ture with satisfactory accuracy. The slight deviation seen inthe result can principally be attributed to inherent differencesin both the spatial position and the region for sensing be-tween the thermocouple and the cold wire.

On the basis of our experience, we may conclude thatthe time constants thus estimated will be reliable when R.0.99.

ACKNOWLEDGMENTS

The authors would like to thank S. Nagaya for his assis-tance in constructing the experimental apparatus. This workwas partially supported by a Grant-in-Aid for Scientific Re-search from the Ministry of Education, Science, Sports andCulture of Japan ~Grant No. 09650235!.

APPENDIX: FREQUENCY RESPONSE ANDCOMPENSATION OF A FINE-WIRE RESISTANCETHERMOMETER

The frequency response of a fine-wire resistance ther-mometer ~cold wire! is given by an analytical formula.18 Fig-ure 8 shows a bode diagramgain h and phase lag f (f.0) versus frequencyfor the cold wire in Fig. 2. The coldwire is placed in an air flow whose mean velocity and meantemperature are 4 m/s and 300 K, respectively. In Fig. 8, thetime constant of the prong ~supporting needle! is assumed tobe 1 s.18 The gain h indicates a stepwise decrease in a low-frequency region around 0.1 Hz because of the thermal iner-tia of the prong, and then holds a constant value lower thanunity, which is due to the heat conduction in the axial direc-tion of the cold wire. In a frequency region higher than 20Hz, h begins to fall again because of the thermal inertia of

18,2527

FIG. 7. Signal traces at y514 mm (Rmax method!.

Tagawa, Shimoji, and Ohtaboth the stub and the wire itself. In reality, the bodediagram will be influenced by velocity fluctuation and uncer-

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This arttainty in the time-constant value of the prong. However, botheffects are negligible in the present experiment.

On the basis of the bode diagram shown in Fig. 8, wemay compensate the cold-wire response. The procedure canbe summarized as follows:

~1! The time-series data of 2N samples from the coldwire are transformed into frequency components by the fastFourier transform ~FFT!. In the following, the real andimaginary components thus obtained are denoted by xR andxI , respectively. The total number of samples was 8192 (N513) in this study. To avoid obtaining a false spectrum dueto discontinuity at both ends of the time-series data, we haveapplied a cosine-type data window to fluctuating parts of the100 samples from the ends.

~2! The measurements of mean velocity and mean tem-perature ~Fig. 3! permit us to calculate the gain h and thephase f in the same way as in Fig. 8. Then, a compensationfunction for the cold-wire response, A , is given by

A5exp~ jf!/h , ~A1!where j denotes the imaginary unit.

~3! Using Eq. ~A1!, response compensation in the fre-quency domain is given by (xR1 jxI)A . Then, the inverseFFT of this will output the compensated cold-wire measure-ment, i.e., a reference fluid temperature, Tg . After this stage,we exclude the 100 samples at both ends from further datacompilation so that we can diminish the influence of the datawindow. Since the present cold-wire measurement is some-what affected by the electric noise and the quantized errordue to the A/D conversion, we have damped frequency com-ponents higher than 1 kHz using a finite impulse response~FIR! digital low-pass filter.35 This operation has little influ-ence on the rms temperatures shown in Fig. 6. It is notedhere that a compensation scheme for the thermocouple re-sponse by Bradley et al.36 provided useful information on aFFT-based compensation technique.

Procedures ~1!~3! enable us to obtain the referencefluid temperature. In the present experiment, the compensa-tion of the cold-wire response increased the rms temperatureby 20% in comparison with the uncompensated one. This isprimarily due to restoration of the low-frequency compo-nents of the cold-wire signal. As seen from Fig. 8, the sys-

FIG. 8. Frequency response of cold wire (d53.1 mm).

Rev. Sci. Instrum., Vol. 69, No. 9, September 1998tematic ~bias! error will be introduced into the cold-wiremeasurement because of the deterioration in the response to

icle is copyrighted as indicated in the article. Reuse of AIP content is subje177.20.130.9 On: Tue, 1low-frequency temperature fluctuations, unless the length-to-diameter ratio l/d ~l is the length of the sensing part! is largerthan at least 1000. On the other hand, temperature measure-ment by a thermocouple of l/d.200 is almost free from thiskind of error, since the thermocouple will inherently providepoint measurement and hardly suffer from the heat loss dueto heat conduction along the wire axis.17 Thus, the frequencyresponse of a fine-wire thermocouple can be simply ex-pressed as the first-order lag system given by Eq. ~1!,6,19 andcompensation of the response only to high-frequency tem-perature fluctuations is therefore necessary. A fine-wire re-sistance thermometer ~cold wire! generally shows a muchfaster response than a thermocouple. However, great cautionis needed with regard to its quantitative accuracy in the fluc-tuating temperature measurement, since a cold wire does notalways show a proper response to every frequency compo-nent of fluctuating temperature.

NOMENCLATUREd: wire diametere: mean square value of the difference between two

compensated temperatures, Tg1 and Tg2 @Eq. ~2!#G: time derivative of hot-junction temperature 5dT/dtm: length of data window @Eq. ~12!#R: correlation coefficient between Tg1 and Tg2 @Eq. ~4!#T: hot-junction ~raw/uncompensated! temperatureTg : compensated temperaturet: timeU: flow velocityy : coordinate axis ~Fig. 1!

Greek symbols and othersDT21 : difference between two hot-junction tempera-

tures 5T22T1t : thermocouple time constant @Eq. ~1!#( ): time average( )8: fluctuation component^p ,q&: mathematical operator [p1q22p2q11,2: thermocouple 1, thermocouple 2i ,n: ith, nth in time-series data

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3378 Rev. Sci. Instrum., Vol. 69, No. 9, September 1998 Tagawa, Shimoji, and Ohta

This article is copyrighted as indicated in the article. Reuse of AIP content is subje177.20.130.9 On: Tue, 1ct to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:0 Dec 2013 18:23:53