Applied Acoustics 129 (2018) 248–257

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Applied Acoustics

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Measurement of boiling liquid levels by decomposition of sound wavesin a waveguide

http://dx.doi.org/10.1016/j.apacoust.2017.08.0050003-682X/� 2017 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Indian Institute of Technology Kharagpur, India.E-mail address: snehasingh.iitkgp@gmail.com (S. Singh).

S. Singh ⇑, A.R. MohantyDepartment of Mechanical Engineering, IIT Kharagpur, Kharagpur 721302, India

a r t i c l e i n f o

Article history:Received 22 May 2017Received in revised form 18 July 2017Accepted 1 August 2017Available online 12 August 2017

Keywords:WaveguideDecompositionResonanceModesBoilingBubbles

a b s t r a c t

Many industrial processes require knowledge of the level of hot bubbling liquid in a closed vessel.However, usually conventional instrumentation is unsuitable because of extremely high temperatures,and hot poisonous gases generated by the thermo-chemical processes of the boiling liquid. This paperproposes a novel method to detect such boiling liquid levels by monitoring the boiling/bubbling noisein the vessel using a waveguide. The principle of this method is that the axial modes of the air columnand the liquid column in the closed vessel would change with change in liquid height. The sound pressurewaves produced in the vessel due to bubbling and propagating along the vessel’s axial direction can becaptured by decomposition using two microphones in a waveguide co-axially attached to the vessel.These decomposed waves would show peaks at resonance frequencies corresponding to the vessel’s axialmodes, which would be used to calculate the liquid height. The proposed method was verified throughextensive finite element simulations and experiments of boiling water over a wide range of conditions.The boiling water levels were correctly measured in each condition with an average accuracy of 98.8%.Thus, this waveguide system can continuously monitor boiling liquid levels based on the incident wavefrequency and amplitude. Such a system has wide industrial applications, particularly in steel plants,where knowing the amount of molten steel during oxygen lancing faces many challenges.

� 2017 Elsevier Ltd. All rights reserved.

1. Introduction

The phenomenon of nucleate pool boiling has numerous appli-cations such as in power electronic and microelectronic coolingsystems [1,2], in industrial boilers for power generation [3], chem-ical processing [4], steel making [5,6], research nuclear reactors[7–9]. Further, this phenomenon is extensively studied in nucleatepool boiling experiments to understand how bubbling phe-nomenon is affected by different parameters such as the liquid vol-ume, the liquid microstructure, and type of heating surface [10,11].In all these applications, the level or volume of the liquid is animportant parameter that needs to be monitored continuouslythroughout the process. However, due to the hazardous conditionsin and around vigorously boiling and bubbling liquids [6,9], cur-rently there is no sensor that can monitor the boiling and bubblingliquid levels continuously throughout the boiling process.

For example, in steel plants, the process of basic oxygen steel-making requires the knowledge of the accurate molten steel vol-ume throughout the process [6,12,13]. However, continuousmonitoring of the molten steel is difficult as oxygen lancing gener-

ates extremely high temperatures and poisonous gases in the ves-sel that render the conventional instrumentation unsuitable tomeasure the steel volume [6,12,13]. Similarly, in boiling waternuclear reactors, it is challenging to continuously monitor anddetect the volumes of boiling water, and mass of steam produced.This is primarily because of the thermohydraulic and neutronicinstabilities and large modal oscillations during the boiling stage[9]. Moreover, heavy mass flow of steamwould damage any instru-mentation placed near or attached to the reactor. In such systems,liquid volume is usually measured before starting the boiling pro-cess or, much later after the process has ended and the liquid hascooled down and the end product is already produced. However,the liquid volume changes throughout the process due to phasechange during boiling, and/or due to reactions that produce liquidproducts in process plants. A measurement technique that canmonitor the changing liquid levels throughout the boiling processwithout any sensor being attached to or near the boiling vesselwould be very useful for such industrial and research applications.

Current techniques for measuring liquid levels are optical-fibersensors [14,15], ultrasonic transducers [16], and capacitative sen-sors or electrodes [17,18], but these are not applicable to vigorousboiling liquids. Capacitive sensor-based methods require thesensor to be in contact with the liquid, and optical fiber-based

L

a

Liquid (CL)

Gas (CG)

h

L

a

L

a

(a) (b) (c)

Liquid (CL)Gas (CG)

zrθ

Fig. 1. Analogy of a closed cylindrical vessel as a duct.

S. Singh, A.R. Mohanty / Applied Acoustics 129 (2018) 248–257 249

methods require fiber tip to be in close proximity to the liquid.Thus, they are unsuitable in hazardous boiling conditions such asin process plants, nuclear reactors and steel making industry.Moreover, optical fiber tip placed closed to the liquid interfacewould be inaccurate in boiling conditions that lead to hot poi-sonous gases and fumes. Ultrasonic transducers use acoustic signaltime of flight method [16], and sometimes the transmitter ismounted on long waveguides. This method can be more suitablein boiling liquid conditions. However, it needs an external driverand the measurement sensitivity in limited by the driver frequency.

This paper introduces a novel technique to detect boiling liquidlevels continuously that does not require any sensor or transmitterto be placed near the vessel and does not require any external dri-ver. The proposed technique measures the boiling liquid level bymonitoring the bubbling noise through a waveguide connected tothe vessel. The principle of this technique is that a change in theboiling liquid height in a closed vessel will change the length ofthe air column and the liquid column in the closed vessel, whichin turn will change their axial normal modes. When the liquid isundergoing nucleate boiling the rigorous bubbling phenomenonwill excite the liquid and air columns. The acoustic waves thus pro-duced will show sharp resonance peaks at the axial modal frequen-cies of the vessel, which will be a function of the liquid level. Thesewaves can be measured by wave decomposition using a waveguideattached coaxially to the vessel through an acoustically transpar-ent film. Since, waveguides by construction have less sound atten-uation and very low heat conduction per unit length, the sensorsmounted at its other end would be protected from high tempera-ture and fumes of boiling liquids. The resonance frequency of thedecomposed incident wave spectra measured on the waveguidewould be an indicator of the liquid height. To the best of ourknowledge, this paper is the first ever application of wave decom-position theory into liquid level measurement. Other novelty ofthis paper is the use of nucleate bubbling as the source of acousticexcitation for resonance-based measurement. The following sec-tions discuss the theory of this method and test it through finiteelement simulations and experiments.

2. Theory

The present model for boiling liquid height detection is basedon the prerequisite that at the time of measurement the liquid isat the stage of nucleate boiling and there is significant bubblingwithin the liquid. This bubbling or boiling is equivalent to a ran-dom noise that excites the liquid and gas columns within the ves-sel over a wide range of frequencies that at least cover the firstacoustic axial mode of the vessel. The theoretical development ofthis method is discussed in details below:

2.1. Resonance frequencies of a vessel containing hot bubbling liquid

Although the proposed method is applicable to any vesselshape, for sake of demonstration a cylindrical vessel is chosen. Acylindrical closed vessel of length ‘L’ and radius ‘a’ is a right cylin-drical rigid walled duct. The pressure P in a right cylindrical ductcan be represented as a superposition of three standing waves Z,R and H in the axial direction (z), radial direction (r), and circum-ferential direction (h) respectively, and is given by the following setof equations [19]:

Pðr;h; zÞ ¼ RHZ; Z ¼ cos kzlz;H ¼ cosðmhþ clmnÞ;R ¼ JmðkmnrÞ; l;m;n ¼ 0;1;2; . . . ð1Þwhere, J and c are Bessel functions of the first and second kindrespectively. A cylindrical duct has (l,m,n) resonant modes corre-sponding to the axial, radial, and circumferential directions. For

the cylindrical duct with both ends closed, the boundary conditionsare velocity nodes or pressure antinodes at the rigid ends of thecylinder. This gives the following set of equations for normal axialmodes of the cylinder and the corresponding natural axial modalfrequencies, fzl.

sin kzlz ¼ 0; at z ¼ 0; L ) fzl ¼ c2L

� l ð2Þ

Similarly, functions R andH have pressure antinodes at walls ofthe cylinder (at r = a), which gives the following set of equations forthe natural frequencies of the cross-sectional modes, fmn:

kmna ¼ j0mn ) fmn ¼ j0mn �c

2pa; j0mn ¼ 0;1:84;3:05;3:83; . . .

ð3ÞHere, j0mn is the nth extremum of the mth Bessel function of the

first kind.For a cylindrical duct with top end open and bottom end closed,

the cross-sectional modes remain the same as given in Eq. (3).However, for axial modes the boundary conditions are pressureantinode (velocity node) at bottom end and pressure node (veloc-ity antinode) at top end of the cylinder. Thus, the natural axialmodal frequencies, fzl are given by the following equation:

cos kzlL ¼ 0; ) fzl ¼ c4L

� ð2lþ 1Þ ð4Þ

Fig. 1 shows a cylindrical vessel containing boiling liquid. Let cLand cG be the speed of sound in the liquid and the gas media of thevessel respectively, and qL and qG be their specific densities. At thestage of nucleate boiling, the hot boiling liquid and the vaporsabove the liquid in a rigid closed vessel after attaining thermalequilibrium will be at the liquid’s saturation temperature. There-fore, for the presented problem, cG and cL are the speed of soundfor the gas and liquid column at the liquid’s saturationtemperature.

A closed cylindrical vessel when empty or completely filledwith hot liquid is acoustically equivalent to a rigid-walled bothend closed right cylindrical duct with speed of sound as cG and cLrespectively. When the vessel is partially filled, the vessel’s liquidcolumn is acoustically equivalent to a rigid-walled one end openand one end closed cylindrical duct with speed of sound as cL.Now, considering that qL >> qG, the vessel’s gas column (abovethe liquid) is acoustically equivalent to a rigid-walled both endclosed cylindrical duct with speed of sound as cG. From Fig. 1 andEq. (2), when the vessel is empty then the first axial modal fre-quency is cG/(2 L), and when the vessel is completely filled withthe liquid then the first axial modal frequency is cL/(2 L). However,when the vessel is partially filled with the liquid to a height h, thenreferring to Fig. 1 the first axial modal frequency of the gas columnis cG/(2(L�h)) (from Eq. (2)), and of the liquid column is cL/(4 h)(from Eq. (4)). In this case, since cG < cL, the vessel’s first axial modeis due to the gas column until the liquid length increases to a valuegiven by Eq. (5).

cL4h

6 cG2ðL� hÞ ) h P

cL2cG þ cL

L ð5Þ

250 S. Singh, A.R. Mohanty / Applied Acoustics 129 (2018) 248–257

Therefore, if the vessel is excited by the bubbling phenomenonthen the resultant sound waves produced in the vessel and propa-gating along the axial direction would have their first resonancefrequency as given by the following function:

f zl ¼

cG2ðL�hÞ ; h 6 cL

2CGþcLL

cL4h ;

cL2cGþcL

L 6 h < LCL2h ;h ¼ L

8>><>>: ð6Þ

The first cross-sectional modal frequency of the vessel is givenby Eq. (3) with j0mn = 1.84. This frequency corresponds to the gascolumn when vessel is empty or partly filled and to the liquid col-umn when vessel is full. If the vessel is excited by the bubblingphenomenon then the resultant sound waves produced in the ves-sel and propagating along the radial or circumferential directionwould have their first resonance frequency given by followingfunction:

f mn ¼0:293�CG

a ; h < L0:293�CL

a ;h ¼ L

(ð7Þ

Therefore, the propagation vector kzl of the sound waves in theclosed vessel is a simple and direct indicator of the liquid level.However, the total sound pressure within the vessel is a 3D wavewhich will shows resonance due to (l,m,n) mode which is a resul-tant of the axial, radial and circumferential modes of the vessel.The resonance frequencies of this 3D wave would not follow a con-sistent relationship with the liquid height. Therefore, it is proposedto capture only the wave propagating along the vessel’s axial direc-tion (here, Z) as a plane wave using a waveguide.

X

x2x1

Pi(x,t)Pr(x,t)

Sound Source

s

2 1

S22 S11

Fig. 2. Schematic of decomposition in the waveguide.

2.2. Waveguide for capturing the axial modes of the vessel

Let aW and Lw be the radius and length of the waveguide, and cthe speed of sound in the waveguide respectively. Waveguides areconstructed to have aW << Lw, thus the first cross-sectional modalfrequency 0.293 c/aW is much higher than the axial modal fre-quency 0.5 c/Lw. Therefore, below the limiting frequency of0.293 c/aW, irrespective of the noise source at one end of thewaveguide only plane waves propagate along the axis of thewaveguide [20]. Thus, waveguides are widely used as efficientmeans of transmitting and capturing plane waves [20]. Moreover,waveguide is constructed of a rigid heavy tube with a fine internalsurface which gives less sound attenuation to the propagatingwaves. Because of the high density, the thermal conduction perunit length is poor (high heat capacity per unit length (qcDT)),therefore sensors mounted at its one end would be protected fromhigh boiling temperatures at the other end.

A waveguide connected coaxially with the boiling vessel willallow plane waves to propagate with the propagation vector ‘k’along the axis of the waveguide, i.e. of the vessel. This plane wavewill constitute of the axial modes of the vessel while the vessel’sorthogonal modes will be diminished, and thus will show reso-nance at frequencies dependent on the liquid height. However,the total pressure at any point inside the waveguide will alsodepend on the reflected waves based on the waveguide termina-tion condition. It is the incident wave, from vessel to waveguidetermination, which is a more accurate indicator of liquid height,and this incident wave spectrum will show resonance at frequen-cies that are a function of the liquid height as given by Eq. (6). Boil-ing liquid levels in closed vessels can be determined by monitoringthe peaks in the incident wave spectrum of the waveguideattached coaxially to the vessel. This incident wave could beobtained by wave decomposition using two microphones alongthe axis of the waveguide.

2.3. Decomposition to obtain incident wave spectrum

To obtain incident wave spectrum, the wave decompositionalgorithm by Seybert and Ross [21,22] is used. This methodrequires two microphones that are placed inside and along the axisof the waveguide near its termination. Fig. 2 shows the schematicof the incident wave decomposition in the waveguide.

If the spectrum of sound pressure measured in the waveguide atmicrophone locations, x1 and x2 are denoted as S11 and S22, thenwith the assumption of no flow condition in the waveguide, thedecomposition equation gives the incident wave spectrum, SII asfollows:

SII ¼ S11 þ S22 � 2C12 cosðksÞ � 2Q12 sinðksÞ4 sin2 ks

; k ¼ 2pfdc

; s ¼ x1 � x2

ð8Þwhere, C12 and Q12 are real and imaginary parts of the cross spec-trum between the pressures at x1 and x2, respectively. These canbe obtained in terms of the transfer function H12 and the phase dif-ference h12 between microphones at x1 and x2 as follows:

C12 ¼ jH12jS11 cosðh12Þ;Q12 ¼ jH12jS11 sinðh12Þ ð9ÞThus, this waveguide measurement system requires two micro-

phones with known spacing, and the signal data acquired are thespectrum at each microphone location, and transfer function andphase difference between the two microphone locations.

2.4. Excitation in a close vessel due to boiling noise

Since the method presented in this paper uses the noise emittedfrom the boiling/bubbling liquid as the input excitation for liquidand gas columns in a closed vessel, a preliminary measurementwas carried out to verify if the boiling/bubbling phenomenon hasa broadband spectrum and thus excites all frequencies within thehearing range. Noise was measured from an open electric kettlefilled with water boiling at the developed nucleate boiling stage.Fig. 3 shows the noise spectrum of water boiling in the kettle. Asseen, the power spectral density of the boiling noise is found tobe a broadband white noise spectrum but with a few low fre-quency peaks within 1000 Hz.

In the phase of saturated nucleate boiling, the vapor bubbles areformed at the heating surface, they grow and combine to formlarge bubbles and gas columns that rises to the surface and col-lapses. The magnitude of the boiling noise is determined by thenumber of vapor bubble formation, growth, and collapse. Duringthe vigorous boiling phenomenon the bubble pulsation noise arecoupled with rapid formation of bubbles, chaotic oscillations dueto liquid movement and shock waves emitted from bubble collapse[23]. The combined effect of the boiling process has been found tobe a white noise in previous studies [7,23,24], and the broadbandspectrum of Fig. 3 is in agreement with the above explanation.The low frequency peaks in the boiling noise spectrum are likelydue to the sound produced by kettle vibration at its structuralmode(s). Similar observations have been found in previous studies

0 500 1000 1500 2000 2500 3000 3500 4000

-10

0

10

20

30

40

50

60

70

80

Frequency (Hz)

Sou

nd p

ress

ure

leve

l (dB

)

Fig. 3. A typical noise spectrum of water boiling in an open electric kettle (L=0.120, a=0.06).

S. Singh, A.R. Mohanty / Applied Acoustics 129 (2018) 248–257 251

[24]. Thus, if the structural modes of the vessel are eliminated thenthe boiling noise is a white noise that excites all frequencies withinthe hearing range. It should be noted that this measurement is notpart of the actual boiling experiment and was only done to under-stand the type of acoustic excitation produced by the boiling phe-nomenon. Fig. 3 only shows the boiling noise spectrum, which inturn can be used as an input excitation to drive the liquid andgas columns in closed boiling vessel to their resonance mode(s).Since here the boiling noise was measured in an open vessel with-out using a waveguide or a decomposition algorithm, the spectrumdoes not capture the acoustic axial modes of the liquid/gas col-umns of the kettle by plane wave propagation through a waveg-uide. Thus, Fig. 3 does not show resonance at frequencypredicted by Eq. (6).

2.5. Model for boiling liquid level measurement in closed vessels

Based on the above discussion, a measurement system isdevised to monitor hot bubbling liquid levels in closed vessels asshown in Fig. 4. Here, a waveguide is attached, via an acousticallytransparent heat resistant mylar film, coaxially to the top of aclosed boiling liquid vessel. This system will only propagate planewaves generated from the axial modes of the vessel driven by theboiling noise. The resonance frequencies of the incident wave spec-trum in the waveguide would be same as the axial modal frequen-cies of the vessel. The range of the analysis frequency, fa

L

Boiling Liquid

a

Gas

h

LW

aWMicrophone at x2

x Microphone at x1

Boiling sound

Fig. 4. Equipment setup for boilin

corresponds to the range of the first axial modes of the vesseland is given as:

cG2L

6 fa 6 max2cG þ cL

4L;cL2L

� �ð10Þ

From Eq. (6), the height of the boiling liquid in terms of the firstresonance frequency fr of the decomposed incident wave spectrumis given as follows:

h ¼L; f r ¼ cL

2LcL4f r

; 2cGþcL4L 6 f r

2f r�L�cG2f r

; f r 6 2cGþcL4L

8>><>>: ð11Þ

3. Testing the setup for signal acquisition and processing

Before testing this method through simulations and boilingexperiments, preliminary tests were performed on an impedancetube with two-microphone set-up. This was done to verify if thedecomposition algorithm was correct and also if the signal acquir-ing setup was working accurately. Fig. 5 shows the result of thesetests. Firstly, a sample of 50 mm thick wool with rigid backing wastested at the waveguide termination. The sound absorption coeffi-cient for the sample was calculated using the decomposition equa-tions given in Eqs. (8) and (9), and compared with the coefficientcalculated using the standard ISO-10534-2 transfer function

Dual channel FFT analyzer

Decomposi�on algorithm

Incident wave spectrum

Liquid level

g liquid level measurement.

)b()a(

0 1000 2000 3000 4000

0.0

0.2

0.4

0.6

0.8

1.0

Soun

d ab

sorp

tion

coef

ficie

nt

Frequency (Hz)

Transfer function method Decomposition method

0 1000 2000 3000 4000

-100

-50

0

50

100

Mec

hani

cal r

eact

ance

(Ray

l)

Frequency (Hz)

Experimental Theoretical

Fig. 5. Results from the verification of the decomposition algorithm in an impedance tube. (a) Comparison of sound absorption coefficient of 50 mm wool sample with rigidbacking. (b) Comparison of impedance of a 150 mm rigid end tube termination (experimental vs theoretical values).

252 S. Singh, A.R. Mohanty / Applied Acoustics 129 (2018) 248–257

method [20] (see Fig. 5a). Secondly, a rigid closed tube of length150 mm was attached at the termination. The theoretical valueof the specific acoustic impedance of this end tube is given by –j�qc�cotkL where L is the tube length measured from the termination(x=0), and q and c are the density and speed of sound in the fluidmedia inside the tube [19]. The impedance obtained from thedecomposition algorithm was compared with the theoretical value(see Fig. 5b).

The results show that both the sound absorption coefficient andthe specific acoustic impedance obtained from the decompositionalgorithm are in very good agreement with the theoretical values.Thus, the signal acquiring set up was correctly configured and thedecomposition algorithm worked accurately. The same equipmentand algorithm were later used for actual boiling experiments.

4. Simulations

The proposed method for boiling liquid level measurementneeds to be extensively tested for different sized vessels, at differ-ent liquid heights, and with different waveguide termination con-ditions, to understand the effects of these parameters on thespectra obtained at the waveguide. Performing such extensive boil-ing experiments is very difficult due to resource and time con-straint. Therefore, firstly the proposed method was testedthrough finite element simulations. Based on the learning gainedfrom simulation results, actual experiments were performed forverification.

4.1. Procedure

Finite element simulations were performed in ANSYS Work-bench 15 using the ACT Acoustics extension [25]. A cylindrical ves-sel containing boiling water with a waveguide coaxially attachedto its top as proposed in Fig. 4 was modeled. The dimensions ofthe waveguide were Lw = 1.25 m, aw =0.035 m. The speed of soundand density of the acoustic media within the vessel were taken ascG = 391 ms�1, qG = 0.928 Kgm�3, cL = 1543 ms�1 and qL = 958.4Kgm�3 for the air and water column respectively at a temperatureof 100 �C. The air column in the waveguide was given a linear tem-perature gradient from 100 �C at the vessel-waveguide connectionto 20 �C at the microphone location x1. The speed of sound anddensity of air column in the waveguide varied based on the

temperature. A harmonic normal surface velocity excitation of1 mm/s was given to the vessel base along the z direction, tosimulate the excitation due to boiling with vessel base as the heat-ing surface. Harmonic response was obtained at two microphonessituated at a distance of x1 = 0.085 m and x2 = 0.050 m from thewaveguide termination with microphone spacing as, x1�x2 =s = 0.035 m. Extensive simulations were performed for variousvessel sizes (L = 0.4 m, 0.7 m, 1 m and a = 0.036 m) and at varyingwater heights (h = 0.25 L, 0.5 L and 0.75 L). Simulations were alsoperformed for a vessel with L = 0.195 m, a = 0.1325 m,h = 0.031 m, 0.062 m and 0.074 m, to match the conditions of theactual boiling experiments presented in Section 5. The spectraobtained in the waveguide were analyzed in the frequency rangegiven in Eq. (10). Fig. 6 shows a typical FEM model.

4.2. Results and discussions

Fig. 7 show the incident wave spectrum obtained from thesesimulations. Table 1 shows the boiling water heights calculatedfrom the first measured resonance frequency of decomposed inci-dent wave spectrum in the waveguide using Eq. (11), and theerrors in the measurement.

Overall, simulation results show that the first resonance fre-quency, fr, of the decomposed incident wave measured using thewaveguide within the analysis frequency range of Eq. (10), cor-rectly measures the liquid height, using Eq. (11). The average errorin measurement is 0.6%. However, it should be noted that in mostcases this small error is due to the frequency resolution of the exci-tation, which in this case is 10 Hz. For example, for simulationnumber 3, the frequency corresponding to 0.1 m water height is651.7 Hz. However, the harmonic excitation was not provided at651.7 Hz, but at frequencies 650 Hz followed by 660 Hz. Therefore,in this case, the method correctly shows peak frequency at 650 Hz.Similarly, for simulation number 6, 9, and 11, the correct frequencyis 372.4 Hz, 260.7 Hz and 514.3 Hz, and the method correctlyfound the nearest peak frequency, i.e., 370 Hz, 260 Hz, and510 Hz respectively.

In the presented study, simulation conditions 6 and 9 show avery high amplitude (> 1000 Pa) at 1220 and 680 Hz respectivelyas their first peak. These could be due to waveguide modes where680 Hz and 1220 Hz peak corresponds to the 5th harmonic and9th harmonic of the waveguide (natural frequency of the

Fig. 6. FEM model of the vessel attached with the waveguide.

S. Singh, A.R. Mohanty / Applied Acoustics 129 (2018) 248–257 253

waveguide = n� c/(2Lw); Lw = length of waveguide).The effect ofthe waveguide could be compensated for theoretically if thetemperature dependent thermal conductivities are known at everypoint on the waveguide to calculate accurate speed of sound atevery point. It is expected that in the actual boiling experimentsthis problem would not occur. For simulations, only peaks withinthe audible range (<200 Pa) were analyzed. The large peaks insimulations 6 and 9 were greater than 1000 Pa and therefore notused for liquid height calculation. Overall, simulations showthat boiling liquid height can be correctly measured using thedecomposed incident wave resonance peak for vessel of any size.

5. Experiments

5.1. Procedure

An experiment was performed on an aluminum alloy pressurecooker with L= 0.195 m, a= 0.1325 m, and LW= 1.25 m. This vesselwas put on a noiseless 1000 W electric heater and the waveguidewas attached to the top by drilling a hole on the pressure cookerlid and attaching a mylar film to stop steam flow to the waveguide.Water was put at heights of 0.022 m, 0.031 m, 0.062 m and0.074 m from the cooker base and heated to its boiling point.Fig. 8(a) shows the setup for boiling water experiment. The signalwas measured by the two microphones during the bubbling phe-nomenon. Here, the bubbling noise of the boiling water was theexcitation. The magnitude of bubbling noise was low; thereforethe setup was kept inside a semi-anechoic enclosure of soundabsorbing jute felts to acquire better quality signals. Two B&K4136, ¼” pressure field microphones, phase calibrated as per ISO10534 were used to measure the sound pressures at two fixed loca-tions on the walls of the waveguide, x1 = 0.085 m and x2 = 0.050 mand x1�x2 = s = 0.035 m. The microphone signals and the transferfunction between these signals were measured using an OROS 8channel OR25 FFT analyzer.

Before the experiments, experimental modal analysis was con-ducted on the measurement setup attached to the vessel. For this arandom noise excitation in the frequency range of 0–2000 Hz wasgiven to the vessel wall using a B&K 4824 electromagnetic exciter

with a B&K 8001 impedance head to measure the force. The vibra-tion response was measured by PDV-100 Laser Vibrometer. Thiswas done to obtain the structural modes of the vessel and theattached measurement setup, so that these modes could beavoided while analyzing the decomposed wave spectrum obtainedfrom the boiling experiment. Fig. 8(b) shows the setup for experi-mental modal analysis.

5.2. Results and discussions

Fig. 9 shows the results from experimental modal analysis.Fig. 10 shows the amplitude of the pressure waves at point x1and x2, and the amplitude of the incident wave after decomposi-tion (using Eqs. (8) and (9)) for the four different water heights.The structural modes occurred at 60 Hz, 182.5 Hz, 722 Hz,1735 Hz, and 1913 Hz (refer Fig. 9). However, none of these struc-tural modes interfered with our main analysis as the decomposedincident wave did not show resonance at any of these frequencies(refer Fig. 10). Table 2 compares the level of boiling water in theclosed vessel as measured from the decomposed incident wavespectrum with the level measured using a 1 mm least count steelruler on heated water just before beginning the experiment. It isfound that the incident wave accurately predicted the water heightfor all the test conditions with an average error of 3.3%. However, itshould be noted that the error is likely to be less because measur-ing water height from a ruler before beginning the experiment is initself not an accurate method. In fact, the incident wave peak fre-quency, theoretically, is a more accurate measure.

5.3. Effect of decomposition

Experimental results, presented in Fig. 10 clearly show that theindividual pressures measured on the waveguide do not correctlypredict the boiling water height at levels of 0.022 m, 0.062 m and0.074 m. This highlights the necessity of incident wave decomposi-tion using two microphones instead of one. In real world boilingconditions, the pressure measured at individual points in thewaveguide would depend on the microphone location, waveguidetermination condition, and/or environment noise just outside the

(a) L = 0.195 m (b) L = 0.4 m

1000 1500 2000 2500 3000 3500 4000

0

50

100

150

0

50

100

150

0

50

100

150

1000 1500 2000 2500 3000 3500 4000

Frequency (Hz)

h=0.031 m

Inci

dent

pre

ssur

e am

plitu

de (P

a)

h=0.062 m

h=0.074 m

1190 Hz

1470 Hz

1610 Hz

600 900 1200 1500 1800 2100

0

50

100

150

0

50

100

150

0

50

100

150

600 900 1200 1500 1800 2100

Frequency (Hz)

h=0.25LInci

dent

wav

e am

plitu

de (P

a)

h=0.5L

h=0.75L

1290 Hz

980 Hz

650 Hz

(c) L = 0.7 m (d) L = 1 m

200 400 600 800 1000 1200

0

50

100

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0

50

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200 400 600 800 1000 1200

Frequency (Hz)

h=0.25 LInci

dent

wav

e am

plitu

de (P

a)

h=0.5L

h=0.75L

730 Hz

550 Hz

370 Hz

300 450 600 750 900 1050

0

50

100

150

0

50

100

150

0

50

100

150

300 450 600 750 900 1050

Frequency (Hz)

h=0.25 LInci

dent

wav

e am

plitu

de (P

a)

h=0.5L

h=0.75L

260 Hz

390 Hz

510 Hz

Fig. 7. Spectrum of the decomposed incident pressure amplitude in the waveguide attached coaxially to a cylindrical vessel containing boiling water. (a) L=0.195 m,a=0.1325 m. (b) L=0.4 m, a=0.1325 m. (c) L=0.7 m, a=0.1325 m. (d) L = 1 m, a=0.1325 m.

254 S. Singh, A.R. Mohanty / Applied Acoustics 129 (2018) 248–257

waveguide termination. However, in all conditions, the decom-posed incident wave will correctly measure the liquid height. Tofurther demonstrate this, the same cylindrical vessel was modeledwith L = 0.195 m, a = 0.1325 m, and h = 0.031 m. The harmonicexcitation was given to FEM model from 1000 Hz to 4000 Hz, asgiven by Eq. (10). The waveguide was terminated with a closedtube of length 0.0221 m. Therefore, the total closed tube lengthat x2 corresponded to 0.0721 (0.0221 + 0.05) m. The impedance

of this tube is -jcot(kl) [19] where l = 0.0721 m. The maximumimpedance occurs at odd multiples of c/4 l. Therefore, the reflectedwave at x2 will have its minimum at 1190 Hz. Fig. 11 show the har-monic response of this waveguide termination condition. Thedecomposed wave shows peak at 1190 Hz which using Eq. (11)correctly gives the water height as 0.031 m, but the total pressureat x2 has no peak at 1190 Hz due to the effect of termination impe-dance, instead its first peak occurs at 1850 Hz.

Table 1Prediction of water height using the waveguide setup in the simulations.

Simulation No. Vessel length (mm) Actual water height (mm) Resonance frequency fr (Hz) Measured water height (mm) Measurement error

1 195 31.0 1190 31.0 0%2 62.0 1470 62.0 0%3 74.0 1610 73.6 �0.54%

4 400 100.0 650 99.0 �1%5 200.0 980 200.0 0%6 300.0 1290 299.0 �0.33%

7 700 175.0 370 172.0 �1.7%8 350.0 550 345.0 �1.4%9 525.0 730 528.0 0.6%

10 1000 250.0 260 248.0 �0.8%11 500.0 390 500.0 0%12 750.0 510 756.0 0.8%

(a) (b)

Fig. 8. Experimental set-up. (a) Setup for measuring the boiling water noise spectra using a waveguide coaxially attached to the vessel. (b) Setup for experimental modalanalysis of the vessel and the waveguide.

0 500 1000 1500 2000

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

Velo

city

per

uni

t for

ce (m

/N-s

)

Frequency (Hz)

Velocity per unit force (m/N-s)

60 Hz

182.5 Hz

722 Hz 1735 Hz 1913 Hz

Fig. 9. Result of experimental modal analysis of the experiment setup.

S. Singh, A.R. Mohanty / Applied Acoustics 129 (2018) 248–257 255

6. General discussions

6.1. Advantages and limitations of the proposed technique

The average measurement error in 13 FEM simulations and 4experiments is 1.2%. The proposed technique correctly measuresthe boiling water levels in each of the 17 conditions with an aver-age accuracy of 98.8%. Theoretically, and as demonstrated from theresults the error in measuring the liquid height is only limited bythe frequency resolution of the signal acquiring technique. Withthe help of modern data acquisition at intervals of 5 Hz, the errorin axial modal frequency can go to a maximum of 4 Hz. Usingthe minimum axial modal frequency, f, for a vessel L as given byEq. (10), the maximum possible error in boiling liquid measure-ment is given by:

Error% 62ðfþ4ÞL�cG

2ðfþ4Þ � 2fL�cG2f

L� 100% ¼ 100

1þ cG8L

% ð12Þ

For the case of boiling water, the levels can be measured withhigh degree of accuracy (error% < 5%) for any vessel length up to2.5 m.

The other advantages of this method is that is does not require ahigh temperature/pressure sensor and any cheap microphone can

(a) (b)

(c) (d)

1000 1500 2000 2500 3000 3500 40000.000

0.004

0.008

0.012

0.000

0.004

0.008

0.012

0.000

0.004

0.008

0.012

1000 1500 2000 2500 3000 3500 4000

P1 a

mpl

itude

(Pa)

Frequency (Hz)

P1

P2 a

mpl

itude

(Pa) P2

Pi a

mpl

itude

(Pa) Pi1130 Hz

1000 1500 2000 2500 3000 3500 4000

0.000

0.003

0.006

0.009

0.000

0.003

0.006

0.009

0.000

0.003

0.006

0.009

1000 1500 2000 2500 3000 3500 4000

P1 a

mpl

itude

(Pa)

Frequency (Hz)

P1

P2 a

mpl

itude

(Pa)

P2

Pi a

mpl

itude

(Pa) Pi1165 Hz

1000 1500 2000 2500 3000 3500 4000

0.000

0.001

0.002

0.0030.000

0.001

0.002

0.0030.000

0.001

0.002

0.0031000 1500 2000 2500 3000 3500 4000

P1 a

mpl

itude

(Pa)

Frequency (Hz)

P1

P2 a

mpl

itude

(Pa)

P2

Pi a

mpl

itude

(Pa) Pi

1470 Hz

1000 1500 2000 2500 3000 3500 40000.000

0.001

0.002

0.003

0.000

0.001

0.002

0.003

0.000

0.001

0.002

0.003

1000 1500 2000 2500 3000 3500 4000

P1 a

mpl

itude

(Pa)

Frequency (Hz)

P1

P2 a

mpl

itude

(Pa)

P2

Pi a

mpl

itude

(Pa)

Pi

1610 Hz

Fig. 10. Wave decomposition of spectra obtained from the experiment, L = 0.195 m, a = 0.1325 m, LW = 1.25 m at water heights of (a) 0.022 m (b) 0.031 m (c) 0.062 m, (d)0.074 m.

Table 2Prediction of water height using the waveguide setup in the experiments.

Experiment No. Water height measured from ruler (mm) Resonance frequency fr (Hz) Water height measured from incident wave (mm) Measurement error

1 22.0 1130 22.0 0%2 31.0 1165 27.2 �12.3%3 62.0 1470 61.8 �0.3%4 74.0 1610 73.6 �0.5%

256 S. Singh, A.R. Mohanty / Applied Acoustics 129 (2018) 248–257

be used for this measurement. The only requirement is a high fre-quency resolution of the data acquisition system which most mod-ern day equipments offer. Moreover, boiling liquid levels can be

monitored continuously throughout the boiling process withoutany interruption. Since the waveguide only captures the axialmodes and any orthogonal modes don’t interfere with

1190 Hz

0

20

40

60

80

100

120

950 1450 1950 2450 2950 3450 3950

Pres

sure

am

plitu

de (P

a)

Frequency (Hz)

P1

P2

Pi

Fig. 11. Effect of waveguide termination impedance on a vessel (L = 0.195 m,a = 0.1325 m, h = 0.031 m) and terminated by a rigid end tube of length 0.0221 m.

S. Singh, A.R. Mohanty / Applied Acoustics 129 (2018) 248–257 257

measurement, this method does not depend on the vessel shapeand can be successfully applied to any shape vessel.

The limitation of this method is that it depends on the rigorousbubbling and hence is applicable only for process involving nucle-ate boiling, which is the most common form of boiling in industrialprocess. The other limitation is that resonance modes due to struc-tural vibrations or waveguide resonance may interfere with thespectrum analysis. This can be minimized by using a rigid connec-tion of the waveguide to the vessel and the rigid vessel mount tostop structural vibrations.

6.2. Future recommendations

Future studies need to be conducted to test this method forother boiling liquids. Studies can also be conducted to test thismethod on chemical processes that lead to change in the chemicalcompositions of the liquid and/or leads to poisonous gases. How-ever, conducting experiments for such hazardous conditions wouldalways be a challenge.

7. Conclusions

This paper aimed at devising a simple and novel technique todetect boiling liquid levels in a closed vessel by monitoring thebubbling/boiling noise in the vessel using a waveguide and decom-posing the incident wave in the waveguide using two-microphones. The resonance frequency of the decomposed inci-dent wave is used to calculate the boiling liquid height. Finite ele-ment simulations and boiling water experiments showed that theincident wave spectrum in the waveguide correctly measure theboiling liquid level for various vessel sizes, various liquid heightsand with different waveguide termination, at a high accuracy of98.8%. Overall, the proposed method has huge potential to measurethe changing liquid levels continuously throughout the boilingprocess without any interruption, far away from the hazardousboiling condition using cheap sensors and without external

excitation. The proposed technique has wide industrialapplications, such as in oxygen steel making, chemical processing,industrial boilers for power generation, electronic cooling systems,and research nuclear reactors, where high temperature, pressure,poisonous gases, and thermo-hydraulic instabilities within theboiling vessel do not allow any sensors to work accurately on ornear the vessel.

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