PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen,
Germany

Acoustic design principles for energy efficient excitation of a high intensity
cavitation zoneÖrjan JOHANSSON 1; Taraka PAMIDI1; Vijay SHANKAR1, Torbjörn LÖFQVIST1
1

Luleå University of Technology, Sweden

ABSTRACTEnergy-efficient process intensification is a key aspect for a sustainable industrial production. To improveenergy conversion efficiency high intensity cavitation is a promising method, especially in cases where thematerial to be treated is valuable and on the micro meter scale. Transient collapsing cavitation bubbles gives
powerful effects on objects immersed in fluids, like cellulose fibers, mineral particles, enzymes, etc. The
cavitation process needs optimization and control, since optimal conditions is a multivariate challenge. This
study focuses on different design principles to achieve high intensity cavitation in a specific volume in acontinuous flow. The investigation explores some potential design principles to obtain energy efficient
process intensification. The objective is to tune several different resonance phenomena to create a powerfulexcitation of a flowing suspension (two-phase flow and cavitation bubbles). The reactor is excited bysonotrodes, connected to two coupled resonant tube structures, at the critical frequency. Finally cavitation
bubbles are initiated by a flow through a venturi nozzle. The acoustically optimised reactor geometry ismodelled in Comsol Multiphysics®, and excited by dedicated ultrasound signals at three different
frequencies. The effect of the high intensity cavitation is experimentally evaluated by calorimetric method,
foil tests and degree of fibrillation on cellulose fibers.
Keywords: Structural acoustics, Ultrasound, Hydrodynamics, Cavitation

1. INTRODUCTION

Process industries are cornerstones in today’s industrialized society for the manufacturing ofvarious goods and products used in our day to day life. Process intensification is a strategy developed
to satisfy these demands and at the same time improve production capabilities. One promising example
is the implementation of devices using the principle of hydrodynamic and acoustic cavitation.
High-intensity cavitation in the ultrasonic range can change the physical and chemical properties of awide range of substances and hence, improve the production rate and quality.
Pulp and paper production and extraction of metals from minerals are two industrial sectors thatneed to meet new demands regarding energy efficiency. Special focus is on fibrillation which is themost energy demanding processes within pulp and paper industry, with an energy efficiency of 1%.
Leaching of minerals characterized by high activation energy is another energy demanding process.
Minerals that fall into this category are tetrahedrite, nickel laterites, scheelite, wolframite, etc., thesetypes of minerals are typically leached in autoclaves at elevated pressures and temperatures making
them highly energy intensive.
There are two main reasons for the limited use of dedicated devices for hydrodynamic and acoustic
cavitation in process industry: (i) The method requires extensive optimization that depends on
multiple process parameters and (ii) Problems in the implementation on a larger scale with respect to,stability and robustness, energy efficiency and high flow rates.
This study focuses on the methodology for the design and optimization of a scalable flow throughreactor. In the applications mentioned above, with respect to high-power ultrasound, various reactor
concepts exist (1 - 5). Focusing on energy efficiency and a scalable flow through concept has governed
1orjan.johansson@ltu.se
taraka.pamidi@ltu.se
vijay.shankar@ltu.se
torbjorn.lofqvist@ltu.se

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the design ideas towards a double tube type of reactor, combining both hydrodynamic and acoustic
cavitation. The outer tube is made of stainless steel. The two-phase flow that needs to be processedflows through the inner tube of plastic material containing the venturi. Before entering the reactor,
bubbles are initiated when the flow passes through the optimized nozzle (venturi). The water volume
between these tubes is filled with degassed water. The goal of optimization is to create high vibration
amplitude in the tube wall, coupled to a resonant mode in the enclosed water volume causing ahigh-pressure zone in the center of the tube. Zones with high sound pressure variation are directly
related to high cavitation intensity.

2. Acoustic and hydrodynamic design methodology
2.1 Acoustic design principles
The objective is to design and match different resonance phenomena to achieve a flow throughreactor resilient for high intensity acoustic cavitation. The idea is to fully couple a number of resonant
structures, to generate an efficient power transmission between electrical input and high intensity
cavitation output. In order to achieve the same, different wave types need to be analyzed and controlled.
One fundamental assumption is the pressure that builds up inside the tube, is maximized, if thestanding waves in the water column are excited by a combination of bending wave and breathing mode
resonances in the tube walls.
The design strategy starts from classic linear structural acoustic theory, followed by a stepwise andmultivariate approach using multiphysic simulation software. Start values for optimization are linked
to: a) Wave speeds in fluids and solid materials, b) Fluid eigenmodes in a cylindrical volume, c)Longitudinal resonances in a shell structure, d) Bending wave resonances in a free cylindrical shelland e) Critical frequency of a bending wave in a cylindrical shell structure.
The design parameters are:I.
Temperature (T),II.
The diameter of fluid volume (D i) or tube mean diameter (D m),
III.
Length of fluid volume (Lf ),IV.
Wall thickness of the surrounding tube (h),V.
Length of the tube (Ls ) andVI.
Fluid and physical properties of the inner tube. ** The goal is to design the reactor, so that it is as robust as possible with respect to losses in the central volume.

Varying the design parameters I to VI in three levels, 729 combinations need to be considered.
However, initially three of the design variables were kept fixed. Temperature was set to 20 °C,
diameter Di and water volume length L f, are set to values that give appropriate mode shapes of thecontained volume at the target frequency and temperature. However, in the practical application thesevariables are dependent on the effective wave speed of the contained fluid and the cavitation effects.
a) Wave speeds in fluids and solid materials
The speed of sound in water is dependent on temperature and the amount of gas and particles in thewater. However, since the design of the reactor allows the water jacket volume to be degassed by theultrasound only the temperature dependence is considered. The inner volume where solvents andmixtures flow through is controlled by variable boundary conditions and loss factors. Compression
speed of sound in solids and fluids is defined by the Bulk of modulus of elasticity (K) and the fluid
density (ρ), which both are temperature dependent. Stainless steel (K =163E9 Pa) is approximately 80
times harder to compress than water, K= 2.20E9 Pa at 20°C. The speed of sound in water varies to agreater extent with temperature than stainless steel. In water at 20°C, speed of sound c = 1480 m/s, but
the temperature dependence is useful for tuning the final response. In the tube structure, thelongitudinal speed of sound needs to be corrected since the wall thickness is much smaller than the
wavelength. In plate and shell structures, the quasi-longitudinal wave speed is defined by Equation 1.
𝐸𝐸
𝑐𝑐𝐿𝐿 = �𝜌𝜌(1 − 𝑣𝑣 2)(1)b) Fluid eigenmodes in a cylindrical volume
For the reactor type aimed for, the geometry of the fluid volume is defined so that two independent
mode shapes can co-exist at the same eigenfrequency. Like a pure mode in the radial direction axial
and a pure mode in axial direction. Positive interference between these two modes will get maximum949

amplification. But also benefit from the excitation of the related complex mode at a higher secondary
frequency.
The inner diameter (Di ) of the tube is the first parameter to be defined. If D i = 90 mm then the pureradial cross-sectional eigenmode (l = 0, m = 0, n = 1, β 0,1 = 3.83) occurs at around 20 kHz as inEquation 2.
2𝑙𝑙𝑓𝑓
𝑐𝑐 2𝛽𝛽𝑚𝑚,𝑛𝑛 2𝑓𝑓𝑙𝑙,𝑚𝑚,𝑛𝑛 = ��� +� �
𝜋𝜋𝜋𝜋𝑖𝑖𝐿𝐿𝑓𝑓
2(2)

The mode shape generates maximum pressure at the center and along the walls as shown in Figure
1. The two wavelength radial eigenmode will be at around 37 kHz. The pure radial and axial modes are
the least sensitive to the losses that occur in the central region where an inner tube in soft plastic
material will be located. However, real boundary conditions will alter the eigenfrequency e.g. causedby resonances in the tube structure and due to changes in the effective fluid wave speed due tocavitation.
At 20°C and 20 kHz the wavelength in water is 74 mm. The length of the water column (L f) isdefined to create a pure axial standing wave (m = 0, n = 0) at the target frequency see, Equation 4. Two
options exist: (i) lf defined as an even number, i.e. full wavelengths and (ii) lf defined by an odd number,i.e. half wavelength, The benefit of an even mode number is the symmetric field and stronger pressurebuild up at high cavitation intensity. However, there are multiple competing modes in the frequency
range of interest. Some frequencies may not contribute to the objective and need to be avoided.
The best result is can be achieved by superposition of a pure axial mode and the first radial mode,due to expected losses along the reactor centerline, both by the presence of the plastic tube and thechanged wave impedance due to high cavitation activity in the two-phase flow inside the tube.

Figure 1: Different types of eigenmodes in a circular cross-section of a water filled tube in stainless steel.
c) Longitudinal eigenmodes in a cylindrical shell structure
The tubes structural modes are coupled to the contained fluid of the tube. A key frequency (fO ) isrelated to the breathing mode of the tube, a longitudinal eigenmode in the circumference of thecylinder (mx=1, lz =0), as described in Equation 3. The breathing mode is strongly coupled to the watercolumn inside the tube. The idea is to enhance the breathing mode and suppress the axial longitudinal
eigenmodes, by mistuning of the tube length (L S≠ lz λS /2). fO =16.6 kHz for a cylinder in stainless steel
with D M=100mm. fO < fUS to maximize the acoustic coupling between structure and fluid (6).

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𝐸𝐸
𝑚𝑚𝑥𝑥 2𝑙𝑙𝑧𝑧 2
𝑓𝑓𝑛𝑛,𝑙𝑙 = �
��
�
+
�
� �
2𝐿𝐿𝑆𝑆
𝜌𝜌(1 − 𝑣𝑣 2 ) 𝜋𝜋𝐷𝐷𝑀𝑀(3)d) Bending wave eigenmodes in a cylindrical shell with free boundaries
Bending waves exist in plate or shell-like structures and is the wave type that radiates sound mostefficiently. The bending wave phase speed (c B) is frequency dependent. Equation 4, defines c B for
specific eigenmodes of a thin walled cylinder (h<<D M ) with finite length (L S) and free boundaries.
2𝜋𝜋𝜋𝜋𝑐𝑐𝐵𝐵 =22(4)��𝜋𝜋𝑙𝑙𝑧𝑧 � + �2𝑚𝑚𝑥𝑥 �
𝐿𝐿𝑠𝑠
𝐷𝐷𝑀𝑀
The idea is to excite bending waves in the tube wall in three points, normal to the wall surface, forming
an equal-sided triangle seen in cross-section. The circumference of the neutral layer is divided by three(m x=3) defines a tangential wavelength (λ x). An even integer of lz will give a symmetrical mode along
the tube.

e) Critical frequency of a bending wave in a cylindrical shell structure
The design strategy was to set the target resonance frequency of the tube structure (fUS= 20.0 kHz)
close to the critical frequency (fc ). At fc, c B equals the fluid wave speed. The critical bending
wavelength λ B then needs to be equal or greater than 74 mm. The bending wave in a tube travels
normally in spiral form, and by unfolding the cylinder λ B can be decomposed in axial (λz ) andtangential components (λ x), see Equation 5. However, due to the difference in acoustic impedances andthe bending wave radiation, the boundary was assumed to be defined by the mean diameter of the tube
(DM). Another aspect of importance is that the bending wave speed (c B) also depends on material
properties (ρ, E, ν) and the wall thickness (h), see Equation 6, valid if h << λ B and f > fO.
𝜆𝜆𝐵𝐵 =

𝜆𝜆𝑥𝑥 𝜆𝜆𝑧𝑧

�𝜆𝜆2𝑥𝑥 + 𝜆𝜆2𝑧𝑧

𝐸𝐸ℎ2
𝑐𝑐𝐵𝐵2 = 2𝜋𝜋𝜋𝜋�
12𝜌𝜌(1 − 𝑣𝑣 2 )

(5)

(6)

A bending wave speed (c B) of 1480 m/s at fUS=20.0 kHz requires a wall thickness of 11.5 mm(stainless steel). By that, the assumption of a thin structure is not valid, and c B need to be corrected dueto the inertia of rotating segments and Poisson effects (6).
To summarize aspects of start values regarding tube structure dimensions. If the circumferential
bending mode number (mx) is set to 3, h = 11.5 mm and D M = 101.5 mm, λx = 106.3 mm. With respect to
λ B = 74 mm then λz = 103.0 mm. If lz =8 the tube length LS will be 412 mm, and Equation 4 give f3,8 =20028 Hz. However, these start values are limited to the eigenmodes of an uncoupled tube structure.
Simulating the forced pressure response of the complete reactor with respect to fluid loading anddifferent loss factors, require a multi-physical FE-modelling approach.
2.2 Hydrodynamic design principles
Several principles for hydrodynamic cavitation have been investigated (6 – 8, 15). Shankar (7) has
shown the potential of CFD-optimization with the purpose of significantly lowering the energy
consumption in case of fiber refining. The idea comes from the fact that cavitation has been noted tocontribute to the fibrillation and processing of fibers already in the disc refiners in place today.
The design should induce hydrodynamic cavitation when the pulp stream flows through a venturi
nozzle. By this, an increase in flow speed and hence the dynamic pressure is achieved and at the sametime the static pressure is reduced, as both mass and energy must be conserved in the passage throughthe nozzle. The cavitation bubbles, thus induced, will then collapse, in the greater area of the venturi
and the static pressure increases. The idea is to focus the bubble collapse on the surface of the fibers toachieve fibrillation and intensified processing (15). Therefore, the venturi nozzle needs to be carefully
designed, so that the cavitation can be controlled and used in order to concentrate the processing
energy on to the fibers and provide effective refinement of the fibers in the flow (13). To improve
process efficiency further the hydrodynamic cavitation is combined with acoustic control of the951

bubbles collapse. So instead of having to pressurize or heat the entire stream, the same similar effects
can be achieved by inducing cavitation in the flow, where the only energy consumption correlates tothe pressure drop over the cavitation zone. The disadvantage is that it is hard to control and therefore
has been considered too complicated to be used on an industrial scale, but with the rapid developmentof CFD technology and engineering design in the past decades, the potential of hydrodynamic hasreceived ever more interest, in particular from chemical and process industries (9).

3. Numerical Simulation

Optimization of the geometrical design requires a combination of numerical modeling andexperimental verification. The software Comsol Multiphysics®, used during the numerical analysis,
has inbuilt defined modules to connect and optimize the vibration response of solid materials, dynamic
behavior of piezoelectric materials, structure vibrations and coupling to the surrounding fluid, bothoutside and inside of the closed volumes of the reactor. It is also possible to modify the numericalmodel with respect to the nonlinear response caused by cavitation (11).
In short, the system's linear response is tweaked with the assumption of a global loss factor
continuously determined by means of experiments. To speed up the numerical calculation procedure
two important simplifications applied corresponds to: 1) Modeling the loss factors and impedance
characteristics of the inner-tube by a complex valued Young modulus and 2) Handling the harmonic
response caused by cavitation by analyzing pressure responses mode shape at specific harmonics.
High cavitation intensity requires a matching between the mechanical components, liketransducers – tube reactor – and the acoustic wave propagation in the water volume inside the reactor
tube. Customizing the tube thickness, inner diameter, length, and boundary conditions, created acoupled resonance inside the reactor tube. This forms a standing wave, linking structural vibrations to
the wave propagation in water volume as shown in Figure 4. During the process of numerical
optimization, eigenmode analysis is first used to balance the contribution from different eigenmodes.
When matching of modes are satisfactory, the force response is then calculated and analyzed withrespect to vibration displacement and pressure distribution. Since resonance frequencies of thesonotrodes shifts down in frequency when fed with high power, the same is compensated by adjusting
the sonotrode geometrical properties in the numerical simulation model.

Figure 2: The reactor model with three attached sonotrodes and the inner tube along the center line.

4. Experimental Setup

High energy efficiency of a hydrodynamic and acoustic cavitation reactor concept require
extensive experimental tuning in order to: (i) Obtain impedance matching between components, (ii)
Equalize sonotrodes resonant frequencies, (iii) Minimize coupling losses and (iv) Determine therequired electrical power to reach optimum cavitation intensity. The loss factor is determined as thesum of all losses in the system. Losses occur due to geometrical deviations, the mismatch between thesonotrodes and tube reactor, and friction between the sonotrode masses and the piezoelectric discs.
Verification of the reactor design has been executed by measuring frequency response and modeshapes in the center of the tube. Signal conditioning and analysis was conducted using a PC-based
software and hardware (CLIO) for frequency response measurements in time and frequency domain at
192 kHz sampling rate. Pressure responses were measured at 4 - 400VA electric input power. A
pressure sensor (Dytran model 2200V1) and a miniature shock type accelerometer (PCB 353M15)952

were connected to the analysis system by a signal conditioning amplifier (B&K NEXUS 2692). The
pressure transducer has an upper-frequency limitation of 300 kHz. Pressure response was alsomeasured using an oscilloscope with a bandwidth of 1 MHz.
Measuring the impedance during chirp-excitation at various levels enabled tuning and matching of
the sonotrodes to the specific target frequency, slightly above the eigenfrequency of the tube structure.
The sonotrode resonance frequency response varies with the electrical input power. Both as free andwhen attached to the tube structure. The optimum input power was to some extent balanced by thechanges in electrical impedance. Finally the effect of the high intensity cavitation was experimentally
evaluated by calorimetric method, foil tests and degree of fibrillation on cellulose fibers.

5. Results and Discussions

The most critical step in the acoustic modeling was to model the boundary conditions in aphysically realistic way. Critical aspects are the attachment of the end caps to the tube reactor, and theconnection of the sonotrodes to the tube wall. The length of the tube structure in relation to the watercolumn inside the tube had a major impact on the pressure response, and therefore variable in the
prototype. Sonotrode responses (defined by the piezoelectric material, the length, stiffness and mass)required extensive tuning to get best possible results. Each sonotrode was tuned separately to the same
frequency (impedance minimum) and then tested together to ensure an equal input power distribution.
The numerically optimized reactor design deviated from the start values defined by structural acoustic
theory. The largest deviations were related to wall thickness and the tube structure length.
The final design was validated by pressure measurements, foil tests, and calorimetric tests. Single
frequency power conversion efficiency was 35% in stationary mode. Double frequency excitation inflow through mode was, however, superior especially when the flow conditions were optimized.
Figure 3 shows a comparison between the simulated and measured frequency response in the center ofthe prototype reactor. The smoothed and compressed spectrum in case of swept sine is due to losses
and a disturbed field. The calculated mode shapes were verified by testing the erosion rate ofaluminum foil in pure water. Erosion patterns correspond to a superposition of the simulatedmodeshapes of sound pressure variation in the water volume.

Figure 3: Modeled pressure response compared to measured response at high power excitation (200 W).
A good reactor design provides high gain while it is relatively robust to geometric deviations due tomanufacturing and assembly. The results of the sensitivity analysis showed that the lengths of thewater column and impedance properties of the inner tube are the most critical parameters. In reality the
response became more robust due to higher losses. The most important aspect related to optimal input
power level and excitation frequencies with respect to the properties of the flow through solvent.
Figure 4, presents the simulated vibration and pressure modeshapes at three resonance frequencies.
However, excitation of the sonotrodes with a high power pure sine at the resonance frequency createsa broad band harmonic pressure response (14), affected by the reactor volume resonances. Cavitation
intensity is also temperature dependent. Pressure measurements in the center of the reactor showed953

that the sound intensity increases by 10 dB when the water temperature increased from 21 to 55°C. Foil
tests verified that the erosion rate increased in a similar way (12).
Foil-tests carried out in a fiber suspension (1% conc of wet fibers) showed that the erosion rate
decreases significantly compared to tests in pure water. One reason is the attenuation of the resonant
gain, which increases electrical impedance and reduces input electric power, and thus the soundpressure and cavitation intensity. The main reason for reducing erosion rate is that the fiber wallabsorbs the cavitation energy in the form of mechanical processing of the fiber wall and friction losses.

Figure 4. Vibration and pressure modeshapes at target frequencies: 22 kHz; 37 kHz; and 53 kHz.
The excitation signal and duty cycle has a profound impact on cavitation intensity. The pure sine,with onset and off set makes a difference. A duty cycle of around 30/70 gives maximum energy
efficiency (10). With respect to energy efficiency and robustness, a narrow chirp signal has shown tobe promising. A chirp signal is of special interest when resonance peaks shift in frequency. The chirp
signal characteristics produce an onset/offset response since electrical impedance varies.
In the case of two-phase and non-Newtonian flow, the nozzles designed for hydrodynamic
cavitation also gave a much better mix of the pulp, and sedimentation was avoided. In case of fiber
treatment, the duty cycle of acoustic cavitation was 10/90 (circulating system). The duty-cycle was not
optimized but produced better results than in case of a stationary system with acoustic cavitation only.
Figure 5(a) shows the effect on the fiber walls mechanical properties in case of two frequency
excitation and a top-down flow through a venturi type of nozzle (12, 13). With respect to modifying theabsorption properties of the fiber wall an optimum was identified at an input power level of 186
kWh/bdt. At 568 kWh/bdt fibers were damaged, which was observed by SEM analysis.

Figure 5 (a) and (b): Effects of hydrodynamic and acoustic cavitation. a) Treatment of cellulose
fibers - Black curve represents no cavitation, only flow through, and b) Recovery rate of tungsten Best result was achieved at 80°C with bottom up flow, a new nozzle design (M. 3) and acoustic
excitation by 23 and 40 kHz at 350 VA.
When recovering tungsten from scheelite ore concentrates, leached by sodium hydroxide solution
at 80°C, the duty cycle was 20/80. In the experimental validation, both flow direction and the nozzle
design had a significant impact on the results. Figure 5b shows the recovery rate of tungsten by954

hydrodynamic and acoustic cavitation (HAC) compared to traditional chemical leaching. At 60°C,
acoustic cavitation only, increased recovery rate from 18% to 23%. Best result was obtained using
bottom up flow, a new nozzle design, and acoustic excitation at 23 kHz and 40 kHz. The HAC
combination increased the recovery rate, in comparison to traditional chemical leaching, from 26% to54 % during 5h exposure. The new nozzle design has probably an even greater potential, since theeffect of the secondary excitation signal became weak, due to impedance mismatch. A shift inresonance frequency at 40 kHz, caused by a more turbulent flow, decreased the power input.

6. Conclusions

The developed reactor concept is based on the principles of hydrodynamic and acoustic cavitation.
The final design is the result of an iterative optimization process using multi-physics simulations andexperiments. The fundamental design principles were inspired by previous research (1 - 6 and 8 - 11).
The aim of this investigation has been, to define criteria for a robust, energy efficient and scalable
design to be applicable in the process industry. High cavitation intensity is generated in the center ofthe reactor where the solvents to be processed flow through. The degassed water volume between the
tube wall in stainless steel (excited by piezo electric driven sonotrodes) and plastic inner tube, acts aseffective wave carrier of high intensity ultrasound. Optimized vibration patterns allows pressure
maximum to appear in the center of the inner tube, which impedance characteristics can vary within arather big range without losing its basic performance. Also, cavitation efficiency is to a great extent
dependent on the flow characteristics and properties of the flow through medium.

ACKNOWLEDGEMENTS

The project was funded by Swedish Energy Agency and VINNOVA, and has been supported by ÅF,
SCA, Stora Enso, Holmen, Innventia AB, TU Dresden and Mid Sweden University.

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