ultrason malakle

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Chemical Engineering Science 59 (2004) 299311

www.elsevier.com/locate/ces

Investigations in mass transfer enhancement in textiles with ultrasound

V.S. Moholkar, M.M.C.G. Warmoeskerken

Textile Technology Group, Department of Chemical Engineering, University of Twente, P.O. Box 217, NL - 7500 AE Enschede, Netherlands

Received 13 November 2002; received in revised form 2 September 2003; accepted 23 September 2003

Abstract

A novel application of ultrasound is for the intensication of wet textile treatments, in which mass transfer in the inter- and intra-yarn

pores of the textile is the basic physical mechanism. This paper describes a simple methodology for the estimation of mass transfer

enhancement in ultrasonic textile treatments. For this study, washing of EMPA 101 fabric, soiled with carbon soot and olive oil, is selectedas a model process. In the absence of precise knowledge of the convection velocity resulting due to transient cavitation, a semi-empirical

method is used to estimate mass transfer enhancement. The experimental soil removal rate during model process is determined by precise

time-controlled ultrasonic treatment of the textile, with the source of cavitation nuclei located close to the textile surface. The mass

transfer in the textile during the model process is found to occur in two distinct steps, characterized by two dierent convective diusion

coecients. This eect is explained in terms of uneven soil distribution in the inter- and intra-yarn region. The mass transfer enhancement

factor, dened as ratio of convective diusion coecient to molecular diusion coecient of soil particles, is found to be in the range

10002000. In addition, it is found that the mass transfer enhancement increases with acoustic pressure amplitude during textile treatment.

A qualitative estimate of the convection velocities generated in the vicinity of the bubble is provided using numerical simulations of

bubble dynamics equation.

? 2003 Elsevier Ltd. All rights reserved.

Keywords: Cavitation; Ultrasound; Process intensication; Bubble dynamics; Acoustics; Convection

1. Introduction

Wet textile nishing is dened as the treatment of

textile materials with chemicals in order to manipu-

late the properties of the textiles such as color, design,

hydrophobicity, etc. Wet textile nishing basically involves

transport of solid/liquid substances across the textiles with

water as the medium. More precisely, mass transfer in

the inter- and intra-yarn pores of the textiles is the basic

phenomenon in wet textile nishing. Current wet textileprocesses suer from two major drawbacks: large process

times and low energy eciency. Improvement of these pro-

cesses basically involves intensication of mass transfer in

textile materials. More detailed discussion about mass trans-

fer in textile materials is given in Appendix A. Ultrasound

as a means of intensication of wet textile processes has

been attempted by several researchers in past few years (for

Corresponding author. Current address: Karishma, 2/A, Pramod

Nagar, Vijapur Road, Solapur 413 004, Maharashtra, India.

E-mail address: [email protected] (V.S. Moholkar).

example McCall et al., 1998; Thakore et al., 1988;

Yachmenev et al., 1998, 1999; Rathi et al., 1997). In-

spite of encouraging results on laboratory-scale studies, the

ultrasound-assisted wet textile processes have not been im-

plemented on industrial scale yet. Two major factors that

have contributed to this eect are: lack of precise knowl-

edge about the physical mechanism of the ultrasonic mass

transfer enhancement in textiles and inherent drawbacks

of the ultrasonic processors, such as directional sensitivity,

erosion of sonicator surface and non-uniform volumetricenergy dissipation. It was recently shown by us (Moholkar

and Warmoeskerken, 2002; Moholkar, 2002) that transient

cavitation in the medium (i.e., water) in the close vicinity

of the textile surface is the basic physical mechanism of

ultrasonic mass transfer intensication in textiles.

In this paper, we present results of further research in this

subject, i.e., quantication of the ultrasonic mass transfer

enhancement in textiles. The phenomena of transient cavita-

tion near a solid surface, such as textile, is extremely com-

plex and, hence, it is very dicult to determine exactly the

micro-convection velocity produced due to bubble motion.

For this reason, we take a semi-empirical approach, combin-

0009-2509/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ces.2003.09.018
mailto:[email protected]:[email protected]


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300 V.S. Moholkar, M.M.C.G. Warmoeskerken/ Chemical Engineering Science 59 (2004) 299 311

ing the experimental results and theoretical model, for the

quantication of mass transfer enhancement.

2. Model wet textile process

So as to estimate the degree of intensication of masstransfer in the textile with ultrasound, a model wet textile

process with a model fabric and a model diusing sub-

stance, which acts as a monitor for mass transfer, needs to

be selected. A proper study of the ultrasonic mass trans-

fer enhancement will require that the model fabric and

mass transfer monitor possess the following properties and

characteristics:

The monitor for the transport (in the form of a particleor a molecule) should be chemically inert.

The monitor should not form any strong chemical or

physical bond with the bers of the textile that can hinderits transport across the fabric.

The concentration per unit area of the monitor in the fab-ric should be constant, so that the samples used for dif-

ferent experiments should have the same initial concen-

tration of the monitor diusing out.

The monitor should be completely reversible. A completeremoval of the monitor from the textile should be possible

and the original textile should be recovered after total

extraction of the monitor.

The original diusion rate of the monitor should be slowenough to be accelerated by ultrasound. If the diusion

coecient of the monitor is already very fast, then the

acceleration due to ultrasound may not be discernible. The size of the monitor particles should be smaller than

the intra-yarn pores of the textile (typically 5 m or so),

that will basically ensure the presence of the monitor in

these pores. Another advantage of having a very small

size of the monitor is the separation of the secondary

eect of ultrasound that is hypothesized to be responsible

for the intensication of the textile processes, viz. particle

size reduction (Klutz, 1997). Due to very small initial

size, any further reduction in the size of the monitor will

not occur and, hence, the above-mentioned secondary

eect can be isolated.

The fabric used in the model process should preferably bea plain weave (with both inter- and intra-yarn porosity)

that oers a simple geometry for the transport of the

monitor.

In view of these requirements, we have selected EMPA

101 fabric (manufactured by ETH, Zurich) as the model

fabric for the experiments, with the model process being

washing of this fabric with ultrasound. This is a plain weave

cotton fabric, mass per unit area 100 g m2, soiled with

carbon soot and olive oil. Its SEM images are shown in

Fig. 1. Fig. 1D shows the cross-section of the EMPA 101

fabric. The presence of the carbon particles inside the yarn

is clearly visible, although in quite small concentration. This

indicates that the soil is unevenly distributed in the inter-

and intra-yarn space.

EMPA 101 fabric satises all of the requirements of the

model fabric and monitor described above except that the

carbon particles are not free to be transported by convec-

tion, but adhere to the ber along with the olive oil. Theseparticles, therefore, need to be loosened from the surface of

the bers before being transported in the medium. This ne-

cessitates pretreatment of the textiles before the ultrasound

treatment such as soaking in a detergent solution. It could

be inferred from Fig. 1 that the carbon particles are very

small, and hence, an exact determination of their size is dif-

cult. However, an approximate particle size that could be

assumed for the purposes of calculations would be 0:1 mor so (Koster, personal communication).

3. Quantication of mass transfer enhancement: conceptof convective diusion coecient

As stated earlier, it was shown by us (Moholkar and

Warmoeskerken, 2002) that (high-energy) transient bubble

motion near the textile surface is responsible for the cre-

ation of strong convection in the close vicinity of the textile,

which enhances the mass transfer in the textile. However,

the micro-mechanism of this process is not clear. It is not

known by which mechanism, viz. shock wave, high-velocity

micro-jet or oscillatory spherical velocity eld generated

due to radial bubble motion or by combination of all three

possibilities, the convection responsible for the mass trans-fer enhancement is caused. A conventional approach to the

modeling of the mass transfer enhancement due to convec-

tion, is to solve the mass balance equation after substitution

of the uid velocity. Most of the bubble dynamics equations

available so far apply for a single bubble oscillating in an

innite medium. However, in practical situations, such as

the current study, the net physical/chemical eect is a result

of multi-bubble phenomena. A comprehensive model that

describes dynamics of thousands of bubbles, with strong in-

teraction between them, is not developed yet. Therefore, an

exact determination of the convection velocity is not possi-

ble. We, therefore, adopt a semi-empirical method to esti-

mate the mass transfer enhancement factor by an order ofmagnitude.

The carbon particles in the soil in the model fabric are

not soluble in the medium; however, due to their very

small size (0:1 m), they can be approximated as rigidspheres diusing in the continuum of the washing medium.

For this situation, the most common basis for the deter-

mination of the diusion coecient is the StokesEinstein

equation

D =T

6dp: (1)


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V.S. Moholkar, M.M.C.G. Warmoeskerken/ Chemical Engineering Science 59 (2004) 299 311 301

(A)

(B)

(C)

(D)

Fig. 1. SEM images of the model EMPA 101 fabric (SEM machine: Jeol Inc., Model GSM 5800, Accelerating voltage: 5 kV, Working distance: 10 mm):

(A) surface of the fabric; (B) a single yarn; (C) ber of the fabric: carbon soot on the surface of the fabric is clearly visible; (D) cross-section of a

single yarn: some carbon particles are present inside the yarn. (Cross-section of the yarn was obtained by rst soaking the textile in a monomer solution

of epoxy resin, while taking care to avoid entrapment of air bubbles. It was then put in an oven, where the monomer is converted into a cross-linked

polymer with the textile entrapped in-between. Thus, in this procedure the textile was basically frozen in the resin matrix. Next, slices of the resin matrix

were obtained with a super-cut machine (Reichert Jung Inc., Model 2050)).

If the liquid, in which the particle is suspended, is moving

due to convection currents set by density or pressure dier-

ences or due to external energy input from sources such as

a mechanical stirrer, the diusion coecient of the particle

is much higher than that predicted by the StokesEinstein

equation. We characterize this enhanced mobility of the par-

ticle due to the bulk uid motion as convective diusion co-

ecient. The convective diusion coecient is basically a

synonym for the widely used term eddy diusion in trans-

port phenomena, which represents solute transfer due to tur-

bulent uid motion. However, due to lack of knowledge of

the exact magnitude of the uid velocity created by the tran-

sient motion of the bubbles, which will help determine the

Reynolds number for the uid motion in the close vicinity

of the textile and, hence the ow regime, we have used the

term convective diusion coecient.

3.1. Description of the fabric geometry

As described in Fig. 9 (Appendix A), a textile is com-

prised of yarns, which in turn are made of bers. In order

to propose a mathematical model for the diusion process

in the textile, it is necessary to assign a suitable geometry

to the textile. For this purpose, many possibilities exist. For

example: (1) a porous plate (representing the textile as a

whole); (2) a porous cylinder (representing a single yarn in

the textile). In both the cases, a homogeneous porosity in

the structure is assumed.

In order to make a proper choice for the textile geom-

etry, one needs to take into account the relative sizes of

the source of convection (i.e., the bubbles) and the struc-

ture through which the mass transfer occurs due to the con-

vection (i.e., the textile). The typical diameter of a yarn is


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200 m, as is evident from Fig. 1B. The radius of bub-bles that undergo transient motion near the textile surface

(for 25 kHz frequency and acoustic pressure amplitudes

1.21:5 bar) oscillates between a few microns, and perhaps

100 m (Moholkar, 2002). An intense spherical velocity

eld is created in the close vicinity of the bubble during

compression. During this phase, the porous structure, i.e., asingle yarn in the textile, tends therefore to be larger than

the dimensions of the source of convection, i.e., the bub-

ble. This circumstance justies considering the textile as

a porous plate. However, while analyzing the eects of

ultrasound treatment on large portion of the textile (or on

a macroscopic scale), the dual porosity of the textile (viz.,

inter- and intra-yarn porosity) needs to be taken into account.

3.2. Approach

A ratio of the convective diusion coecient and molec-

ular diusion coecient (obtained from StokesEinsteinequation) gives the mass transfer enhancement factor due

to ultrasound. For the estimation of convective diusion co-

ecient, we combine experimental and theoretical methods

in three steps as follows:

1. Use an approximate method of cavitation nucleation near

the fabric surface for the experimental determination of

the rate of soil removal for dierent periods of ultrasound

irradiation. The soil concentration in the fabric can be

calculated from its reectance using the KubelkaMunk

theory, which is discussed subsequently.

2. Solve the diusion equation for a plane sheet to determine

the concentration proles of the diusing substance fordierent time intervals, and obtain an expression for the

total amount of diusing substance transported across the

plate per unit time.

3. Using experimentally measured values of soil removal

from the model fabric for dierent time periods of ul-

trasonic treatments and the theoretical expression for the

soil removal (obtained in the previous step) to nd out

the convective diusion coecient for soil under the ef-

fect of ultrasound irradiation.

4. Experimental

4.1. Experimental system

The experimental system had two main components: an

experimental cell and an ultrasound horn along with a signal

generator and amplier. A schematic diagram of the set-up

is shown in Fig. 2.

The experimental cell: The experimental cell was made

of three detachable glass rings and a lid made of Teon.

The height of the two rings was 15 mm, which is equal to

one-quarter of the wavelength of 25 kHz ultrasound (i.e.,

60 mm) in water, while the height of the third ring was

Signal

Generator and

Amplifier

OscilloscopeCurrent and Voltage

Monitoring Unit

Ultrasound horn and

experimental cell with textile

Timer

switch

Fig. 2. Schematic diagram of the experimental set-up.

60 mm. The cell was mounted on a stainless-steel bot-

tom (thickness: 51 mm) with four vertical bars that act as

support for the glass rings placed above each other. The

stainless-steel bottom of the experimental cell acts as a

rigid reector for the ultrasound waves. The cell had cali-

brated distance marks on it to measure the distance betweenthe ultrasound horn tip and the rigid bottom. The textile

sample could be placed in-between the glass rings. Rubber

gaskets were placed between the textile and glass rings in

order to avoid leakage. At the bottom of the cell, a special

arrangement was made to place the hydrophone (Bruel &

Kjer Ltd., Type 8103) for the measurement of the acous-

tic pressure amplitude. The output of the hydrophone was

transformed into proportional voltage by a charge amplier

(Nexus Range, Model 2690). This voltage was monitored

on a digital oscilloscope (Tektronics Ltd., Model 430A).

The ultrasound unit: The ultrasound unit comprised of a

special-made ultrasound horn with a central resonance fre-quency of 25 kHz, when vibrating in air. The horn was

driven by a signal generator (Hewlett-Packard Inc., Model

3324A) and a radio frequency amplier (ENI Inc., Model

2100L). The amplier could supply a maximum of 200 W

of electrical power for a large frequency range (10 kHz

1 MHz). The output power of the amplier could be con-

trolled by adjusting the voltage of the input signal to the

amplier. The voltage and current supplied to the ultrasound

horn were monitored using a voltage probe (Tektronics Ltd.,

Model 6138A) and a current clamp (Farnell Inc., Model

PR-20). The ultrasound horn was mounted onto the shaft of

a laboratory jack and the experimental cell was placed on

its base. The base of the jack could be raised or lowered to

adjust the distance between the bottom of the cell and the tip

of the horn. A timer switch was added between the signal

amplier output and ultrasound horn. This switch connects

the amplier output to the ultrasound horn only for a speci-

ed time period, thus, controlling the time of the ultrasonic

textile treatment accurately.

4.2. Source of cavitation nucleation

The cavitation-nucleation in the medium is an important

factor for any cavitation-aided physical or chemical pro-


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cess, because the physical or chemical eect depends, to a

signicant extent, on the population and the size distribu-

tion of these nuclei. The time of ultrasound exposure is also

an important factor in these processes. During the process,

the initial nuclei population and their size distribution un-

dergo changes due to fragmentation of the bubbles during

radial motion and re-growth (and collapse) of the daughterbubbles. Therefore, for ultrasound processes of longer du-

ration, as compared to the period of acoustic cycle, the ef-

fect of initial nuclei population and size distribution is likely

to be smoothed out. However, for a short ultrasound expo-

sure, the initial nuclei population and size distribution is of

crucial importance because these parameters are not likely

to change much during the exposure. Thus, for experiments

aimed at determining the kinetics of cavitation-aided phys-

ical or chemical processes, the source of cavitation nuclei,

which determines the initial population and size distribu-

tion of the cavitation nuclei, is a parameter of paramount

importance.

In case of bubble activity near a solid surface, the extent of

convection produced by the bubble depends not only on the

population and the size distribution of the cavitation nuclei

but also on their distance from the surface. Depending on

the stando factor, which is the ratio of the distance of the

bubble center from the textile and the radius of the bubble

at beginning of collapse, the bubble may undergo dierent

kinds of motions (Naude and Ellis, 1961; Tomita and Shima,

1986; Blake et al., 1986; Phillip and Lauterborn, 1998). For

large stando factors (3), the bubble shape during the ra-

dial motion remains spherical without appreciable deforma-

tion and, thus, the velocity eld produced around it is also

spherically symmetric. For smaller stando factors (1:5),the bubble undergoes deformation producing a high-velocity

micro-jet directed towards the boundary, when the bound-

ary is suciently rigid. In addition, the distribution of the

cavitation nuclei over the surface of the textile is also an

important factor because the dynamics of a single bubble

is inuenced signicantly by the interaction with adjacent

bubbles.

The nuclei close to the textile surface, before the start of

ultrasound irradiation, can create the convection for soil re-

moval almost immediately after ultrasound exposure. The

nuclei, initially away from the textile, may not help in im-

mediate removal of soil because of the rapid reduction inthe intensity of velocity eld away from the center of the

bubble. However, during ultrasonic irradiation, circulatory

ows are set up in the system due to the absorption of the

momentum of the acoustic waves by the liquid. Due to these

ows, a bubble initially present away from the fabric can

migrate towards the fabric surface (if it does not collapse

before reaching the fabric), and create nucleation near tex-

tile that could assist the soil removal. However, in this case,

the rate of soil removal is determined by the time for bub-

ble migration and the intensity of convection created by the

bubble. This obviously can lead to signicant errors in the

calculations of rate of soil removal, especially for smaller

periods of ultrasound exposure. Thus, the choice of source

of nuclei is of paramount importance in the accurate mea-

surement of rate of soil removal.

Conventionally, a suspension of polystyrene latex parti-

cles is used to provide nuclei in the medium (e.g., Holland

and Apfel, 1990). The air pockets entrapped in this suspen-

sion can provide nuclei for cavitation in the medium. How-ever, these air pockets are distributed all over the medium

along with the polystyrene particle suspension. Therefore,

for the present study with very short periods of ultrasound

exposure, the nucleation by polystyrene latex suspension

may not give good results, for the reasons explained earlier.

Instead, particles with surface crevices that can entrap air

pockets, placed on the fabric surface itself, can provide nu-

cleation at desired location for production of instantaneous

convection with ultrasound exposure without any time-lag

eects arising out of factors just discussed.

Pumice stones, which are widely used for providing nuclei

for boiling, can be used for the present purpose. Since these

stones are commercially available in large sizes (3 mm

or so), they may give rise to non-uniform nucleation. How-

ever, by reducing their sizes by grinding, restricting the size

range after screening using sieves of certain mesh sizes, and

adding an exact quantity of stones to the medium each time,

this problem could be overcome to some extent. It must be

noted that having uniform particle sizes may not be of much

use as far as the uniformity of nucleation is concerned, be-

cause nuclei for cavitation are mainly contributed by the

gas pockets trapped in surface crevices of the particles, and

not by the particles themselves. In view of the above con-

siderations, we have used pumice stones of size range of

200400 m as a source of nuclei in the present experi-ments. If added in large amounts, the pumice stones can al-

ter the power consumption of the system due to reduction in

system impedance as a result of large bubble population in

the system that increases the compressibility of the medium

(Moholkar, 2002). Therefore, we have used a very small

amount (20 mg) of pumice stones for the nucleation in each

experiment. For these conditions, it was found that the power

consumption of the ultrasound horn remained unchanged.

4.3. Experimental procedure

4.3.1. Characterization of the ultrasound eld

According to the analysis of the pressure and velocity

variation in the standing wave eld given by Pierce (1989),

the locations of pressure minimum occur at a distance of

=4, 3=4, 5=4, etc., while the locations of pressure max-

imum occur at a distances of =2, , 3=2, etc., from the

rigid surface, for plane-wave reection from it. In order to

conrm this theoretical result, pressure measurements were

done in the experimental cell using a very small (4 mm) un-

calibrated hydrophone at two locations viz., at distances =4

and =2 from the rigid bottom of the cell with degassed wa-

ter as the medium. The distance between the tip of the horn


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8

4

0

8

4

0

FourierCoefficients

FourierCoefficients

0 10 20 30 40 50 60 40 80 90 100Frequency [kHz]

0 10 20 30 40 50 60 40 80 90 100

Frequency [kHz]

(A)

(B)

Fig. 3. Characterization of the standing wave eld generated in the

experimental cell: (A) acoustic spectrum at pressure antinode; (B) acoustic

spectrum at pressure node.

and the rigid bottom of the experimental cell was adjusted

to 60 mm, which is the wavelength of 25 kHz ultrasound

in water. The ultrasound horn was driven at 25 kHz with a

power input of 20 W. For this power input, it produced an

acoustic wave with pressure amplitude 1:3 bar, which isabove the transient cavitation threshold for bubbles of sizes

510 m at 25 kHz frequency (Moholkar, 2002). Fig. 3

shows the FFT of the pressure signals (duration: 12 ms;

sampling frequency: 2:5 MHz) obtained at distances =4 and

=2 from the rigid reector. The amplitude of the funda-mental peak in the acoustic spectrum corresponding to the

driving frequency (indicative of the acoustic pressure am-

plitude) at location =2 is much higher than the amplitude

of the corresponding peak at location =4. In addition, the

subharmonic peak, indicative of transient cavitation, is quite

prominent in the acoustic emission spectrum at location

=2, while it is completely absent in the acoustic emission

spectrum at location =4. Thus, the characterization of the

acoustic wave eld in the experimental cell approximately

conrms the theoretical prediction of the location of the pres-

sure node and the pressure antinode. Another feature of the

acoustic wave phenomena, which is clear from these results

is that the resultant acoustic pressure amplitude at =4 is not

zero, but has some positive residual value. A possible reason

for this is that due to presence of small bubble fraction the

wavelength of sound is not exactly 60 mm, but somewhat

less, which causes a shift in the position of pressure node.

4.3.2. Washing experiments

Investigations of Moholkar and Warmoeskerken (2002)

revealed that the ultrasonic washing eciency depends on

the gas content of the fabric and the washing medium.

A maximum washing eciency (indicative of maximum

mass transfer) was obtained with a degassed washing

medium and non-degassed fabric, with the fabric positioned

at a pressure antinode. In this study, we use the same condi-

tions for the experiments. In this case, no cavitation nuclei

are contributed by the medium. The circular grooves in the

tip of the horn, which can entrap small amount of air that

can provide alternate source of nuclei in the medium, were

lled with silicon rubber. Hence, pumice stones are the onlysource of cavitation nuclei in the system. Eight specic time

intervals ranging from 0.1 to 2 s were selected for ultra-

sound irradiation. For each interval, three experiments were

conducted with dierent pieces of the model fabric to assess

reproducibility of results. Power input to the ultrasound

horn was 20 W at 25 kHz frequency. At this power input,

the ultrasound horn produces an acoustic wave with a pres-

sure amplitude of1:3 bar. The model fabric was soakedin the detergent solution (1:75 g l1 of sodium dodecyl

benzene sulfonate) for 5 min before ultrasonic treatment.

The model fabric was then xed between the glass rings of

the experimental cell in a water bath to avoid entrapment of

air beneath the fabric. In each experiment, 250 ml degassed

water was used as the washing medium. The dissolved oxy-

gen content of the washing medium was lowered to 2 ppm

using a chemical method (van der Vlist et al., 1994). The

time of ultrasound treatment was 3 min. After ultrasound

treatment, the model fabric was removed from the cell and

was dried in air. In order to assess the eect of the acoustic

pressure amplitude on the convective diusion coecient,

the set of experiments at 20 W power input was repeated

at power input of 15 W, for which the ultrasound horn

produces a wave with pressure amplitude of1:1 bar.

4.4. Data analysis

The quantication of the washing eect can be done by

measuring the reectance of the ultrasound-treated textile

sample. In order to determine the rate of soil removal, we

need to convert the reectance measurements into soil con-

centrations. For this purpose, KubelkaMunk theory is of-

ten used, which is the simplest tool for the description of

the optical properties of a turbid medium that absorbs and

scatters light (Kubelka and Munk, 1931). The mathemati-

cal expression of the KubelkaMunk theory that relates the

absorption coecient (K), scattering coecient (S) and the

reection coecient of the layer (Rf) is

K

S=

(Rf 1)2

2Rf: (2)

For particles larger than the wavelength of light, the scat-

tering coecient is inversely proportional to the diameter.

If the distribution of particle diameters is very small, then

one can approximate: S1=dp. The KubelkaMunk theorycan be coupled with the LambertBeer law to convert the

remission function into absolute concentration. In case of

negligible scattering coecient, the relationship between


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absorption coecient K and concentration of soil C is

K = Cln(10) = nC; where n = ln(10): (3)

We make two simplifying assumptions: (1) the diameters of

all carbon particles in the soil over textile are the same; (2)

the scattering coecient of carbon particles is negligible.

In the present study, it is not possible to determine n,since the manufacturer of EMPA 101 model fabric does

not supply the soil separately. Therefore, the soil concentra-

tions need to be dened on a relative basis. The reectance

of untreated EMPA 101 textile is 16%, which gives K=S

value of 2.205, corresponding to 100% soil concentration

in fabric. The reectance of completely washed EMPA 101

fabric is 70%, which gives K=S = 0:064, correspondingto 0% soil concentration in fabric (note that scattering co-

ecient is cancelled out in the calculations of relative soil

concentrations due to assumption 1). The fraction of origi-

nal soil removed from the fabric after ultrasound treatment

for dierent time periods (Mt) is estimated by assuming a

linear relation between K and Mt with: K= 2:205 at Mt = 0;K = 0:064 at Mt = 1, respectively.

5. The mathematical model

As discussed earlier, we approximate the geometry of the

textile as a porous plate. One-dimensional unsteady state dif-

fusion (in absence of velocity) in a porous plate is described

by Ficks second law

@C

@t= D

@2C

@x2: (4)

Let lxl denote the boundaries of the plate with thick-ness 2l. The solutions of Eq. (4) are listed by Carslaw

and Jaeger (1959) for dierent boundary conditions. For the

conditions

C = C1; for x = l;l;

C = C0; for t= 0;

the concentration of the diusing substance in the plate is

(Crank, 1975)

C C1C0 C1

= 1 4

n=0

(1)n

2n + 1exp

D(2n + 1)22t

4l2

cos

(2n + 1)x

2l

: (5)

The total amount of substance, which has left the plate after

a time t(Mt), and the corresponding amount after innite

time (M) are related by

Mt

M= 1

n=0

8

(2n + 1)22exp

D(2n + 1)22t

4l2

: (6)

Concentration prole for Fourier numbers 0.1: It is

convenient to introduce the Fourier number dened by

Fo =Dt

l2: (7)

For times such that Fo0:1, the concentration prole

can be approximated by a more convenient expression. The

detailed theoretical analysis of this (which is an extension

of the penetration theory) can be found in several references

such as Beek et al. (1999) and Janssen and Warmoeskerken

(1997). We give herewith only the nal result. For longer

contact times, the average concentration in the porous plateis

C1 C

C1 C0=

8

2exp(2Fo): (8)

This means that plot of logarithm of the expression on LHS

of Eq. (8) against Fourier number should yield a straight

line with slope 9:86.

5.1. Bubble motion near the fabric

The bubble motion near a solid boundary is a multi-faceted

phenomenon. The kind of radial motion the bubble under-

goes in the vicinity of the solid boundary, whether symmet-

ric or asymmetric, depends on the physical characteristics

of the boundary (such as elastic modulus) and also on the

stando factor.

Recently, Brujan et al. (2001) have reported high-speed

imaging studies of a vapor bubble dynamics near an elastic

boundary. They conclude that for boundaries with very low

elastic modulus, the bubble motion is asymmetric (with for-

mation of a jet) only for stando factors smaller than 1. For

cases where stando factor1, the bubble motion remains

symmetric as if occurring in an innite uid. Cotton bers

have a very low static elastic modulus (Morton and Hearle,

1997). It is likely that, at 25 kHz, the elastic modulus is stillrelatively small and, therefore, the cotton fabric acts as a soft

boundary for a bubble oscillating in its vicinity. Thus, we

present simulations of the bubble motion and the spherical

velocity eld generated around it using the Gilmore bubble

dynamics equation (Gilmore, 1954)

R

1

U

c

d2R

dt2+

3

2

1

U

3c

dR

dt

2

=

1 +

U

c

H +

U

c

1 +

U

c

dH

dR: (9)

H is the free enthalpy on the surface of the bubble

H =n

n 1

A1=n

0

P0 +2

R0

R0

R

3

2

R+ B

(n1)=n

[P + B](n1)=n

: (10)

A, B and n are constants (For water, A = 3001 atm,

B = 3000 atm and n = 7). P, the pressure in the bulk

liquid driving the bubble motion is

P = P0 PA sin(2ft): (11)


8/13


The velocity in the bulk liquid at a distance rfrom the bubble

center is

u(r; t) =R2

r2

dR

dt

: (12)

5.2. Adoption of the diusion model for the present study

While adopting the above models to explain the results

of the present experiments, we need to take into consider-

ation the possible discrepancies between the experimental

and theoretical results, which could occur as a result of mis-

matches between the assumptions made in the model and

the practical situation. One such mismatch is the geometrical

dierence between the model porous plate and the textile.

As discussed earlier, the textile is modeled as a porous plate

on the basis of the dierences between sizes of an individ-

ual bubble and that of an individual yarn. However, during

the experiments, we will be determining the rate of soil re-

moval from the textile on a macroscopic scale. We will bequantifying the soil transfer rates by measurement of the re-

ectance of the textile. The reectance is measured over a

large area of the textile (and not for a single yarn). There-

fore, this measurement technique will quantify the soil trans-

fer not only in the intra-yarn pores but also in the inter-yarn

pores. The reectance of the fabric is a function of the av-

erage soil concentration in the textile. Images of the EMPA

101 model fabric shown in Fig. 1 reveal that the soil is dis-

tributed unevenly over the textile surface, with the major

fraction of the soil present in the inter-yarn pores. As dis-

cussed in Appendix A, due to geometrical constraints, the

intensity of the convection resulting from the radial bub-ble motion in the inter-yarn region is much higher than in

the intra-yarn region. Thus, the soil removal process is ex-

pected to have two steps, with dierent soil removal rates:

(1) rate for smaller time period of ultrasound irradiation

during which only inter-yarn soil is removed; (2) rate for

longer time period of ultrasound irradiation during which

both inter- and intra-yarn soil is removed. It is obvious that

soil removal rate for step 1 is higher than step 2. This could

give rise to a dierence between the theoretical results pre-

dicted by the models with assumption of constant diusion

coecient with homogeneous porosity in the sheet and the

experimental results.

6. Results and discussion

The results of the experiments done with 20 and 15 W

power input to the ultrasound horn are shown in Figs. 4 and

5, respectively. It can be seen that the washing eect for

both power inputs shows a gradual rise with time of ultra-

sound irradiation. However, for shorter ultrasound irradia-

tion the rate of soil removal from the fabric is higher than for

longer ultrasound treatments. This result may perhaps be in-

terpreted from the discussion given in previous section. The

0.5

0.4

0.3

0.2

0.1

0.0

0 0.5 1 1.5 2

Time (sec)

Reflectance

0.00

0.25

0.50

0.75

1.00

Fractionoforiginalsoil

removed

0 0.5 1 1.5 2

Time (sec)

(A)

(B)

Fig. 4. Experimental results for fabric treatment at 20 W power input:

(A) reectance of the fabrics treated for dierent time periods; (B) soil

removal from the textile with time.

0.00

0.10

0.20

0.30

0.40

Reflectance

0.00

0.25

0.50

0.75

1.00

Fractio

noforiginalsoil

removed

0 0.5 1 1.5 2

Time (sec)

0 0.5 1 1.5 2

Time (sec)

(A)

(B)

Fig. 5. Experimental results for fabric treatment at 15 W power input:

(A) reectance of the fabrics treated for dierent time periods; (B) soil

removal from the textile with time.

results are consistent with the hypothesis that, for shorter ul-

trasound irradiation, only the soil on the fabric surface and

in the intra-yarn pores is removed, while for longer irradia-


9/13


0.00

0.25

0.50

0.75

1.00


removed

Experimenal D = 7.5e-10 D = 2.0e-9

0.00

0.25

0.50

0.75

1.00


removed

D = 5e-10 D = 1e-9 Experimental

0 0.5 1 1 .5 2

Time (sec)(A)

(B)

0 0.5 1 1 .5 2

Time (sec)

Fig. 6. Features of mass transfer in fabric under ultrasonic treatment: (A)

20 W power input; (B) 15 W power input. (Values ofD indicate diusion

coecient in m2 s1 for theoretical calculation of the soil removal curve

using Eq. (6).)

tion soil in the intra-yarn pores is also removed. However,

since the intra-yarn soil forms only a small fraction of the

total soil in the textile, the reectance of the fabric does not

rise signicantly for longer ultrasound irradiation. A com-

parison of Figs. 4B and 5B reveals that larger power input,which means cavitation at larger pressure amplitude, results

in faster and higher removal of soil from the fabric.

Some experimental errors, however, need to be taken into

account. In Fig. 5A, the reectance of the fabric does not

show a consistent rise with time of irradiation. Between 0.5

and 1 s, it decreases slightly, indicating lesser soil removal

in 1 s ultrasound irradiation than that for 0:5 s. This is in

contradiction with the theoretical speculation that soil re-

moval should increase with time of ultrasound irradiation.

In addition to duration of ultrasound irradiation, the soil re-

moval from the fabric depends on several other factors such

as size distribution and population of bubbles, the locationof bubble from fabric surface. Since we used a crude method

of providing cavitation nucleation near textile surface, it is

likely that the above-mentioned factors varied signicantly

in consecutive experiments. Thus, we attribute the anomaly

of lesser fabric reectance for 1 s ultrasound treatment than

0:5 s to the error in having exactly same kind of nucleation

produced near the textile surface in consecutive experiments.

A comparison of the theoretical and experimental soil re-

moval rate is shown in Fig. 6A and B for 20 and 15 W

power input, respectively. It can be seen that a single value

of the diusion coecient is inadequate to reproduce the

data. To the extent that an eective diusion coecient

y = -0.4999x - 0.4671

-1.60

-1.20

-0.80

-0.40

0.0 0.5 1.0 1.5 2.0 2.5

Time (sec.)

Time (sec.)

ln

(C)

y = -0.1963x - 0.5805

-1.2

-0.8

-0.4

0

0.0 0.5 1.0 1.5 2.0 2.5

ln(C)

(A)

(B)

Fig. 7. The logarithmic average soil concentration for longer contact time

(Eq. (8)): (A) soil concentration for 20 W power; (B) soil concentration

for 15 W power. The slope of the line (= 2D=l2) indicates the

convective diusion coecient.

can be dened, the data imply that it decreases with time.

The initial portions of the curves are described reasonably

well by Eq. (6) with D = 2 10

9 m2 s

1 at 20 W andD = 1 109 m2 s1 at 15 W. Thus, the diusion coef-cient is found to increase with the acoustic pressure am-

plitude, as expected. If the process can be described by a

superposition of two dierent diusive processes, charac-

terized by dierent diusion coecients, at large times the

solution will be dominated by the slower diusive process

and, therefore, one can estimate this smaller diusion co-

ecients from the large-time behavior of the data. On the

basis of this argument, one deduces from the data the values

D = 7:5 1010 m2 s1 at 20 W and 5 1010 m2 s1 at15 W. The shift between the two mass transfer rates occurs

at approximately 0:4 s (referring to Fig. 6A), which corre-sponds to a Fourier number of0:03 as calculated with thesmaller diusion coecient. A similar analysis for Fig. 6B,

where the shift between mass transfer rates occurs at 0:6 s,reveals that the Fourier number at which this shift occurs

also is 0.03.

Fig. 7 shows plots of variation in average soil concentra-

tion in the fabric for longer contact time using Eq. (8). The

plots are shown for t0:5 s for 20 W, and for t0:6 s for

15 W power input. The value of convective diusion coe-

cient obtained from the slopes of the plots, assuming fabric

thickness of 100 m, are 5 1010 m2 s1 for 20 W and31010 m2 s1 for 15 W. These values fairly match with


10/13


4

2

0

5

2

-5

2.5

1.25

0

1.5

0

-1.5

0 1 2 3 4 5

Time (No. of Acoustic Cycles)

0 1 2 3 4 5


0 1 2 3 4 5


0 1 2 3 4 5


Velocity(m/s)

Velocity(m/s)

Ratio(R/R0)

Ratio(R/R0)

(A)

(B)

Fig. 8. Simulations of radial dynamics and spherical velocity eld for

oscillations of 10 m bubble near textile surface. The distance of bubble

center from textile: 50 m, frequency: 25 kHz: (A) acoustic pressure

amplitude = 1:3 bar; (B) acoustic pressure amplitude = 1:1 bar.

the values of convective diusion coecients given in Fig. 6,

calculated from Eq. (6).

Mass transfer enhancement: Assuming that the carbonparticles in the soil on model fabric are spherical and have

size 0:1 m, the molecular diusion coecient obtainedfrom StokesEinstein equation is 3 1012 m2 s1.This diusion coecient needs further correction for the

porosity and tortuosity of the fabric. The corrected diu-

sion coecient, per denition given in Appendix A, is:

3 1013 m2 s1. The ratio of convective diusion coef-cient and the molecular diusion coecient indicates that

the enhancement factor is in the range 10002000.

As mentioned earlier, a direct estimation of the convec-

tion velocities near the textile surface is not possible in the

present experiments. However, simulations of the bubble dy-

namics equation can give us an order of magnitude estimate

of the convection velocities. Fig. 8 shows simulations of the

radial dynamics and the spherical velocity eld of a 10 m

bubble driven at dierent acoustic pressure amplitudes. The

other parameters for the simulation are given in the gure

caption. It can be seen that the bubble creates an oscillatory

velocity eld around it with dierent magnitudes of the ve-locities in the two directions: towards and away from the

bubble center. The (absolute) magnitude of the velocity eld

in any direction and, hence, the overall convection gener-

ated due to radial bubble motion increases with the acoustic

pressure amplitude. A typical estimate of the time scale of

particle motion will be d2p=D. With substitution of the

representative values dp0:1 m andD11013 m2 s1,

we nd = 0:1 s. From Fig. 8, it could be perceived that

the time scale of radial motion of the bubble is same as that

of the acoustic wave period, which is 40 s for 25 kHz fre-

quency. Therefore, the eective motion of the particle re-

sulting due to convection created by a single radial bubblemotion is negligible. The net motion of the soil particle un-

der the convection created by the bubble motion is, thus, a

function of the time-averaged velocity over several acoustic

cycles. The estimation of this velocity is highly complicated

because it will not only be a function of the bubble popula-

tion but also of the spatial distribution of the bubbles from

the textile surface and the strong interaction between adja-

cent bubbles. Since the purpose of the simulations presented

here is to give an approximate estimate of the convection

created by the radial bubble motion, we have not incorpo-

rated such complications in our analysis.

7. Conclusion

This paper describes a semi-empirical methodology for

a quantitative estimate of the mass transfer enhancement in

the ultrasonic textile treatments. In absence of precise quan-

tication of the convection created due to transient bub-

ble motion near the textile surface, we dened a convective

diusion coecient, which is representative of the mass

transfer enhancement due to ultrasound. The geometry of

the textile was modeled as a porous plate. The experimen-

tal results were correlated to the theoretical values of soil

removal with time using solutions of the diusion equa-tion. The soil removal from the textile showed two distinct

regimes with two dierent convective diusion coecients.

This result was explained in terms of the non-uniform ini-

tial distribution of soil in the textile and the dual porosity of

the textile; inter- and intra-yarn porosity. Textile treatments

at higher acoustic pressure amplitudes were found to give

higher diusion coecients. The order of magnitude of the

mass transfer enhancement for the model wet textile pro-

cess, which was dened as the ratio of convective diusion

coecient to molecular diusion coecient, is in the range

10002000. This methodology can also be applied for other

ultrasonic wet textile treatments; however, the mass trans-


11/13


fer enhancement factor may change from system to system

depending on the properties of the textiles and those of the

ultrasound system. We hope that this study will contribute

towards the design and scale-up of ultrasonic wet textile

processes.

Notation

c velocity of sound, m s1

C concentration of diusing substance, kg m3

or kmol m3

df diameter of a ber, m

dp particle radius, m

dy diameter of a yarn, m

D diusion coecient, m2 s1

De eective diusion coecient, m2 s1

f frequency of the acoustic wave, Hz

H free enthalpy on the bubble surface, J kg

1

k1 mass transfer coecient for intra-yarn

region, m s1

k2 mass transfer coecient for inter-yarn

region, m s1

k3 mass transfer coecient for the boundary

layer, m s1

K absorption coecient, m1

Kinter-yarn permeability between the yarns, m2

Kintra-yarn permeability between the bers, m2

K0 Kozenys constant, dimensionless

K0 Kozenys constant, dimensionless

KT overall mass transfer coecient in the textile,m s1

l half-thickness of the porous plate, m

M amount of diusing material transported in

innite time, kg or kmol

Mt amount of diusing material transported in

time t, kg or kmol

PA pressure amplitude of the acoustic wave, Pa

P pressure in the medium at innity, Pa

P0 ambient (atmospheric) pressure, Pa

r radial coordinate, m

R bubble radius, m

Rf reection coecient, dimensionless

R0 initial bubble radius, mS scattering coecient, m1

t time, s

T temperature of the medium, K

u velocity in the medium, m s1

U bubble wall velocity, m s1

v mean uid velocity, m s1

x distance coordinate, m

Greek letters

tortuosity factor, dimensionless

f porosity between the bers, dimensionless

n Naperian molar absorption coecient,

dimensionless

y porosity between the yarns, dimensionless

polytropic constant of the bubble contents,

dimensionless

Boltzmann constant, J K1

wavelength of the acoustic wave, m viscosity of the medium, Pa s

L density of liquid, kg m3

0 density of the liquid in undisturbed state,

kg m3

surface tension, N m1

Acknowledgements

The authors are grateful to Stork Brabant B.V., Nether-

lands for the funding of the project at the University of

Twente.

Appendix A. Transport phenomena in textile materials

A.1. Fluid ow through the fabric

For a uid owing through the textile, the inter- and

intra-yarn regions form two parallel paths. The relative con-

tribution of the ows in these two regions can be judged by

the ratio of the permeabilities of these regions. The perme-

abilities of the inter- and intra-yarn regions can be written

as (Gooijer, 1998)

Kintra-yarn =1

16K0

3f

(1 f)2d2f; (A.1)

Kinter-yarn =1

16K0

3y

(1 y)2d2y: (A.2)

We make an assumption of constant K0 for the inter- and

intra-yarn pores on basis of similar porosities of an individ-

ual yarn and the overall textile (Gooijer, 1998). The ratio

of Eqs. (A.1) and (A.2) with representative values of the

parameters: f = 0:35 0:45; y = 0:4 0:5; dy = 250 m;df = 15 m gives

Kintra-yarn

Kinter-yarn =

1

200 to

1

2000 : (A.3)

The ratio of Kintra-yarn and Kinter-yarn indicates that the per-

meability between the yarns is far higher than the perme-

ability in the yarns. Therefore, most of the ow will follow

the path of least resistance (between the yarns), and practi-

cally no ow will occur through the yarns. The ow through

the inter-yarn region may, however, penetrate an individ-

ual yarn to a small extent. This can create a thin convective

shell near the periphery of the yarn. Nonetheless, the cen-

tral core of the yarn still remains a stagnant core zone with

no ow inside. Therefore, the mass transfer in this region

occurs only by the slow process of diusion.


12/13


TEXTILE

YARNS

FIBRES

INTRA YARN

PORE

INTER YARN PORE

Convective shell

in the yarn

Stagnant

Intra-yarn

region

Boundary layer

YARN BULK

C

r

Inter-yarn region

(A)

(B)

Fig. 9. The mass transfer mechanism in the textiles: (A) schematic

diagram indicating the general structure and dual porosity of the textiles;

(B) the three stages of mass transfer in textile materials and the typical

concentration proles of the diusing substance.

A.2. Mass transfer in the fabric

For washing, the mass transfer in these textile materials

can be divided into three distinct steps, which occur in series,

as shown in Fig. 9:

1. mass transfer from the stagnant intra-yarn region to the

inter-yarn region through the convective shell on the pe-riphery of the yarn;

2. mass transfer from the inter-yarn region to the boundary

layer between the bulk liquid and the textile surface;

3. mass transfer from the boundary layer to the bulk liquid.

In order to assess the relative contribution of each of these

steps to the overall mass transfer process, one needs to have

an insight into the nature of uid ow through the textiles.

In order to nd the overall mass transfer coecient (KT) in

the textile, we specify k1, k2 and k3 as the individual mass

transfer coecients for the rst, second and the third step of

mass transfer mentioned above. KT can be determined using

familiar relationship in the mass transfer theory1

KT=

1

k1+

1

k2+

1

k3: (A.4)

We make a simplifying assumption that the mass transfer

in the third step (from the boundary layer to the bulk) is

very fast compared to the other steps. Now to estimate the

relative magnitudes of k1 and k2 we use the mass transfercorrelations developed for a packed bed. Several correlations

for the mass transfer coecients are listed by Janssen and

Warmoeskerken (1997). A single yarn can approximately

be described as a cylinder. To estimate the inter-yarn mass

transfer coecient, we choose the following correlation for

the ow past long cylinders perpendicular to the ow:

Sh= 0:42Sc1=5 + 0:57Re1=2 Sc1=3; 1Re104;

Sc0:7; Pe 1: (A.5)

The non-dimensional numbers are dened as

Re =L vdy

; Sh =

kdy

D; Pe =

vdy

D: (A.6)

To estimate the inter-yarn mass transfer coecient, we

substitute the following representative values in the above

correlation: v =2 cm s1; dy =250 m; L =1000 kg m3;

D = 1 109 m2 s1; = 0:5 and Sc 1000. Substitut-ing the above values, we nd that Sh 25 and from thedenition of the Sherwood number it follows that k2 1:4102 cm s1. To estimate the mass transfer coecientinside the yarn, we make use of the analysis given by Matsui

et al. (1978), who have shown that for longer contact times

[Dt=d2y ] the Sherwood number 5:8. Due to the porousstructure of the yarns, the eective diusion coecient in-

side the yarn is smaller than the actual diusion coecient.Therefore, we use the correction given by Rietema (1976)

to estimate the eective diusion coecient in a porous

medium

De =y

2D: (A.7)

Substituting representative values of y and as 0.4 and 2,

respectively, gives De 11010 m2 s1. The mass trans-

fer coecient inside the yarn (k1) is then calculated from

the Sherwood number as 4 104 cm s1. The ratio ofk1 and k2 gives us an idea of the mass transfer process in the

textiles: 97% of the mass transfer resistance is determined

by the diusion process inside the yarns. As such, the diu-

sional mass transfer in the yarn is the rate-controlling step inthe overall mass transfer in the textile. This also implies that

enhancement of the mass transfer in the textile would ne-

cessitate conversion of the intra-yarn diusion process into

the faster convection.

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