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Artikel: A mathematical model of cleaning processes
Auteur: DurrH ;Grasshoff—A Tijdschrift: l «NS ID «. SUREAClANlS DﬁlﬁRGﬁNlSRItl EUR PHYSIK, CHﬁMIﬁ Jaar: 2001 Vol. 38 Pagina(s): 1 4 8 — 1 5 7 Aﬂevering: 3
Plaatsnr.: 167 F Uitsluitend VOOI‘ eigen gebruik / for own use only UND APPUCAT'ON 7 ,, i H. DUrr, Ess/ingen and A. Grthofﬁ Kid/Germany A mathematical model of cleaning processes Soil removal from solid or textile surfaces using mechanical en
ergy, heat and aqueous solutions of chemical cleaning agents is
a complex procedure. Owing to the number and variety of influ
encing technical, physical and chemical factors, modelling the
soil removal kinetics using a simple mathematical equation ap
pears to be almost impossible. But thorough analyses of numer—
ous experiments showed that there exists a mathematical mod—
el that correlates with the experimental data with a coefficient of
correlation higher than 0.99. The model is equivalent to the two—
parameter Weibull distribution, widely used in reliability and life
time testing of technical systems. Cleaning characteristics fitting
the new model can therefore be interpreted as the "lifetime dis
tribution of the soil subjected to a specified cleaning procedure".
The paper describes the new model, its stochastic background,
advantages and successful applications: bleaching process
in washing machines, dust removal from floor carpets using
electrical vacuum cleaners, and cleaninginplace of deposits
encrusted on milk heat exchanger surfaces using chemical
agents. Ein mathematisches Modell vom Reinigungsprozess. Die
Reinigung von festen Oberfl'achen und Textilien unter Einsatz
von Wéirme, mechanischer Energie sowie wassrigen Losungen
von Reinigungschemikalien ist ein komplexer Vorgang. lnfolge
der zahlreichen und unterschiedlichen technischen, physika
lischen und chemischen Einflussfaktoren, schien es kaum
miiglich, die Reinigungskinetik in ein einfaches mathematisches
Modell zu fassen. Die Analyse zahlreicher Experimente hat nun
gezeigt, dass ein solches mathematisches Modell existiert und
die Korrelationskoeffizienten zwischen den Messwerten und
der Modellgleichung tiber 0,99 liegen. Das Modell entspricht
mathematisch der zweiparametrigen WeibullVerteilung, die im
Rahmen von Zuverlassigkeits und Lebensdauertests technischer
Systeme haufig eingesetzt wird. Reinigungscharakteristiken, die
dem neuen Modell folgen, konnen demnach interpretiert wer
den als ,,Lebensdauerverteilungen der Verschmutzung unter
der Beanspruchung durch ein definiertes Reinigungsverfahren”.
Der Beitrag beschreibt das neue Modell, seine stochastische
Begriindung, Vorteile, sowie die erfolgreiche Anwendung fiir
Bleichprozesse in Haushaltswaschmaschinen, die Staubauf
nahme von Teppichbtiden durch Staubsauger und die ClPReini gung hartna'ckiger Ablagerungen auf W'a'rmetauscherfléichen von
Milcherhitzern unter Einsatz wassriger Reinigungschemikalien. 1 Introduction Soil removal from solid or textile surfaces using mechanical
energy, and aqueous solutions of chemical cleaning agents is
a complex procedure. It involves a number of physical and
chemical parameters such as mechanics, cleaning time,
composition, concentration and temperature of the cleaning
agents, soil properties, and — last but not least — the material
and structure of the soiled surfaces. To improve the result
and efﬁciency of practical cleaning processes, a mathemati cal model would be helpful. This would allow quantitative
comparison of different procedures. 148 Numerous attempts have been made at modelling the
cleaning kinetics using mathematical formulae. As discussed
[1], neither the “simple” exponential formula of Schlilﬁler [2]
nor the twoparameter approximation of Foremska [3] covered
the spectrum of cleaning processes or had a reasonable phy
sical interpretation, respectively [1]. The model presented here was initially discovered when
analysing the efﬁciency of household vacuum cleaners at re
moving dust from ﬂoor carpets [4, 5]. As it was also suitable
for describing bleaching intensity after several washing
cycles in household washing machines [4], the use of the
model for wet cleaning processes was tested. In addition,
when removing solid deposits encrusted on heat transfer
surfaces of milk heat exchangers with chemical cleaning
agents [1]. the resulting coefficients of correlation between
the experimental data and the values calculated with the
model were above 0,950. Despite this positive "numerica"
aspect of the model. an important question still remained
unanswered at that time. Was there a technological back
ground to explain the good ﬁt of the model for the different
processes analysed? New studies led to an explanation. based
on the stochastic character of the processes ﬁtting the model
[6]. The derivation of the present model, its theoretical back
ground. and the resulting type categorization of cleaning pro
cesses are given in this paper. 2 The model
2.] Cleaning characteristic According to the model [1] the remaining soil r (t) after the
affecting time t is: r(t) : e“"(‘i‘lkz remaining soil (1) where
T= time constant: theoretical time to reduce remaining soil
to 1/e a 0.368 or to reach (1 ~ 0.368) n 63.2% soil removal,
R= slope of the cleaning characteristic. A detailed explana
tion of this parameter is given in the section "Cleaning
characteristic grid".
The cleaning efﬁciency  or quantitative soil removal 5(t)
— can now be calculated as: s(t) = 1 —— r(L): ratio of soil removal. (2) Using the formulae (1) and (2) the complete equation for the
soil removal s(t) — the cleaning characteristic — is: s(t) = 1 — all)... (3) Fig. 1 a shows four ideal cleaning characteristic examples ac
cording to the model on the (linear) "soil removal 5 versus
time tgrid” to demonstrate the effect of the time constant T
and the interesting spectrum of slope R: three have the same
time constant Ta 10 min. but varying slopes R= 0.4, 1.0, «:3 Carl Hanser Publisher, Munich Tenside Surf. Det. 38 (2001) 3 and 2.7, while the last has the time constant T= 20 min and
slope R = 2.7. Both characteristics with the slope R = 2.7
have a point of inﬂection indicating the maximum cleaning
rate of the process. The general formula for the time coordinate ii of the
point of inﬂection is [1]: t, = r (jig—1)” (4) Only cleaning characteristics with a slope R> 1.0 have a
point of inﬂection. 2.2 Reference time A useful quantity to specify a particular characteristic is the
reference time, deﬁned as the time to reach a speciﬁed clean
ing result, e. g. 52 99% or r: 1% remaining soil, respec
tively. With R r99 = 0.01 s edit) (5)
it follows that: tgg = T(ln100)1/R: reference time (99% soil removal) (6) The advantage of the reference time is that it contains both
parameters of the characteristic: Tand R. The second factor
of the formula we called the “slope factor”, as it only depends
on the slope R of the characteristic: fgg = (1n100)1/R: slope factor (99% soil removal). (7) To specify a reasonable percentage of soil removal e. g. 99,
95, or 80 %, the particular cleaning procedure has to be taken into consideration. The examples in section 5 offer some cri
teria. 2.3 Cleaning characteristic grid One of the striking advantages of the cleaning characteristic
is that in a loglog versus log plot the characteristic is repre
sented as a straight line [1]. As derived in [1] the equation can
ﬁnally be written as [1, equations (14 c, 14 e)]: lads 1/1) == R lair} +1g(1s e)  R em
= R lg(t} + constant. In a grid with the coordinate axes
lg (lg 1/1) and
1s {t} the cleaning characteristic is a straight line with the slope R.
(The curly brackets here and in (8) indicate that the numeri
cal value of the time t and the time constant Tis meant. [1]) Fig. 1b shows the four ideal cleaning characteristics of
Fig. 1 a on the "cleaning characteristic grid". The meaning
of the slope R is obvious. I ordinate:
I abscissa: 3 Theoretical basis of the model 3.1 The first approach
The idea of searching for a mathematical model of the clean ing process arose when analysing the dust removal efﬁciency
of electric vacuum cleaners [4]. According to International Tenside Surf. Det. 38 (2001) 3 H. D'Lirr and A. Grthoff: A mathematical model of cleaning processes r I 15 20 2 Time in min )
iea.2%‘ Soil removal in "la —) 30 Figurila) 1 a Four ideal cleaning characteristics (linear grid, Weibull grid see
Fig.1 ) 99 0 III'AJH
4‘ 70 III, A”
f 50 Kim“
'7; 30 E III A a) r
' V
s f"
m . 1 2 5 10 20 50 100 Time In min 9
   53.2% Figure 1 b Four ideal cleaning characteristics (Weibull grid, linear grid see
Fig. la) Standard IEC 312 [7], dust removal from carpets is tested
using a sand soil of speciﬁed particle size distribution. As a
first approach the spectrum of particle size distributions was
therefore studied and checked for its match with the results
of practical vacuum cleaning tests. The best correlation be—
tween the measured and calculated dust removal was
achieved with the RRSBdistribution [4], the name of which
honours the four persons connected with its development:
Rosin, Rammler, Sperling, and Bennett. The fraction of parti
cles passing sieve width dis 0(a) : 1 — 6(5)": RRSB~distribution [8] (9) where d = particle size or sieve width, respectively, d’ = particle size corresponding to D (d) = 0.632 = 63.2 %,
n = slope of the RRSBdistribution on the RRSBgrid. The similarity with the equation of the cleaning characteris
tic (3) is obvious. The main reason for starting to search again for an expla
nation of the theoretical or technological background was the
good fit of the model with wet cleaning [1, 6] and bleaching
processes [4] as well, although these processes have no con
nection at all with particle size distribution. 3.2 The Weibull process New studies [6] led to the Weibull distribution widely used in
the statistical analysis of reliability and in lifetime testing of 149 H. Dun and A. Grthoff: A mathematical model of cleaning processes m... .r... technical systems such as machines and components. The
cleaning characteristic is quite similar to the Weibull distribu—
tion: in the twoparameter notation of failure probability F (t)
at time tis [9]: F(t) : 1 —— e—(illll: failure probability (10) where the parameters 0 = characteristic lifetime: corresponding to 63.2 % failure
probability, 3 = failure slope. Compared with other distributions used in reliability analy
sis, the Weibull distribution is of outstanding importance: it
is — in simple terms ~ the mathematical expression of the
principle "The strength of a chain cannot be better than the
strength of its weakest link.” For example, the Weibull distri
bution describes the failure probability of a technical assem
bly exactly in theoretical terms when the assembly consists of
an (in a borderline case) inﬁnite number of components and
the failure is caused by the weakest component [10]. The ﬁrst
practical ﬁndings on this subject were explored in 1939 by
Weibull [11], and — mainly the stochastic aspects — e. g. in
1943 by Gnedenko [12]. For more detailed information about
extreme order statistics, statistical reliability theory, and life
time testing methods the reader is referred to some basic lit
erature, e. g. [13, 14, 15]. As a consequence, the following interpretation of the
cleaning process can be stated. if a cleaning process is ex
pressed by the new model with good fit, it indicates that this
process I has a stochastic character, and i does not progress evenly but at any time is determined by
its "weakest link" or in other words: the soil most easily
removable at the particular moment. Neither conclusion contradicts present ideas of the cleaning
process. On the contrary, they include aspects previously ex
cluded just because of their complexity. The cleaning characteristic according to the new model
now can be interpreted as the "lifetime distribution of
the soil subjected to a speciﬁed cleaning procedure". Such
cleaning processes we will therefore call "Weibull pro
cesses". 4 Type categorization of cleaning characteristics According to the range of slope values, a type categorization
of cleaning processes can be developed. The distinctive fea ture is the "relative cleaning rate" derived from the “cleaning
rate”. 4.1 Cleaning rate The cleaning rate 6(t) is the time derivative of the soil re
moval 5(t). According to equation (19b) in [1] we obtain: 6(t) = r(t) R#1: cleaning rate. ( 11) Fig. 2 shows the cleaning rates for the four ideal characteris
tics in Fig. 1 a in a linear grid. Both characteristics with slope
R = 2.7 have a maximum cleaning rate, the time coordinate of which is identical with that of the point of inﬂection in the
cleaning characteristics of Fig. 1 a. The cleaning rate is equivalent to the density function of the twoparameter Wei
bull distribution. WWWW _\
N iné
_L
O OJ NhO) “ Cleaning rate in %lm 0 5 10 15 2O 25 30
Time in min 9 Figure 2 Cleaning rate for the four ideal cleaning characteristics in Fig. i 4.2 Relative cleaning rate As the cleaning rate according to (11) is at any time propor
tional to the remaining soil r(t), it is possible to calculate a
cleaning rate related to the remaining soil, in short a “relative
cleaning rate" /)(t): )(t)  (t) — R (will relative cleanin rate (12)
’ " r(t) " r T ' g The relative cleaning rate is equivalent to the hazard or fail
ure rate in lifetime testing terminology. For a sloPe R=1
only — equivalent to the "simple" exponential function — the
relative cleaning rate is constant: [)(R = 1) = 1/T: constant (no function of time). (13) Fig. 3 shows the relative cleaning rates for the four ideal
cleaning characteristics in Fig. 1a in a linear grid, and three basic types of cleaning processes can easily be identi
ﬁed: l The “exponential” type with slope R = 1: constant relative
cleaning rate (CRCR). Because of its independence from
time, this type — using the reliability terminology — can be
characterised as a memoryless or historyless process: the
cleaning time already elapsed has no inﬂuence on soil
removal capability and there is neither an increase/
improvement nor a decrease/reduction in the relative
cleaning rate. This seems to be the most valid explana
tion as to why the exponential cleaning process
model succeeded in only a few cases when used in the
past Rel. cleaning rate in 1lmin —> Time in min 9 Figure 3 Relative cleaning rate for the four ideal cleaning characteristics in
Fig. i Tenside Surf. Der. 38 (2001) 3 MP??— WW l The type with slope R< 1: characterised by a monoto
nously decreasing relative cleaning rate (DRCR). The
longer the cleaning time that has elapsed, the lower the
relative cleaning rate and the more difﬁcult it is to re
move the remaining soil. As will be shown in section 5,
the previously analysed process of dust removal from
ﬂoor carpets using household vacuum cleaners and
bleaching processes in household washing machines be
long to this type. I The type with slope R>1: characterised by a monoto
nously increasing relative cleaning rate (IRCR). The "his
tory’ of the cleaning process has a positive inﬂuence on
the relative cleaning rate. It is not surprising that it is
mainly the wet cleaning processes analysed that belong
to this type (soaking effect). Interesting features of the cleaning process become apparent
when plotting the relative cleaning rate versus soil removal
[16, 17]. As already published in [1], with the new model it
is possible to calculate the relative cleaning rate as a function
of the soil removal 5: wetter In this context the time coordinate is not apparent any more. (14) 5 Examples In the following examples, the parameters time constant T
and slope R of the characteristics have been determined
using linear regression analysis between the measurements
and the characteristic according to equation (8 a). In the ﬁg
ures, the symbols represent the measured values and the
lines the characteristics calculated with the parameters
found. The tables additionally give the slope factor, reference
time and coefficient of correlation rc (index c in order to
avoid confusion with the symbol r for remaining soil). 5.] Bleaching performance of washing detergents 5.1.1 Experimental Bleaching intensity was tested after a number of washing
cycles using dyed “bleaching test fabrics” of Characteristic Bleaching fabric type  w = 25
Parameters  Cycle constant H. DUrr and A. Grthoff: A mathematical model of cleaning processes EMPA type: Immedial blackdyed cotton, and El
WFK type: Immedial greendyed cotton. After 5, 10, 20 and 25 washing cycles, one 15 cm long section
of the 15 cmx75 cm test fabric was cut off and the reﬂec
tance was measured according to the publication EC 456 [18]. The bleaching intensity BI after w washing cycles was
then calculated according to: Xw —Xs BI = 1000 ——————
w AXo—Xs (15) where
X0 = reﬂectance of the undyed test fabric (white),
X3 = reﬂectance of the dyed test fabric before washing, Xw=reﬂectance of the dyed test fabric after it) washing
cycles. Two heavyduty powder detergents (code 1 and code 2) with
bleach activator were tested in the 60 °C programme without
prewash in a household washing machine [4]. 5.1.2 Results Table 1 includes all measurements and results of the regres’
sion analysis and Fig. 4 the four bleaching intensity charac
teristics on the Weibull grid: WFK and EMPA test fabrics
washed with both detergents. The symbols represent the
measured bleaching intensities (detergent 2: ﬁlled symbols),
the lines the characteristics calculated with the parameters
found (detergent 2: broken lines). With respect to the inde
pendent variable “Washing cycles" (symbol w), the parameter
“time constant” had to be changed to "washing cycle con
stant” (symbol W). The coefﬁcient of correlation rc is above 0.9995, thus sup
porting the good fit of the model for this kind of bleaching
process. Slope R in all cases is about 0.8, indicating that the
processes belong to the type with monotonously decreasing
relative cleaning rate (DCRC). Because of the decreasing re
lative cleaning rate, it does not make much sense to increase
the number of washing cycles far above 25, in View of the ad
ditional testing expenditure. The reference cycles are there
fore calculated for 80 % bleaching intensity. There is a remarkable difference between detergents 1
and 2: the cycle constant for detergent 2 is about 10% smal Blcaching fabric type Table I Bleaching effect in washing machine: 2 detergents, WFK and EMPA bleaching fabrics Tenside Surf. Bat. 38 (2001) 3 151 H. Dilrr and A. Grallhoft: A mathematical model of cleaning processes (0 UIVCDLO
OO (JO
OO Bleaching intensity in % —> 100 1 2 5 10 20 50 Number of washing cycles —)
:—   63.2%l Figure 4 Bleaching intensity characteristics of two detergents on WFK and
EMPA bleaching test fabrics (Weibull grid; symbols represent the measure
ments, lines the calculated characteristics) ler than the value for detergent 1, indicating that detergent 2
has a higher bleaching performance. The inﬂuence of the
test fabric type (WF K and EMPA) does not seem important;
according to the cycle constant, the bleaching effect on the
WFK fabric is proportional (about 65 %) to the effect on the
EMPA fabric. Although the slopes of the four characteristics vary be
tween 0.76 and 0.83, the inﬂuence of fabric type and deter
gent d0es not seem to be signiﬁcant; particularly because of
the similar slope factors for the WFK and EMPA fabrics, the
assumption had already been made in [4] that the slope is
mainly a parameter of the washing process in general, e. g.
the washing machine, water, and washing programme were
kept constant during these tests. 5.2 Dust removal from carpets using household vacuum
cleaners 5.2.1 Experimental The method of measuring the dust removal efﬁciency of
household vacuum cleaners is speciﬁed in detail in the pub
lication IEC 60 312 [19]. The speciﬁcation covers the follow
ing: l Carpet: e. g. wool pile of speciﬁed height (recommended
for international comparison), but also other types of na
tional interest. I Dust: mineral dust with speciﬁed particle size distribu
tion, quantity e. g. 125 g/m2, dust embedded using the
standardised roller (3.8 kg, 50 mm diameter, 380 mm
width). I Test area: constant length of 700 mm and width with re
sPect to cleaning head of the vacuum cleaner (cleaning
head width ~20 mm), long side in the direction of the
carpet pile (ﬁrst stroke). l Cleaning procedure: to (forward) and fro (backward)
strokes (cleaning cycles) at 0.5 m/s speed and over 1.2 m
length, thus providing a starting and braking distance of
20 cm. I The dust removal 5 is determined by the increase in va
cuum cleaner or ﬁlter weight after a speciﬁed number
of cycles divided by the weight of the dust initially distrib
uted on the carpet. The following examples include modiﬁcations made with re
spect to checking the inﬂuence of particular parameters of
the vacuum cleaning procedure. m 152 \l
O Floor nozzle DA: “againsl'r I
A.‘
r. I
2 3 4 5 6 8 10 Stroke number >
   83.2%l 01
O (. Dust removal in % 9 _\
O
._x Figure 5 Dust removal "stroke" characteristics (Weibull grid; symbols repre
sent the measurements, lines the calculated characteristics) 5.2.2 Results The usual carpet cleaning procedure in practice and accord
ing to IEC 60 312 [19] consists of a sequence of cycles each
including a forward and backward stroke. During a single cy
cle, the stroke direction changes not only with respect to the
actual “front” edge of the cleaning head but also with respect
to the carpet pile, e. g. its inclination. It cannot be assumed,
therefore, that the cleaning characteristics of the forward and
backward strokes are identical. Fig. 5 shows — on the Weibull grid ~ the inﬂuence of dust
removal when using strokes only in one direction, e. g. “in”
or “against” the direction of pile inclination. The results
(5 test series, each with measurements after n= 1, 2, 3, 4, 5
and 6 strokes) are taken from the doctoral thesis of Gerber
(Tables 34 and 35 in [20], see also Table 1 in [5]). Test para
meters:  Cleaning head: ﬂoor nozzle DA
I Particle size: 20 to 600 um (Sc)
l Dust quantity. 50 g/m2. As the ﬁrst stroke is different from the following (no preced
ing contact of floor nozzle with carpet), the dust removal
after the first stroke is not included in the regression analysis
used for calculating the parameters N (stroke constant) and
R for the characteristic in Table 2. Fig.5 shows that in fact
the dust removal measured (symbols) after the ﬁrst stroke
has about 2 and 4 % distance from the calculated characteris
tics (straight lines). The resulting coefﬁcients of correlation
are 0.998 and 0.999, thus supporting the good fit of the new
model for dust removal by household vacuum cleaners as
well. The resulting slopes — 0.45 "in" and 0.32 “against” direc
tion of pile inclination — indicate that both procedures belong
to the type with monotonously decreasing relative cleaning
rate (DRCR). With each additional cleaning stroke, the dust
removal becomes signiﬁcantly smaller. It is reasonable,
therefore, to calculate the reference stroke number for 80%
dust removal and not for a higher value. Fig. 6 and Table 3 show the results of dust removal tests
when the cleaning head is moved in the usual cycles but with
the stroke direction at right angles to the direction of carpet
manufacture (see also [4], Figs. 1a, b and 3, Tables 1 and 2).
The tests were performed on two carpets (made from wool
and polyamide fibres), using both cleaning heads of the clea
ner provided for dust removal from ﬂoor carpets, i. e. the
floor nozzle and — motordriven  “power” brush. Dust Tenside Surf. Bat. 38 (2001) 3 Characteristic Strokes (floor nozzle on carpet) ‘in 1 ‘against' direction of pile inclination Parameters"  Stroke constant  Slope  Slope factor (80%)  Reference strokes (80%)
Coefﬁcient of corre rc 0.9991 0.9981
lation” ” calculated without results of ﬁrst stroke Table 2 Vacuum cleaning ~ Influence of stroke direction on dust removal
(floor nozzle) removal was measured after 6 = l, 2, 5 and 10 cycles (at least
3 measurements). The coefficient of correlation is above 0.99. This high val
ue is extremely surprising, since any cycle characteristic in
principle consists of two basic characteristics: one for the for
ward, the other for the backward stroke. As already published
in [5] (equations 9, 10 and 11), each cycle 1:, therefore, has its
individual slope RC: Rc = R’ —— bc AR where (16a) AR: R’—— R” (16b)
1 JL b, =—2—G—é:—1—. (16c)
lgc~ 1 Characteristic
Cleaning with Floor nozzle  L‘ = 10 . 3
Parameters — Reference cycles (80%)
Coefﬁcient of correlation Table 3 Vacuum cleaning w Inﬂuence of carpet type, cleaning head (ﬂoor nozzle, power brush) on dust removal Tenside Surf. Bat. 38 (2001) 3 Power brush H. Dijrr and A. Gralihoff: A mathematical model of cleaning processes E as r l ""'  I I I Dust removal in % —> IIIHIII 1O . I I
1 2 3 4 5 6 v 8 0
Cycle number 9
3   63.2% Figure 6 Dust removal "cycle" characteristics (Weibull grid; symbols repre
sent the measurements, lines the calculated characteristics) The cycledependent factor b, has the following tendency. n 2ncl cycle I); = 0.415
r 8‘h cycle b8 = 0.483
r 32“d cycle b32 = 0.504. When the starting stroke characteristic (R') has the greater
slope, the slope of the cycle characteristic according to equa—
tion (16 :1) becomes smaller with each additional cycle. For
AR=0, only, the cycle characteristic has the same slope
RC: R’ (forward stroke characteristic) = R“ (backward stroke
characteristic) . For both carpets, using the power brush (broken lines in
Fig. 6) provides the better coefﬁcient of correlation 0.9996
and 0.9998 compared to the use of the ﬂoor nozzle: 0.9937
and 0.9981. This leads to the conclusion that this particular
power brush does not react as sensitively to a change in
stroke direction as the floor nozzle used, with the conse
quence that forward and backward strokes have the same or
at least rather similar slopes. When using the floor nozzle, the situation is remarkably
different. The soil removal measured at the ﬁrst and the
tenth strokes (Table 3: italic ﬁgures) lies slightly below the
calculated values, While the other measurements lie above
the regression line. Imagining that the measured values in
Fig. 6 (not ﬁlled symbols) were joined up, a slightly bent Carpet A (PA velour) Carpet B (wool velour) Cleaning with Floor nozzle Powar brush 153 H. Dun and A. Grthoff: A mathematical model of cleaning processes curve would result for both carpets (carpet A: square symbol,
carpet B: triangular symbol), allowing the conclusion that the
slope of the starting (forward) stroke characteristic must be
greater than the slope of the backward stroke. The slope seems to be speciﬁc to a particular carpet: in
the case of carpet A, R=0.36 (no difference between the
use of the ﬂoor nozzle and power brush) while the carpet B
slope is about 0.32. With both carpets, the power brush
(Fig. 6, broken lines) has a signiﬁcantly lower cycle constant
than the ﬂoor nozzle (Table 3). Because of the constant slope,
the cycle constant C seems to represent the essential para
meter for distinguishing between the dust removal efﬁciency
of various appliances (vacuum cleaners, cleaning heads) on
the same carpet (see also [4]). 5.3 Milk heat exchanger cleaning 5.3.] Experimental The removing of solid deposits encrusted on the heat trans
fer surfaces of milk heat exchangers using chemical cleaning
agents is a complex procedure. To study the cleaning process,
a laboratoryscale cleaningin—place (CIP) test rig was de
signed, which allowed cleaning trials to be carried out under
specified process conditions (temperature, ﬂuid dynamics,
time) that were based on values found in practice in cleaning
dairy plant heat exchangers. Circular stainless steel plates
50 mm in diameter, previously soiled using a specified heat—
ing procedure with fresh raw milk, were inserted into a rec
tangular flow channel. The cleaning solutions under test
were pumped in a circuit and deposit removal was visually
monitored through the transparent front on the flow channel
and recorded with a video camera. Single digitised frames
showing different stages of soil removal were transmitted to
a computer. With the aid of image processing software, a de
ﬁned area of interest on the screen could be differentiated
into clean (metallic) or still soiled sections. At the end of a
cleaning trial — usually 30 min — an Excel plot of soil removal
(remaining soilcovered area versus time) was obtained [1,
16,17]. Comparative trials have been performed with different
chemicals. For example in [1] the new model was used in
testing the deposit removal efﬁciency of the additives EDTA
and RVO®. In all cases, the reference cleaner used was an
aqueous solution of sodium hydroxide (5.0 g/l) in tap water
(17° German hardness) without any additives at a tempera
ture of 65°C and with a mean ﬂow velocity of 1 m/s. The
cleaning process using this reference cleanser without any
additives is analysed as follows: 5.3.2 Results Fig. 7 a shows the cleaning characteristic on the Weibull grid.
Although the coefﬁcient of correlation (0.9936) seems to be
rather impressive (Table 4: column "initial analysis”), it is ob
vious that there exists a discrepancy between the measured
process (symbols) and the characteristic calculated with the
parameters found (straight line). The regression line gives
the path in general, but does not represent the real process
in detail, which obviously involves some subprocesses (ap
parent in the graph from the wavy path of the symbols). As long ago as 1983 it was shown [21], by measuring the
ﬂow resistance in the ﬂow channel, that removal of the de
posits using alkaline detergents in fact takes place in several
partial phases. The starting time and duration of the separate
phases, caused e. g. by swelling or removal of single layers of
the soil, could be pinpointed exactly by the differential pres. 154 sure curve. Now the question was whether the new model
would allow these subprocesses to be identified and quanti
fled. The methods for analysing reliability and lifetime data in
clude procedures for identifying the subdistributions within
a mixed lifetime distribution [22] but even more elaborate
rules are used for analysing mixed particle size distributions
[23]. Following these rules for mixed RRSB distributions 4‘ 99
32 90
.s 70
1g 50
a 30
'6
5 10
E
9
{'3‘
U) 1 1 2 5 10 20 50 100
Time in min >
   63.2% Figure 7 a Milk heat exchanger cleaning: "simple" regression characteristic
(Weibull grid; symbols represent the measurements, line the calculated char—
acteristic) I 2 (13.5 %): T = 20 min, 1
1 2 5 10 20 50 100
Time in min ) 63.2% Figure 7 b Milk heat exchanger cleaning: sub and mixed characteristics
(Weibull grid; symbols represent the measurements, lines the calculated
(sub)characteristics) Soil removal (area) in % ) Time in min —)
' mam, Figure 7 c Milk heat exchanger cleaning: mixed characteristic (linear grid;
symbols represent the measurements, line the calculated characteristic) Tenside Surf. Bat. 38 (200]) 3 ., A1? w H. Diirr and A. Grthoff: A mathematical model of cleaning processes WWW W Characteristic Measurements Initial analysis (regression
line) .0
\J 00
y. 30
o \l is“
o
N
4:.
o .. to N ._.
2" >1 S" )5 P i
Ln \l c o J: 19.2
43.1
71.1
86.6
91.3
96.4 S r
I=18min .r I=24min HHHHIIE 46.9
64.8
79.7
90.0 I=27 min .r Parameters
is  Reference time (95%) m5 Coefﬁcient of correlation
 Weibull grid r 0.9936 — Linear grid (percent) — 0.9936 1) Measurements at 1 min intervals but reported data at 3 min intervals, only. 2.75 26.4 Combination of subprocesses (mixed) Subprocess number 4. Combination (weighting factor of subprocess 1: 6.5%.
2: 13.5%, 3: 73%, 4: 7%) w
o v
to so to
La.) N
.~.o
MN Ln 4: 1*" 1“ . m o
Lu
\I
tax a
P‘ on
mu
49".
o~ b) “\l
.wsa
mu
_
C)
u M
LI!
.5 0.9984 0.9994 to
.0
o
'3 0.
.0 . LII
N
O
N
M
0 Table 4 Milk heat exchanger cleaning — Cleaning characteristic under reference conditions (NaOH 5.0 g/l, no additives, 65 °C, 1.0 m/s) with Sshaped characteristic on the RRSB grid (Fig. 41 in
[23]), four subprocesses of the Weibull type were identiﬁed. In a second step the weighting factors were determined
(thin regression lines) for optimum ﬁtting with the soil re
moval measured. The method requires the following proce
dures: l Calculation of the cleaning rates of the measured soil re
moval data and the four subprocesses found. 1 Determination of the weighting factors necessary to give
an overall cleaning rate that is an optimum ﬁt with the
measured cleaning rate, using multiple regression analy
sis between the measured data and the values for the four
subprocesses. I Changing from the overall cleaning rate characteristic to
the overall (mixed) soil removal characteristic: the soil re
moval characteristic is the cumulative cleaning rate char
acteristic. The results are shown in Fig. 7b and the parameters and
characteristics of the subprocesses are reported in the sec—
ond part of Table 4. In Fig. 7b the improved ﬁt of the mixed
characteristic (bold curve) with the measurements (symbols)
is obvious: while the initial analysis resulted in differences
between measured and calculated soil removal ranging from
+10% (13 min result) to «7% (21 min result), these differ
ences are reduced to a range of about +2 % to —3 %. Parallel
with this, the coefﬁcient of correlation is improved from
0.9936 to 0.9984 (Weibull grid), and 0.9994 (linear grid), re
spectively. Analysis of the cleaninginplace of milk heat exchanger
surfaces with the new model provides the following insights. When using the reference cleaner, the following phases can
be distinguished: l One Weibull process (subprocess 2) starting almost from
the beginning and active until the end of the 30min
cleaning period (time constant T: 20 min). Its soilre Tenside Surf. Det. 38 (2001) 3 moval efﬁciency is not predominant: 13.5 % of the total
cleaning rate. 11 Three successive Weibull processes (subprocesses 1, 3,
and 4) all characterised by remarkably increased slopes —
from R = 4.5 (subprocess 1) to 25 (subprocess 4) — com
pared with the slope R = 2 of subprocess 2. Subprocess
1 is not predominant (only 6.5 % of the overall cleaning
rate) but — as we know — essential for starting the clean
ing process. Additional test runs with the aim of measur
ing the mass~related soil removal showed that about 65 %
of the soil mass is removed after the ﬁrst ﬁve minutes
(time constant T = 4.5 min, slope R = 0.7). When this
process  after about 7 min — is over, the most efﬁcient
cleaning phase starts: subprocess 3, contributing 73%
to the (arearelated) overall cleaning rate. When this pro
cess is ﬁnished, a fourth and ﬁnal subprocess starts:
time constant T: 25 min, slope R= 25. representing
about 8 % of the overall cleaning rate. Finally, Fig. 7c shows the measured soil removal values
(symbols) and the mixed characteristic (curve) on the linear gnd. 6 Conclusions Numerous experiments with very different cleaning proce
dures support a new approach to understanding cleaning
kinetics: the cleaning characteristics tting the new model —
Weibull processes — can be interpreted as the “lifetime distri
bution of the soil subjected to a speciﬁed cleaning proce
dure”. The cleaning characteristic (soil removal s(t) versus
cleaning time t) is expressed by the Weibull function: s(t) = 1 a an)" (3) The parameters time constant Tand slope R characterise a
particular cleaning process. According to the processes ana 155 H. Diirr and A. Grthoff: A mathematical model of cleaning processes lysed (see also [1]), the parameters seem to have particular
connections with the technological (technical, physical and
chemical) process parameters: I The slope R seems to be speciﬁc to the type of soil, soiled
surface and cleaning appliance, I While the time constant T (in other cases: stroke constant
N, or cycle constant W, or C, respectively) seems to be an
indicator for the tool (ﬂoor nozzle or power brush clean
ing head) or the cleaning agent used. There is no doubt that the most important achievement of. the new model is to provide a mathematical reference for
studying, analysing and improving cleaning processes. The
detailed advantages of the new model can be summarised
as follows: I It covers the cleaning process completely. from the begin ning until the end of soil removal. Its mathematical structure is relatively simple. It provides a quantity to condense — e. g. for reasons of
comparison — the cleaning characteristic into one single
value: the reference time or the reference cycle/stroke
number, respectively. I It offers — in general — effective methods for quick, quan
titative analysis of household and industrial cleaning pro
cesses. I On the Weibull grid the cleaning characteristic is repre
sented as a straight line. Only three data points are there—
fore required to check whether a characteristic ﬁts the
model or not, and — due to the log{t} abscissa of the grid
— optimum t steps for measurements follow a geometric
progression, e. g. 1, 3, 9, 27 min. I There are two welltried basic procedures for determin
ing whether the parameters ﬁt a set of measured “time
—~ soil remov ” pairs: the linear regression method and
graphical determination on the Weibull grid. For this pur
pose, commercial Weibull grid paper has a scale for deter
mining the slope R. I The slope R classiﬁes the type of process: R = 1.0 char—
acterises processes with a constant relative cleaning rate,
well known as “historyless” processes of the “simple
exponential" type, while characteristics with R < 1.0
have monotonously decreasing and characteristics with
R > 1.0 monotonously increasing relative cleaning rates;
the latter indicates that the “history” of the cleaning pro
cess has a positive inﬂuence on the relative cleaning
rate. I Using the tools developed for analysing mixed or com
posed lifetimes and particle size distributions of the Wei
bull or RRSB type with the new model, it is possible to
identify and specify subprocesses within a cleaning pro
cess consisting of several parallel or successive subpro
cesses and to quantify their contributions to the “overall”
cleaning rate. I Last but not least, the model is new only with respect
to its application for cleaning and similar processes
but it has already been widely employed and long estab
lished in other applications, e. g. the statistical analysis
of reliability and lifetime data. An extensive number of
references from this ﬁeld can now be used for a new
subject: the theory and practice of cleaning processes. While the new model undoubtedly improves our insight into
cleaning processes, it also offers a number of new and inter
esting topics for ﬁirther investigation: I Probably not all cleaning processes are of the Weibull type
but what other cleaning processes might follow the new model? For example, not all cleaning procedures enable
complete soil removal. After a certain cleaning time has
elapsed, soil removal cannot be further improved and re
mains at s < 100 %. Do we ﬁnd Weibull processes among
such cleaning processes? 1 In this paper no conﬁdence intervals have been calcu
lated: neither for the estimated parameters nor for the
predicted soil removal. It is recommended that in future
possible conﬁdence intervals should be determined for
the estimated cleaning characteristic. 1 Reliability theory distinguishes between estimation meth
ods for parameters achieved in lifetime tests running un
til all samples in the test have failed, and methods to be
used for “censored tests", ending after a speciﬁed time or
when a speciﬁed number of samples has failed. Is it ne
cessary to follow this discrimination when analysing
cleaning processes of the Weibull type? In 1951, W. Weibull published his pioneering paper “A statis
tical distribution function of wide applicability’ [24]. The re
sults presented in our paper support the assumption that we
can now add a new and important ﬁeld of application: clean—
ing technology. References Dr‘Jrr, H.,' Ora/lhofﬁ A.: Trans IchemE 77 (C) (1999) 1 14. SchlUB/er, H../.: Brauwissenschaft 29 (1976) 263. Foremska, E.: Tenside Surf. Det. 27 (1990) 202. Dr'Jrr, H.: Hauswirtsch. u. Wissensch. 43 (1995) 64. Diirr, H.: Hauswirtsch. u. Wissensch. 43 (1995) 200. DL’J‘rr, H.,' Wi/dbrerr, 6.: SOFW—Joumal 126(11) (2000) 54. Publication EC 312 (1981). Standard DIN 56145 (1976). Standard DIN 5535022 (1987). Bensche, 3.; Lechner, 0.: Zuverlassigkeit im Maschinenbau — Ermittlun von Bauteil und SystemZuverlassigkeiten. 2"" edition, 42 pp. Springer, Berin, Heidelberg, New York (1999). 11. Weibull, W.: lng. Handl.Vetenskaps Akad. 151 (1939) 1. 12. Gnedenko, B. W.: Ann. Math. 44 (1943) 423. 13. Galambos, 1.: The as mptotic theory of extreme order statistics. John Wiley 8.
Sons, New York, Chichester, Brisbane, Toronto (1978). 14. (Gumbel, E. 1.: Statistics of extremes. Columbia University Press, New York 1958). 15. Bain, L. 1.: Statistical analysis of reliability and lifetesting models — Theory and
methods. Marcel Dekker, Inc. New York and Basel (1978). 16. Grail/70ft A.: Tenside Surf. Det. 27 (1990) 130. 17. Gin/Shelf, A. ; PotthoIf—Karl, Ii: Tenside Surf. Det. 33 ‘(1 996) 278. 18. Publication IEC 456 (1974). 19. Publication IEC 60 312 (1998). 20. Gerbes, 8.: Modelluntersuchungen zurSau reinigungtextiler Bodenbelage.
Thesis Technische Universitat Milnchen, Fa ultat ltir Landwirtschaft und
Gartenbau (1986). 21. Ora/Sheff, A.: Kieler Milchwirtschaftliche Forschungsberichte 35 (4) (1983)
493. 22. Deutsche Gesellschaft for Quolildt (ed): Das Lebensdauernetz — Erlauterun
gen und Handhabung. Beuth, Berlin (1975) 23. 23. Bale], W.: Einfuhrung in die Korngrollenmefltechnik. 3'd edition, 58pp. Sprin
ger, Berlin, Heidelberg. New York (1971). 24. Weibull, W.:Trans. ASME, Series E: J. of Appl. Mechanics 18 (1951) 293. QPPNP‘P‘PP‘NT‘ Received: Sept. 20, 2000
Revised: Feb. 21, 2001 1 Correspondence to Horst Dﬁrr (Dipl.—Ing.) ipi—lnstitut filr Produktforschung und Information
Rontgenstr. 1 73730 Esslingen Germany email: [email protected] The authors of this paper Academically trained engineer (Dipl.ln .) Horst DU” graduated in mechanical engi
neerin from TU Fridericrana in Karlsru e in 1961. After some years as head of the
researc department of a grinding machine manufacturer, he took over the house
hold appliances laboratory of the ifw—lnstitut fL‘rr Warenprtifung near Stuttgart, the Tenside Surf. Det. 38 (2001) 3 if?! ﬁrst privately owned institute for comparative testing of consumer goods in Ger
man . In 1971, he and two colleagues established the ipi—lnstitut ft‘tr Produkt
forsc ung und Information (ipi—lnstitute for Product Research and Information) —
now in Esslingen ~ also working in the field of market research and product devel
opment. Since 1976, he has been lecturin in household technology and engineer—
ing at the University of Hohenheim. He ist e coauthor of four books and has writ— ten numerous papers, mainlyI concerning the methodology of comparative testing
and the techno ogy of house old appliances. Dr.—ng. Albrecht Graflhoff graduated in brewery engineering from the TU Berlin in
1967. This was followed by five years industrial experience in the manufacture of
brewery machines. Since 1972, he has been a member of the scientific staff at the A Multitalented Tensiometer The K100 Tensiometer is the latest new development from
KRUSS. As a further development of the K12 Tensiometer the K100 offers the wellknown and extremely wide range of
measuring methods: I Surface and interfacial tension measurements
(plate, ring with corrections according to Hawkins 8: lor
dan, Huh 8c Mason, Zuidema 8: Waters) 1 Contact angle measurements (advancing and retreating
angles) l c.m.c. measurement I Surface energy determination, dynamic adsorption mea
surements 1 Density determination and sedimentation measurements On top of all this, the combination of the K100 with the new
“LabDesld’ Windows software from KRUSS provides you
with maximum ﬂexibility —— a new scripting language incor
porated in the software means that measuring procedures
can easily be adapted to your special applications. The further development of the wellknown design used
from the K11 onwards with its spacious and wellilluminated
sample chamber allows quick and easy sample changing.
The automatic balance locking protection system also makes
changing samples easier while at the same time protecting
the highresolution motionless force measuring system. The
K100 has a precise highdynamic drive for positioning the
sample at speeds from 0.099 to 450 min/min; this greatly
accelerates the measuring procedure. The extremely high
measuring accuracy of 0.001 mN/m with a distance resolu—
tion of 0.11m makes the K100 into an outstanding tensi
ometer. For special applications a sample holder can be used
which is thermostattable up to 130°C, a builtin stirrer with Tenside Surf. Ber. 38 (2001) 3 H. Di'irr and A. Grthoff: A mathematical model of cleaning processes Institute for Process Engineering at the Federal Dai Research Centre Kiel. in 1978
he obtained his doctorate from the TU Berlin in the ield of process engineering with
a thesis entitled: "Uber den Eintrag von Sauerstoff in wttllrige Fermentationsmedien
mit Hilfe selbstansaugender Begasungsriihrer' — ’The introduction of oxygen into
aqueous fermentation media by selfpriming aeration stirrers”. Since 1978, he has
been Involved in the cleaninginplace (ClP) of food manufacturing plants. In 1991/
92, he participated in a research project on: "Cleaning of milkin pipelines with a
two«phase liquid~air flow" at the University of Wisconsin, Madison USA. He has writ—
ten numerous papers and lectured widel on the design and cleanabiiity of food
manufacturing plants, the fundamentals o deposit formation in milk heat exchang
ers and the fundamentals of the CIP process. out magnetic field inﬂuences on either the sample or the
measuring system as well as the possibility of working under
protective gas and for connecting an automatic dosing sys
tem; these are included as standard. KRUSS GmbH Advancing Surface Science
Borsteler Chaussee 85—99 a
22453 Hamburg Germany TEL: +49~4051 44 01—0 Fax: +4940S 11 60 49
Email: [email protected] ...
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