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Unformatted text preview: \ Bibliotheek T8 R \ m ’52 Delft . Prometheusplein 1 Postbus 98 2600 MG DELFT Telefoon: 015 - 2784636 Fax: 015 - 2785673 Email: [email protected] D3‘1111130 1—dec—0 3 Bonnummer: 7 2 6 9 3 8 Aan: T.N.O. TECHN. PHYSISCHd. DI fiNSl POSTBUS 155 2600 AD DELFT NfiDfiRRAND Tav: A. van de Runstraat Aantal kopieén: 10 Uwreferentie(s): 008.05105/01.01 Artikelomschrijving bij aanvraagnummer: 72 6938 Artikel: A mathematical model of cleaning processes Auteur: Durr-H ;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 cleaning-in-place 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 Weibull-Verteilung, 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 ClP-Reini- 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 co-ordinate 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 log-log 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 co-ordinate 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 -) i---ea.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 RRSB-distribution on the RRSB-grid. 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 two-parameter 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 co-ordinate 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 two-parameter Wei- bull distribution. WWWW _\ N iné _L O OJ N-h-O) “ 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 black-dyed cotton, and El WFK type: Immedial green-dyed 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 heavy-duty 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 — motor-driven - “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 cycle-dependent 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 laboratory-scale cleaning-in—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 soil-covered 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 sub-processes (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 sub-processes to be identified and quanti- fled. The methods for analysing reliability and lifetime data in- clude procedures for identifying the sub-distributions 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 sub-processes (mixed) Sub-process number 4. Combination (weighting factor of sub-process 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 S-shaped characteristic on the RRSB grid (Fig. 41 in [23]), four sub-processes 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 sub-processes 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 sub-processes. 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 sub-processes 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 cleaning-in-place 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 (sub-process 2) starting almost from the beginning and active until the end of the 30-min cleaning period (time constant T: 20 min). Its soil-re- Tenside Surf. Det. 38 (2001) 3 moval efficiency is not predominant: 13.5 % of the total cleaning rate. 11 Three successive Weibull processes (sub-processes 1, 3, and 4) all characterised by remarkably increased slopes — from R = 4.5 (sub-process 1) to 25 (sub-process 4) — com- pared with the slope R = 2 of sub-process 2. Sub-process 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: sub-process 3, contributing 73% to the (area-related) overall cleaning rate. When this pro- cess is finished, a fourth and final sub-process 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 well-tried 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 sub-processes within a cleaning pro- cess consisting of several parallel or successive sub-pro- 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 55350-22 (1987). Bensche, 3.; Lechner, 0.: Zuverlassigkeit im Maschinenbau — Ermittlun von Bauteil- und System-Zuverlassigkeiten. 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 life-testing 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 e-mail: [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 co-author 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 well-known 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 well-known design used from the K11 onwards with its spacious and well-illuminated 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 high-resolution motionless force measuring system. The K100 has a precise high-dynamic 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 built-in 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 self-priming aeration stirrers”. Since 1978, he has been Involved in the cleaning-in-place (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: +49~40-51 44 01—0 Fax: +49-40-S 11 60 49 E-mail: [email protected] ...
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