Modelling attenuation and velocity of ultrasonics in reconstituted milk powder
MModelling attenuation and velocity ofultrasonics in reconstituted milk powder
Bernard LacazeTØSA, 9 Bd de la Gare, Toulouse, France e-mail : [email protected] 6, 2020
Abstract
In the context of food quality control, ultrasonics provide proven meth-ods which are able to replace manual controls. The latter are subject tothe lack of objectivity of human judgement. Automatic control increasesreliability and reduces costs.This paper revisits data coming from ultrasonics through reconstitutedmilk powder. Two characteristics are studied using (cid:133)ve productions of awell known manufacturer. Measured attenuation and dispersion of ultra-sonics are explained through stable probability laws and random propaga-tion times. We have proved elsewhere that this model is available in manypropagation problems, in acoustics, ultrasonics and in the electromagneticworld. keywords: ultrasonics , instant milk powder, linear (cid:133)ltering, stableprobability law, random propagation times. When milk powder is mixed with water, the result is a liquid which homogeneityis blurred by liquid-fat aggregates at the surface and small particles of proteingel in the bulk. To evaluate the impact of these (cid:135)aws on consumers, visual re-constitution tests (RT) are carried out. People give a rating to some number ofsamples coming from di⁄erent sources. In [1] (forming the basis of this article),(cid:133)ve samples were examined, from (cid:133)ve di⁄erent milk powders, by an undeter-mined number of people. Each of them received two notes between 0 and 5, onefor the surface appearance (ST) and the other for the bulk appearance (BT).The table below gives the test results. BT : : ST : : : : Table 1: visual reconstitution testThe numbering of samples was done a posteriori, from the worst (powders 1 and2 are not acceptable) to the best ( 4 and 5 are acceptable) and 3 is questionable.1t is clear that BT and ST scores are linked: a high BT goes with a high STand likewise for middle and low scores. This is corroborated by the estimationof the correlation coe¢ cient equal to 0.9 (but limited to 5 samples). We knowthat its maximum value is 1, which implies a linear relation between both rowsof table 1. Noneless, people agreed with the opinion that this kind of test isnot very reliable, too dependent of the mood of examiners, and due to " thelack of objectivity of the visual inspection step ". This leads to replacing peopleby machines and we can expect that methods based only on physical and/orchemical properties are better, at least stable and reliable (and less expensive).Among them, ultrasonics provide methods of quality control in many man-ufacturing processes, and particularly in food production [2], [3]. In particular,paper [1] studies the propagation of ultrasonics through media providing bothvisual tests ST and BT.
Measurements of attenuation and velocity of ultrasonics in the frequency band3-7MHz are reported in [1]. Here, (cid:133)gures 1a, 1b, 2a, 2b utilize (cid:133)gures 3 to 6of [1]. The (cid:133)rst two (cid:133)gures 1a, 1b, are linear regressions about velocities, buttheir reliability is questionable due to non-negligible curvatures (for instance forsamples 2, 3, 4). Figures 2a, 2b express ln att as function of ln f ( f in MHz) ; inaccordance with the usual attenuation in f (cid:11) : Each (cid:133)gure contains 5 curves, and data of [1] show that it is reasonableto approximate with straight lines (in spite of doubts about velocity v ( f ) ).Authors assert that " these experiments were repeated with another set of powderfrom another factory and similar results were obtained" . Relative positions oflines in (cid:133)gures 2a and 2b are in accordance with the order in table 1. This isincitative to replacing visual tests by measures of attenuation of ultrasonics. Onthe contrary, lines of velocities v ( f ) in (cid:133)gures 1a, 1b intersect, which prohibitsthe same procedure.In this paper, we propose a model of propagation where v (cid:0) ( f ) is a linearfunction of f (cid:11) (cid:0) or ln f (when (cid:11) = 1) : (cid:11) is the parameter de(cid:133)ned below in (1) and it characterizes the attenuation shape. Experiments in [1] are in accordancewith this model (see (cid:133)gures 3a, 3b, 3c, 3d). Equivalently, the complex gain of thelinear invariant (cid:133)lter F which summarizes the propagation is the characteristicfunction of a stable probability law (see sections 2.3 and 3.1). The fact thatone of parameters is close to (cid:6) in all cases, shows that the causality propertyis well approached (section 3.2).Finally, we explain why random propagation times following stable proba-bility laws provides a good framework for low power ultrasonics used in foodindustry and particularly dairy products. In ultrasonics, it is wellknown that attenuations are most of the time in theform (see [4] to [9]) att [ f ] = c (2 (cid:25)f ) (cid:11) (1)2 and c are characteristics of the crossed medium. (cid:11) = 2 for water and at-mosphere [10], [11] (it is the highest value) but (cid:11) can take any value between0 and 2. Figures 2a and 2b deduced from (cid:133)gures 5 and 6 of [1] illustrate thisproperty for milk powder under the linear form (with respect to ln f )ln att [ f ] = (cid:11) ln f + ln c + (cid:11) ln (2 (cid:25) ) (2)despite gaps close to 7MHz. Table 2 gives estimations of ln att [ f ] for eachsample BT and ST ln att [ f ] BT ST :
86 ln f + 3 :
24 1 :
13 ln f + 2 :
512 0 :
87 ln f + 3 :
31 ln f + 2 :
63 ln f + 3 :
27 1 : f + 2 :
734 0 :
85 ln f + 3 :
45 0 :
85 ln f + 3 :
55 0 :
77 ln f + 3 :
59 0 :
87 ln f + 3 : Table 2: equations of lines in (cid:133)gures2a and 2b about attenuations.Equality (1) is an empirical relation which is simpler than an approximationby a polynomial of degree larger than 1. Physical justi(cid:133)cations are not givenfor approximation (1) ; except in rare cases, for instance (cid:11) = 2 for atmosphereand water, or (cid:11) = 1 = for high frequency propagation through electrical cables [ ] ; [ ] : Lines of (cid:133)gures 2a and 2b have slopes ( (cid:11) values) around 1 and di⁄erentenough values of c which lead to separate curves without intersection. Theirrespective positions are in accordance with visual tests.The concordancy between the relative place of curves in (cid:133)gures 2a, 2b, andthe rank in visual tests cannot be due to chance. The probability of such acoincidence is equal to 1/120 for 5 products (like here), 1/24 for 4 or 1/720 for6. Consequently, replacing visual tests by measure of attenuation is not a veryrisky gamble. Figures 1a and 1b are obtained from data in (cid:133)gures 3 and 4 of [1]. They arelinear regressions giving velocity v ( f ) (in m:s (cid:0) ) as a function of frequency f (in MHz) : According to the authors, the velocity dispersion is measured withan accuracy of 0.1-0.2 m:s (cid:0) which seems optimistic to me. Linear regressionsdo not take into account curvatures in data of [1], which can be hidden byinaccuracies of measurements. Moreover, the propagation time on the unitdistance is v (cid:0) ( f ) : It is a quantity as interesting as the celerity v ( f ) , whichjusti(cid:133)es the research of regressions of v (cid:0) ( f ) .For reasons explained below, it is of interest to highlight a linear dependencebetween v (cid:0) ( f ) and f (cid:11) (cid:0) or ln f (when (cid:11) = 1) ; where (cid:11) is the slope in (2) : Wewill accept the following equalities v (cid:0) ( f ) = (cid:26) (cid:13)f (cid:11) (cid:0) + m; (cid:11) = 1 (cid:13) ln f + m; (cid:11) = 1 (3)which link attenuation and dispersion through (cid:11): Figures 3a, 3b, 3c, 3d provethis property for data in [1]. Fig3a corresponds to BT1 ( (cid:11) = 1 : ; Fig3b to3T3 ( (cid:11) = 1) ; Fig3c to ST1 ( (cid:11) = 1 : and Fig3d to ST5 ( (cid:11) = 0 : : Table 3contains the whole set of cases ( f in MHz and results in (cid:22) s.m (cid:0) ) . v (cid:0) ( f ) BT ST : f (cid:0) : + 405 (cid:0) : f : + 4352 21 : f (cid:0) : + 401 (cid:0) :
71 ln f + 4213 (cid:0) : f + 427 (cid:0) : f : + 4314 19 : f (cid:0) : + 403 13 : f (cid:0) : + 4075 11 : f (cid:0) : + 412 21 : f (cid:0) : + 401 Table 3: estimations of parameters (cid:13) and m of equality (3) Formulas (3) are linked to causality (Kramers-Kronig relations)[7], [14], [15],and to stable probability laws [13], [16], [17], see section 3. Tables 4 and 5summarize both linear forms for BT and ST.
BT (cid:11) c (cid:13) m :
86 5 : : :
87 5 : : : (cid:0) : :
85 6 : : :
77 8 : : Table 4: parameters valuesfor bulk samples, m in (cid:22) s.m (cid:0) ST (cid:11) c (cid:13) m :
13 1 : (cid:0) : : (cid:0) :
71 4213 1 : : (cid:0) : :
85 6 :
05 13 : :
87 5 :
64 21 : Table 5: parameters values forsurface samples, m in (cid:22) s.m (cid:0) Tables 4 and 5 give no obvious information about the quality of samples (exceptperhaps the parameter c for ST ) , though they summarize data of [1]. In thecase of ST samples, values of (cid:11); c; are very di⁄erent for ST1, ST2, with respectto ST4 and ST5, which could justify a split in two groups. But values for ST3are questionable about a membership to the (cid:133)rst group. Let g ( t ) = exp [2 i(cid:25)f t ] be the monochromatic wave at the frequency f > : The crossing of reconstituted milk provides the same kind of wave with someamplitude and some phase depending on the frequency f . It is equivalent toconsider a Linear Invariant Filter (LIF) F such as (for a unit thickness) F [ g ] ( t ) = exp (cid:20) (cid:0) att [ f ] + 2 i(cid:25)f (cid:18) t (cid:0) v ( f ) (cid:19)(cid:21) (4)4here v (cid:0) ( f ) is the propagation time in seconds and attf is given in Neper(for a distance of one meter). This means that the result is the monochromaticwave at the frequency f delayed by v (cid:0) ( f ) and weakened by exp [ (cid:0) att ( f )] : Therefore, (cid:0) att ( f ) is the neperian logarithm of the weakening (used in [1]).Using (2) and (3) ; the complex gain (or frequency response) F of F is[18], [19] F ( f ) = (cid:26) exp (cid:2) (cid:0) c (2 (cid:25)f ) (cid:11) (cid:0) i(cid:25)f (cid:0) m + (cid:13)f (cid:11) (cid:0) (cid:1)(cid:3) ; (cid:11) = 1exp [ (cid:0) c (2 (cid:25)f ) (cid:11) (cid:0) i(cid:25)f ( m + (cid:13) ln f )] ; (cid:11) = 1 : (5)If g ( t ) = exp [2 i(cid:25)f t ] is the input of the LIF F , F [ g ] ( t ) = F ( f ) e i(cid:25)ft is theoutput of F . In the case of complex gains as (5) ; we will explain (see section 3)why the most favourable situations correspond to parameters linked as (cid:13) (cid:24) = c (2 (cid:25) ) (cid:11) (cid:0) tan (cid:25)(cid:11) for (cid:11) = 1 and (cid:13) (cid:24) = (cid:0) c(cid:25) for (cid:11) = 1 : To summarize, the propagation of ultrasonics through diluted milk powder canbe modelled by a LIF F of complex gain F ( f ) de(cid:133)ned by (5) from parameters ( (cid:11); c; (cid:13); m ) which characterize the attenuation att [ f ] and the propagation time v (cid:0) ( f ) : In section 3 and 4 below, we explain links with stable probability lawsand with random propagation times.
Let assume that A ; A ::: are independent (real) random variables (r.v) followingsome probability law L . It is a stable probability law if linear combinations ofthe A n follows the same law L (excluding location and scale parameters). TheGaussian law is stable, and it is the only one with two (cid:133)nite moments (the meanand the variance). The real r.v A follows a stable law of parameters ( (cid:11); c; (cid:12); m ) if its characteristic function has the shape ( ! > E (cid:2) e (cid:0) i!A (cid:3) = exp [ (cid:0) im ! (cid:0) c j ! j (cid:11) (1 + i(cid:12)(cid:18) ( ! ))] (cid:18) ( ! ) = (cid:26) tan ( (cid:25)(cid:11)= if (cid:11) = 1(2 =(cid:25) ) ln ! if (cid:11) = 1 (6)with < (cid:11) (cid:20) ; (cid:0) (cid:20) (cid:12) (cid:20) ; c > ; real m ; and a continuation for ! < ; using the Hermitian symmetry [16], [20], [21]. Stable laws have probabilitydensities, and are unimodal. The probability density (cid:22) m ( x ) corresponding tothe characteristic function E (cid:2) e (cid:0) i!A (cid:3) veri(cid:133)es ([16] th.3.2.2) (cid:22) m ( x ) = 12 (cid:25) Z E (cid:2) e (cid:0) i!A (cid:3) e i!x d!: (7)Clearly, (5) and (7) are confused when ( ! = 2 (cid:25)f ) ; (cid:26) c(cid:12) (2 (cid:25) ) (cid:11) (cid:0) tan ( (cid:25)(cid:11)=
2) = (cid:13); m = m if (cid:11) = 1 (cid:25) c(cid:12) = (cid:13); m + (cid:25) c(cid:12) ln (2 (cid:25) ) = m if (cid:11) = 1 : (8)5onsequently, e⁄ects of ultrasonics on milk powder can be explained in the con-text of stable probability laws provided that parameters stay within acceptablelimits. It is the case for (cid:11) ( : < (cid:11) < : see tables 4 and 5), and for c ( c > : Table 6 gives values of (cid:12) computed from measured values of (cid:11); c; (cid:13) (see tables 4and 5). For BT (bulk samples), we (cid:133)nd (cid:11) (cid:20) ; j (cid:12) j > : with one value at 1.08which may be lowered to 1. For ST (surface samples), values of (cid:12) are larger (forthe (cid:133)rst three), up to exceeding the limit j (cid:12) j = 1 . (cid:12) BT :
91 1 : (cid:0) :
86 0 :
96 0 : ST : (cid:0) :
26 1 :
12 0 :
73 1 : Table 6: values of (cid:12) from data in [1]Estimation errors of parameters (cid:11); c; (cid:13); m can explain anomalous values of (cid:12):
A stable law is symmetric (with respect to m ) when (cid:12) = 0 : On the other hand,the stable law is one-sided only when (cid:11) < ; (cid:12) = (cid:6) : For (cid:11) < ; (cid:12) = 1 wehave (cid:22) m ( x ) = 0 for x < m [16] : Elsewhere, (cid:12) is the parameter which rulesasymmetry. The larger is j (cid:12) j ; the larger is the asymmetry of the probabilitydensity (cid:22) m ( x ) : For (cid:11) > (resp. (cid:11) = 1) ; the causality is approached when (cid:12) is close to 1 (resp. close to -1). When (cid:12) > , the impulse response (cid:22) m ( x ) loses the positive character of a probability density, but the near causality isnot a⁄ected.In practical situations, parameters are estimated, and the true value of (cid:12) isnever reached (except by chance). So, we never (cid:133)nd exactly (cid:12) = (cid:6) and a strictcausality. A positive value of (cid:12) , when (cid:11) = 1 ; is the sign of a probability lawwith a fat tail on the positive axis, and weak probability on the negative axis. Apositive m moves probability masses towards the right, and then lightens themon the negative axis. The addition of both properties leads to an approximatecausality [8].When we approximate the complex gain (5) with the stable law (7) ; rela-tion (8) links the parameters (cid:13) and (cid:12) through (cid:11) and c: Data in [1] show thatestimations of (cid:12) remain in a neighbourhood of (cid:6) (table 6) : Equivalently, from (9) ; we always have (cid:13) close to c (2 (cid:25) ) (cid:11) (cid:0) tan (cid:25)(cid:11) for (cid:11) = 1 and (cid:0) c=(cid:25) for (cid:11) = 1 : This means that the attenuation (governed by (cid:11); c ) approximately determinesvariations of the celerity (governed by (cid:11); (cid:13) ) and vice versa. This property islinked to the Kramers-Kr(cid:246)nig relations [7], [14], [15].Equivalently, from (8) ; the (cid:133)lter F de(cid:133)ning the ultrasonic propagation iscausal only when (cid:11) < ; c (2 (cid:25) ) (cid:11) (cid:0) tan ( (cid:25)(cid:11)=
2) = (cid:13): (9)The strict causality is obtained at conditions which cannot be veri(cid:133)ed becausethe parameters are computed from experiments which contain errors, and thenthe value (cid:12) = (cid:6) cannot be exactly obtained. Noneless, the value of (cid:22) ( x ) isclose to 0 for x < x ; where (cid:0) x is close to a few units. Here, m is always in theorder of 400 ( v (cid:0) in (cid:22) s.m (cid:0) ) , which implies that (cid:22) m ( x ) = (cid:22) ( x (cid:0) m ) is alwaysnegligible for x < : We can consider that the causality condition is ful(cid:133)lled [8].Figures 4a, 4b, 4c, show probability densities (cid:22) of stable probability lawswhich are the Fourier transforms (7) of (6) . They are matched to cases ST2 ( (cid:11) = ; c = 2) , BT2 ( (cid:11) = 0 : ; c = 5) , ST1 ( (cid:11) = 1 : ; c = 1 : ; with (cid:12) = 0 : ; : ; : The Fourier transforms are also given for values (cid:12) = 1 : ; : ; but they are nolonger probability densities (they take negative values). To take into accountthe value of m or m (around 400) is equivalent to move curves towards theright by approximately 400 units. We see the in(cid:135)uence of c; which is a scaleparameter (the curves for ST1 are more distinct). When it increases, it spreadand separates the di⁄erent curves. In this section, we explain why LIF in ultrasonics can often be replaced byrandom propagation times [8], [9], [17]. We consider the random process U = f U ( t ) ; t R g de(cid:133)ned by U ( t ) = e i! ( t (cid:0) B ( t )) (10)where B ( t ) = f B ( t ) ; t R g is a real process such that both following charac-teristic functions (in the probability sense) do not depend on t (cid:26) ( ! ) = E (cid:2) e (cid:0) i!B ( t ) (cid:3) (cid:30) ( (cid:28) ; ! ) = E (cid:2) e (cid:0) i! ( B ( t ) (cid:0) B ( t (cid:0) (cid:28) )) (cid:3) : (11)They de(cid:133)ne the probability laws of the random variables (r.v.) B ( t ) and B ( t ) (cid:0) B ( t (cid:0) (cid:28) ) : The independence with respect to t of these laws implies a stationaritystronger than the usual second order stationarity. U models a monochromatic wave of frequency f = ! = (cid:25) which has crosseda medium on some distance : B models the time spent by the wave. Because B has a random character, U is no longer monochromatic. It is not di¢ cult toprove the following formulas [8], [13], [22] U ( t ) = G ( t ) + V ( t ) ; G ( t ) = ( ! ) e i! t (cid:25)s V ( ! ) = R h (cid:30) ( (cid:28) ; ! ) (cid:0) j ( ! ) j i e (cid:0) i ( ! (cid:0) ! ) (cid:28) d(cid:28) (12) G is deterministic. It is the monochromatic wave which has been attenuatedand dephased by ( ! ) : V is a zero-mean random process, stationary (in thewide sense), and with the spectral density s V ( ! ) : The integral in (12) is wellde(cid:133)ned provided that (cid:30) ( (cid:28) ; ! ) ! (cid:28) !1 j ( ! ) j fast enough (this means thatthe random variables B ( t ) and B ( t (cid:0) (cid:28) ) are almost independent for large (cid:28) ).In a signal theory context, G is the output of the linear invariant (cid:133)lter (LIF)with complex gain (or frequency response) ( ! ) : G is monochromatic, butaccompanied by a process V with band spectrum, which can be considered asan additive noise. The total power of U is one, and then the transformationfrom the monochromatic wave e i! t to U veri(cid:133)es the theorem of energy balance:the whole transmitted energy is retrieved through the addition "signal" G +"noise" V .A liquid or a gas is made of molecules which move inde(cid:133)nitely, the di⁄erencecoming from the type of links between molecules. The power is transmittedby random shocks. The result has to contain a random part even when thepower to be transmitted is deterministic. V represents this random part whichis not taken into account in measurements, because of a power spectrum whichis too spread out (with respect to the frequency window of devices). Properties7hich de(cid:133)ne the process are the time between molecular shocks, the length oftrajectories, or the number of shocks by time unit. These events are so numerousthat the result V is like a noise with very high frequencies : This is why devices,which have a limited frequency window (cid:133)tted to G , cannot see V : Let(cid:146)s assume that B ( t ) follows a stable law ( (cid:11); c; (cid:12); m ) as A (section 3.1).The probability law of B ( t ) is (cid:22) m ( x ) de(cid:133)ned by (6) and (7) : From (12) ; G is the output of a LIF of complex gain ( ! ) : As explained in section 3, G isa correct model for the studied propagation, and also U=G+V, because V cannot be viewed by devices.The same model of propagation can be used for other frequency bands andother media, for instance acoustics [23], or light propagation [24], or propagationthrough electrical cables [13]... In some cases, the G part disappears, the V partis observed, but it is possible that both parts G and V remain visible [25]. Ultrasonics provide methods of quality control in many manufacturing processes,and particularly in food production [2], [3]. The beam attenuation and/or itsvelocity are measured and they give information which may be used to monitorfactories. A conclusion of paper [1] about reconstructed milk is:".. the measurement of the ultrasonic attenuation coe¢ cient can be very wellcorrelated to visual scores, for both the ST and RT " and"
Ultrasonic velocity cannot be correlated to a visual score ..."Authors added:"
These experiments were repeated with another set of powder from anotherfactory and similar results were obtained ".Both citations prove that ultrasonics in some frequency band provide goodinformation from the attenuation, and that a plan with (cid:133)ve products is su¢ cient.We have shown that they allow to characterize the linear (cid:133)lter which modelsthe propagation.Table 1 of [1] is the result of visual examinations of diluted milk powders.Powders are ranked from the worst to the best, and results are more or lessidentical for bulk and surface appearances. Ultrasonics in the 3-7MHz bandprovide values of attenuations and velocities. Graphs of ln att in function ofln f are in accordance with visual examinations (see (cid:133)gures 2a, 2b), and de(cid:133)neregression lines of slope (cid:11) with : < (cid:11) < : (see table 2) and of parameter c in accordance with the usual relation att ( f ) = c (2 (cid:25)f ) (cid:11) .It is not the case for velocities (see (cid:133)gures 1a, 1b), perhaps due to the choiceof coordinates ( f; v ( f )) to perform regression lines. The system of coordinates( f (cid:11) (cid:0) ; v (cid:0) ( f ) or (cid:0) ln f; v (cid:0) ( f ) (cid:1) when (cid:11) = 1 ; is the right choice for regressionlines (see (3) and (cid:133)gures 3a to 3d). Parameters of both systems are closelylinked through the values of (cid:11); c and (cid:12) (which is close to (cid:6) : The previous considerations allow to characterize the propagation by anequivalent linear invariant (cid:133)lter F and a system of parameters ( (cid:11); c; (cid:13); m ) (for-mulas (4) and (5)) which are given in tables 4 and 5 from data of tables 2 and3. Actually, the complex gain F ( f ) (formula (5)) is the characteristic functionof a stable probability law of parameters ( (cid:11); c; (cid:12); m ) (section 3). This strongproperty is derived from both linearities of ln att with ln f and of v (cid:0) ( f ) with f (cid:11) (cid:0) (or ln f ) ; (cid:11) being the slope of the (cid:133)rst line. Values of (cid:12) are given in table8. They are close to (cid:6) ; and this parameter is closely linked to the causality of(cid:133)lters (section 3-2). Moreover, we show that random propagation times providethe correct framework taking into account the random character of propagationdue to irregular motion of molecules. References [1] S. Meyer, V. S. Rajendram, M. Povey,
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