Characterization of cane sugar crystallization using image fractal analysis

Journal of Food Engineering 100 (2010) 77–84

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Journal of Food Engineering journal homepage: www.elsevier.com/locate/jfoodeng

Characterization of cane sugar crystallization using image fractal analysis Oscar Velazquez-Camilo a, Eusebio Bolaños-Reynoso b, Eduardo Rodriguez a, Jose Alvarez-Ramirez a,* a b

Division de Ciencias Basicas e Ingenieria, Universidad Autonoma Metropolitana-Iztapalapa, Apartado Postal 55-534, Mexico D.F. 09340, Mexico Division de Estudios de Posgrado e Investigacion, Instituto Tecnologico de Orizaba, Av. Tecnologico No. 852, Orizaba, Veracruz 94320, Mexico

a r t i c l e

i n f o

Article history: Received 3 December 2009 Received in revised form 18 March 2010 Accepted 19 March 2010 Available online 3 April 2010 Keywords: Cane sugar Crystallization Fractal patterns Lacunarity

a b s t r a c t Automated image analysis has emerged as a useful tool for quality evaluation and inspection of food processes and products. Image analysis techniques are aimed to the extraction of features for quantifying texture, shapes and distributions of irregular geometries recasted on a microscopy image. The monitoring of crystal growth evolution in traditional industrial processes commonly relies on the visual expertise of long-term trained operators, which limits seriously the automated operation of the process. The objective of this study was to investigate the potential usefulness of fractal metrics; namely Fourier analysis fractal dimension and lacunarity using images, as quantitative descriptors of crystallization evolution. To our knowledge this is the first reported use of lacunarity for the characterization of images of crystallization images from direct samples of crystallization slurries. Fractal dimension and lacunarity increase with the crystallization time. Increased fractal dimension was related to the formation of large clusters in the image, and was taken as an indicative of the amount of formed crystals. On the other hand, lacunarity is an index of non-uniformity of particles on the image, such that lacunarity can be considered as an indicator of the crystal shape and size diversity. In an overall sense, the results showed that fractal analysis can be incorporated as a complementary tool for monitoring the evolution of cane sugar crystallization process. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Crystallization is a solid–liquid separation process where molecules are transferred from a solute dissolved in a liquid phase to a solid phase through two steps: nucleation and crystal growth. The formation of secondary nuclei is unwanted in many crystallizations, especially sugar crystallization, because it widens the range of the crystal size distribution. A broader size distribution range means the product of the crystallizer is out of commercial specification. As a consequence, sugar with small-size crystals can lead to unsatisfactory rheological properties when used as an additive in different food processes. In this way, the formation of a sugar product with reduced tails in the crystal size distribution is of prime importance for the food industry (Chen and Chou, 1993). A typical industrial crystallization process for cane sugar involves many steps for obtaining a final commercial product, including seeding, cooling and filtration. The main step of the process is the crystallization phase where the liquor is subjected to cooling and evaporation by means of vacuum. Many variables can have determinant effects in the quality of the crystals, including temperature, pressure, agitation and impurities content (Sangwal, 2007). Although some properties like temperature and solute concentration are usually relatively easy to measure and quantify, * Corresponding author. E-mail address: [email protected] (J. Alvarez-Ramirez). 0260-8774/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2010.03.030

these variables are not sufficient to monitor the advance of the crystallization. As a consequence, motivated by the demand of more and better products, the usage of modern technologies for the design of operating policies and the monitoring of the crystal growth evolution has been incorporated (Sutradhar, 2004). These technologies include the usage of feedback control strategies based on on-line and virtual measurements (Bolaños-Reynoso et al., 2008). Recently, image acquisition by high-speed camera and electronic microscopes (Calderon-De-Anda et al., 2005; Wang et al., 2007) has been considered for improving the performance of feedback controllers and monitoring of crystallization from solutions. Images provide a close view of the crystallization evolution by estimating dominant crystal shape, crystal size distribution (CSD) and possible secondary nucleation and attrition phenomena. Several applications of image analysis for monitoring different crystallization oriented to food processing have been tested both for monitoring and control of the crystallization process. Multi-scale segmentation methods have shown to be accurate to analyze effectively the on-line images of different crystal morphological forms, and of varied qualities (Calderon-De-Anda et al., 2005). Combined on-line video microscopy and X-ray diffraction for monitoring the polymorphic transformation of l-glutamic acid has been also tried (Dharmayat et al., 2006). The ability of image analysis for evaluating the effects of additives in batch cooling crystallization was evaluated (Qu et al., 2006). The usage of wavelets for automatic classification of crystallized proteins has shown promising results

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(Watts et al., 2008). Summing up, recent results have shown that the use of image analysis has an important potential for monitoring the evolution of crystallization processes in the food industry. The image analysis commented above focused on the detection of shape and size for individual crystals by means of segmentation techniques. However, to obtain accurate results, segmentation techniques require that crystals appear isolated on the image, which require preprocessing (e.g., dilution and dispersion) of the sample taken from the crystallization slurry. However, images taken directly (i.e., in-line) from the crystallization slurry show a complex, quite irregular accommodation of crystals as shown in Fig. 1 for four different crystallization times for cane sugar crystallization in a small pilot plant (Bolaños-Reynoso et al., 2008). For relatively small times (about 10 min), the contour of individual crystals can be still distinguished. However, the density of formed crystallized mass increases with time, so that individual crystals are not further isolated, presenting an irregular packing with overlapping of small and large particles. These features limit the application of segmentation techniques as individual crystal contours cannot be easily extracted from a cluster. The complex pattern presented by images from the crystallization slurry reflect important aspects of the crystallization evolution, such as the amount and diversity of formed crystallized mass. The traditional cane sugar industry have recognized this when monitoring crystallization evolution by visual inspection of samples taken directly from the crystallization slurry. However,

this monitoring is based on the expertise of a crystallization operator with a trained eye along many working years. In this way, oriented to an automation of the cane sugar process, it is desirable to dispose of image analysis techniques applied for extracting information on the crystallization dynamics from samples of the crystallization slurry. Rather than being a drawback, the complex patterns displayed on these images should be considered as a source of information related to the cane sugar product. Hence, it seems to be natural the use of techniques from the analysis of complex systems for delineating a method for analyzing the pattern displayed by images like the ones shown in Fig. 1. This work explores the use of fractality methods (Mandelbrot, 1983; Zheng et al., 2006) for the analysis of images taken directly from the crystallization slurry. The idea is to extract some fractality indices indicating a directionality of the crystallization dynamics and, in this way, overcome most of the drawbacks of traditional methods, e.g., human inspectors and instrumental techniques in the monitoring of cane sugar crystallization process. The usage of fractal analysis in the food industry is not new, dating back from early years of the applications of fractality concepts. Haussdorf fractal dimension of digital images for food quality evaluation and inspection was initially explored (Liao et al., 1990; Dziuba et al., 1999). Fourier power spectrum methods were proposed for the assessment of the jaggedness of the stress-strain relationship of two kinds of puffed extrudates stored under different humidity conditions (Barrett et al., 1992) and in the characterization of rug-

Fig. 1. Sequence of selected images taken from slurry samples for cane sugar: (a) 10, (b) 20, (c) 55 and (d) 83 min. For small crystallization times, crystals are isolated and display irregular shape. For larger times, crystal aggregates appear as a consequence of high densities of formed crystals. Overlapping of small crystals over larger ones is also apparent.

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gedness changes in tapped agglomerated food powders (Barletta and Barbosa-Cánovas, 1993). Evaluation of micro-structural changes in food surfaces using fractal image texture analysis were also explored (Quevedo et al., 2002). Fractality was used in the quantification of the fractal dimension of fat crystal networks (Tang and Marangoni, 2006, 2008). Fractal and lacunarity analysis to examine the structure of heat-induced gels of plasma proteins showed that fractal measures can be related to macroscopic properties such as water holding capacity (Davila and Pares, 2007). The evaluation of senescent spotting in banana (Quevedo et al., 2008) and the texture appearance of pre-sliced pork ham images were described using fractal metrics (Valous et al., 2009). The images exhibited in Fig. 1 show that sugar crystals are accommodated in a complex, fractal-like, geometry, resembling the texture of food surfaces like fat crystal networks and caseinate formulations. In this way, the objective of this study was to investigate the potential usefulness of fractal metrics; namely Fourier analysis fractal dimension and lacunarity using images, as quantitative descriptors of crystallization evolution. To our knowledge this is the first reported use of lacunarity for the characterization of images of crystallization images from direct samples of crystallization slurries. Studies in this line should lead to a more systematic automated operation of industrial cane sugar crystallizer (Sangwal, 2007; Bolaños-Reynoso et al., 2008). 2. Methodology 2.1. Experimental equipment The lab-scale crystallizer was designed and constructed in the Instituto Tecnologico de Orizaba to perform basic research on crystal growth and to design operating conditions for the local cane sugar industry (Bolaños-Reynoso et al., 2008). A detailed description

of the equipment is made according to the schematic diagram in Fig. 2. The batch crystallizer was a 12.77 L stainless steel vessel with four vertical wall baffles of 17 cm (wide) by 3.5 cm (length), agitation arrow of 39 cm (length), and an 11.10 L heating–cooling jacket equipped with a steam boiler (model MBA9 of SUSSMAN). The agitation system for closed tank is model NSDB of HP, direct transmission of 1750 rpm (1 phase, 60 cycles), 110 VCA totally closed, without ventilation, stainless steel 316 with a stainless stell 4 in bridle. The hydraulic pump is a QB60 of Clean Water Puma model; power, 1/2 HP for 127 VCA (1 phase), 3450 rpm, 35 L/min of capacity with 1 in of input and output. The vacuum pump is a FE-1400 model of Felisa, power 0.33 HP and 0–60 Hz of frequency. Heat insulation for high temperature was made with glass fiber, 1 1/2 in. of thickness (18 kg of density) and steam barrier of aluminum foil paper; finished with aluminum blade of caliber 26, smooth, bevelled and it holds with galvanized wire; capacity of 80–150°C. Table 1 presents a detailed description of the electronics and instrumentation devices used in the vacuum batch crystallizer Virtual Instrument. Program ‘‘vacuum_regulation.vi” developed for this work using LabVIEW 7.1 (National Instruments de Mexico S.A. de C.V) is an interface of supervisory control and data acquisition (SCADA) that takes as its basis the program called ‘‘SCADA.vi”, whose aim is to acquire both vacuum pressure and temperature data and control the process through the dynamic regulation profiles of vacuum pressure and agitation trajectories. Program ‘‘vacuum_regulation.vi” consists of two parts: (a) a frontal panel, which is a virtual control board that contains both control and indicators buttons for each function, and (b) the block diagram that is integrated by the ‘‘G” graphical programming code of the developed system. These computer programs simplify the study of dynamic behavior of the cane sugar crystallization process (Bolaños-Reynoso et al., 2008).

PP Host computer M VT OS TJ2

PVV Condenser PT1 TJ1 PSV

VP

HT

MV TT

PT2

agitator

Steam generator

250

Control Panel

FBP: Feeding boiler Pump HT: Humidity trap M: Motor (variable agitation ) MV: Manual valve OS: Optic sensor PP: Peristaltic pump PSV: Proportional steam valve PVV : Proportional vacuum valve PT1-2: Pressure Transmitter TJ1-2: Thermocouple J type TT: Thermodynamic trap VP: Vacum pump VT: Vacuum transmitter WT: Water Tank

FBP

Fig. 2. Experimental set-up for the cane sugar crystallization process.

WT

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Table 1 Electronics and instrumentation devices of vacuum batch crystallizer. Quantity

Devices

1

Stainless steel crystallizer of 12.77 L, heating-cooling jacket of 11.10 L, four vertical wall baffles of 17 cm (wide) by 3.5 cm (length), agitation arrow of 39 cm (length) Agitator for closed tank, model NSDB of 1/8 HP, direct transmission of 1750 rpm (1 phase, 60 cycles), 110 VCA totally closed, without ventilation, stainless steel 316 with bridle of 4 in (diameter) in stainless steel, with agitating arrow of 26 in (length) and 1/2 in of diameter in stainless steel 316. Velocity investor (driver) integrated with rank from 0 to 1750 rpm Impeller marine type of 3 in (diameter) with three blades of stainless steel 316 Impeller flat palettes type of 3 in (diameter) of stainless steel 316 Thermocouple J type. Temperature from 0°C to 760°C, wire length: 1 m Hydraulic pump, model QB60 of Clean Water Puma. Power: 1/2 HP for 127 VCA (1 phase), 3450 rpm, 35 L/min of capacity with 1 in of input and output Thermo-wells of 1 in (NPT inlet diameter) in stainless steel Heat insulation for high temperature made with glass fiber: 1 1/2 in of thickness (18 kg of density) and steam barrier of aluminum foil paper, finished with aluminum blade of caliber 26, smooth, bevelled and it holds with galvanized wire, capacity of 80°C–150 °C Vacuum pump, model FE-1400 of Felisa, power 0.33 HP and 0–60 Hz of frequency Velocity investor (driver), model cat 1305-AA04A-ES-HA2 series C of Allen-Bradley, 0.75 kW/1 HP Steam boiler, model MBA9 of SUSSMAN. Maxim pressure: 100 Psi. Work voltage: 240 VAC. Control voltage: 120 VAC Thermodynamic trap of 1/2 in NPT Galvanized pipe system of 1/2 in (diameter) for the circulation of heating-cooling water Pressure controller valve (Norgren) of 1/2 in, with 21 kg cm2 of maxim input and 9 g cm2 of maxim unloading Stainless steel condenser, input of 1/2 in and output of 1/4 in, with 27 in (length) by 4 in (diameter) Plastic tank of 1100 L, with spiral cove Stainless steel humid trap, with 13 in (length) by 3 in (diameter). Input, output and purge of 1/2 in

1

1 1 2 1 2 1 1 1 1 1 1 1 1 1 1

2.2. Image acquisition system The crystallizer was equipped with a professional compound microscope: 48923-30; trinocular Cole Palmer and a monochrome camera with video RS-170. Lens: 0.19 mm by pixel. National Instruments, Inc. For the pseudoline measurement at every sampling time of the experimental runs and the corresponding crystal analysis through captured images, the software IMAQ Vision Builder (National Instruments, Inc.) was used. The technique consists of acquiring an image using a monochromatic camera with video RS-170 and 60 Hz crisscross (8 bits of resolution) and handling the light beam from a microscope trinocular. The sample is transported from crystallizer to the imaging acquisition system by means of a peristaltic pump running at 1500 rpm. The camera captures an image square that is to be processed and cleaned. This avoids undesirable light variations. The latter is achieved through the threshold technique that allows obtaining only an image in gray scale (Bolaños-Reynoso et al., 2008). Table 2 provides details of the electronics devices, acquired in National Instruments de Mexico S.A. de C.V., for data and images acquisition system. 2.3. CSD estimation Five different regions of an image are isolated to be independently analyzed for size distribution estimation and fractal quanti-

Table 2 Electronics devices for data and images acquisition system. Quantity

Devices

3

Data acquisition hardware: PCI-6229M, PCI-6023E and PCI-6025E. National Instruments, Inc. Images acquisition hardware: PCI-1407. National Instruments, Inc. Shielded Carriers modules: SC-2345. National Instruments, Inc. Shielded I/O Connector Block for DAQ: SCB-68. National Instruments, Inc. Signal conditioning modules: 2 SCC-TC02 and 2 SCC-CI20. National Instruments, Inc. Shielded Terminal Block: SCB-100. National Instruments, Inc. Vacuum pressure transmitter, model 07356-02. Cole Parmer. Professional compound microscope: 48923-30; trinocular. Cole Palmer. Monochrome camera with video RS-170. Lens: 0.19 mm by pixel. National Instruments, Inc.

1 1 1 2 1 1 1 1

fication purposes. Relevant areas are isolated to be independently analyzed, and black pixels groups representing individual crystals are counted. The black pixels are compared with acceptation limits to decide if the object is present or not according to binary images (background) Then, a threshold technique approach, similar to that reported in further multiscale segmentation image approach, was used to compute the CSD features (Calderon-De-Anda et al., 2005). The measurements and analysis of particles were carry out against a previous calibration through a Neubauer’s recount camera (simple calibration) in order to get a direct conversion from 1 pixel (one pixel side) to 200 lm (length). A pixel is defined as the smallest homogeneous unit in color that is part of the digital image. The pixels appear as small squares in white, black, or gray shades. In this work, a microscope with a 10 ocular lens with a 40 objective and an E square from Neubauer’s camera from 50 lm away, were used. This is equivalent to 20,000 lm ð10  40  50Þ per 100 pixels (length of pixel side). Thus, 1 lm is equal to 0.005 of the length of a pixel side. 2.4. Operating conditions A cane sugar solution was prepared with 2922.7 g of commercial sugar and 9235.9 g of water (75.962° Brix to the saturation temperature of 70°C) inside the crystallizer, seeding sugar crystal particles with diameter average Dð4; 3Þ of 195.3 lm in % volume and standard deviation S(4, 3) of 19.52 lm, during the first four minutes of processing (69°C and 72.8 kPa of vacuum pressure). The agitation velocity was set at 250 rpm and remained constant throughout the batch time (90 min). An average constant evaporation (0.2431 g water/sec) took place at 65°C and 72.8 kPa of vacuum pressure for 30 min, to favor the water retirement, stabilize the batch and benefit the seeded crystals growth. This step is taken to minimize the spontaneous nucleation. After 30 min of processing, one dynamic regulation profile of vacuum pressure was applied during 60 min, causing a cooling trajectory by adiabatic evaporation from 65°C to 40°C (see Fig. 3.a) (Bolaños-Reynoso et al., 2008). 2.5. Supersaturation conditions The saturation zones were built according to experimental data and modeled according to the expressions previously reported (Quintana et al., 2004), which have a rigorous support based in first

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90

(a)

70

55

45

50

30

45 15

40 35 0

20

40

60

80

3

Density, ρ (gr/cm )

60

60

Vaccum Pressure (KPa)

65

Temperature, T (oC)

(b)

1.37 75

Experimental trajectory

1.36

Unstable zone

1.35

etas nd m

o Sec

1.34

Fir

sta eta st m

ble

z one

z on

e

Unsaturation zone 1.33

0 100

40

50

60

70

Temperature, T (oC)

Time (minutes)

400

4000

(c)

210

(d)

350 Diameter, D(4,3) (μm)

Mass of Formed Crystals (gr)

table

3000

2000

1000

180

300

150

250

120

200 90

150

60

100

30

50 0

0 0

20

40

60

80

0

100

Standard Deviation, σ

75

Time (minutes)

20

40 60 Time (minutes)

80

0 100

Fig. 3. Evolution of the crystallization process as monitored by in-line and out-line traditional variables: (a) temperature and vacuum pressure; (b) experimental trajectory compared to the stability diagram of the cane sugar; (c) amount of formed crystals contained in the crystallization slurry; (d) mean diameter Dð4; 3Þ.

principles approach (kinetics parameter for growth and nucleation rates). The zones are governed by the following equation considering the following limits:

C i ¼ SC i





Brixi 100  Brixi



of the image can be related to the evolution of the crystallization process. To address this question, a quantification of the image fractality should be made with a suitable analysis method applied for two-dimensional arrays.

ð1Þ 3.1. Fractal dimension o

where C i is solute parts/100 solvents parts, Brixi is solution concentration at specific temperature point, and SC i is specific saturation coefficient to each line: unsaturated (90), saturate (100), maximum limit to first metastable zone limit (120), maximum limit to intermediate zone (130). Hence, the supersaturation is given by S ¼ C exp  C eq , where C exp is experimental concentration at g sugar/g water, which is obtained in pseudoline by means of a peristaltic pump and refractometer (°Brix hand-held refractometer Atago) and C eq is equilibrium concentration as a function of temperature is given by C eq ¼ 0:0007T 2 þ 0:264T þ 60:912 (Bolaños-Reynoso et al., 2008). 3. Fractal analysis methods Fig. 1 illustrates images for four different crystallization times; namely, 10, 20, 55 and 83 min. The images show a set of small and large crystals generated by the effects of nucleation and growth from the saturated liquid solution. The stochastic distribution of the crystals on the surface is apparent, suggesting a fractal structure. It is also observed that, as the crystallization time is increased, the crystal size and number are also increased with an apparent increment of the image complexity. This suggests that the structural properties of the crystal aggregate is a function of the crystallization time. In this way, an interesting question is if the fractality

Images in Fig. 1 show a certain degree of self-similarity (Mandelbrot, 1983) induced by the packing and spatial distribution of large and small-size crystals. It apparent that the roughness, as opposed to smoothness, of the images seen as surfaces increases with the crystallization time. The fractal dimension for a rough surface is higher than the dimension for a smooth one. In this work, the fractal dimension of the gray-scale images was estimated from the power-law scaling behavior of the Fourier spectra (Russ, 1994). Briefly, the fractal dimension determination is made as follows. After taking the fast Fourier transform of the region of interest in the horizontal directions, the average horizontal power spectrum Pðf Þ of the surface is a function of the frequency f and satisfies the following relationships:

P / f 2b

ð2Þ

where b ¼ 2H þ 2 and H is the Hurst exponent (Mandelbrot, 1983; Quevedo et al., 2002). The parameter b can be computed from the slope of a logðPÞ versus logðf Þ plot. The fractal dimension is estimated from b as

Df ¼

6þb 2

where b is the slope of the linear descending part of the plot.

ð3Þ

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3.2. Lacunarity In a restrictive sense, it is a measure of the lack of rotational or translational invariance or radial symmetry in an image (Allain and Cloitre, 1991). The name lacunarity is from the Latin lacuna for lack, gap or hole (Mandelbrot, 1983). Hence, a fractal is said to be lacunar if the gaps in an object are wide, i.e., if they have large intervals, holes or voids. In a more general sense, lacunarity is a measure of the non-uniformity heterogeneity of structure or the degree of structural variance within an object. Lacunarity concepts are increasingly being used for characterizing inhomogeneous structures in food products (Davila and Pares, 2007; Valous et al., 2009; Lobato-Calleros et al., 2009). A brief description is given as follows (Lobato-Calleros et al., 2009). The basic lacunarity algorithm is implemented along the following steps (Allain and Cloitre, 1991). A square structuring element or moving window of side length r pixels is placed in the upper left-hand corner of the image of side length T pixels, such that r < T. The algorithm records the number or mass m of pixels that are associated with the image underneath the moving window. The window is then translated by r pixels to the right and the underlying mass is again recorded. When the moving window reaches the right-hand side of the image, it is moved back to its starting point at the left-hand side of the image and is translated by one pixel downward. The computation proceeds until the moving window reaches the lower righthand edge of the image, at which point it has explored every one 2 of its NðrÞ ¼ ðT  r þ 1Þ possible positions. This procedure produces a frequency distribution of the box masses nðm; rÞ. This frequency distribution is converted into a probability distribution Q ðm; rÞ by dividing by the total number of boxes NðrÞ of size r. The first and second moments of this distribution are now determined:

Z 1 ðrÞ ¼ Z 2 ðrÞ ¼

X X

mQ ðm; rÞ m2 Q ðm; rÞ

ð4Þ

The lacunarity for the box size r is now defined as LðrÞ ¼ Z 2 ðrÞ=Z 1 ðrÞ2 . This calculation is repeated over a range of box sizes, ranging from r min  0:05T to some fraction of T, usually T=2. Note that LðrÞ > 1, indicating that as LðrÞ ! 1 the image approaches a homogeneous structure. In contrast, if the lacunarity changes with the scale r, the fractal image is not translational homogeneous. Commonly, fractional random structures display scaling behavior, meaning that a specific property is decreasing or decreasing according to a power-law behavior (Mandelbrot, 1983). Power laws appear widely in physics, biology, earth and planetary sciences, economics and finance, computer science, demography and the social sciences (Newman, 2005). Hence, it is very likely that lacunarity can also present a power-law behavior. Along this line, the lacunarity structure is scaling if, over some range of scales ðr1 ; r 2 Þ, the lacunarity function LðrÞ follows a power-law

KðrÞ ¼ aK r k ;

r 2 ðr 1 ; r2 Þ

ð5Þ

where aK is the lacunarity degree and k is the lacunarity scaling exponent. As the lacunarity exponent k describes the rate at which lacunarity decays with the scale, this parameter is a measure of the image sharpness. That is, the larger the value of k, the sharper the lacunarity decays with scale increments. If k ! 0, the image texture measured in terms of lacunarity is homogeneous. 4. Results and discussion 4.1. Crystallization dynamics Fig. 3.a shows the crystallization temperature and the applied vacuum as a function of time for the experimental run. For times

smaller than 30 min, the temperature is maintained about 70°C, although a small decrement is observed as the effect of the vacuum transient. After 30 min, the crystallizer temperature is decremented continuously to about 40°C in the end of the experimental run. Fig. 3.b displays the experimental trajectory with respect to the stability diagram of cane sugar. It is shown that, except at the vacuum transient observed in Fig. 3.a, the trajectory is maintained within the metastable zone. This guarantees that the crystallization process is stable with a reduced impact of secondary nucleation. Fig. 3.c shows the mass of formed crystals where a sustained increments is observed. Finally, Fig. 3.d shows the diameter Dð4; 3Þ obtained from the CSD as described in Bolaños-Reynoso et al. (2008). The diameter increases continuously from the start up of the crystallization process where crystal seeds are added up to 70 min where the diameter achieves a maximum value. After this time, the average diameter shows a slight decrement caused by crystal attrition induced by the large amount of formed crystals, agitation effects and possibly irregular fractal shapes. 4.2. Images during crystallization Fig. 1 shows examples of images acquired during the crystallization process for four sampling times; namely, 10, 20, 55 and 83 min. As described in Section 2, microscopy images were obtained from the slurry to monitor the advance of the crystallization process. As expected, the amount of crystals increases with time covering up the image surface. For small times, as shown in Fig. 1.a and b, individual crystals are isolated, so that their shape and lengths can be obtained by means of, e.g., segmentation methods. However, for large crystallization times (Fig. 1.c and d), the amount and size of formed crystals is large, such that there is not sufficient space for displaying individual crystals. Here, the images exhibit clusters of small and large crystals, leaving only a small fraction of empty space. These crystal clusters are packed in a complex and quite irregular manner, even showing overlapping effects where small crystal are mounted on large ones. The difference in the pattern for different times is apparent and, given the irregular nature of the crystal arrangement, the use of fractality methods seems natural. 4.3. Fractal dimension A regularization dimension was calculated for each group of samples and for each time of the test. Two different fractal behaviors have been observed, for low and for high frequencies (see Fig. 4). Therefore, upper and lower fractional dimensions have been calculated for each sample. Roughly speaking, the lower fractional dimension measures the irregularity of the whole formed mass of crystals since it corresponds to the low frequencies, and the upper fractional dimension measures the irregularity of the individual crystals and in the medium surrounding particles since it corresponds to the high frequencies. Error bars taking into account the standard deviation have been calculated for each sampling time. As it can be seen in Fig. 5, the error bars had approximately similar small lengths in all considered cases, which testified of the good representation of the data and the relevance of the results. The fractal dimension for high frequencies is higher than for low frequencies, indicating that the distribution of individual crystals onto the image is more irregular than that for the formed crystal mass as this is dominated by large crystals. In both cases, although clearer for lower frequencies, the fractal dimension decreases with time, reflecting the fact that as the crystallization time advances the image geometry is dominated by a higher amount of large crystals covering the image. In the limit as the whole sample is covered with formed crystals, the image luminance is dominated by the black, rendering a regular dimension of two.

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der, reflected by a large fractal dimension, was induced by mechanical attrition provoked the large density of formed crystals. 0.01

1E-3 0.2

0.25

0.3

0.35

0.4

0.45

0.5

Frequency, f (1/pixel) Fig. 4. Smoothed fast Fourier power spectra for a typical image. The presence of a crossover separating low- and high-frequency scaling behavior is apparent.

2.50

Df,low Fractal Dimension, D f

2.45

Df,up

2.40 2.35 2.30 2.25 2.20 2.15 0

20

40

60

80

100

Time (minutes) Fig. 5. Evolution of lower and upper fractal dimensions showing a long-term decrement with time. A brief increment is observed at about 30 min, and this increment can be related to a change in the operation mode.

For both low and high frequencies, the corresponding fractal dimension showed an increment at about 30 min. After some minutes, the fractal dimension dynamics recover the decreasing trending. This effect is more pronounced for the high-frequency fractal dimension, which recast the geometric configuration of small-size crystals. Interestingly, this temporal fractal dimension increment coincides with the change in the operation mode when, after 30 min of processing, one dynamic regulation profile of vacuum pressure was applied during 60 min, causing a cooling trajectory by adiabatic evaporation from 65°C to 41°C (see Fig. 3.a). Given the relatively low density of formed crystals, it is apparent that the decrement of regularity in the image was not caused by mechanical effects (e.g., attrition), but by a secondary nucleation effect by the sudden cooling of the crystallization slurry. Finally, the low-frequency fractal dimension exhibits a slight increment for large times. Fig. 1.d for 83 min shows the presence of small crystals with quite irregular shape and dimensions. On the other hand, Fig. 3.d suggests that the increment of the fractal dimension could be associated to the decrement of the mean crystal diameter Dð4; 3Þ. Secondary nucleation for these large time can be discarded given that the operation trajectory moves along the stable region (see Fig. 3.b). It is apparent that this decrement of or-

Lacunarity measures for the images in Fig. 1 and different box sizes are shown in Fig. 6. For all cases, lacunarity decreases with the box size, meaning that non-uniformity is more pronounced for small scales where small-size crystals dominates the image pattern. For large box sizes, the lacunarity approaches the unity, reflecting that for large scales the image is basically uniform. In turn, this approach to uniformity can be reflecting the packing of small-size crystals within the voids between large-size crystals (see Fig. 1.d). On the other hand, lacunarity increases with crystallization time, suggesting that broader crystal size distributions induced by non-uniform growth of crystals due maybe to diffusional limitations (Solomatov, 1995; Miyashita et al., 2005). The evolution of lacunarity with crystallization time was tracked with the lacunarity scaling exponent k. Similar to the fractal dimension, the behavior of lacunarity is scale-dependent, showing a crossover that separates two different scaling behaviors. For small scales (i.e., high frequencies) the lacunarity scaling exponent is, in general, larger than for large scales (i.e., low frequencies). This suggests that the higher irregularity of the crystal distribution for high frequencies, as detected by larger fractal dimensions in Fig. 5, can be related to large lacunarity induced by broad distributions of small- and large-size crystals. In other words, for images from crystallization process, it is apparent that fractal dimension and lacunarity are correlated in the sense that fractality is induced by the irregular and non-uniform distribution of crystals on the image workspace. Fig. 7 shows the lacunarity scaling exponent for up and low frequencies. Both lacunarity indices increas with time, indicating an important growth of the luminance gradients on the crystal distribution pattern. This means that the larger the crystallization time is, the lacunarity for large frequencies is more important than for low frequencies. In turn, this can be taken as an indicative of the dominance of large-size crystals growth and the formation of small-size crystals by secondary nucleation at time 30 min and mechanical attrition after 60 min. In fact, similar to fractal dimension, lacunarity is able to detect changes in the crystallization pattern as indicated by an important increment in the lacunarity exponent at about 30 min and a decrement of it at about 60 min.

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5. Conclusions It has been shown that fractal analysis can be used for in-line monitoring the evolution of cane sugar crystallization from images taken directly from the crystallization slurry. While the monitoring of traditional cane sugar processes commonly relies on the expertise of long-standing operators, automated image analysis becomes an additional tool for assessing the progress of crystal size distributions. Fractal analysis provides indices of order that can be recovered from images and these indices are indirect measures of the regularity and homogeneity of cane sugar crystals. In this way, fractal dimension and lacunarity of crystallization slurry images can be used as tracking indices for detecting changes in the crystal growth process, such as secondary nucleation and mechanical attrition. It should be remarked that direct computation of crystal size distribution parameters (e.g., mean crystal size) cannot be avoided in practice as these parameters are close measures of the quality of the final product. However, fractal analysis is an inexpensive technique that can complement traditional in- and off-line measurements. References Allain, C., Cloitre, M., 1991. Characterizing the lacunarity of random and deterministic fractal sets. Physical Review A 44, 3552–3558. Barletta, B.J., Barbosa-Cánovas, G.V., 1993. Fractal analysis to characterize ruggedness changes in tapped agglomerated food powders. Journal of Food Science 58, 1030–1046. Barrett, A.M., Normand, M.D., Peleg, M., Ross, E., 1992. Characterization of the jagged stress–strain relationships of puffed extrudates using the fast Fourier transform and fractal analysis. Journal of Food Science 57 (1), 227–235.

Bolaños-Reynoso, E., Xaca-Xaca, O., Alvarez-Ramires, J., Lopez-Zamora, L., 2008. Effect analysis from dynamic regulation of vacuum pressure in an adiabatic batch crystallizer using data and image acquisition. Industrial & Engineering Chemistry Research 47, 9426–9436. Calderon-De-Anda, J., Wang, X.Z., Roberts, K.J., 2005. Multi-scale segmentation image analysis for the in-process monitoring of particles shape with batch crystallizers. Chemical Engineering Science 60, 1053–1065. Chen, J.C.P., Chou, Ch.-Ch., 1993. Cane Sugar Handbook: A Manual for Cane Sugar Manufacturer and Their Chemists, twelfth ed. John Wiley & Sons, New York. Davila, E., Pares, D., 2007. Structure of heat-induced plasma protein gels studied by fractal and lacunarity analysis. Food Hydrocolloids 21, 147–153. Dharmayat, S., Calderon-De-Anda, J., Hammond, R.B., Lai, X., Roberts, K.J., Wang, X.Z., 2006. Polymorphic transformation of l-glutamic acid monitored using combined on-line video microscopy and X-ray diffraction. Journal of Crystal Growth 294, 35–40. Dziuba, J., Babuchowski, A., Smoczynski, M., Smietana, Z., 1999. Fractal analysis of caseinate structure. International Dairy Journal 9, 287–292. Liao, K., Cavalieri, P., Pitts, J., 1990. Hausdorff dimensional analysis and digital imaging based quality inspection. Transactions in Agriculture 33, 298–304. Lobato-Calleros, C., Recillas-Mota, M.T., Espinosa-Solares, T., Alvarez-Ramirez, J., Vernon-Carter, J., 2009. Microstructural and rheological properties of low-fat stirred yoghurts made with skim milk and multiple emulsions. Journal of Texture Studies 40, 657–675. Mandelbrot, B.B., 1983. Fractal Geometry of Nature. W.H. Freeman Freeman & Co, New York. Miyashita, S., Saito, Y., Uwaha, M., 2005. Fractal aggregation growth and the surrounding diffusion field. Journal of Crystal Growth 283, 533–539. Newman, M., 2005. Power laws, Pareto distributions and Zipf’s law. Contemporary Physics 46, 323–351. Qu, H., Louhi-Kultanen, M., Kallas, J., 2006. In-line image analysis on the effects of additives in batch cooling crystallization. Journal of Crystal Growth 289, 286– 294. Quevedo, R., Carlos, L.G., Aguilera, J.M., Cadoche, L., 2002. Description of food surfaces and micro structural changes using fractal image texture analysis. Journal of Food Engineering 53, 361–371. Quevedo, R., Mendoza, F., Aguilera, J.M., Chanona, J., Gutierrez-Lopez, G., 2008. Determination of senescent spotting in banana (Musa cavendish) using fractal texture Fourier image. Journal of Food Engineering 84, 509–515. Quintana, H.P., Bolaños-Reynoso, R.E., Miranda, C.B., Salcedo, E.L., 2004. Mathematical modeling and kinetic parameter estimation in batch crystallization. AIChE Journal 50, 1407–1417. Russ, J., 1994. Fractal Surfaces, first ed. Plenum Press, Rayleigh, North Carolina, New York and London. Sangwal, K., 2007. Additives and Crystallization Processes: From Fundamentals to Applications. John Wiley and Sons, England. Solomatov, V.S., 1995. Batch crystallization under continuous cooling: analytical solution for diffusion limited crystal growth. Journal of Crystal Growth 148, 421–431. Sutradhar, B.C., 2004. Coping with crystallization problems: there are many angles to consider when troubleshooting your crystallization process. This primer reviews the common problems that can arise, and offers. Chemical Engineering 3, 46–52. Tang, D., Marangoni, A.G., 2006. Computer simulation of fractal dimensions of fat crystal networks. Journal of the American Oil Chemists’ Society 83 (4), 309–314. Tang, D., Marangoni, A.G., 2008. Fractal dimensions of simulated and real fat crystal networks in 3D space. Journal of the American Oil Chemists’ Society 85, 495– 499. Valous, N.A., Mendoza, F., Sun, D.-W., Allen, P., 2009. Texture appearance characterization of pre-sliced pork ham images using fractal metrics: Fourier analysis dimension and lacunarity. Food Research International 42, 353–362. Wang, X.Z., Calderon-De-Anda, J., Roberts, K.J., 2007. Real-time measurement of the growth rates of individual crystal facets using imaging and image analysis. A feasibility study on needle-shaped crystals of L-glutamic Acid. Chemical Engineering Research and Design 85, 921–927. Watts, D., Cowtan, K., Wilson, J., 2008. Automated classification of crystallization experiments using wavelets and statistical texture characterization techniques. Journal of Applied Crystallography 41, 8–17. Zheng, C., Sun, D.-W., Zheng, L., 2006. Recent developments and applications of image features for food quality evaluation and inspection: a review. Trends in Food Science & Technology 17, 642–655.