Fractal Behavior in the Clarification Process of Cane Sugar Production

Abstract

Cane sugar production is an important industrial process. One of the most important steps in cane sugar production is the clarification process, which provides high-quality, concentrated sugar syrup crystal for further processing. To gain fundamental understanding of the physical and chemical processes associated with the clarification process and help design better approaches to improve the clarification of the mixed juice, we explore the fractal behavior of the variables pertinent to the clarification process. We show that the major variables in this key process all show persistent long-range correlations, for time scales up to at least a few days. Persistent long-range correlations amount to unilateral deviations from a preset target. This means that when the process is in a desired mode such that the target variables, color of the produced sugar and its clarity degree, both satisfy preset conditions, they will remain so for a long period of time. However, adversity could happen, in the sense that when they do not satisfy the requirements, the adverse situation may last quite long. These findings have to be explicitly accounted for when designing active controlling strategies to improve the quality of the produced sugar.

1. Introduction

For centuries, sugar has been a highly regarded and widely traded commodity, for the simple reason that sweetness is an essential ingredient of life. Before 1990, sugar beet accounted for about 40% of the total sugar production in the world. Since then, cane sugar market has been rapidly expanding, due to the low cost in sugarcane production. According to the US Department of Agriculture, the global cane sugar production reached 117.5 million tons in 2009, amounting to 78.33% of the total sugar production. Among the major cane sugar producer and consumer, China ranks the 3rd, accounting for 7% of the world totalproduction.

In general, cane sugar manufacturing consists of six phases: milling, clarification, evaporation, crystallization, centrifuging, and drying so as to obtain the white sugar, brown sugar, and other products. In particular, the juice from the milling workshop is called mixed juice.

Besides water, sugarcane, and reduced sugar, the mixed juice also contains many organic and inorganic nonsugar components, such as colloidal substances, inorganic salts (iron, magnesium, aluminum, calcium, etc.), and pigments. While these nonsugar components are residual nutrients in the sugarcane, they are detrimental to the sugar production. For example, flavonoids, multiacid, and organic-acid can make the mixed juice appear dark brown. Since the nonsugar components affect not only the appearance and color, but also the concentration of the sugar (i.e., reflected as the sweetness of sugar by a consumer), they have to be carefully removed. The purpose of the clarification process is to remove as many nonsugar components as possible, improve the purity of the juice, and reduce its viscosity and color values. This is critical for providing high-quality, concentrated sugar syrup crystal to the boiling stage. Therefore, the clarification stage is a key process in cane sugar production.

There have been efforts to improve the clarification of the mixed juice by using neural network-based prediction schemes [1, 2]. Such approaches amount to treating the clarification process as a black or gray box and thus do not yield any understanding of the basic physics and chemistry involved in clarification. To help gain more fundamental understanding and guide the design of better approaches to improve clarification, in this paper, we carry out fractal analysis of the key variables pertinent to the clarification process.

Among the types of activity that characterize complex systems, the most ubiquitous and puzzling is perhaps the appearance of 1/𝑓^𝛼 noise, a form of temporal or spatial fluctuation characterized by a power-law decaying power spectral density. A subclass of such processes, denoted as 1/𝑓^2𝐻+1, is called processes with long-range correlations (or long memories) characterized by a Hurst parameter 𝐻. Depending on whether 0<𝐻<1/2, 𝐻=1/2, or 1/2<𝐻<1 [3], they are said to have antipersistent correlations, memoryless or only short-range correlations, or persistent long-range correlations. Prominent examples of such processes include vision [4], finance [5], DNA sequences [6–10], human cognition [11] and coordination [12], posture [13], cardiac dynamics [14–17], and the distribution of prime numbers [18], to name but a few. In this paper, we wish to explore whether fractal theory can shed new light on the clarification process of cane sugar production.

The remainder of the paper is organized as follows. In Section 2, we provide some details about the cane sugar clarification process, explain the various variables to be analyzed here, and discuss the challenges of analyzing those variables. In Section 3, we carry out fractal analysis of these variables using the key concept in random fractal theory, the Hurst parameter, which characterizes the basic correlation structure of the signals. Concluding discussions are presented in Section 4.

2. Clarification of Mixed Juice in Cane Sugar Production

Sugar clarification is a key process in cane sugar production. It is characterized by two important indices called color value and clarity degree. Whether the color value and the clarity degree can achieve the desired or preset values can critically affect the quality of the cane sugar and the revenue of a factory.

Many methods have been developed for clarifying the mixed juice. The two major ones are the carbonation method and the sulphitation method. The former uses the milk of lime and carbon dioxide as main detergent on the mixed juice. Besides high cost, the accompanying pollution by carbonated mud is hard to deal with. Therefore, its usage has been substantially reduced over the years.

Unlike the carbonate method, the sulphitation method adds clarifying agent SO₂ in addition to lime and phosphoric acid. Due to the absorption effect of calcium sulfite (which is the product of lime and SO₂), together with the inhibition of pigment formation due to SO₂, the sulphitation method is significantly better than the carbonate method. In the following we will focus on the sulphitation method for the clarification of the mixed juice.

The sulphitation method for clarification is schematically shown in Figure 1. It is a complicated physical chemical process, involving solvents such as SO₂, milk of lime and phosphoric acid. Roughly, the process can be divided into four stages, cane juice phosphorus-preliming, first heating, SO₂ sulphitation-neutralization and settlement after 2nd-heating. More specifically, in the preliming stage after the first heating to 55~65°C, milk of lime and phosphoric acid are added to the mixed sugar cane juice. In the next sulphur fumigation process, SO₂ gas enters the chemical process pipeline. At the end of the pipeline, milk of lime is further added for achieving neutralization, since phosphoric acid sulfite reacts with calcium hydroxide to form calcium phosphate and calcium sulfate. Then, the 2nd heating is carried out so that sugar cane colloids of mixed juice can not only fully condense in the precipitator, but also (1) accelerate precipitation, (2) decrease juice viscosity, and (3) facilitate the precipitated particles to sink fast. Then, the clarified juice can come out of the settler from the top of each later, while the mud juice is discharged into the vacuum suction filter by gravity. Filtered juice can then be directly incorporated into the clear juice heating device and sent to the evaporation device to condense into syrup. Therefore, controlling the process parameters steadily is the key to improve the quality of clarified juice and sugar.

Figure 1 

Schematic of the clarification stage of cane sugar production. Variables 𝑥𝑖,𝑖=1,2,…,11, are analyzed here.

In this paper, time series of the 11 variables indicated in Figure 1 will be analyzed using random fractal theory. The data were measured in an advanced cane sugar production factory in Guangxi Zhuang Autonomous Region of China. Guangxi is the largest sugar producer in China, amounting to more than 60% of the total sugar in the country. The data were collected with a time interval of 2 hours and lasted for the whole season (about 4 months). These variables are grouped into three groups, with the last two variables, 𝑥₁₀, color value, and 𝑥₁₁, the clarity degree, being the target variables. The time series data of these 11 variables are shown in Figures 2, 3, and 4. As can be seen, the major challenge in analyzing these data is nonstationarity; that is, the statistical moments, including mean and variance, vary with time. Understandably, this may be a salient feature for variables that are measured over a very long period of time (which is about 4 months here).

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Figure 2 

Signals 𝑥1 to 𝑥4.

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Figure 3 

Signals 𝑥5 and 𝑥6.

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Figure 4 

Signals 𝑥7 to 𝑥11.

3. Fractal Analysis of the Clarification Process

In this section, we first overview the basics of fractal processes with long memory and then briefly describe two methods for detecting fractal structures from data with trends. One is the seminal detrended fluctuation analysis (DFA) [19]. The other is adaptive fractal analysis (AFA) [20–23]. Since AFA provides additional advantages over DFA to deal with signals with arbitrary trends, we only present results of AFA for the variables pertinent to the clarification process of cane sugar production here.

3.1. Basics of Fractal Processes with Long Memory

As we have mentioned, fractal processes with long memory are characterized by the Hurst parameter 𝐻. To better understand the meaning of 𝐻, it is useful to mathematically be more precise. Let {𝑥₁,𝑥₂,…,𝑥_𝑛} be a stationary stochastic process with mean 𝑥 and autocorrelation function of the type 𝑟(𝑘)∼𝑘^2𝐻−2,as𝑘⟶∞, (1) where 0<𝐻<1 is the Hurst parameter. When 1/2<𝐻<1, ∑_𝑘𝑟(𝑘)=∞, leading to the term long range correlation. {𝑥₁,𝑥₂,…,𝑥_𝑛} is often called an increment (or noise) process. Its power spectral density (PSD) is 1/𝑓^2𝐻−1. Its integration, 𝑢(𝑖)=^𝑖_𝑘=1𝑥_𝑘−𝑥,𝑖=1,2,…,𝑛, (2) is called a random walk process having PSD 1/𝑓^2𝐻+1. Being 1/𝑓 processes, they cannot be aptly modeled by Markov processes or ARIMA models [24], since the PSD for those processes is distinctly different from 1/𝑓. To adequately model 1/𝑓 processes, fractional order processes have to be used. The most popular is the fractional Brownian motion model [3].

To deepen our understanding of the Hurst parameter, let us smooth {𝑥₁,𝑥₂,…,𝑥_𝑛} using nonoverlapping windows to yield a new time series 𝑋_𝑡^(𝑚)=𝑥_{𝑡𝑚−𝑚+1}+⋯+𝑥_𝑡𝑚𝑚,𝑡≥1. (3) It can be proven that the variance of the new time series is given by [25] var𝑋^(𝑚)=𝜎²𝑚^2𝐻−2, (4) where 𝜎² is the variance of original stochastic process {𝑥₁,𝑥₂,…,𝑥_𝑛}. Equation (4) offers an excellent means of understanding 𝐻. For example, if 𝐻=0.50, 𝑚=100, then var(𝑋^(𝑚))=𝜎²/100. When 𝐻=0.75, in order to have var(𝑋^(𝑚))=𝜎²/100, then we need 𝑚=10⁴, which is much larger than 𝑚=100 for the case of 𝐻=0.50. On the other hand, when 𝐻=0.25, if we still want var(𝑋^(𝑚))=𝜎²/100, then 𝑚≈21.5, much smaller than 𝑚=100, the case of 𝐻=0.50. An interesting lesson from such a simple discussion is that if a time series is short while its 𝐻 is close to 1, then smoothing is not a viable option for reducing the variations there.

There are many excellent methods for estimating 𝐻 [25, 26]. In the next two subsections, we describe the methods that are most promising for detecting fractal variations in variables pertinent to the clarification process of cane sugar production.

3.2. Detrended Fluctuation Analysis (DFA)

Denote a time series of interest by 𝑥(1),𝑥(2),𝑥(3),…; DFA works as follows [19]. First divide a given time series of length 𝑁 into ⌊𝑁/𝑙⌋ nonoverlapping segments, each containing 𝑙 points. Then, define the local trend in each segment to be the ordinate of a linear least-squares fit or best polynomial fit of the time series in that segment; this is schematically shown in Figure 5 using a linear trend as an example. Finally, compute the “detrended walk,” denoted by 𝑥_𝑙(𝑛), as the difference between the original “walk” 𝑥(𝑛) and the local trend. One then examines the following scaling behavior: 𝐹_𝑑(𝑙)=^𝑙_𝑖=1𝑥_𝑙(𝑖)²∼𝑙^2𝐻, (5) where the angle brackets denote ensemble averages of all the segments.

Figure 5 

A schematic showing local detrending in the DFA method.

Note that the best linear or polynomial fits of DFA may have large discontinuities at the boundaries of adjacent segments. This is clearly shown in Figure 5. Such discontinuities prevent DFA from effectively removing a complicated trend such as the 11-year cycle in sunspot numbers [27]. As a result, DFA alone may not be effective enough to detect fractal variations from data with trends.

3.3. Adaptive Fractal Analysis (AFA)

AFA is built upon the adaptive detrending algorithm [28, 29]. It fixes the potential problem of DFA by finding a globally smooth trend signal. Because of this, AFA has additional advantages over DFA [20, 21]. For example, AFA can deal with arbitrary, strong nonlinear trends while DFA cannot [23, 27], AFA has better resolution of fractal scaling behavior for short time series [22], AFA has a direct interpretation in terms of spectral energy while DFA does not [23], and there is a simple proof of why AFA yields the correct 𝐻 while such a proof is not available for DFA.

To find the global trend, the method first partitions a time series into segments (or windows) of length 𝑤=2𝑛+1 points, where neighboring segments overlap by 𝑛+1 points, and thus introducing a time scale of ((𝑤+1)/2)𝜏=(𝑛+1)𝜏, where 𝜏 is the sampling time. For each segment, we fit a best polynomial of order 𝑀. Note that 𝑀=0 and 1 correspond to piecewise constant and linear fitting, respectively. Denote the fitted polynomial for the 𝑖th and (𝑖+1)th segments by 𝑦^(𝑖)(𝑙₁), 𝑦^(𝑖+1)(𝑙₂), 𝑙₁,𝑙₂=1,…,2𝑛+1, respectively. Note that the length of the last segment may be smaller than 2𝑛+1. We define the fitting for the overlapped region as 𝑦^(𝑐)(𝑙)=𝑤₁𝑦^(𝑖)(𝑙+𝑛)+𝑤₂𝑦^(𝑖+1)(𝑙),𝑙=1,2,…,𝑛+1, (6) where 𝑤₁=(1−(𝑙−1)/𝑛), 𝑤₂=(𝑙−1)/𝑛 can be written as (1−𝑑_𝑗)/𝑛,𝑗=1,2, where 𝑑_𝑗 denotes the distances between the point and the centers of 𝑦^(𝑖) and 𝑦^(𝑖+1), respectively. This means that the weights decrease linearly with the distance between the point and the center of the segment. Such a weighting ensures symmetry and effectively eliminates any jumps or discontinuities around the boundaries of neighboring segments. In fact, the scheme ensures that the fitting is continuous everywhere, is smooth at the nonboundary points, and has the right- and left-derivatives at the boundary. The method can effectively determine any kind of trend signal. Those for the 11 variables studies here are shown as red curves in Figures 2–4.

Now we can describe AFA. If we start from an increment process, 𝑥(1),𝑥(2),…. Similar to DFA, we first construct a random walk process using (2). If the original data can be considered as a random walk-like process, which is true for EEG [25, 30, 31], and sea clutter radar returns [26, 32, 33], then this step is not necessary. However, for ideal fractal processes, there is no penalty if this is done, even though the process is already a random walk process.

Next, for a window size 𝑤, we determine, for the random walk process 𝑢(𝑖) (or the original process if it is already a random walk process), a global trend 𝑣(𝑖),𝑖=1,2,…,𝑁. Here, 𝑁 is the length of the random walk process. The residual, 𝑢(𝑖)−𝑣(𝑖), characterizes fluctuations around the global trend, and its variance yields the Hurst parameter 𝐻 [23]: 𝐹⁽²⁾(𝑤)=1𝑁^𝑁_𝑖=1(𝑢(𝑖)−𝑣(𝑖))^21/2∼𝑤^𝐻. (7)

3.4. Fractal Variations in the Clarification Process of Cane Sugar Production

Let us first focus on the group 1 variables, 𝑥_𝑖,𝑖=1,2,3,4, which represent corrected brix, corrected polarimeter, pH value of the mixed juice, and the total dosage of P₂O₅, respectively. Their AFA curves are shown in Figures 6(a)–6(d). We observe that there are two scaling regions for 𝑥₁: on time scales shorter than 2⁵×2=64 hours, the Hurst parameter is 0.74. On time scales greater than 64 hours, the Hurst parameter becomes 1.42. This means that, for time scales smaller than about 3 days, the variation in 𝑥₁ has persistent long-range correlations. On time scales longer than about 3 days, the signal becomes very non-stationary—𝐻>1 is often associated with non-stationarity [25, 26]. The behavior of 𝑥₂ is very similar to that of 𝑥₁. Indeed, one can clearly observe that the signal of 𝑥₂ looks very similar to that of 𝑥₁. However, the behaviors of 𝑥₃ and 𝑥₄ are quite different from those of 𝑥₁ and 𝑥₂. In fact, they have better fractal scaling behavior (i.e., straighter linear relations in log₂𝐹(𝑤) versus log₂𝑤 plots) and stronger persistent correlations.

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Figure 6 

Adaptive fractal analysis (AFA) of signals 𝑥1 to 𝑥4: log2𝐹(𝑤) versus log2𝑤 for (a) 𝑥1, (b) 𝑥2, (c) 𝑥3, and (d) 𝑥4.

Next, we consider the variables 𝑥₅ and 𝑥₆, the neutral pH value, and the sulphur fumigation intensity. Their AFA curves are shown in Figure 7. They also have two scaling regions, and persistent correlations on short time scales (about 64 hours for 𝑥₅ and 128 hours for 𝑥₆). On longer time scales, the correlation in 𝑥₅ becomes more persistent (𝐻 changes from 0.74 to 0.88), while 𝑥₆ becomes non-stationary, since 𝐻 becomes larger than 1.

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Figure 7 

Adaptive fractal analysis (AFA) of signals (a) 𝑥5 and (b) 𝑥6.

Third, we consider 𝑥_𝑖,𝑖=7,8,…,11, which represents the corrected brix, corrected polarimeter, the pH value for the clarified juice after the 2nd heating, the color value, and the clarity degree, respectively. Their AFA curves are shown in Figure 8. We first examine 𝑥_𝑖,𝑖=7,8,9. It is instructive to compare the curves in Figures 8(a)–8(c) with those in Figures 6(a)–6(c). In particular, we observe that, for the brix and polarimeter, on shorter time scales, the correlations become more consistent, since now 𝐻 becomes bigger. However, on longer time scales, 𝐻 becomes only slightly larger than 1. Therefore, the degree of nonstationarity has decreased. The behavior of the pH value becomes more complicated than that in Figure 6, since there are two scaling regions in Figure 8(c), but there is only one scaling behavior in Figure 6(c).

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Figure 8 

Adaptive fractal analysis (AFA) of signals (a) 𝑥7, (b) 𝑥8, (c) 𝑥9, (d) 𝑥10, and (e) 𝑥11.

Finally, we examine the AFA curves for the two most important variables, color value 𝑥₁₀, and clarity degree 𝑥₁₁, which are shown in Figures 8(d) and 8(e). We observe that, for 𝑥₁₀, the first scaling, with 𝐻=0.87, is up to a time scale 2⁷×2=256 hours. Beyond that time scale, 𝐻 becomes 1; therefore, the signal becomes marginally nonstationary. The clarify degree, on the other hand, has an 𝐻=0.73 for time scales up to about 64 hours, and 𝐻=0.63 for longer time scales.

In summary, we have observed that persistent correlations up to a few days are salient features of the 11 variables that are most important for the clarification process of cane sugar production.

4. Concluding Discussions

Cane sugar production is an important industrial process. One of the most important steps in cane sugar production is the clarification process, which provides high-quality, concentrated sugar syrup crystal for further processing. To gain fundamental understanding of the physical and chemical processes associated with the clarification process and help design better approaches to improve the clarification of the mixed juice, in this paper, we have examined the fractal behavior of the 11 variables pertinent to the clarification process. We have shown that they all show persistent long-range correlations, for time scales up to at least a few days. Persistent long-range correlations amount to unilateral deviations from a preset target. This means that when the process is in a desired mode such that the target variables, color of the produced sugar and its clarity degree, both satisfy preset conditions, they will remain so for a long period of time. However, adversity could happen, in the sense that when they do not satisfy the requirements, the adverse situation may last quite long. These findings have to be explicitly accounted for when designing active controlling strategies to improve the quality of the produced sugar.

Conflict of Interests

The authors declare no conflict of interests.

Acknowledgment

This work (X. Yu and X. Liao) was partially supported by the Chinese Natural Science Foundation project number 50965003.