Chinese Name: Flocculation Stage: Coagulation and Flocculation Theory Generation: American French mathematician Mandelbrot simulation model: Introduction, overview, simulation model, calculation method, application, research prospect, references, introduction Coagulation and flocculation are two important stages of coagulation process, and the perfection of flocculation process directly affects the treatment effect of subsequent treatment (precipitation and filtration). However, the structure of floc is complex, fragile and irregular. In the past, due to the lack of suitable research methods, only the input of coagulant and the coagulation effect of effluent were considered, and the coagulation system was regarded as a "black box" without in-depth research. Even considering the microscopic process, all colloidal particles are only abstracted as spheres, which are explained by the existing colloid chemistry theory and chemical kinetics theory [1], and the conclusions are quite different from the characteristics of colloids and flocs actually observed in the experiment. Although some researchers introduce particle coefficient to modify the theoretical derivation and form the final mathematical expression, the theoretical and experimental results are still difficult to agree. The introduction of fractal theory fills the blank of floc research method. As a new research method of flocculation, fractal theory inspires researchers to further understand the structure, coagulation mechanism and dynamic model of floc. Summarize the emergence of 1. 1 fractal theory.
1975 [2] B.B.Mandelbrot, a French-American mathematician, put forward a new method, which can be used to describe and calculate the properties of rough, broken or irregular objects, and created the word fractal to describe them.
Fractal refers to an irregular, chaotic and complex system, but its part is similar to the whole, and its self-similarity and scale invariance are its important characteristics. The formation process of the system is random, and the dimension of the system can be not an integer but a fraction [3]. Its external features are generally fragile, irregular and complex, while its internal features are self-similar and self-affine. Self-similarity is the core of fractal theory, which means that the local shape is similar to that of the whole, that is, the local shape of an object is the same or similar to that of the whole after being enlarged in the same proportion in all directions. Self-affine means that although local and global fractals are different, they are not only similar, but also overlap after stretching and compression.
Fractal theory injects new contents into the concepts of part and whole, disorder and order, finiteness and infinity, simplicity and complexity, certainty and randomness, which enables people to explore the essential relationship behind these complex phenomena with new concepts and means.
Fractal characteristics of 1.2 floc
Floc growth is a stochastic process with nonlinear characteristics. If the floc breakage is not considered, the conventional flocculation process is that the initial particles are superimposed by linear random motion to form small groups, and the small groups collide and aggregate into larger groups, and then further aggregate and gradually grow into large flocs. This process determines that flocs have self-similarity and scale invariance in a certain range, which are two important characteristics of fractal [4], that is, the formation of flocs has fractal characteristics. Simulation model 2. 1 floc fractal structure model
In order to better understand the formation process of floc and predict it as much as possible, many floc structure models have been put forward after a lot of research.
2. 1. 1 Early Floc Structure Model
The earliest model [5] is a three-layer model proposed by Vold through computer simulation: (see figure 1[4]) initial particles, flocs and floc aggregates. Floc structure consists of a central core and a set of rough surfaces formed by tentacles (protrusions) extending outward. Floccules are formed by the random movement of initial particles, regardless of the internal recombination process. The further aggregation of flocs forms a three-stage aggregation structure, which leads to rapid sedimentation and visible suspended particles. The further analysis of its structural characteristics shows that the floc density gradually decreases outward with the center, and the empirical formula of floc density changing with particle size is derived from this.
Sutherland criticized the random characteristics of particle aggregation in Vold floc model [6]. He thinks that the main mechanism of floc growth is not the collision of single particles, but the collision and aggregation between clusters containing different numbers of particles, which seems more logical. Because in fact, the collision of initial particles is very important only in the process of smaller cluster formation. Compared with Vold model, Sutherland model (see Figure 2[4]) forms more porous and looser structures with lower density. With the increase of particle size, its density decreases and its porosity increases. When the floc grows, the internal reorganization of the structure will also occur. When cocurrent flocculation occurs during suspension mixing, the aggregation conditions of flocs will change. The shear force of fluid will destroy the structure of floc, resulting in the formation of floc with characteristic particle size under certain conditions Sutherland model is only suitable for flocs with particle size less than several microns.
The complex structure of floc makes it very difficult to describe it quantitatively. The early model quantitatively analyzed and described the floc structure from different angles, which involved fractal characteristics to some extent, but it was not widely used because it did not summarize the concept of classification.
2. 1.2 development of floc structure model
The initial particle considered in the early model is a uniform sphere with a single particle size, but this is not always the case. Good-arz-Nia established a new model [7], in which the initial particle size distribution is based on a standard normal distribution, which is elliptical initial particles with different axial-diameter ratios, and the structure is composed of chains formed by the initial particles. The calculated floc particle size is not much different from that of floc with single particle size distribution. The volume of floc is relatively small. This is because the existence of small particles can fill the gaps between particles, resulting in denser flocs.
In Vold model and Sutherland model, the motion of particles and clusters is linear, excluding Brownian motion, which is inconsistent with the actual situation. Amp Sander modified this [8], and they set a number of seed particles as growth points. Add other particles at random positions and walk randomly until they reach the position adjacent to the seed particles and adhere to each other to form a growth group, and then continue to add particles until a sufficiently large floc is formed.
Francois & ampampVanHaute put forward a floc structure model, which has four layers [7]: initial particles, flocs, flocs and floc aggregates. Different from the previous model, this model thinks that the binding bonds of different secondary flocs are elastic and variable. In the elastic model, the shear force of the fluid can penetrate all the particles in the floc. The structure pattern of multi-layer flocs is consistent with the fractal structure characteristics of flocs, but with the formation of different clusters, the fractal dimension of flocs will change accordingly.
2.2 Dynamic growth model of floc fractal structure [9]
With the gradual deepening of the research on fractal growth process, various dynamic growth models have been put forward, which can be basically summarized into three categories, namely:
1) diffusion-limited aggregation model, referred to as DLA model;
2) Ballistic aggregation model, abbreviated as BA model;
3) Reaction-limited aggregation model, abbreviated as RLA model.
These three models can be divided into two parts, monomer aggregation and cluster aggregation. In DLA model, monomer aggregation is called Witten-Sander model, and group aggregation is called cluster aggregation model with limited diffusion rate, which is called DLCA model for short. Correspondingly, there are Vold mode and Sutherland mode in BA mode. There are Eden model and reaction-limited clustering (RLCA) model in RLA model. The parameter of fractal system characterized by calculation method is FractalDimension, which corresponds to the irregularity and complexity of fractal or the degree of space filling measurement. Because of the different research objects, there are many different definitions of dimensions. There are four commonly used fractal dimensions of particle morphology: d, D 1, D2 and Dk. D, D 1, D2 and Dk are obtained from the relationships between area and perimeter, length and perimeter, length and area, and area and rank, respectively. The mathematical relationship is as follows:
p∝AD/2； p∝LD 1； a∝LD2； NR(a & gt； A)∝A–Dk/2 .
Where p is the perimeter, a is the area, l is the maximum length of the particle, and Nr is the area of a (a >; The quantity or order of flocs in a). The instantaneous changes of D, Dk and D2 are consistent with the observed changes of particle morphology and can be quantified, but D 1 does not have this feature [10].
At present, there are generally two methods to calculate fractal dimension: computer simulation of floc growth process and experimental direct determination. Based on the formation mechanism of floc, computer simulation was widely used in 1970s and 1980s. With the development of science and technology, it is possible to measure fractal dimension directly with advanced instruments. At present, there are many methods used, such as image method, particle size distribution method, light scattering method and sedimentation method. 3. 1 computer simulation [8]
The computer simulation of floc growth process should choose the appropriate dynamic model and structural model according to the actual situation. There are two specific simulation methods: grid simulation and non-grid simulation.
Grid simulation is carried out in grid plane (two-dimensional) or cubic grid space (three-dimensional) with peripheral boundary conditions. The so-called periodic boundary refers to the re-entry of particles from a symmetrical place when they overflow the grid boundary during the movement.
Non-grid simulation is carried out in a continuous finite space, which is different from grid simulation in grid length. Meshless simulation is measured by particle size, and the position of each particle or group is determined by its centroid.
Because the two methods adopt different frames and floc shapes, the particles in the floc simulated by grid lines are square (two-dimensional) or cubic (three-dimensional); The particles in the floc obtained by non-grid simulation are round (two-dimensional) or spherical (three-dimensional), and the smoothness of the floc is better than that by grid simulation.
3.2 Direct determination
3.2. 1 image method [1 1, 12]
Through photomicrography, the flocs in the water are magnified and photographed, and the images of flocs are analyzed by computer image processing software, so that the projected area a, perimeter p and maximum length l of flocs in a certain direction can be measured, and one-dimensional and two-dimensional fractal dimensions can be obtained according to the following relationship:
P∝LD 1( 1)
A∝PD2 or A∝LD2(2)
Generally, the three-dimensional fractal dimension can not be obtained directly by image method, and it needs to be transformed. One method is to calculate the diameter dp (equivalent diameter) of a circle with the same area according to the projected area, and then convert it into the volume V of a sphere, and calculate D3 according to the following formula:
V∝PD3 or V∝LD3(3)
However, some studies believe that the deviation of the three-dimensional fractal dimension calculated by this method is large, so it is suggested to convert an ellipse with the same size as the projected area into an ellipsoid, and then calculate it by Formula (3). Mirror image method is a widely used fractal dimension calculation method at present.
3.2.2 Particle size distribution method [13]
This method, also known as double slope method, is obtained by measuring the slopes of the cumulative particle concentration distribution curve N(L) with the characteristic length l (generally the maximum length in a certain direction) and the distribution curve N(v) with the floc volume as the parameter under the same conditions.
The length and volume distribution functions are as follows:
N(L)=ALLSL(4)
N(V)=AvvSv(5)
In the formula, SL and Sv are the exponents of the length and volume particle distribution curves, respectively, and AL and Av are constants. Because it is a cumulative distribution curve under the same conditions, there are:
N(L)=N(v)(6)
Then: ALLSL=AvvSv(7)
It is generally believed that flocs are composed of primary particles. Using the initial particle length L, shape coefficient α, density ρ and packing coefficient β, the volume V is expressed as:
v=m/ρ=ψD/3αL3-DLD(8)
Substitute Formula (8) into Formula (7) as follows:
ALLSL=Av(ψD/3αL3-D)SvLDSv(9)
(9) The L-term indices on both sides of the formula should be equal, then:
D=SL/Sv
If the particle distribution curve with length and volume as parameters is known, the fractal dimension can be calculated according to the above formula according to the slope of the curve.
3.2.3 Other methods [14]
Settling method is to calculate the fractal dimension by measuring or calculating the relationship between settling velocity u and floc characteristic length l, u∝LD. This method is suitable for the case that flocs are dense and not easy to break.
The light scattering method is based on the relationship between the scattered light intensity I(q) and the light wave vector Q = | q | d, and the fractal dimension is obtained by the small angle X-ray scattering method. This method is based on Rayleigh scattering, and the deviation is large when the floc size is too large.
The fractal dimension of fast flocculating floc model measured by static light scattering method is 1.75 ~ 1.80, and that of fast flocculating floc measured by sedimentation method is 1.65 ~ 1.70. The size of bridging floc measured by static light scattering method is 2. 12, while that measured by sedimentation method is 1.8 1[3]. Among them, the light scattering method is better for the determination of small and loose flocs, and the sedimentation method is better for the determination of large and dense flocs.
In addition, there are methods to find fractal dimension by changing observation scale, finding fractal dimension according to correlation function and finding fractal dimension according to frequency spectrum. 4. The relationship between the application of1fractal parameter and coagulation effect
Some researchers have verified the relationship between fractal parameters of floc and coagulation effect through experiments. The research on coagulation control by Chang Ying [15] and Li Meng [16] shows that the fractal dimension of floc has a good correlation with the turbidity of settled water for different raw water turbidity. The experiment of Lu et al [17] discussed the relationship between floc structure and fractal dimension under different dosage, stirring conditions and sedimentation time, and found that the fractal dimension of floc was higher when the flocculation effect was better. Fractal dimension is very sensitive to reflect the degree of flocculation change of flocs, and different fractal dimensions can be used to characterize the self-similar fractal characteristics of flocs formed under different conditions. Therefore, the growth of floc can be controlled by measuring fractal dimension.
4.2 Application example
In the process of water treatment, the fractal characteristics of flocs play an important role in regulating the migration and removal of particles. For example, Li Dongmei et al. [18] measured the floc dimension in the electron microscope photograph (Figure 1) in the experimental study of bridging flocculation of high-concentration suspended solids represented by the Yellow River sediment, and found that in the middle and early stage of slow flocculation (flocculation time 180 seconds), the "fractal dimension" of floc reached the maximum and the structural compactness was the best.
A— Flocculation time is 10s (taken at the end of rapid mixing); B—— flocculation time 50s (obtained during slow mixing); C—— Flocculation time 180s (value taken when stirring slowly); D—— Flocculation time is 600 seconds (taken at the end of slow stirring); E— Shoot after stirring stops 15s; Local scanning photo of F-A diagram (magnified 5000 times)? A ~ E are micrographs of flocs (magnification is 180 times).
The experimental results also show that (1) floc structure gradually transits from DLCA mode with low fractal dimension to RLCA mode with high fractal dimension, and finally tends to a relatively stable shape. ⑵ The evolution of fractal structure of flocs leads to significant differences in internal permeability of flocs. When D3 >: 2, the larger D3, the higher the floc settling speed; When D3 research looks forward to the traditional flocculation theory, it provides a basic framework for simulation calculation. Further study of flocculation mechanism combined with fractal theory can deepen the understanding of its process and connotation. The change of fractal dimension of floc in coagulation process can be used to predict the turning point of different floc structures, and further study the influencing factors of floc formation, and put forward the best coagulation control conditions. However, the research on flocculation mechanism is still in its infancy. Although many coagulation kinetic models have been produced, they cannot fully describe the actual situation of coagulation process due to the limitation of models based on micro-appearance. Researchers' understanding of coagulation mechanism and dynamic process is still limited to the discussion of flocculation process in simple system, and the research of complex system process needs to be further deepened. References [1] Lu, Tang Youyao. Determination of floc fractal and its fractal dimension during flocculation [J]. Journal of Huazhong University of Science and Technology (Urban Science Edition), 2003,20 (3): 46-49.
[2] Zhang Yue Zhang. Scientific and Philosophical Basis of Fractal Theory [J]. Social Science Research, 2005(5):8 1-86.
Wang yijiu, you Development of fractal theory and its application in coagulation [J]. Journal of Tongji University, 2003,31(5): 614-618.
Wang Xiaochang Bao Dan Ren Xian. On the morphology and density of flocs from the fractal structure [J]. Journal of Environmental Science, 2000,20 (3): 257-262.