A prediction method for particle size distribution after powder grinding based on Markov chain

Abstract

The present invention discloses a method for predicting the particle size distribution after powder grinding based on Markov chain, comprising the following steps: (1) dividing the particle size range of the powder to be predicted; (2) converting the measured particle size distribution result into the mass proportion of particles in each sub-range in step (1), and the proportion of the mass of particles in all sub-ranges to the total mass of the powder constitutes a row vector; (3) grinding the powder and measuring the row vector of the mass proportion of each sub-range after each grinding; (4) fitting to obtain the state transition matrix after one grinding by solving a constrained least squares problem; (5) multiplying the initial particle size distribution matrix by the state transition matrix on the right for the number of grinding times, or multiplying the particle size distribution matrix under the current grinding by the state transition matrix on the right for the difference in the number of grinding times to obtain the particle size distribution matrix after multiple grindings of the powder. The powder processing enterprise does not need to measure the particle size after each grinding, which improves the production efficiency.

Description

Technical Field

[0001] The present invention relates to the technical field of materials science and engineering applications, and particularly to a method for predicting the particle size distribution after powder grinding based on a Markov chain. Background Art

[0002] Powders refer to fine particulate substances. Powders are used in a large number of industries such as ceramic manufacturing, metal processing, spraying, and medicine. The particle size and particle size distribution of powders are one of the most basic properties of powders, and changes in their particle size distribution will have a great impact on physical and chemical properties such as density, fluidity, and density. R & D personnel related to powder manufacturing will measure and preprocess the particle size of powders as soon as they obtain the raw materials.

[0003] Most of the preprocessing of powders is to grind and refine powder particles using various grinders, and measure the particle size distribution again after grinding several times. Measurement and grinding are repeated until the particle size distribution of the powder meets the production requirements. Measuring the particle size distribution of powders takes a certain amount of time. If there is a method to accurately predict the particle size distribution of powders after grinding for a certain time or a certain number of times, the production efficiency of powder processing enterprises will be greatly improved. Summary of the Invention

[0004] Aiming at the above existing technical problems, the object of the present invention is to provide a method for predicting the particle size distribution after powder grinding based on a Markov chain. Based on the Markov model, calculate the transition matrix of a single grinding of a certain material powder on a certain grinder. According to this transition matrix, the particle size distribution of the material powder after multiple grindings on this grinder can be accurately predicted. Powder processing enterprises do not need to measure the particle size after each grinding, which speeds up the production efficiency.

[0005] The technical solution of the present invention is as follows:

[0006] A method for predicting the particle size distribution after powder grinding based on a Markov chain of the present invention includes the following steps:

[0007] (1) Divide the particle size range of the powder to be predicted according to the particle size measurement range of the used powder particle size measuring instrument;

[0008] (2) Convert the particle size distribution result given by the particle size measuring instrument into the mass proportion of particles in each sub-interval in step (1). The proportion of the mass of particles in all sub-intervals to the total mass of the powder constitutes a row vector;

[0009] (3) Grind the powder, and according to the definition in step (2), measure the row vector of the mass proportion of each sub-interval after each grinding to obtain the particle size distribution matrix before and after grinding;

[0010] (4) The state transition matrix after the first grinding is obtained by solving the least squares problem with constraints.

[0011] (5) Using the state transition matrix obtained in step (4), the initial particle size distribution matrix is right-multiplied by the state transition matrix for the number of grinding times, or the current particle size distribution matrix during grinding is right-multiplied by the state transition matrix for the difference in the number of grinding times to obtain the particle size distribution matrix after multiple grindings of the powder.

[0012] Optionally, in step (1), let the total number of intervals be N, where N is a natural number greater than or equal to 1. The range of the k-th sub-interval is (x k-1 , x k , with x0 = 0 and x N being the upper limit of the particle size. The interval division is as follows:

[0013] {(0, x1], (x1, x2], (x2, x3], …, (x N-1 , x N )}.

[0014] Optionally, in step (2), the proportion of the mass of the particles in the N sub-intervals to the total mass forms a 1*N row vector X 1×N , and the i-th element in the first row of the row vector X 1×N is:

[0015]

[0016] Where in formula (1), represents the mass of the particles in the interval (x i-1 , x i , and i ≤ N.

[0017] Optionally, if the sedimentation method is used to measure the particle size distribution of the powder, the measurement result is the mass proportion, and the row vector X 1×N is given by the measuring instrument.

[0018] Optionally, if a laser particle size analyzer is used to measure the particle size distribution, the measurement result is the proportion of the number of particles. The directly measured proportion of the number of particles needs to be converted to the mass proportion X 1×N according to the following formula:

[0019]

[0020] Where in formula (2), represents the number of particles, or the proportion of the number of particles, in the interval (x i-1 , x i , and i ≤ N.

[0021] Optionally, in step (3), let the measurement result of the particle size distribution before the first grinding be denoted as The particle size distribution result after the Mth grinding is denoted as Define the particle size distribution result matrix X before grinding M×N , and the particle size distribution matrix Y after grinding M times M×N :

[0022]

[0023] Where is the jth column of matrix Y M×N , and 1 < j < N.

[0024] Optionally, step (4) includes first constructing an intermediate matrix B i,N×N , E N(N+1)×1 , F MN×1 to calculate the step of the N 2 -dimensional column vector , and then the matrix is rearranged to obtain the state transition matrix A for one grinding N×N .

[0025] Optionally, the matrix sequence B of N rows and N columns i,N×N ,

[0026]

[0027] where i is equal to 1 to N, I (i-1)×(i-1) represents the identity matrix with (i - 1) rows and columns, and 0 (N-i)×(N-i) represents the zero matrix with (N - i) rows and columns;

[0028] The matrix of N(N + 1) rows and N 2 columns

[0029]

[0030] The matrix of NM rows and N 2 columns

[0031]

[0032] The N 2 -dimensional column vector E N(N+1)×1 , with the first N elements all being 1 and the subsequent elements all being 0:

[0033]

[0034] The N-dimensional column vector F MN×1 , F MN×1 is rearranged from the elements of each column in Y M×N :

[0035]

[0036] Optionally, calculate N 2 dimensional column vector By solving the following least squares problem with constraints

[0037] Get:

[0038]

[0039] Where the first min{*} represents the minimum objective of the least squares method; the part after s.t is the constraint condition.

[0040] Optionally, the state transition matrix A for one-time grinding in step (5) N×N , is a matrix After rearrangement, we get

[0041] To, matrix A N×N The element in the i-th row and j-th column is:

[0042]

[0043] Let the initial particle size distribution measurement be a row vector Or the particle size distribution measurement after M times of powder grinding be a row vector Then the particle size distribution after R times of powder grinding Estimated value of:

[0044] Or

[0045]

[0046] Compared with the prior art, the advantages of the present invention are:

[0047] The state transition matrix for single-time grinding of a certain material powder entering the grinding equipment is calculated through the particle size distribution results before and after M times of grinding of the material powder; after obtaining the state transition matrix, the particle size distribution after different grinding times of a new batch of the same material powder can be accurately predicted; powder processing enterprises do not need to measure the particle size after each grinding, which improves production efficiency; when powder processing enterprises find that the initial particle size distribution of the purchased raw materials changes, they can also adjust the grinding times according to the estimated results. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] The present invention will be further described below in conjunction with the drawings and embodiments:

[0049] Figure 1 It is a schematic diagram of the powder mass transfer in a divided particle size interval for the prediction method of the particle size distribution after powder grinding based on the Markov chain in the embodiment of the present invention (the i-th interval range (x i-1 , xi The mass m(x of the particles inside i-1 , x i is partially transferred to the j-th interval (x j-1 , x j , and the mass of the transferred part is A(i, j) times the mass of the original interval). Detailed implementation manners

[0050] To make the objectives, technical solutions and advantages of the present invention clearer and more understandable, the present invention will be further described in detail below in combination with specific implementation manners and with reference to the accompanying drawings. It should be understood that these descriptions are merely exemplary and are not intended to limit the scope of the present invention. In addition, in the following descriptions, descriptions of well-known structures and technologies are omitted to avoid unnecessarily confusing the concepts of the present invention.

[0051] It should be noted that when the powder is ground, the grinding effects on particles of different particle sizes are different, which are reflected in different converted masses and finally generated particle sizes. The loss of each grinding by the grinder is very small, and there are enough particles in the powder. The mathematical model of the Markov chain believes that the ratio of conversion from one range of particle size to another is constant. That is, if we divide the particle size of the particles into multiple intervals in advance, the transfer of the powder mass inside the interval is fixed, as shown in the attached Figure 1 drawing. Assume that the powder particle size distribution is divided into N intervals, and the range of the i-th interval is (x i-1 , x i . The mass m(x of the particles inside i-1 , x i is partially transferred to the j-th interval (x j-1 , x j , and the mass of the transferred part is A(i, j) times the mass of the original interval. This A(i, j) is constant, and all elements A(i, j) form the state transition matrix A N×N . Therefore, to estimate the particle size distribution after grinding the powder in the embodiment of the present invention, the state transition matrix A needs to be obtained first N×N .

[0052] More specifically, a method for estimating the particle size distribution after grinding a powder based on a Markov chain in an embodiment of the present invention includes the following steps:

[0053] (1) Divide the particle size intervals of the powder to be estimated according to the particle size measurement range of the used powder particle size measuring instrument; specifically, assume that the total number of intervals is N, a natural number where N≥1, and the range of the k-th sub-interval is (x k-1 , x k , x0 = 0, x N is the particle size upper limit, and the interval division is as follows:

[0054] {(0, x1], (x1, x2], (x2, x3], …, (x N-1 , x N )}。

[0055] (2) Convert the particle size distribution results given by the particle size measuring instrument to the mass fraction of the particles in each sub-interval in step (1). The proportion of the mass of the particles in all sub-intervals to the total powder mass forms a row vector. Specifically, the particle size distribution referred to in the present invention means the percentage of the particle mass in the sub-interval to the total mass. The percentage of the mass of the particles in N sub-intervals to the total mass forms a 1*N row vector X 1×N , that is, the row vector X 1×N The i-th element in the first row is:

[0056]

[0057] Where in formula (1), represents the mass of the particles in the particle size range (x i-1 , x i , and i ≤ N.

[0058] It should be noted that if the sedimentation method is used to measure the particle size distribution of the powder, the measurement result is the mass fraction, and the row vector X 1×N is given by the measuring instrument.

[0059] If a laser particle size analyzer is used to measure the particle size distribution, the measurement result is the particle number fraction. It is necessary to convert the directly measured particle number fraction to the mass fraction X according to the following formula 1×N :

[0060]

[0061] Where in formula (2), represents the number of particles, or the particle number fraction, in the particle size range (x i-1 , x i , and i ≤ N.

[0062] (3) Grind the powder. According to the definition in step (2), measure the mass fraction row vector of each sub-interval after each grinding to obtain the particle size distribution matrix before and after grinding; specifically, let the particle size distribution result measured before the first grinding be denoted as The particle size distribution result after the M-th grinding is denoted as Define the particle size distribution result matrix X before grinding M×N , and the particle size distribution matrix Y after grinding M times M×N :

[0063]

[0064] Where is the matrix YM×N the j-th column, where 1 < j < N.

[0065] (4) Fit to obtain the state transition matrix after the first grinding by solving a least squares problem with constraints; specifically, first construct the intermediate matrix B i,N×N , E N(N+1)×1 , F MN×1 to calculate the N 2 -dimensional column vector in the steps, and then re-arrange by the matrix to obtain the state transition matrix A of the first grinding N×N . More specifically,

[0066] a sequence of N×N matrices B i,N×N ,

[0067]

[0068] where i ranges from 1 to N, I (i-1)×(i-1) represents the identity matrix of size (i - 1), and 0 (N-i)×(N-i) represents the zero matrix of size (N - i);

[0069] a matrix of size N(N + 1)×N 2 columns

[0070]

[0071] a matrix of size NM×N 2 columns

[0072]

[0073] the N 2 -dimensional column vector E N(N+1)×1 , with the first N elements all being 1 and the subsequent elements all being 0:

[0074]

[0075] the N-dimensional column vector F MN×1 , F MN×1 is obtained by re-arranging the elements of each column in Y M×N :

[0076]

[0077] Calculate the N 2 -dimensional column vector obtained by solving the following least squares problem with constraints:

[0078]

[0079] Among them, the first min{*} represents the minimum objective of the least squares method; the content after s.t is the constraint condition.

[0080] (5) Through the state transition matrix obtained in step (4), multiply the initial particle size distribution matrix by the state transition matrix on the right for the number of grinding times, or multiply the particle size distribution matrix under the current grinding by the state transition matrix on the right for the difference in the number of grinding times to obtain the particle size distribution matrix after the powder is ground multiple times.

[0081] Specifically, the state transition matrix A for one grinding N×N , is the matrix rearranged to obtain, the matrix A N×N The element in the i-th row and j-th column is:

[0082]

[0083] Let the initial particle size distribution measurement be a row vector or the particle size distribution measurement after the powder is ground M times be a row vector Then the predicted value of the particle size distribution after the powder is ground R times:

[0084] or

[0085]

[0086] Exemplarily, assume that the results of the initial particle size distribution measurement in N intervals are a row vector The state transition matrix of this material on a certain grinder is A N×N (already obtained in advance). The method of the present invention gives the predicted value of the particle size distribution after grinding one more time:

[0087]

[0088] The predicted value of the particle size distribution after grinding R more times:

[0089]

[0090] The following takes the sedimentation method for measuring the particle size distribution as an example to estimate the grinding particle size distribution of the powder (taking feldspar powder as an example, feldspar powder is an important powder for ceramic production and needs to be dispersed in water and ground multiple times in a sand mill) for a detailed description.

[0091] Example 1

[0092] In the embodiment of the present invention, the interval division takes N = 10 as an example. Specifically,

[0093] (1) Divide N = 10 particle size intervals according to the particle size measurement range of the particle size distribution measuring device,

[0094] {(0, x1], (x1, x2], (x2, x3], …, (x N-1 , +∞)}

[0095] = {(0, 1], (1, 1.5], (1.5, 2], (2, 3], (3, 5], (5, 10], (10, 20], (20, 30], (30, 50], (50, 100)}

[0096] The unit of each sub - interval is μm.

[0097] (2) The particle size measuring instrument is an existing conventional particle size analyzer, and the sedimentation method is used to measure the particle size distribution. The result of the first measurement by the particle size analyzer is:

[0098]

[0099] Among them, the first element 7.2 / 100 represents the proportion of the powder mass in the interval (0, 1] measured by the particle size analyzer before the first grinding of feldspar. By analogy, it will not be elaborated.

[0100] (3) Taking the number of grinding times M = 4 as an example, the particle size distribution result matrix X before 4 grindings M×N is:

[0101]

[0102] The particle size distribution result matrix Y after 4 grindings M×N is:

[0103]

[0104] (4) Calculate the state transition matrix of one grinding by the method of constructing the intermediate matrix described above. It should be noted that in this step, according to the known particle size distribution matrices X before several grindings M×N and the particle size distribution matrix Y after grinding M×N fit to obtain the state transition matrix. Mathematically, it is to solve the following least - fitting problem with constraints:

[0105]

[0106] where A N×N is the transfer matrix to be fitted. Mathematically, there is no general method to solve the above problem. The present invention creatively converts the above least - fitting problem into a least - squares fitting vector problem. The above least - fitting problem is equivalent to the least - squares problem:

[0107]

[0108] Among them, the additional matrices C, D, E, F introduced (i.e., the intermediate matrix B above i,N×N , E N(N+1)×1 , F MN×1 ). The solved vector G (i.e., the above ) can derive the state transition matrix A (i.e., the above A N×N ). Among them, it is necessary to solve a least squares problem with constraints. Commonly used numerical calculation software has this calculation package. Exemplarily, in the embodiments of the present invention, the lsqlin(*) function in the commercial software MATLAB is used, and finally the state transition matrix A N×N is:

[0109]

[0110] Among them, the physical meaning of A(2,1)=0.238 is that the powder mass in the second interval (1, 1.5] um has a proportion of 0.238 transferred to the powder in the first interval (0, 1] um. That is, the elements in the state transition matrix A N×N describe the transformation of the powder grinding becoming finer. In the embodiments of the present invention, the state transition matrix A N×N is a lower triangular matrix, and the physical meaning is that the powder will only become finer and will not agglomerate after grinding; at the same time, the sum of all elements in any row of the state transition matrix A N×N is 1, and the physical meaning is mass conservation during grinding. In the embodiments of the present invention, introducing the intermediate matrix and solving the least squares problem with constraints in step (4) is to make the state transition matrix A N×N have the above two properties.

[0111] (5) The state transition matrix calculated in step (4) is used to right-multiply the initial particle size distribution matrix by the number of grinding times or right-multiply the current particle size distribution matrix under grinding by the difference in the number of grinding times to obtain the particle size distribution matrix of the powder after multiple grindings, so as to accurately predict the particle size distribution of the subsequent such materials after grinding on this grinder.

[0112] To verify the accuracy of the estimation method in the embodiments of the present invention, in step (5), we use the measured value after the fourth grinding to estimate the particle size distribution after the fifth grinding. Specifically:

[0113]

[0114] In an alternative embodiment, the initial particle size distribution measurement row vector is right-multiplied by the matrix state transition matrix A N×N five times to obtain the estimated distribution of the fifth time:

[0115]

[0116] In order to verify the error between the prediction method and the actual measurement of the conventional method to verify the accuracy and reliability of the prediction method of the embodiments of the present invention, in the embodiments of the present invention, the particle size distribution after the fifth grinding measured by the conventional method is as follows:

[0117]

[0118] It can be seen from this that the prediction method of the embodiments of the present invention has a small difference from the actual measurement results, and the absolute deviation of the mass fraction is within 0.5%, thus verifying that the prediction method of the embodiments of the present invention is accurate and reliable.

[0119] One of the innovations of the present invention is to model the powder grinding with a Markov chain model. The second innovation is to give the calculation steps of the state transition matrix. The state transition matrix A N×N has special properties: First, it is a lower triangular matrix, and its elements are greater than or equal to 0. The physical meaning is that the powder will only become finer and will not agglomerate after grinding; then the state transition matrix A N×N has a row sum of 1, indicating that the physical meaning is mass conservation during grinding. By introducing an intermediate matrix and a least squares problem with constraints, this is to make the state transition matrix A N×N have the above two properties.

[0120] It should be understood that the above specific embodiments of the present invention are only used for exemplary illustration or explanation of the principle of the present invention, and do not constitute a limitation to the present invention. Therefore, any modifications, equivalent replacements, improvements, etc. made without departing from the spirit and scope of the present invention shall be included within the protection scope of the present invention. In addition, the appended claims of the present invention are intended to cover all changes and modifications that fall within the scope and boundaries of the appended claims, or equivalent forms of such scope and boundaries.

Claims

1. A method for predicting the particle size distribution after powder grinding based on Markov chain, characterized in that, It includes the following steps: (1) Divide the particle size range of the powder to be estimated according to the particle size measurement range of the powder particle size measuring instrument used; let the total number of intervals be N, where N is a natural number greater than or equal to 1, and the range of the k-th sub-interval is (x k-1 , x k , where x0 = 0 and x N is the upper limit of the particle size. The interval division is as follows: ； (2) Convert the particle size distribution results given by the particle size measuring instrument into the mass proportion of the particles in each sub-interval in step (1). The proportions of the particles in all sub-intervals in the total powder mass form a row vector; (3) Grind the powder. According to the definition in step (2), measure the row vector of the mass proportion of each sub-interval after each grinding to obtain the particle size distribution matrix before and after grinding; Let the particle size distribution result measured before the first grinding be denoted as , and the particle size distribution result after the Mth grinding be denoted as . Define the particle size distribution result matrix before the Mth grinding as , and the particle size distribution matrix after grinding M times as : ； ； where is the j-th column of the matrix and 1 < j < N; (4) The state transition matrix after the first grinding is obtained by solving the least squares problem with constraints, including first constructing the intermediate matrix , , , , to calculate the -dimensional column vector . After that, the state transition matrix of the first grinding is rearranged from the matrix ; A sequence of N×N matrices , ； where i ranges from 1 to N, denotes an identity matrix with rows and columns, denotes a zero matrix with rows and columns; Row matrix of : ； Row matrix of : ； Column vector of dimension , with the first N elements all being 1 and the subsequent elements all being 0: ； Column vector of dimension , is formed by rearranging the elements of each column in ； Calculation column vector of dimension , obtained by solving the following least squares problem with constraints: ； Where the first min{*} represents the minimum objective of the least squares method; the content after s.t is the constraint condition; (5) Through the state transition matrix obtained in step (4), multiply the initial particle size distribution matrix by the state transition matrix on the right for the number of grinding times, or multiply the current particle size distribution matrix under grinding by the state transition matrix on the right for the difference in the number of grinding times to obtain the particle size distribution matrix of the powder after multiple grindings; State transition matrix of primary grinding , which is a matrix reordered to obtain a matrix The element in the i-th row and j-th column is: 。 2. The prediction method for particle size distribution after powder grinding based on Markov chain according to claim 1, characterized in that In step (2), the proportion of the mass of the particles in the N sub-intervals to the total mass forms a 1×N row vector , the row vector The i-th element in the first row is: （1） In formula (1), represents the mass of particles with a particle size in the range, .

3. The prediction method for particle size distribution after powder grinding based on Markov chain according to claim 2, characterized in that If the sedimentation method is used to measure the particle size distribution of the powder, the measurement result is the mass fraction, and the row vector is given by the measuring instrument.

4. The method for predicting particle size distribution after powder grinding based on Markov chain according to claim 2, characterized in that, If a laser particle size analyzer is used to measure the particle size distribution and the measurement result is the proportion of the number of particles, the proportion of the number of particles directly measured needs to be converted into the proportion of mass according to the following formula : （2） In formula (2), represents the number of particles in the range, or the proportion of the number of particles, .

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