Method for estimating particle size distribution after powder grinding based on Markov chain

Abstract

The invention discloses a Markov chain-based pre-estimation method for particle size distribution after powder grinding. The method comprises the following steps of: (1) dividing a particle size interval of powder to be pre-estimated; (2) converting the measured particle size distribution result into the mass ratio of the particles in each sub-interval in the step (1), wherein the ratio of the mass of the particles in all the sub-intervals to the total powder mass forms a row vector; (3) grinding the powder, and measuring to obtain a mass ratio row vector of a subinterval after each grinding; (4) fitting by solving a constrained least square problem to obtain a state transition matrix after primary grinding; and (5) enabling the initial particle size distribution matrix to pass through the grinding frequency right multiplication state transition matrix or enabling the current ground particle size distribution matrix to pass through the grinding frequency difference right multiplication state transition matrix so as to obtain the particle size distribution matrix after the powder is ground for multiple times. A powder processing enterprise does not need to measure the particle size after grinding every time, and the production efficiency is improved.

Description

Technical Field

[0001] The invention relates to the field of material science and engineering application technology, and in particular to a method for estimating particle size distribution of powder after grinding based on a Markov chain. Background Art

[0002] Powder refers to fine granular 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. Changes in its particle size distribution will have a great impact on physical and chemical properties such as density, fluidity, and compactness. R&D personnel related to powder manufacturing will measure the particle size and pre-treat the powder as soon as they get the raw materials.

[0003] Most of the pretreatment of powders is to use various grinders to grind the powder particles into fine particles, and re-measure the particle size distribution after grinding several times. The measurement and grinding are repeated until the powder particle size distribution meets the production requirements. The measurement of powder particle size distribution takes a certain amount of time. If there is a method to accurately estimate the particle size distribution of powder after grinding for a certain time or a certain number of times, it will greatly improve the production efficiency of powder processing companies. Summary of the invention

[0004] In view of the above-mentioned technical problems, the purpose of the present invention is to provide a method for estimating the particle size distribution of powder after grinding based on a Markov chain. Based on the Markov model, the transfer matrix of a single grinding of a certain material powder on a certain grinder is calculated. According to this transfer matrix, the particle size distribution of the material powder after multiple grindings on this grinder can be accurately estimated. Powder processing companies do not need to measure the particle size after each grinding, which speeds up production efficiency.

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

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

[0007] (1) According to the particle size measurement range of the powder particle size measuring instrument used, the particle size range of the powder to be estimated is divided;

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

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

[0010] (4) fitting the state transfer matrix after one grinding by solving the least squares problem with constraints;

[0011] (5) The state transfer matrix obtained in step (4) is used to right-multiply the initial particle size distribution matrix by the number of grinding times or to right-multiply the particle size distribution matrix of the current grinding by the difference number of grinding times to obtain the particle size distribution matrix of the powder after multiple grindings.

[0012] Optionally, in step (1), let the total number of intervals be N, a natural number ≥ 1, and the range of the kth sub-interval (x k-1 ,x k ], x 0 =0,x N is the upper limit of particle size, and the interval is divided as follows:

[0013] {(0,x 1 ],(x 1 ,x 2 ],(x 2 ,x 3 ],…,(x N-1 ,x N )}.

[0014] Optionally, in step (2), the proportion of the mass of particles in the N subintervals to the total mass constitutes a 1*N row vector X 1×N , the row vector X 1×N The i-th element in row 1 is:

[0015]

[0016] In formula (1), Indicates the particle size in (x i-1 ,x i ] is the mass of particles in the interval, i≤N.

[0017] Optionally, if the sedimentation method is used to measure the powder particle size distribution, the measurement result is the mass percentage, and the row vector X 1×N 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 particle number percentage. The directly measured particle number percentage needs to be converted to the mass percentage according to the following formula: 1×N :

[0019]

[0020] In formula (2), Indicates the particle size in (x i-1 ,x i ]The number of particles in the interval, or the proportion of particles, i≤N.

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

[0022]

[0023] in is the matrix Y M×N The jth column of <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 N 2 Dimensional column vector steps, followed by the matrix Rearrange to get the state transfer matrix A of one grinding N×N .

[0025] Optional, N-row and N-column matrix sequence B 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, 0 (N-i)×(N-i) represents a zero matrix with (Ni) rows and columns;

[0028] N(N+1) rows, N 2 Matrix of columns

[0029]

[0030] NM line, N 2 Matrix of columns

[0031]

[0032] N 2 dimensional column vector E N(N+1)×1 , the first N elements are all 1, and the remaining elements are all 0:

[0033]

[0034] N-dimensional column vector F MN×1 , FMN×1 Y M×N The elements in each column are rearranged:

[0035]

[0036] Optionally, calculate N 2 Dimensional column vector By solving the following constrained least squares problem

[0037] get:

[0038]

[0039] The first min{*} represents the minimum objective of the least squares method; the one after st is the constraint condition.

[0040] Optionally, the state transition matrix A of a grinding in step (5) is N×N , is the matrix Rearrange

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

[0042]

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

[0044] or

[0045]

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

[0047] The state transfer matrix of a single grinding of a material powder entering the grinding equipment is calculated by calculating the particle size distribution results of a certain material powder before and after M times of grinding. After the state transfer matrix is ​​obtained, the particle size distribution of a new batch of the same material powder after grinding different times can be accurately predicted. Powder processing companies do not need to measure the particle size after each grinding, which speeds up production efficiency. When powder processing companies find that the initial particle size distribution of the incoming raw materials has changed, 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 accompanying drawings and embodiments:

[0049] Figure 1Schematic diagram of powder mass transfer within a divided particle size range of a method for estimating the particle size distribution of powder after grinding based on a Markov chain according to an embodiment of the present invention (i-th interval range (x i-1 ,x i ]Internal particle mass m(x i-1 ,x i ] part is transferred to the jth interval (x j-1 ,x j ], the mass of the transferred part is A(i,j) times the mass of the original interval). DETAILED DESCRIPTION

[0050] In order to make the purpose, technical scheme and advantages of the present invention clearer, the present invention is further described in detail below in conjunction with specific embodiments and with reference to the accompanying drawings. It should be understood that these descriptions are only exemplary and are not intended to limit the scope of the present invention. In addition, in the following description, the description of well-known structures and technologies is omitted to avoid unnecessary confusion of the concept of the present invention.

[0051] It should be noted that when the powder is ground, particles of different sizes are subjected to different grinding effects, which is reflected in the different conversion mass and the final particle size. The loss of the grinder each time 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 particle size range to another is constant. That is, if we divide the particle size into multiple intervals in advance, the transfer of powder mass within the interval is fixed, as shown in the attached manual. Figure 1 Assume that the powder particle size distribution is divided into N intervals, and the range of the i-th interval (x i-1 ,x i ]Internal particle mass m(x i-1 ,x i ] part is transferred to the jth interval (x j-1 ,x j ], 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) constitute the state transfer matrix A N×N Therefore, the estimation of the particle size distribution of the powder after grinding in the embodiment of the present invention requires first finding the state transfer matrix A N×N .

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

[0053] (1) According to the particle size measurement range of the powder particle size measuring instrument used, the particle size interval of the powder to be estimated is divided; specifically, let the total number of intervals be N, a natural number N ≥ 1, and the range of the kth sub-interval (x k-1 ,x k], x 0 =0,x N is the upper limit of particle size, and the interval is divided as follows:

[0054] {(0,x 1 ],(x 1 ,x 2 ],(x 2 ,x 3 ],…,(x N-1 ,x N )}.

[0055] (2) The particle size distribution result given by the particle size measuring instrument is converted into the mass percentage of the particles in each sub-interval in step (1), and the mass percentage of the particles in all sub-intervals to the total powder mass constitutes a row vector. Specifically, the particle size distribution referred to in the present invention refers to the percentage of the mass of the particles in the sub-interval to the total mass. The percentage of the mass of the particles in N sub-intervals to the total mass constitutes a 1*N row vector X 1×N , that is, the row vector X 1×N The i-th element in row 1 is:

[0056]

[0057] In formula (1), Indicates the particle size in (x i-1 ,x i ] is the mass of particles in the interval, i≤N.

[0058] It should be noted that if the sedimentation method is used to measure the powder particle size distribution, the measurement result is the mass percentage, and the row vector X 1×N 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 percentage. The directly measured particle number percentage needs to be converted into the mass percentage according to the following formula: 1×N :

[0060]

[0061] In formula (2), Indicates the particle size in (x i-1 ,x i ]The number of particles in the interval, or the proportion of particles, i≤N.

[0062] (3) Grind the powder, and according to the definition in step (2), measure the mass proportion row vector of the 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 recorded as The particle size distribution result after the Mth grinding is recorded as Define the particle size distribution matrix X before grinding M×N , after grinding M times, the particle size distribution matrix Y M×N :

[0063]

[0064] in is the matrix Y M×N The jth column of <j<N。

[0065] (4) The state transfer matrix after one grinding is fitted by solving the least squares problem with constraints; specifically, the intermediate matrix B is first constructed. i,N×N , E N(N+1)×1 , F MN×1 To calculate N 2 Dimensional column vector steps, followed by the matrix Rearrange to get the state transfer matrix A of one grinding N×N More specifically,

[0066] The matrix sequence B with N rows and N columns i,N×N ,

[0067]

[0068] Where i is equal to 1 to N, I (i-1)×(i-1) represents the identity matrix with (i-1) rows and columns, 0 (N-i)×(N-i) represents a zero matrix with (Ni) rows and columns;

[0069] N(N+1) rows, N 2 Matrix of columns

[0070]

[0071] NM line, N 2 Matrix of columns

[0072]

[0073] N 2 dimensional column vector E N(N+1)×1 , the first N elements are all 1, and the remaining elements are all 0:

[0074]

[0075] N-dimensional column vector F MN×1 , F MN×1 Y M×N The elements in each column are rearranged:

[0076]

[0077] Calculate N 2 Dimensional column vector By solving the following constrained least squares problem:

[0078]

[0079] The first min{*} represents the minimum objective of the least squares method; the one after st is the constraint condition.

[0080] (5) The state transfer matrix obtained in step (4) is used to right-multiply the initial particle size distribution matrix by the number of grinding times or to right-multiply the particle size distribution matrix of the current grinding by the difference number of grinding times to obtain the particle size distribution matrix of the powder after multiple grindings.

[0081] Specifically, the state transition matrix A of a grinding N×N , is the matrix Rearrange to get the matrix A N×N The element in row i and column j is:

[0082]

[0083] Let the initial particle size distribution measurement be the row vector Or the particle size distribution of the powder after grinding M times is measured as a row vector The particle size distribution of the powder after grinding R times is Estimated value:

[0084] or

[0085]

[0086] For example, assume that the result of the initial particle size distribution measurement on N intervals is the row vector The state transfer matrix of the material on a grinding machine is A N×N (already obtained in advance). The method of the present invention gives the particle size distribution after grinding once more: Estimated value:

[0087]

[0088] Particle size distribution after regrinding R times Estimated value:

[0089]

[0090] The following is a detailed explanation of the estimation of the grinding particle size distribution of powder (taking feldspar powder as an example. Feldspar powder is an important powder in ceramic production and needs to be dispersed in water and ground multiple times in a sand mill before use) by measuring the particle size distribution by sedimentation method.

[0091] Example 1

[0092] In the embodiment of the present invention, the interval division is taken as N=10. Specifically,

[0093] (1) According to the particle size measurement range of the particle size distribution measuring device, N = 10 particle size intervals are divided.

[0094] {(0,x 1 ],(x 1 ,x 2 ],(x 2 ,x 3 ],…,(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 um.

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

[0098]

[0099] The first element 7.2 / 100 represents the mass ratio of feldspar powder in the interval (0,1] measured by the particle size analyzer before the first grinding, and so on.

[0100] (3) For example, the number of grinding times M = 4, the particle size distribution matrix X before 4 grindings M×N for:

[0101]

[0102] Particle size distribution matrix Y after 4 grindings M×N for:

[0103]

[0104] (4) Calculate the state transition matrix of a grinding by constructing the intermediate matrix described above. It should be noted that in this step, the particle size distribution matrix X before several grindings is known. M×N and the particle size distribution matrix Y after grindingM×N The state transfer matrix is ​​obtained by fitting, which mathematically solves the following constrained minimum fitting problem:

[0105]

[0106] Among them A N×N is the transfer matrix to be fitted. There is no general method to solve the above problem in mathematics. The present invention creatively transforms the above minimum fitting problem into a least squares fitting vector problem. The above minimum fitting problem is equivalent to the least squares problem:

[0107]

[0108] The additional matrices C, D, E, and F introduced here (i.e., the intermediate matrix B mentioned above) i,N×N , E N(N+1)×1 , F MN×1 ). The solved vector G (i.e. the above ) can derive the state transfer matrix A (that is, the above A N×N ). It is necessary to solve the least squares problem with constraints. Common numerical computing software has this computing package. For example, in the embodiment of the present invention, the lsqlin(*) function in the commercial software MATLAB is used to finally solve the state transfer matrix A N×N for:

[0109]

[0110] The physical meaning of A(2,1)=0.238 is that the mass of the powder in the second interval (1,1.5]um is transferred to the powder in the first interval (0,1]um at a ratio of 0.238. That is, the state transfer matrix A N×N The elements in describe the transformation of the powder grinding into fine powder. In the embodiment of the present invention, the state transfer matrix A N×N It is a lower triangular matrix, and its physical meaning is that the powder will only become finer and will not agglomerate after grinding; at the same time, the state transfer matrix A N×N The sum of all elements in any row is 1, which means that mass is conserved during grinding. The introduction of the intermediate matrix and the solution of the constrained least squares problem in step (4) of the present invention is to make the state transfer matrix A N×N Possess the above two properties.

[0111] (5) The state transfer matrix is ​​calculated in step (4), and the initial particle size distribution matrix is ​​multiplied by the state transfer matrix after the number of grindings or the particle size distribution matrix of the current grinding is multiplied by the state transfer matrix after the number of grindings to obtain the particle size distribution matrix of the powder after multiple grindings, so as to accurately predict the subsequent particle size distribution of the material after grinding on the grinding machine.

[0112] To verify the accuracy of the estimation method of the embodiment 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 is measured using the row vector, and the matrix state transfer matrix A is multiplied five times on the right. N×N , and get the fifth estimated distribution:

[0115]

[0116] In order to verify the error between the estimation method and the conventional method to verify the accuracy and reliability of the estimation method of the embodiment of the present invention, the particle size distribution after the fifth grinding was measured by the conventional method in the embodiment of the present invention as follows:

[0117]

[0118] It can be seen that the estimation method of the embodiment of the present invention is slightly different from the actual measurement results, and the absolute deviation of the mass fraction is within 0.5%, thereby verifying that the estimation method of the embodiment of the present invention is accurate and reliable.

[0119] One of the innovations of the present invention is to use the Markov chain model to model powder grinding. The second innovation is to provide the calculation steps of the state transfer matrix. The state transfer matrix A N×N It has special properties: first, it is a lower triangular matrix, and the 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 transfer matrix A N×N The sum of the rows is 1, which means that the mass is conserved during grinding. The intermediate matrix and the least squares problem with constraints are introduced to make the state transfer matrix A N×N Possess the above two properties.

[0120] It should be understood that the above specific embodiments of the present invention are only used to illustrate or explain the principles of the present invention, and do not constitute a limitation of the present invention. Therefore, any modifications, equivalent substitutions, improvements, etc. made without departing from the spirit and scope of the present invention should be included in 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 the equivalent forms of such scope and boundaries.

Claims

1. A method for estimating the particle size distribution of powder after grinding based on Markov chain, characterized in that: The following steps are involved: (1) According to the particle size measurement range of the powder particle size measuring instrument used, the particle size range of the powder to be estimated is divided; (2) converting the particle size distribution result given by the particle size measuring instrument into the mass proportion of the particles in each sub-interval in step (1), and the proportion of the mass of the particles in all sub-intervals to the total mass of the powder constitutes a row vector; (3) grinding the powder, and according to the definition in step (2), measuring the mass proportion row vector of the sub-interval after each grinding, and obtaining the particle size distribution matrix before and after grinding; (4) fitting the state transfer matrix after one grinding by solving the least squares problem with constraints; (5) The state transfer matrix obtained in step (4) is used to right-multiply the initial particle size distribution matrix by the number of grinding times or to right-multiply the particle size distribution matrix of the current grinding by the difference number of grinding times to obtain the particle size distribution matrix of the powder after multiple grindings.

2. The method for estimating the particle size distribution of powder after grinding based on Markov chain according to claim 1, characterized in that: In step (1), let the total number of intervals be N, a natural number ≥ 1, and the range of the kth sub-interval (x k-1 ,x k ],x0=0,x N is the upper limit of particle size, and the interval is divided as follows: {(0,x1],(x1,x2],(x2,x3],…,(x N-1 ,x N )}。 3. The method for estimating the particle size distribution of powder after grinding based on Markov chain according to claim 2, characterized in that: In step (2), the mass of particles in the N subintervals accounts for the proportion of the total mass, forming a 1*N row vector X 1×N , the row vector X 1×N The i-th element in row 1 is: In formula (1), Indicates the particle size in (x i-1 ,x i ] is the mass of particles in the interval, i≤N.

4. The method for estimating the particle size distribution of powder after grinding based on Markov chain according to claim 3, characterized in that: If the sedimentation method is used to measure the powder particle size distribution, the measurement result is the mass percentage, and the row vector X 1×N Given by the measuring instrument.

5. The method for estimating the particle size distribution of powder after grinding based on Markov chain according to claim 3, characterized in that: If a laser particle size analyzer is used to measure the particle size distribution, the measurement result is the particle number percentage. The directly measured particle number percentage needs to be converted into the mass percentage according to the following formula: 1×N : In formula (2), Indicates the particle size in (x i-1 ,x i ]The number of particles in the interval, or the proportion of particles, i≤N.

6. The method for estimating the particle size distribution of powder after grinding based on Markov chain according to claim 5, characterized in that: In step (3), the particle size distribution result measured before the first grinding is recorded as The particle size distribution result after the Mth grinding is recorded as Define the particle size distribution matrix X before grinding M×N , after grinding M times, the particle size distribution matrix Y M×N : in is the matrix Y M×N The jth column of <j<N。 7. The method for estimating the particle size distribution of powder after grinding based on Markov chain according to claim 6, characterized in that: Step (4) includes first constructing the intermediate matrix B i,N×N , E N(N+1)×1 , F MN×1 To calculate N 2 Dimensional column vector steps, followed by the matrix Rearrange to get the state transfer matrix A of one grinding N×N .

8. The method for estimating the particle size distribution of powder after grinding based on Markov chain according to claim 7, characterized in that: The matrix sequence B with N rows and N columns i,N×N , Where i is equal to 1 to N, I (i-1)×(i-1) represents the identity matrix with (i-1) rows and columns, 0 (N-i)×(N-i) represents a zero matrix with (Ni) rows and columns; N(N+1) rows, N 2 Matrix of columns NM line, N 2 Matrix of columns N 2 dimensional column vector E N(N+1)×1 , the first N elements are all 1, and the remaining elements are all 0: N-dimensional column vector F MN×1 , F MN×1 Y M×N The elements in each column are rearranged:

9. The method for estimating the particle size distribution of powder after grinding based on Markov chain according to claim 8, characterized in that: Calculate N 2 Dimensional column vector By solving the following constrained least squares problem: The first min{*} represents the minimum objective of the least squares method; the one after st is the constraint condition.

10. The method for estimating the particle size distribution of powder after grinding based on Markov chain according to claim 9, characterized in that: The state transition matrix A of one grinding in step (5) N×N , is the matrix Rearrange to get the matrix A N×N The element in row i and column j is: Let the initial particle size distribution measurement be the row vector Or the particle size distribution of the powder after grinding M times is measured as a row vector The particle size distribution of the powder after grinding R times is Estimated value: or

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