A method for predicting the permeability of porous ceramics based on fractal principle

Abstract

The present invention discloses a method for predicting the permeability of porous ceramics based on the fractal principle, comprising the following steps: using a laser confocal microscope to characterize the porous ceramic cross section, and performing height value statistics on the three-dimensional morphology of the test area to obtain the mean and variance; binarizing the three-dimensional morphology through a height threshold, and calculating the fractal dimension of the porous ceramic cross section based on the binarization result; obtaining the fractal dimension of the pore area without cross-sectional information by fitting the relationship between the fractal dimension and the height; obtaining the mapping relationship between the permeability and the pore structure parameters based on the gas conservation equation, Darcy's law, Poiseuille's law, etc., bringing in the pore structure parameters to obtain the permeability at different positions; obtaining the permeability of the porous ceramic by obtaining the mean value of the permeability at different positions. The present invention measures the permeability in a non-contact manner to avoid damage to precision components caused by pressure drop experiments, etc. The present invention can be used for porous throttling parts of air bearings.

Description

Technical Field

[0001] The invention relates to a porous ceramic permeability prediction method based on fractal principle, and belongs to the field of porous ceramic permeability measurement. Background Art

[0002] With the rapid development of precision machining, ultra-precision machining, aerospace and other fields, the requirements for bearings are constantly increasing. Traditional rolling bearings are difficult to meet the high-precision requirements between moving pairs, and air bearings have gradually become a more preferred alternative due to their excellent radial and axial rotation accuracy. In gas hydrostatic bearings, small hole throttling or slit throttling solutions are usually used.

[0003] Due to the throttling effect of the throttle, only part of the high-pressure air film inside the air film can provide support function, while the air film away from the throttle has a lower pressure and limited support function. It is also found that the stiffness and stability of the small hole throttling method are mutually restricted. In order to solve this problem, the concept of porous materials is introduced. Porous materials refer to materials with a network structure composed of interconnected, closed or semi-closed holes. By using porous materials, gas can enter the air film through thousands of tiny holes, making the pressure distribution more uniform and providing more reliable support function.

[0004] In order to design and prepare porous throttling parts with qualified current limiting performance, the existing method of measuring the permeability of porous ceramics is through permeability experiments combined with theoretical analysis and numerical calculations. Darcy's law or Darcy-Folkheimer law is generally used to describe the permeability. However, considering that the porous ceramic throttle is an ultra-precision component, it may cause damage to the processed workpiece when performing a pressure drop experiment on it. Therefore, how to provide an efficient non-contact non-destructive testing of porous ceramic throttles is a problem that technicians in this field urgently need to solve. Summary of the invention

[0005] The present invention aims to solve the problem in the prior art that when conducting a permeability test on porous ceramic precision components, a pressure drop test may cause damage to the processed workpiece and low measurement efficiency, and further proposes a porous ceramic permeability prediction method based on the fractal principle.

[0006] The technical solution adopted by the present invention to solve the above problems is:

[0007] A method for predicting the permeability of porous ceramics based on fractal principle comprises the following steps:

[0008] Step 100, characterizing the porous ceramic cross section using a laser confocal microscope to obtain the three-dimensional morphology of the porous ceramic cross section, and performing height value statistics on the three-dimensional morphology of the test area of ​​the porous ceramic cross section to obtain the mean and variance of the height;

[0009] Step 200, binarizing the three-dimensional morphology of the porous ceramic cross section by a height threshold formula, and calculating the fractal dimension of the porous ceramic cross section based on the binarization result by a box counting method;

[0010] Step 300, obtaining the pore area fractal dimension by removing the cross-sectional information through fitting the relationship between the fractal dimension and height of the porous ceramic cross section;

[0011] Step 400, based on the gas conservation equation, Darcy's law, and Poiseuille's law, a mapping relationship between the porous ceramic permeability and the pore structure parameters is obtained, and the pore structure parameters are brought into the permeability of the porous ceramic at different positions;

[0012] Step 500, obtaining the permeability of the porous ceramic by taking the average of the permeabilities of the porous ceramic at different positions.

[0013] Preferably, in step 100, the statistical formula for obtaining the mean and variance is as follows:

[0014]

[0015]

[0016] Where X i is the height, u is the mean, and σ is the standard deviation.

[0017] Preferably, in step 200, the height threshold formula for binarizing the three-dimensional morphology of the porous ceramic cross section is as follows:

[0018]

[0019] Where I(i,j) is the pixel position of the three-dimensional shape and x is the height threshold.

[0020] Preferably, in step 200, the method for calculating the fractal dimension of the porous ceramic cross section is as follows:

[0021]

[0022] Among them, N(λ b ) is the same as the length λ b The cumulative number of boxes intersecting the boxes, the fractal dimension of the pore area can be obtained by fitting the data points with double logarithmic coordinates (lnλ b ,ln N(λ b )) to determine.

[0023] Preferably, in step 300, the steps for obtaining the pore area fractal dimension are as follows:

[0024] Step 301, fitting the fractal dimensions above the mean value u of the height by using Sigmod function fitting to obtain the fractal dimension of the pore area generated by processing information in the cross section;

[0025] Step 302, fitting the fractal dimension below the mean value u of the height by using a polynomial fitting method, to obtain the fractal dimension of the pore area generated by the pore structure in the cross section;

[0026] Step 303 , determining the value of the cross-sectional pore fractal dimension according to the fitting results in step 301 and step 302 .

[0027] Preferably, in step 400, the step of obtaining the permeability of the porous ceramic at different positions is:

[0028] Step 401, improve Poiseuille's law to:

[0029]

[0030] Where b is the major semi-axis of the elliptical capillary, c is the minor semi-axis, μ is the fluid viscosity, and ΔP / L represents the pressure gradient along the length of the capillary;

[0031] Step 402, combining the gas conservation equation, Darcy's law, and Poiseuille's law, the porous ceramic permeability K is obtained as:

[0032]

[0033] Among them, D f is the fractal dimension of the pore area, D T is the zigzag fractal dimension, ε = b / c;

[0034] Step 403, according to the following formula D f Calculate D T .

[0035]

[0036] in, is the mean curvature, is the mean semi-major axis;

[0037] Step 404, determining the maximum semi-major axis and shape factor of the pores according to the u-section, and bringing them into step 403 to calculate the permeability of the porous ceramic.

[0038] Compared with the prior art, the present invention has the following beneficial effects:

[0039] 1. The prediction method of the present invention obtains the mapping relationship between the permeability of porous ceramics and pore structure parameters through the gas conservation equation, Darcy's law, and Poiseuille's law. max , shape factor ε, pore area fractal dimension D f and the zigzag fractal dimension D T ) can predict the macroscopic parameters (permeability) of porous ceramics. Therefore, it can replace the traditional permeability experiment and realize fast and efficient permeability measurement of porous ceramics.

[0040] 2. Because the prediction method of the present invention is a non-contact measurement, it avoids damage to precision components caused by pressure drop experiments, etc. It provides a fast and reliable detection method for key core components in the fields of ultra-precision machining, nano-positioning, etc.

[0041] 3. The mapping relationship between the permeability of porous ceramics and pore structure parameters can provide guidance for the raw material preparation industry. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 is a flow chart of the prediction method of the present invention;

[0043] Figure 2 It is a comparison chart of the permeability experiment and the predicted calculation results in three embodiments of the present invention. DETAILED DESCRIPTION

[0044] The porous ceramic permeability prediction method based on the fractal principle of the present invention is carried out according to the following steps.

[0045] Step 100: characterizing the porous ceramic cross section using a laser confocal microscope to obtain the three-dimensional morphology of the porous ceramic cross section, and performing height value statistics on the three-dimensional morphology of the test area of ​​the porous ceramic cross section to obtain the mean and variance of the height;

[0046] Step 200, binarizing the three-dimensional morphology of the porous ceramic cross section by a height threshold formula, and calculating the fractal dimension of the porous ceramic cross section based on the binarization result by a box counting method;

[0047] Step 300, obtaining the pore area fractal dimension by removing the cross-sectional information through fitting the relationship between the fractal dimension and height of the porous ceramic cross section;

[0048] Step 400, based on the gas conservation equation, Darcy's law, and Poiseuille's law, a mapping relationship between the porous ceramic permeability and the pore structure parameters is obtained, and the pore structure parameters are brought into the permeability of the porous ceramic at different positions;

[0049] Step 500, obtaining the permeability of the porous ceramic by taking the average of the permeabilities of the porous ceramic at different positions.

[0050] In step 100, first, a suitable laser confocal magnification is selected to collect the porous ceramic area according to the size of the porous ceramic pore-forming agent, and 5 to 20 different areas are tested according to the sample size, and the test results are recorded.

[0051] Then, the height value statistics of the three-dimensional morphology of the porous ceramic cross-section test area are performed, and the statistical formula for obtaining the mean and variance of the height is as follows:

[0052]

[0053]

[0054] Where X i is the height, u is the mean, and σ is the standard deviation.

[0055] In step 200, first, the three-dimensional morphology of the porous ceramic cross section is binarized using a height threshold formula, and the height threshold formula is as follows:

[0056]

[0057] Among them, I(i,j) is the pixel position of the three-dimensional shape, and x is the height threshold.

[0058] Then, the method of calculating the fractal dimension of the porous ceramic cross section using the box counting method for the binarized image is as follows:

[0059]

[0060] Among them, N(λ b ) is the same as the length λ b The cumulative number of boxes intersecting the boxes, the fractal dimension of the pore area can be obtained by fitting the data points with double logarithmic coordinates (lnλ b ,ln N(λ b )) to determine.

[0061] In step 300, the step of obtaining the cross-sectional pore area fractal dimension by fitting is:

[0062] Step 301, fitting the fractal dimensions above the mean value u of the height by using Sigmod function fitting to obtain the fractal dimension of the pore area generated by processing information in the cross section;

[0063] Step 302, fitting the fractal dimension below the mean value u of the height by using a polynomial fitting method, to obtain the fractal dimension of the pore area generated by the pore structure in the cross section;

[0064] Step 303 , determining the value of the cross-sectional pore fractal dimension according to the fitting results in step 301 and step 302 .

[0065] In step 400, the steps of obtaining the permeability of porous ceramics at different positions are:

[0066] Step 401, improve Poiseuille's law to:

[0067]

[0068] Where b is the major semi-axis of the elliptical capillary, c is the minor semi-axis, μ is the fluid viscosity, and ΔP / L represents the pressure gradient along the length of the capillary;

[0069] Step 402, combining the gas conservation equation, Darcy's law, and Poiseuille's law, the porous ceramic permeability K is obtained as:

[0070]

[0071] Among them, D f is the fractal dimension of the pore area, D T is the zigzag fractal dimension, ε = b / c;

[0072] Step 403, according to the following formula D f Calculate D T .

[0073]

[0074] in, is the mean curvature, is the mean semi-major axis;

[0075] Step 404, determining the maximum semi-major axis and shape factor of the pores according to the u-section, and bringing them into step 403 to calculate the permeability of the porous ceramic.

[0076] The following is a prediction of the permeability of three specific porous ceramic materials using the prediction method of the present invention. The calculated permeability values ​​obtained are compared with the permeability values ​​obtained from actual experiments to further understand the significance of the present invention. However, they are not intended to limit the present invention.

[0077] Example 1

[0078] First, a porous alumina (Al2O3) ceramic material was prepared through the following preparation process and used as sample 1.

[0079] The preparation process of sample 1 is as follows:

[0080] (1) Using 95-Al2O3 (pure alumina powder doped with 1.5wt.% Yb2O3, 1.5wt.% MgO and 2wt.% SiO2) as raw material;

[0081] (2) adding 20 wt.% polystyrene microspheres with a particle size of 50 μm as a pore former to the above raw materials;

[0082] (3) Using the solid phase sintering method, first, calcination is performed at a low temperature of 600°C for 2 hours, and the heating rate is relatively slow, only 0.3°C / min, to burn off the pore former, thereby forming a porous structure; then, sintering is performed at a high temperature of 1400°C for 2 hours to combine the ceramic particles and improve the density of the material. Thus, the porous alumina (Al2O3) ceramic material of sample 1 is prepared.

[0083] The permeability of the porous ceramic permeability prediction method based on the fractal principle of the present invention is used to calculate the permeability of the prepared sample 1 porous alumina (Al2O3) ceramic material.

[0084] Then, a permeability test was conducted on the prepared sample 1 porous alumina (Al2O3) ceramic material to obtain the experimental results (ie, the permeability results obtained through the experiment).

[0085] The final relevant data are as follows:

[0086] Table 1

[0087]

[0088] Example 2

[0089] First, a porous alumina (Al2O3) ceramic material was prepared through the following preparation process and used as sample 2.

[0090] The preparation process of sample 2 is similar to that of sample 1. The difference lies in the final sintering temperature and time. Sample 2 is sintered at a high temperature of 1450°C for 9 hours.

[0091] The permeability of the porous ceramic permeability prediction method based on the fractal principle of the present invention is used to calculate the permeability of the prepared sample 2 porous alumina (Al2O3) ceramic material.

[0092] Then, a permeability test was conducted on the prepared sample 2 porous alumina (Al2O3) ceramic material to obtain the experimental results (ie, the permeability results obtained through the experiment).

[0093] The final relevant data are as follows:

[0094] Table 2

[0095]

[0096]

[0097] Example 3

[0098] First, a porous alumina (Al2O3) ceramic material was prepared through the following preparation process and used as sample 3.

[0099] The preparation process of sample 3 is similar to that of sample 2. The difference is that different pore formers are used. Sample 3 uses 20 wt.% methyl cellulose as the pore former.

[0100] The permeability of the porous ceramic permeability prediction method based on the fractal principle of the present invention is used to calculate the permeability of the prepared sample 3 porous alumina (Al2O3) ceramic material.

[0101] Then, a permeability test was conducted on the prepared sample 3 porous alumina (Al2O3) ceramic material to obtain the experimental results (ie, the permeability results obtained through the experiment).

[0102] The final relevant data are as follows:

[0103] Table 3

[0104]

[0105]

[0106] The data in Table 1, Table 2 and Table 3 are as follows: Figure 2 shown.

[0107] like Figure 2 As shown in the figure, the calculated flow rates and experimental flow rates for samples 1 and 2 show similar trends. The maximum difference is 8% for sample 1 and 10% for sample 2. A prediction error of less than 10% is acceptable over the entire pressure range. This clearly demonstrates the effectiveness of using this method to replace permeability experiments. In sample 3, the calculated flow rate is consistent with the experimental flow rate in the range of 1 to 3 (×10 5 Pa) within the pressure range, the difference is less than 5%; but when the pressure is increased from 3 (×10 5 Pa) increased to 5 (×10 5 Pa), the phase difference gradually increases to about 17%. This may be due to the different pore structures. The error in the entire pressure difference range is still no more than 20%. This also shows the effectiveness of this method in replacing the permeability experiment.

[0108] The above is only a preferred embodiment of the present invention and does not limit the present invention in any form. Although the present invention has been disclosed as a preferred embodiment as above, it is not used to limit the present invention. Any technician familiar with this profession can make some changes or modify the technical contents disclosed above into equivalent embodiments without departing from the scope of the technical solution of the present invention. However, any simple modification, equivalent replacement and improvement made to the above embodiments without departing from the content of the technical solution of the present invention, based on the technical essence of the present invention, within the spirit and principles of the present invention, still fall within the protection scope of the technical solution of the present invention.

Claims

1. A method for predicting the permeability of porous ceramics based on fractal principle, characterized by: The prediction method comprises the following steps: Step 100, characterizing the porous ceramic cross section using a laser confocal microscope to obtain the three-dimensional morphology of the porous ceramic cross section, and performing height value statistics on the three-dimensional morphology of the test area of ​​the porous ceramic cross section to obtain the mean and variance of the height; Step 200, binarizing the three-dimensional morphology of the porous ceramic cross section by a height threshold formula, and calculating the fractal dimension of the porous ceramic cross section based on the binarization result by a box counting method; Step 300, using the relationship between the fractal dimension of the porous ceramic cross section and the height, obtain the fractal dimension of the pore area by fitting after removing the cross-sectional information; Step 400, based on the gas conservation equation, Darcy's law and Poiseuille's law, a mapping relationship between the porous ceramic permeability and the pore structure parameters is obtained, and the pore structure parameters are brought into the permeability of the porous ceramic at different positions; Step 500, obtaining the permeability of the porous ceramic by obtaining the average permeability of the porous ceramic at different positions; In step 400, the step of obtaining the permeability of the porous ceramic at different positions is: Step 401, improve Poiseuille's law to: Where b is the long semi-axis of the elliptical capillary, c is the short semi-axis, μ is the fluid viscosity, and △P / L represents the pressure gradient along the length of the capillary; Step 402, combining the gas conservation equation, Darcy's law and Poiseuille's law, the porous ceramic permeability K is obtained as: Among them, D f is the fractal dimension of the pore area, D T is the zigzag fractal dimension, ε = b / c; Step 403, calculate D according to the following formula T： in, is the mean curvature, is the mean major semi-axis; Step 404, determining the maximum semi-major axis and shape factor of the pores according to the u-section, and bringing them into step 403 to calculate the permeability at different positions of the porous ceramic.

2. The porous ceramic permeability prediction method based on fractal principle according to claim 1 is characterized in that: In step 100, the statistical formula for obtaining the mean and variance is as follows: Where X i is the height, u is the mean, and σ is the standard deviation.

3. The porous ceramic permeability prediction method based on fractal principle according to claim 2 is characterized in that: In step 200, the height threshold formula for binarizing the three-dimensional morphology of the porous ceramic cross section is as follows: Where I(i,j) is the pixel position of the three-dimensional shape and x is the height.

4. The porous ceramic permeability prediction method based on fractal principle according to claim 3 is characterized by: In step 200, the method for calculating the fractal dimension of the porous ceramic cross section is as follows: Among them, N(λ b ) is the same as the length λ b The cumulative number of boxes intersecting the boxes, the fractal dimension of the porous ceramic cross section can be obtained by fitting the data points (lnλ b ,lnN(λ b )) to determine.

5. The porous ceramic permeability prediction method based on fractal principle according to claim 4 is characterized in that: In step 300, the steps for obtaining the pore area fractal dimension are as follows: Step 301, fitting the fractal dimensions above the mean value u of the height by using Sigmod function fitting to obtain the fractal dimension of the pore area generated by processing information in the cross section; Step 302, fitting the fractal dimension below the mean value of the height by using a polynomial fitting method to obtain the fractal dimension of the pore area generated by the pore structure in the cross section; Step 303 , determining the value of the pore area fractal dimension after removing the cross-sectional information according to the fitting results in step 301 and step 302 .

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