Clinker-free ecological cement mix proportion design method based on particle closest packing theory

Abstract

The invention discloses a clinker-free ecological cement mix proportion design method based on a particle closest packing theory. Through particle size distribution and specific surface area testing, fractal features of powder are extracted based on the fractal theory, and the fractal features are mapped to a Dinger-Funk model to construct theoretical optimal particle size cumulative distribution. Performing weighted stacking on the actual particle size distribution of each raw material according to the mass ratio to form actual cumulative distribution of a mixed system, and further establishing a deviation function between the actual cumulative distribution and the actual cumulative distribution; an optimization model with constraints is established by introducing a Lagrangian multiplier, and the optimal mass ratio of raw materials is automatically obtained by using a numerical method, so that the grain composition of a mixed system is closer to the closest packing state. The clinker-free ecological cement test piece prepared according to an optimization result has relatively good stacking characteristics and mechanical properties. According to the method, parameterization, quantifiability and automation of clinker-free cement proportion design are achieved, the method can adapt to different batches of industrial solid waste raw materials, and the good engineering application value is achieved.

Description

Technical Field

[0001] This invention relates to the field of building materials and solid waste resource utilization technology, and more specifically, to a clinker-free ecological cement mix design method based on the theory of closest particle packing. Background Technology

[0002] Clinker-free eco-cement uses industrial solid waste as its main raw material, eliminating the need for high-temperature calcination and significantly reducing carbon emissions and energy consumption. It represents a crucial direction for the greening and low-carbon transformation of building materials. Finely ground CFB slag and CFB fly ash, typical solid wastes generated by circulating fluidized bed coal-fired power units, are characterized by large production volumes, stable composition, and exploitable activity potential. Mineral powder, silica fume, and desulfurized gypsum, byproducts of the steel industry and flue gas desulfurization units, also offer advantages in terms of large-scale production and stable sources. The synergistic use of these five types of industrial solid waste in the preparation of clinker-free eco-cement not only reduces the use of traditional silicate clinker and alleviates carbon emission pressure, but also effectively solves environmental problems such as land occupation from industrial solid waste storage, secondary dust generation, and leachate pollution, resulting in significant comprehensive resource, environmental, and economic benefits.

[0003] However, the mix design of clinker-free eco-cement currently remains largely based on empirical trial mixing, single-factor testing, or simple orthogonal experiments, lacking scientific and systematic powder gradation analysis and mathematical modeling support. Due to the significant differences in particle size and morphology among finely ground CFB slag, CFB fly ash, mineral powder, silica fume, and desulfurized gypsum, with some raw materials exhibiting multi-peak particle size distributions and others displaying irregular particle shapes, arbitrary mixing often results in loose particle packing structures, high porosity, and insufficient void filling. The numerous interconnected pores within the powder system not only affect density but also lead to insufficient structural strength after cement hardening, excessive water consumption, and increased drying shrinkage, limiting the promotion and use of clinker-free eco-cement in structural engineering. Furthermore, the significant differences in properties between different sources and batches of solid waste make it difficult to establish repeatable and transferable mix design systems using traditional empirical methods, resulting in large fluctuations and insufficient stability in product performance, further impacting the reliability of engineering applications.

[0004] The theory of closest packing of particles is a fundamental theory in the field of powder material design. It can minimize the voids between particles by rationally controlling the proportion of particles of different sizes, thereby increasing the packing density and reducing the porosity of the material. The Dinger-Funk equation, as a classic mathematical model of the closest packing state in a continuous size gradation system, has been applied in material systems such as ceramic powders, dry-mixed mortars, and ultra-high performance concrete, and can be used to guide the optimal construction of particle distribution. However, the application of this theory in clinker-free ecological cement systems is still limited. On the one hand, due to the complex morphology and large particle size range of solid waste particles, traditional gradation models cannot directly reflect their true characteristics; on the other hand, existing studies often use a single index (such as average particle size) for coarse control, which cannot establish a complete powder gradation-performance relationship.

[0005] Meanwhile, the mix design process involves multiple raw materials, multiple constraints, and multi-objective optimization. Relying solely on manual experiments or univariate analysis makes it difficult to obtain globally optimal raw material ratios. The Lagrange parameter method is an effective mathematical tool for solving constrained optimization problems, achieving deviation minimization while ensuring the sum of raw material ratios equals 1; combined with numerical computing platforms, optimization efficiency can be significantly improved. However, in published literature and existing engineering practice, there are no known technical solutions that integrate the theory of closest particle packing, the Dinger-Funk model, and the Lagrange parameter method into a unified mathematical system and apply it to the mix design of clinker-free eco-cement. The lack of scientifically calculable methods means that the gradation optimization of this type of material remains at the level of empirical judgment, failing to meet the requirements of modern clinker-free cement for performance stability, batch consistency, and design accuracy.

[0006] Therefore, there is an urgent need for a clinker-free ecological cement mix design method based on the theory of closest particle packing to solve the above problems. Summary of the Invention

[0007] The purpose of this invention is to solve the technical problems mentioned in the background section and to provide a clinker-free ecological cement mix design method based on the theory of closest particle packing, comprising the following steps: S1: Laser particle size analyzer was used to test the particle size distribution and specific surface area of ​​the finely ground CFB slag, CFB fly ash, mineral powder, silica fume, and desulfurization gypsum, respectively, and the minimum particle size of the mixed system was determined based on the particle size distribution. With maximum particle size ; S2: Based on fractal theory, a log(V)-log(S) scatter plot is constructed by the particle size and volume of the raw material and the specific surface area. The slope is obtained by linear fitting to calculate the fractal dimension of the particles, and the particle size distribution coefficient q of the Dinger-Funk equation is obtained accordingly to form the theoretical optimal particle size cumulative distribution. S3: The cumulative volume fraction of each raw material at different particle sizes is weighted and superimposed according to the mass ratio of the raw materials to form the actual cumulative volume distribution of the mixed system, and the deviation function between the theoretical optimal distribution and the actual distribution is constructed. S4: Under the condition that the total proportion of raw material mass is 1, a constrained optimization model is established by introducing Lagrange parameters, and the particle size distribution data, minimum particle size, maximum particle size and distribution coefficient q of each raw material are imported into MATLAB. The proportion of raw material mass with the smallest deviation is obtained by numerical solution. S5: Prepare clinker-free ecological cement specimens according to the obtained raw material ratio, and test their bulk density and 28-day compressive strength. When the bulk density is not less than 0.75 and the 28-day compressive strength is not less than 42.5 MPa, it is judged to meet the standard. Otherwise, return to step S3 to adjust the parameters and re-optimize.

[0008] As a preferred technical solution of the present invention, the laser particle size analyzer has a testing range of 0.1 to 500 μm, the particle size distribution is recorded as the particle size interval Di and the corresponding cumulative volume fraction, the specific surface area is tested by the Blaine permeability method, and the testing accuracy is ±0.1 m² / kg.

[0009] As a preferred technical solution of the present invention, the theoretical optimal particle size cumulative distribution described in S2 is calculated using the Dinger-Funk equation, the expression of which is: in Theoretically optimal cumulative volume fraction For particle size, Minimum particle size, denoted as the maximum particle size, and q as the particle size distribution coefficient.

[0010] As a preferred technical solution of the present invention, the deviation function described in S3 is constructed using the least squares method, and its expression is as follows: Where E is the distribution deviation value. Number of particle size ranges Let j be the mass percentage of the j-th raw material. For the j-th raw material in terms of particle size The cumulative volume fraction at that location.

[0011] As a preferred technical solution of the present invention, the constrained optimization model described in S4 constructs its objective function by introducing Lagrange parameters, as follows: The constraint condition is 0 < <1 and .

[0012] As a preferred technical solution of the present invention, the MATLAB numerical solution described in S4 includes: calculating the cumulative particle size distribution of each raw material. Particle size Minimum particle size Maximum particle size Import the distribution coefficient q into the MATLAB workspace; define the Lagrangian objective function based on the deviation function and constraints; solve the objective function for each pair of conditions. , , , , And by taking the partial derivative of the input and setting it equal to 0, a system of multivariate equations is constructed; the optimal raw material mass ratio is obtained by using the Newton-Raphson iterative method.

[0013] As a preferred embodiment of the present invention, the convergence accuracy of the Newton-Raphson method is set to... and with Used as the initial iteration value.

[0014] As a preferred embodiment of the present invention, the actual cumulative volume distribution of the mixed system is obtained by the following formula: U mix (D i )= Where Umix(Di) represents the particle size of the mixing system. The cumulative volume fraction at that location.

[0015] As a preferred technical solution of the present invention, the packing density test adopts the drainage method, and the mechanical property test is carried out in accordance with the GB / T17671-2021 standard.

[0016] As a preferred technical solution of the present invention, when the specimen packing density is less than 0.75 or the 28-day compressive strength is less than 42.5 MPa, the optimization solution is re-solved by adjusting the distribution coefficient q, the initial value of the iteration or the iteration step size, so as to obtain the optimal raw material mass ratio that meets the performance requirements.

[0017] Compared with the prior art, the present invention has the following beneficial effects: This invention introduces fractal dimension theory into the characterization of powder particles, establishing a mathematical correlation between raw material particle size distribution, specific surface area, and the optimal Dinger-Funk gradation. This transforms the mix design of clinker-free eco-cement from traditional empirical methods to a parameterized and quantifiable theoretical system. The fractal dimension of the powder is obtained through log(V)-log(S) fitting, and the theoretically optimal particle size distribution curve is determined accordingly. This allows for a precise description of the densest packing state of the particle gradation, effectively solving problems such as the inability to quantify particle gradation and coarse particle size control in traditional methods.

[0018] This invention constructs a deviation function between the theoretical and actual distributions based on the principle of deviation minimization, and establishes a system of multivariate nonlinear equations under mass ratio constraints using the Lagrange multiplier method. The optimal raw material mass ratio is automatically calculated using MATLAB numerical solutions, avoiding the drawbacks of extensive repeated experiments and subjective judgments in traditional batching. The solution process is stable and repeatable, and exhibits high adaptability to different batches and particle size characteristics of industrial solid waste, significantly improving the design efficiency and accuracy of clinker-free cement systems.

[0019] The powder system formed by the optimized mix proportion obtained by this invention can more closely approximate the densest packing state in structure, thereby achieving more uniform particle filling and a denser material skeleton structure. This not only makes the internal pore structure of the clinker-free cement system more rational but also improves the overall performance of the material, while significantly increasing the utilization efficiency of solid waste resources. The method has a clear process, is easy to deploy industrially, and is suitable for embedding into existing production lines, enabling the standardized, intelligent, and green application of clinker-free cement systems. Attached Figure Description

[0020] Figure 1 The flowchart of the method of the present invention includes five steps in sequence: raw material characterization, parameter calculation, model construction, numerical solution, and verification and correction.

[0021] Figure 2 This is a comparison chart of the theoretical distribution curve of the Dinger-Funk system and the actual distribution curve of the mixed system. The solid line represents the theoretical optimal distribution, and the dashed line represents the optimized actual distribution. Detailed Implementation

[0022] To make the objectives, technical solutions, and advantages of this invention clearer, the following description is provided in conjunction with embodiments and appendices. Figures 1-2 The present invention will be further described in detail below. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.

[0023] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments, but the present invention is not limited to these embodiments. Equivalent modifications made by those skilled in the art without departing from the principles of the present invention should fall within the protection scope of the present invention.

[0024] like Figures 1-2As shown, this invention provides a clinker-free ecological cement mix design method based on the theory of closest packing of particles. Its core idea is to utilize the mathematical relationship between the fractal characteristics of particles and the optimal particle size distribution of the Dinger-Funk theory. By constructing a deviation function and using Lagrange constraints for optimization, the five powders of finely ground CFB slag, CFB fly ash, mineral powder, silica fume and desulfurized gypsum are mixed to be as close as possible to the closest packing state, thereby obtaining high packing density and high mechanical properties without using clinker.

[0025] To facilitate understanding by those skilled in the art, the entire process of raw material testing, data processing, formula construction, optimization solution, and specimen verification of the present invention is described in detail below.

[0026] First, the particle size and specific surface area of ​​the finely ground CFB slag, CFB fly ash, mineral powder, silica fume, and desulfurization gypsum need to be tested. The preferred testing range is [insert range here]. The particle size distribution curves of five raw materials were obtained using a μm laser particle size analyzer, and the test results were presented as several characteristic particle sizes. The corresponding cumulative volume fraction is recorded, where The i-th particle size point (in μm) is represented by the cumulative volume fraction, which indicates whether the particle size is less than or equal to the specified value. The cumulative volume fraction (%).

[0027] Simultaneously, the specific surface area of ​​each raw material was measured using the Blaine permeability method, with an optimal accuracy of ±0.1. kg. Based on the particle size range of the five raw materials, determine the minimum particle size of the mixing system. With maximum particle size For example, the minimum particle size among the five raw materials can be used as... The maximum value among the maximum particle sizes can be used as... .

[0028] After obtaining the particle size, volume, and specific surface area, it is constructed as A scatter plot of V-log(S) is used, where V represents the particle volume (either characteristic volume or equivalent volume) and S represents the corresponding particle surface area. Using as the ordinate, Using the x-axis as the horizontal axis, a series of scatter points with a clear linear trend can be obtained. By using MATLAB to perform linear fitting on the scatter plot, the slope K can be obtained. According to fractal theory, the fractal dimension P of the particles can be calculated according to the following relationship: Where P is the fractal dimension of the particle (dimensionless), and K is... V)- S) The slope obtained by fitting (dimensionless).

[0029] Furthermore, the fractal dimension P is converted into the particle size distribution coefficient q in the Dinger-Funk equation using an empirical relation: Where q is the particle size distribution coefficient (dimensionless), representing the curvature characteristic of the theoretically optimal gradation curve, and the constant 1.315 comes from the empirical fitting of the powder's densest packing model. After obtaining... After q, the theoretically optimal cumulative particle size distribution can be calculated using the Dinger-Funk equation. in The theoretically optimal cumulative volume fraction (%) For a certain particle size (μm), and Let be the minimum and maximum particle sizes of the system, respectively, and q be the theoretical particle size distribution coefficient.

[0030] in The theoretically optimal cumulative volume fraction (%) For a certain particle size (μm), and Let be the minimum and maximum particle sizes of the system, respectively, and q be the theoretical particle size distribution coefficient.

[0031] To describe the true particle size distribution of the mixed system, the cumulative volume distribution of each of the five raw materials is needed. Let the j-th raw material (j=1 is ground CFB slag, j=2 is CFB fly ash, j=3 is mineral powder, j=4 is silica fume, j=5 is desulfurized gypsum) have a particle size distribution of... The cumulative volume fraction at that point is Its mass percentage can be directly measured using a particle size analyzer. Let the mass percentage of this raw material be denoted as . ,satisfy Then the actual cumulative volume distribution of the mixed system for: in For the mixed system in terms of particle size The cumulative volume fraction (%) at the specified location. To quantify the difference between the actual distribution and the theoretical optimal distribution, this invention employs the least squares method to construct the deviation function E: Where E is the distribution bias (dimensionless), n is the number of samples of the characteristic particle size, and the term in parentheses represents the difference between the theoretical distribution and the actual distribution in particle size. The error. Because the mass ratio of the five raw materials must meet the following: This invention introduces the above constraints into the objective function, using Lagrange multipliers. Construct the following Lagrange objective function: Where L is the Lagrange objective function. It is a Lagrange multiplier (dimensionless).

[0032] To obtain the optimal raw material mass ratio, it is necessary to calculate L for each of the five mass ratios. Taking the partial derivative with respect to λ and setting it to zero, we obtain the following system of multivariate nonlinear equations: The meanings of each symbol are as follows: The j-th raw material has a particle size Cumulative volume fraction at the location; Theoretical optimal cumulative volume fraction; The mass percentage of the five raw materials; Lagrange multipliers are used to satisfy the constraints; all parameters are dimensionless or expressed as cumulative volume fraction percentages. The above six-variable nonlinear equation system needs to be solved numerically. This invention preferably uses MATLAB's fsolve or a self-written Newton-Raphson iterative method for solving. During the solution process, the particle size points are... Theoretical distribution Actual distribution of each raw material , , Import the workspace with q, and set the initial iteration value to: The iterative convergence condition is set as follows: When the changes in variables obtained from two consecutive iterations are both less than the convergence threshold, the optimal raw material mass ratio can be output. .

[0033] After obtaining the optimal mix proportions, CFB slag, CFB fly ash, mineral powder, silica fume, and desulfurization gypsum were weighed and finely ground according to their respective mass percentages, thoroughly mixed, and clinker-free eco-cement specimens were prepared. The preparation and molding of the mixture were preferably carried out in accordance with GB / T17671-2021 standard, and the bulk density was determined by the drainage method. And test the 28-day compressive strength. When the following conditions are met: If the mix design is satisfactory, the batch can be considered to have met the standards; if not, the q-value should be readjusted or the initial mix design modified based on the performance deviation. Then, the solution is optimized again until the requirements are met.

[0034] To further verify the applicability of the method of the present invention, two different batches of industrial solid waste raw materials were selected for verification.

[0035] In the first batch of raw materials, the specific surface area of ​​finely ground CFB slag is approximately 611 m². 2 / kg, approximately 550m³ of CFB fly ash 2 / kg, mineral powder approximately 420㎡ / kg, silica fume 19000㎡ / kg, desulfurized gypsum approximately 310m³ 2 / kg, with particle sizes ranging from 0.2-250μm, 0.2-290μm, and 0.3-350μm, respectively. μm and 5-100 μm. Determined after particle size analysis. μm, μm. The slope was obtained by fitting log(V)-log(S). The fractal dimension is thus calculated. Then find q By solving the equations using MATLAB, the mass proportions of the raw materials were found to be 25% finely ground CFB slag, 20% CFB fly ash, 40% mineral powder, 5% silica fume, and 10% desulfurized gypsum. Specimens were prepared and tested, and the packing density was approximately 0.84 and the 28-day compressive strength was approximately 47.5 MPa, both of which are far higher than the minimum requirements.

[0036] The second set of raw materials came from another production line and had a wider particle size range, which was calculated to be... μm, μm,log(V)-log(S) fitting slope fractal dimension ,q The optimal mix ratio was determined using the same steps: 22% finely ground CFB slag, 26% CFB fly ash, 38% mineral powder, 8% silica fume, and 6% desulfurized gypsum. Test results showed a bulk density of approximately 0.83 and a 28-day compressive strength of approximately 44.8 MPa, both meeting the requirements.

[0037] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A clinker-free eco-cement mix design method based on the theory of closest particle packing. Includes the following steps: S1: Laser particle size analyzer was used to test the particle size distribution and specific surface area of ​​the finely ground CFB slag, CFB fly ash, mineral powder, silica fume, and desulfurization gypsum, respectively, and the minimum particle size of the mixed system was determined based on the particle size distribution. With maximum particle size ; S2: Based on fractal theory, a log(V)-log(S) scatter plot is constructed by the particle size and volume of the raw material and the specific surface area. The slope is obtained by linear fitting to calculate the fractal dimension of the particles, and the particle size distribution coefficient q of the Dinger-Funk equation is obtained accordingly to form the theoretical optimal particle size cumulative distribution. S3: The cumulative volume fraction of each raw material at different particle sizes is weighted and superimposed according to the mass ratio of the raw materials to form the actual cumulative volume distribution of the mixed system, and the deviation function between the theoretical optimal distribution and the actual distribution is constructed. S4: Under the condition that the total proportion of raw material mass is 1, a constrained optimization model is established by introducing Lagrange parameters, and the particle size distribution data, minimum particle size, maximum particle size and distribution coefficient q of each raw material are imported into MATLAB. The proportion of raw material mass with the smallest deviation is obtained by numerical solution. S5: Prepare clinker-free ecological cement specimens according to the obtained raw material ratio, and test their bulk density and 28-day compressive strength. When the bulk density is not less than 0.75 and the 28-day compressive strength is not less than 42.5 MPa, it is judged to meet the standard. Otherwise, return to step S3 to adjust the parameters and re-optimize.

2. The method according to claim 1, characterized in that, The laser particle size analyzer has a testing range of 0.1 to 500 μm. The particle size distribution is recorded as the particle size interval Di and the corresponding cumulative volume fraction. The specific surface area is tested using the Blaine permeability method, with a testing accuracy of ±0.1 m² / kg.

3. The method according to claim 1, characterized in that, The theoretically optimal cumulative particle size distribution described in S2 is calculated using the Dinger-Funk equation, the expression of which is: ； in Theoretically optimal cumulative volume fraction For particle size, Minimum particle size, denoted as the maximum particle size, and q as the particle size distribution coefficient.

4. The method according to claim 1, characterized in that, The deviation function described in S3 is constructed using the least squares method, and its expression is as follows: ； Where E is the distribution deviation value. Number of particle size ranges Let j be the mass percentage of the j-th raw material. For the j-th raw material in terms of particle size The cumulative volume fraction at that location.

5. The method according to claim 1, characterized in that, The constrained optimization model described in S4 constructs its objective function by introducing Lagrange parameters: ； The constraint condition is 0 < <1 and .

6. The method according to claim 1, characterized in that, The MATLAB numerical solution described in S4 includes: calculating the cumulative particle size distribution of each raw material. Particle size Minimum particle size Maximum particle size Import the distribution coefficient q into the MATLAB workspace; define the Lagrangian objective function based on the deviation function and constraints; solve the objective function for each pair of conditions. , , , , And by taking the partial derivative of the input and setting it equal to 0, a system of multivariate equations is constructed; the optimal raw material mass ratio is obtained by using the Newton-Raphson iterative method.

7. The method according to claim 6, characterized in that, The convergence accuracy of the Newton-Raphson method is set to... and with Used as the initial iteration value.

8. The method according to claim 1, characterized in that, The actual cumulative volume distribution of the hybrid system is obtained by the following formula: ； Where Umix(Di) represents the particle size of the mixing system. The cumulative volume fraction at that location.

9. The method according to claim 1, characterized in that, The bulk density test was conducted using the drainage method, and the mechanical property tests were performed in accordance with the GB / T17671-2021 standard.

10. The method according to claim 1, characterized in that, When the specimen packing density is less than 0.75 or the 28-day compressive strength is less than 42.5 MPa, the optimization solution is re-solved by adjusting the distribution coefficient q, the initial value of the iteration, or the iteration step size to obtain the optimal raw material mass ratio that meets the performance requirements.

Images