Breakage Rate Modeling in Ball Mill Grinding of Calcined Clay and Limestone Mixtures

Abstract

Replacing clinker with mixtures of calcined clay and limestone is one of the most sustainable strategies for decarbonizing the cement industry. However, the kinetic patterns governing the grinding behavior of these materials are not yet fully understood. This study developed a kinetic model based on particle population balance to simulate this process. Experiments were conducted using a standard Bond ball mill, and the samples were characterized by X-ray diffraction. The results show that the grinding of calcined clay and its mixtures with limestone follows first-order kinetics. The proposed model simulates the process with a high degree of accuracy, with residual errors below 1.5% and a coefficient of determination exceeding 99%.

1. Introduction

Portland cement production is fundamental to the construction sector in meeting the levels of urbanization required worldwide today. Future projections suggest that global demand will increase to more than 4.1 billion tonnes in the coming years as infrastructure development continues. Despite its economic and societal importance, cement production is associated with the emission of approximately 600 kg of CO₂ per tonne of cement produced, accounting for around 8–10% of global annual CO₂ emissions [1,2,3].

To achieve the decarbonization of the cement industry, the Global Cement and Concrete Association has set a goal of reducing CO₂ emissions by up to 3.8 gigatons by 2050 [4]. In this regard, the main strategies proposed in the sector to mitigate its aggressive environmental impact consist of improving electrical and thermal efficiency [5], using alternative fuels [6], capturing and storing CO₂ [7] and partially replacing clinker with Supplementary Cementitious Materials (SCMs) [8,9,10], with the latter being one of the most attractive options.

In general, SCMs are considered materials that contribute to the physical and chemical properties of hardened concrete [11,12,13]. To date, the main advances have been achieved through various studies investigating limestone–calcined clay cement (LC3) and the mineral addition limestone–calcined clay (LC2) as a sustainable alternative [14,15,16]. Specifically, the research conducted in recent years on LC2 has focused on:

• Proposing methodologies for the identification and evaluation of new deposits [17].
• Evaluating different replacement levels for Portland cement [18,19].
• Assessing the effects of these substitutions on the final properties of cement [18,20,21,22].
• Evaluation of the principal chemical reactions [23].

To ensure adequate LC2 production, the grinding stage is crucial to the technical–economic indicators [24,25]. In general, studies analyzing the grinding of multicomponent minerals have focused most on determining and modeling the breakage rate, considering both first- and second-order kinetics, to identify key patterns [26,27,28,29,30,31,32]. Equations (1) and (2) present the integrated forms of the models used to analyze particle breakage during grinding, according to the first and second order, respectively, as a function of the rejected percentage.

$$W_{(X,t)}=W_{(X,0)}{e}^{-k_{1}t}$$

(1)

$${\displaystyle \frac{W_{(X,0)}}{W_{(X,t)}}}=1+W_{(X,0)}{k}_{2}t$$

(2)

where

W_{(X, t)} is the weight rejected at x size in time t, % mass.

W_{(X, 0)} is the weight rejected at x size in the feed, % mass.

k₁ is the breakage rate considering first order, min⁻¹.

k₂ is the breakage rate considering second order, % mass⁻¹ min⁻¹.

In recent years, various studies have been developed to improve the understanding of the kinetic behavior of grinding different mineral mixtures; of note are the advances described in relation to the behavior between the specific breakage rate function and variables such as particle size and mill diameter, or its link with powder flow models [28,33,34,35].

To date, studies specifically addressing the kinetic behavior of the grinding of calcined clay and limestone have primarily focused on predicting the co-grinding of calcined clay and clinker in a circuit, using modeling based on N perfect mixers in series [36], and on determining the kinetic regularities governing the grinding of low-quality limestone [37]. However, the kinetic modeling of calcined clay and its mixtures with limestone has not yet been fully investigated.

The objective of this research is to develop a cumulative kinetic model, based on the population balance approach, to simulate the grinding of calcined clay and its mixtures with limestone. This involves characterizing the feed materials (calcined clay and limestone) by X-ray diffraction (XRD), determining the characteristic breakage order of the individual materials and their mixtures, and proposing a mathematical model that simulates the grinding process as a function of both grinding time and limestone content.

2. Materials and Methods

2.1. Materials

The primary materials used in this study were limestone and calcined clay. The clay precursor, sourced from deposits in Moa Bay, Cuba, is rich in kaolinite. To obtain the calcined clay, the raw material was thermally treated at 850 °C for one hour in a laboratory furnace. The limestone is a fine-grained carbonate rock, specifically characterized as calcareous and organodetritic siltstone, originating from deposits located to the northeast of Moa.

2.2. Chemical and Mineralogical Characterization

X-ray fluorescence (XRF) analysis was performed to determine the chemical composition of the calcined clay and limestone using an OLYMPUS Terra 440 spectrometer (Olympus Corporation, Hachioji-Tokio, Japan). The chemical composition of both materials is presented in

Table 1. Chemical composition of calcined clay and limestone (wt., %).

Compounds	Fe2O3	MnO2	Cr2O3	TiO2	SiO2	Al2O3	CaO	K2O	Others	LOI
Calcined clay	10.973	0.125	0.204	0.62	53.50	32.10	0.19	0.28	0.048	1.96
Limestone	1.34	0.04	-	-	3.02	1.77	52.22	0.15	1.44	40.02

. The calcined clay meets the ASTM C 618 requirement for pozzolanic materials [13,38], with the sum of SiO₂, Al₂O₃, and Fe₂O₃ exceeding 70%. The limestone composition, characterized by a high CaO content, is consistent with a material rich in calcium carbonate.

The qualitative mineralogical composition of the samples was obtained by powder X-ray diffraction (PXRD). Diffractograms were acquired using an EMPYREAN diffractometer (Malvern Panalytical, Malvern, UK) equipped with a fine-focus copper tube, a Ni filter, and a PIXcel³D detector. Prior to analysis, samples were gently hand-ground in an agate mortar with isopropanol (wet grinding) and loaded into a non-oriented aluminum sample holder. Measurements were performed over a 2θ range of 4° to 80° with a step size of 0.003° and an integration time of 40 s per step. Phase identification was performed using HighScore Plus v3.0.2 software and the Crystallography Open Database (COD, 2014), applying a routine analysis based on the major chemical constituents of each sample. The main crystalline phases identified in the natural clay (Figure 1a) were kaolinite (Ref. Code: 96-900-9235), hematite (Ref. Code: 96-901-5066), goethite (Ref. Code: 96-900-3080), and quartz (Ref. Code: 96-901-1496). In the calcined clay (Figure 1b), the main crystalline phases detected were hematite (Ref. Code: 96-900-9783) and quartz (Ref. Code: 96-900-7377), consistent with effective calcination, as the kaolinite structure was rendered amorphous.

The limestone (Figure 2) was found to be composed mainly of calcite (Ref. Code: 96-901-6707), along with quartz (Ref. Code: 96-901-2601), montmorillonite (Ref. Code: 96-900-2780), and plagioclase feldspar (Ref. Code: 96-900-1632).

2.3. Equipment and Methodology

To determine the regularities of the kinetic behavior during the grinding of calcined clay and its mixtures with limestone, a ball mill 305 mm long and 305 mm in diameter was used, operating at 91% of the critical speed with a charge of 20,276 kg of steel balls with an appropriate distribution (see

Table 2. Distribution of balls used in the experiments.

Ball Size, mm	Weight by Size, kg
36–38	8.999
29–31	7.304
24–26	0.710
20–22	2.071
15–19	1.192
Total weight, kg	20.276">

). Grinding times of 0.5, 1.0, 1.5, 2.0, 3.0, 4.0, and 5.0 min were evaluated, using the particle size distribution of the unground mixtures as a reference. A constant dry mass of 700 cm³ of compressed material was fed into the mill, ensuring that the maximum particle size was less than 3.15 mm. The 24 samples of each of the evaluated mixtures (with 2 replicates and a degree of homogenization greater than 90%) were ground separately at the different times under study, and were finally analyzed by the wet sieving method to determine their granulometric distribution.

The kinetic parameters of the grinding were obtained from combining the processing of the retained percentages, characteristic of each particle size and grinding time, and the linearization of Equations (1) or (2) (depending on the identified order), according to the methodology described by Menéndez et al. [39].

The relationship between breakage rate and particle size was determined using Equation (3), according to the methodology described by Ciribeni et al. [40], in which, for each batch, different size fractions were ground separately. C and n are constants depending on the features of the mill and ore, which can be obtained by linearizing Equation (3), see Equation (4).

$$k=C{x}^n$$

(3)

$$\ln \left(k\right)=\ln \left(C\right)+n\ln \left(x\right)$$

(4)

3. Results and Discussions

3.1. Granulometric Behavior of Calcined Clay and Its Mixtures with Limestone

Figure 3 shows the particle size distribution of calcined clay samples and mixtures with limestone, ground for 0 to 5 min.

The shape of the curves indicates that fine particles predominate in all samples, regardless of grinding time. In all cases, a consistent trend is observed: as grinding time increases, the cumulative passing percentages increase across all particle size fractions. This behavior is logical and characteristic of an effective comminution process, as reported in other studies [38,40,41,42,43]. The initial data used to perform the kinetic evaluation is provided in the tables in Appendix A.

After the grinding process (see Figure 3), the highest cumulative passing values were observed to be directly associated with increasing limestone content in the samples, indicating that calcined clay is the most difficult mineral to grind. Furthermore, as the proportion of limestone in the mixtures increased, the mill’s ability to generate additional fines decreased. This behavior can be attributed to the substantial fine particle content already present in the initial limestone sample.

3.2. Identification of the Breakage Rate Order

Table 3. R-squared of first- and second-order kinetics for calcined clay samples and their mixtures with limestone.

Particle Size, μm	CC-100		CC75-L25		CC50-L50		CC25-L75
	First Order	Second Order	First Order	Second Order	First Order	Second Order	First Order	Second Order
−3150 +2380	0.9996	0.7539	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
−2380 +1700	0.9880	0.6894	0.9831	0.6636	0.9999	0.7430	0.9996	0.7276
−1700 +1000	0.9986	0.6439	0.9950	0.5761	0.9949	0.5648	0.9499	0.5284
−1000 +710	0.9295	0.7309	0.9943	0.4792	0.9971	0.5129	0.9982	0.6109
−710 +250	0.9613	0.8239	0.9766	0.7735	0.9814	0.7187	0.9952	0.5536
−250 +150	0.9902	0.8910	0.9864	0.8510	0.9784	0.8095	0.9870	0.8151
−150 +90	0.9746	0.9815	0.9772	0.9344	0.9915	0.9339	0.9871	0.8637
−90 +75	0.9607	0.9969	0.9627	0.9357	0.9902	0.9559	0.9887	0.9281
−75 +45	0.9241	0.9919	0.9554	0.9669	0.9838	0.9304	0.9873	0.9134
Average	0.9696	0.8337	0.9812	0.7978	0.9908	0.7966	0.9874	0.7712

presents the coefficients of determination (R²) obtained by simulating the grinding of calcined clay and its mixtures with limestone using first- and second-order kinetic models. A consistent finding is that, for all samples, the highest R² values correspond to the first-order kinetic model for particle sizes larger than 150 μm. For particle sizes smaller than 150 μm in the grinding of the CC-100 sample, the R² values for the second-order model are slightly higher than those for the first-order model. This behavior is reversed as the mass percentage of limestone in the samples increases, which can be attributed to the tendency of limestone to follow first-order grinding kinetics, as reported by Deniz [44] and Lee et al. [37]. Furthermore, it is important to highlight that the behavior observed after the addition of limestone confirms that the grinding of the mixtures exhibits uniform and interference-free behavior, as has been described for other mineral mixtures studied to date [45,46,47]. In general, the average R² values suggest that the first-order kinetic model most accurately simulates the grinding process of the samples under study.

3.3. Determination of the Kinetic Grinding Model Based on Cumulative

Table 4. Breakage rates specific to each particle size in the samples.

Particle Size, μm	CC-100	CC75-L25	CC50-L50	CC25-L75
	k, min−1
−3150 +2380	5.2704	6.1361	6.2858	6.8231
−2380 +1700	4.7894	5.2724	6.1089	6.6356
−1700 +1000	4.1891	4.5729	5.2805	6.0242
−1000 +710	3.0769	3.7361	4.0240	4.1584
−710 +250	0.9853	1.1331	1.2998	1.3990
−250 +150	0.5792	0.6343	0.6936	0.7066
−150 +90	0.3360	0.3600	0.3729	0.3744
−90 +75	0.2786	0.2988	0.3053	0.3069
−75 +45	0.2007	0.2090	0.2145	0.2351

presents the breakage rate values for each particle size of the analyzed samples, if material breakage follows a first-order kinetic model. The rate constant, k, ranges from 0.20 to 6.83 min⁻¹ and shows a systematic decrease with increasing calcined clay content in the mixture and with decreasing particle size. Overall, these results are consistent with expectations and confirm that calcined clay is more difficult to grind than limestone. Moreover, significantly greater energy is required to commute the finest particles, owing to the elimination of fissures commonly observed, as reported by Napier et al. Table 4 presents the breakage rate values for each particle size of the analyzed samples, if material breakage follows a first-order kinetic model. The rate constant, k, ranges from 0.20 to 6.83 min⁻¹ and shows a systematic decrease with increasing calcined clay content in the mixture and with decreasing particle size. Overall, these results are consistent with expectations and confirm that calcined clay is more difficult to grind than limestone. Moreover, significantly greater energy is required to comminute the finest particles, owing to the elimination of fissures commonly observed, as reported by Napier et al. [48] during the grinding of different materials. A study by Guo et al. [49] presents a similar behavior of breakage rates in the Magnetite/Limestone combination.

Based on the data presented in Table 4 and the application of Equation (4), the parameters C and n of the kinetic models for each sample were determined (see Figure 4 and Figure 5). In general, Figure 4 shows that Equation (3) provides a good fit to the breakage rate as a function of particle size, with high coefficients of determination (greater than 97%), consistent with values reported in kinetic studies of the grinding of various materials [38,39,50].

Figure 5 illustrates the behavior of the kinetic parameters C and n as a function of increasing limestone mass fraction. Both parameters follow a linear relationship, with coefficients of determination exceeding 98%. The parameter C decreases from 0.0064 to 0.0059, corresponding to an average reduction of 0.0001406 for each 25% increase in limestone content in the mixture. In contrast, the parameter n increases by an average of 0.0179 for each 25% increase in limestone, within a range of 0.90 to 0.96. Overall, the values of C and n obtained for the different mixtures are consistent with those reported for limestone in previous studies [40,50,51].

Using mathematical models that describe the behavior of parameters C and n as a function of the increasing percentage of limestone in the mixtures, and substituting Equation (3) into Equation (1), we obtain a kinetic model that simulates the grinding of calcined clay and its mixtures with limestone: Equation (5). In general, the validation of the obtained model demonstrates that it can simulate the grinding process of the mixtures under study with a high degree of accuracy, with an average coefficient of determination greater than 99% and an average residual error of less than 1.5%; see

Table A5. Comparison of experimental and model-calculated W(x, t) data for sample CC-100.

Particle Size, mm	Data Type	Grinding Time, min
		0.0	0.5	1.0	1.5	2.0	3.0	4.0	5.0
−3150 +2380	Exp *	9.00	0.69	0.04	0.00
	Cal **	9.00	0.25	0.01	000
−2380 +1700	Exp	18.70	1.52	0.10	0.02	0.00
	Cal	18.70	1.32	0.09	0.01	0.00
−1700 +1000	Exp	36.02	5.04	0.49	0.07	0.00
	Cal	36.02	6.98	1.35	0.26	0.05
−1000 +710	Exp	44.84	11.64	1.86	0.23	0.02	0.01	0.00
	Cal	44.84	13.46	4.04	1.21	0.36	0.03	0.00
−710 +250	Exp	65.02	41.05	24.29	13.07	6.86	2.24	0.93	0.89
	Cal	65.02	40.70	25.48	15.95	9.99	3.91	1.53	0.60
−250 +150	Exp	72.20	52.64	39.79	28.54	21.29	11.15	6.60	4.81
	Cal	72.20	53.75	40.01	29.79	22.17	12.29	6.81	3.77
−150 +90	Exp	77.61	62.18	51.60	41.93	36.31	25.84	20.72	16.38
	Cal	77.61	64.44	53.50	44.42	36.88	25.43	17.53	12.08
−90 +75	Exp	78.71	64.54	55.11	46.26	40.99	31.53	26.36	22.15
	Cal	78.71	67.23	57.42	49.04	41.89	30.56	22.29	16.26
−75 +45	Exp	81.94	69.78	61.00	54.95	49.56	42.31	37.28	33.82
	Cal	81.94	74.19	67.17	60.81	55.06	45.14	37.00	30.33

* Experimental data ** Data calculated using Equation (5).

,

Table A6. Comparison of experimental and model-calculated W(x, t) data for sample CC75-L25.

Particle Size, mm	Data Type	Grinding Time, min
		0.0	0.5	1.0	1.5	2.0	3.0	4.0	5.0
−3150 +2380	Exp	4.80	0.22	0.00
	Cal	4.80	0.08	0.00
−2380 +1700	Exp	11.94	0.56	0.04	0.01	0.00
	Cal	11.94	0.62	0.03	0.00	0.00
−1700 +1000	Exp	23.87	2.68	0.33	0.02	0.00
	Cal	23.87	3.89	0.63	0.10	0.02
−1000 +710	Exp	30.93	6.51	1.24	0.10	0.01	0.00
	Cal	30.93	8.24	2.19	0.58	0.16	0.01
−710 +250	Exp	46.46	29.29	16.54	7.59	3.16	1.15	0.40	0.27
	Cal	46.46	28.03	16.91	10.20	6.15	2.24	0.81	0.30
−250 +150	Exp	52.76	39.04	30.06	20.81	13.66	7.57	3.30	2.76
	Cal	52.76	38.48	28.07	20.47	14.93	7.94	4.23	2.25
−150 +90	Exp	57.49	47.01	40.01	32.70	24.94	20.04	11.79	11.11
	Cal	57.49	47.21	38.77	31.83	26.14	17.63	11.88	8.01
−90 +75	Exp	58.58	48.93	42.63	35.76	28.88	24.94	15.22	15.45
	Cal	58.58	49.60	41.99	35.54	30.09	21.56	15.46	11.08
−75 +45	Exp	61.18	53.37	47.69	43.16	35.97	33.56	24.49	24.13
	Cal	61.18	55.14	49.69	44.79	40.36	32.78	26.63	21.62

,

Table A7. Comparison of experimental and model-calculated W(x, t) data for sample CC50-L50.

Particle Size, mm	Data Type	Grinding Time, min
		0.0	0.5	1.0	1.5	2.0	3.0	4.0	5.0
−3150 +2380	Exp	3.49	0.15	0.00
	Cal	3.49	0.04	0.00
−2380 +1700	Exp	8.55	0.39	0.02	0.00
	Cal	8.55	0.31	0.01	0.00
−1700 +1000	Exp	16.26	1.64	0.10	0.00
	Cal	16.26	2.19	0.29	0.04
−1000 +710	Exp	21.10	3.90	0.33	0.05	0.01	0.00
	Cal	21.10	4.93	1.15	0.27	0.06	0.00
−710 +250	Exp	31.23	17.90	8.12	5.03	1.75	0.53	0.11	0.08
	Cal	31.23	18.11	10.50	6.09	3.53	1.19	0.40	0.13
−250 +150	Exp	37.06	26.92	18.33	15.25	8.60	4.69	1.61	1.50
	Cal	37.06	26.45	18.88	13.47	9.62	4.90	2.50	1.27
−150 +90	Exp	41.58	34.20	27.28	24.20	18.70	13.60	8.53	7.12
	Cal	41.58	33.75	27.40	22.24	18.05	11.89	7.83	5.16
−90 +75	Exp	42.67	36.02	29.72	26.98	22.03	17.27	11.71	10.09
	Cal	42.67	35.79	30.02	25.18	21.12	14.86	10.45	7.35
−75 +45	Exp	45.06	40.22	34.23	32.48	27.93	25.15	20.24	14.70
	Cal	45.06	40.42	36.25	32.52	29.16	23.46	18.87	15.18

and

Table A8. Comparison of experimental and model-calculated W(x, t) data for sample CC25-L75.

Particle Size, mm	Data Type	Grinding Time, min
		0.0	0.5	1.0	1.5	2.0	3.0	4.0	5.0
−3150 +2380	Exp	2.02	0.07	0.00
	Cal	2.02	0.01	0.00
−2380 +1700	Exp	4.97	0.20	0.01	0.00
	Cal	4.97	0.12	0.00	0.00
−1700 +1000	Exp	9.43	0.79	0.08	0.00
	Cal	9.43	1.03	0.11	0.01
−1000 +710	Exp	12.10	1.76	0.21	0.02	0.00
	Cal	12.10	2.45	0.50	0.10	0.02
−710 +250	Exp	19.11	8.84	3.98	1.99	0.89	0.26	0.08	0.02
	Cal	19.11	10.62	5.90	3.28	1.82	0.56	0.17	0.05
−250 +150	Exp	24.04	15.73	10.51	10.53	4.95	2.68	1.30	0.81
	Cal	24.04	16.77	11.69	8.16	5.69	2.77	1.35	0.66
−150 +90	Exp	28.25	22.11	17.97	17.87	12.68	9.40	6.82	4.08
	Cal	28.25	22.65	18.16	14.56	11.68	7.51	4.83	3.10
−90 +75	Exp	29.49	23.80	19.75	19.33	14.36	11.45	9.49	6.33
	Cal	29.49	24.49	20.35	16.90	14.04	9.69	6.69	4.61
−75 +45	Exp	31.50	27.80	25.20	24.63	19.93	15.72	12.60	9.13
	Cal	31.50	28.12	25.09	22.40	19.99	15.92	12.68	10.10

in Appendix A.

$$W_{(X,t)}=W_{(X,0)}{e}^{-\left[\left(0.0064 - 0.0000056〈\%\mathrm{L}〉\right)\right]x^{\left[\left(0.9041+0.0007〈\%\mathrm{L}〉\right)\right]}t}$$

(5)

4. Conclusions

A kinetic mathematical model was developed to simulate the grinding behavior of calcined clay and its mixtures with limestone based on limestone mass fraction, residence time, and particle size. A comprehensive analysis of the results indicates that calcined clay and its mixtures with limestone grind uniformly and without interference. Specifically, this study demonstrates the following:

• Calcined clay and its mixtures with limestone grind according to the first-order cumulative kinetic model, with calcined clay being the most difficult material to grind.
• The specific breakage rate (0.20–6.83 min⁻¹) increases with limestone content and decreases with particle size.
• The relationship between specific breakage rates and particle size can be described by an exponential model with coefficients of determination greater than 97%.
• The parameters C and n were related to the increase in the mass percentage of limestone using linear models with coefficients of determination greater than 98%.
• The proposed kinetic model proved to be robust in simulating the grinding of calcined clay and its mixtures with limestone.