A Mix-Design Method for the Specific Surface Area of Eco-Concrete Based on Statistical Analysis

Abstract

Ecological concrete designed by empirical method does not consider the mesoscopic influence of aggregates, resulting in problems such as low strength, excessive porosity, and poor stability with different gradations, which severely restricts the development and application of ecological concrete. To achieve the refined design of ecological concrete, a mesoscopic specific surface area design method based on statistical analysis is proposed. First, the meso-aggregate model with sub-millimeter precision was established using a high-precision 3D scanner, and CloudCompare was used to calculate the specific surface area of the mesoscopic aggregate model, laying the foundation for the statistical analysis of specific surface area. Second, statistical analysis methods verified that the mean specific surface area of 20 aggregates from a single random sampling reliably estimates the mean of the overall aggregate population. Third, the optimal water–cement ratio was calculated considering the water absorption characteristics and the mortar-wrapping capacity of aggregates; standard cubic specimens were prepared using this optimal water–cement ratio, with aggregates evenly coated with mortar and no obvious mortar settlement. Fourth, the cubic compressive strength of specimens naturally cured for 7 days was tested; experimental results showed that the cubic compressive strength of specimens formed by this project’s design method increased by more than 30% compared to the empirical design method. The results indicate that using the average volume-specific surface area of 20 aggregates to assess the overall average volume-specific surface area of aggregates is both reliable and relatively efficient. Based on the reliable estimation of the overall average volume-specific surface area of aggregates derived from this method, measurements were taken of the thickness of water films adsorbed on dry aggregates and the thickness of mortar coatings on surface-dry aggregates. Further, the optimal water–cement ratio for eco-concrete was deduced, and a comprehensive set of feasible refined methods for eco-concrete mix proportion design was proposed. In contrast to the empirical method, concrete designed via the subject’s methodology exhibits a marked enhancement in compressive strength while retaining favorable pore characteristics—rendering it well-suited for deployment in the slope protection of reservoirs and ponds and thereby facilitating the realization of ecological slope protection functionality.

1. Introduction

With the acceleration of global urbanization and the rapid development of the economy, the concept of ecological environmental protection has increasingly taken root in people’s minds. However, natural geographical issues such as soil erosion have become increasingly severe. In particular, the problem of slope ecological environment caused by human activities has already become a severe challenge for global sustainable development. Against this backdrop, the limitations of ordinary concrete in the application of ecological engineering have gradually become prominent. Unlike ecological concrete, ordinary concrete lacks the characteristics of promoting the exchange of substances and energy between organisms, which is not conducive to the cultivation and development of biodiversity. In contrast, ecological concrete demonstrates significant ecological advantages [1].

As an essential green building material for constructing “sponge cities,” ecological concrete plays a significant role in alleviating urban waterlogging [2], replenishing groundwater resources [3], mitigating the urban heat island effect [4], reducing noise pollution, and purifying water bodies [5]. Among its key performance indicators, permeability and strength are central to the functionality of ecological concrete, with low strength being the main factor restricting its wider application. In recent years, scholars at home and abroad have conducted extensive theoretical and experimental research on the mix proportion design of ecological concrete. These studies mainly focus on permeability characteristics [6,7], purification and filtration performance [5,8,9], the utilization of industrial and agricultural solid wastes [10,11,12], and mesoscopic mechanical properties [13]. The research on ecological concrete follows a spiral ascending path of “practice-theory-tool”. Currently, it has entered the stage of standardized application and presents cutting-edge trends such as biological enhancement and intelligent monitoring. The main challenge still faced by current research is the lack of a unified mix proportion design method, which limits the comparability and promotion between different projects [14].

Research on mix proportioning methods for ecological concrete primarily falls into two categories: the “specific surface area design method” based on “paste coating thickness” and the “absolute volume method” based on “target porosity” [4]. The latter, proposed by Deo in 2011 [15], requires extensive trial mixing, which severely limits its further development. The specific surface area design method, based on paste coating thickness, has gained increasing attention in recent years. In 2008, Li Yankun [16] used spheres to model coarse aggregate particles and calculated the total surface area per unit volume of coarse aggregates, laying a foundation for determining parameters such as paste coating thickness. Subsequently, Zhang Xianchao [17], Nguyen [18], and others proposed optimizing the mix design of ecological concrete from the perspective of paste coating thickness, ultimately developing a complete method using specific surface area parameters. In 2021, Xu Jie [19] pointed out the limitations of the Modern Concrete Mix Design Manual in the empirical mix design of ecological concrete. They also noted that using spheres to model aggregates in the specific surface area method results in considerable error. To address this, they adopted ellipsoids as the modeling basis for pebble aggregates, analyzed the relationship between aggregate specific surface area and paste coating thickness, and proposed a new mix-design method based on ellipsoidal modeling. However, this method is only applicable to pebble aggregates and cannot be generalized to other types of aggregates, thus limiting its applicability beyond laboratory research. In 2022, Yang Lixiang [4] and colleagues used an equivalent sphere model to calculate the specific surface area of aggregates, conducted paste coating experiments, and studied the mix proportion of recycled aggregate ecological concrete, exploring the relationship between paste coating thickness and paste fluidity. In general, the empirical mix proportion calculation methods in modern concrete mix design manuals do not take into account the gradation and the meso-scale effects of aggregates. The specific surface area calculation method based on the thickness of the coated paste has deviations in the calculation of specific surface area. The absolute volume method based on target porosity requires a large number of trial experiments, which is time-consuming and labor-intensive.

Based on the analysis of the current research on mix proportion design methods for ecological concrete, most domestic and international scholars have focused on the promotion and application of the specific surface area design method. Although relevant studies have yielded some results, it is not difficult to observe that most existing methods rely on approximate techniques to estimate aggregate specific surface area, which fail to accurately represent this key parameter. Numerous studies have shown that the shrinkage and creep of recycled aggregate ecological concrete are higher than those of ordinary aggregate concrete. To control the shrinkage and creep of the material, ordinary aggregates rather than the commonly used recycled aggregates were adopted in this study [20,21,22,23].

The calculation and evaluation of specific surface area is a crucial step in ensuring the reliability of the specific-surface-area-based mix-design method for ecological concrete. To address these issues, this paper proposes a specific-surface-area-based mix-design method for ecological concrete using mesoscopic aggregate modeling. The research includes the following components:

• Mesoscopic Model Development: High-precision mesoscopic models of aggregates are created through batch scanning with a 3D scanner. The modeling accuracy reaches sub-millimeter levels, providing a reliable basis for analyzing specific surface area.
• Determination of Aggregate Population Size: A large number of aggregates are randomly and sequentially added to compute the mean specific surface area by volume. It has been determined that at least 150 particles are required for the mean value of specific surface area to stabilize. Therefore, a population of 300 aggregates (well above the minimum) is selected as the representative sample.
• Sampling and Statistical Analysis: The 300 representative aggregates are numbered, meshed, and used to calculate individual specific surface areas. For each sample, 20 non-repeating integers between 1 and 300 are randomly generated to select 20 aggregates, and the average specific surface area is calculated. This sampling process is repeated 100 times. Statistical analysis shows that the distribution of sample means follows a normal distribution, and the 95% confidence interval for the mean specific surface area is obtained. The results confirm that the mean specific surface area of 20 sampled aggregates is a reliable estimator of the population’s mean specific surface area.
• Evaluation of Specific Surface Area and Optimization of Water-to-Cement Ratio: The mean specific surface area from one random sample is used to estimate the aggregate’s surface area. This value is combined with results from a water immersion test (to determine the thickness of the surface water film on the aggregates) and a paste coating test (to measure paste thickness under different water-to-binder ratios) to determine the optimal water-to-cement ratio.
• Comparative Performance Analysis: The compressive strength of ecological concrete specimens prepared using the optimal water-to-cement ratio from this study is compared to specimens designed by empirical methods using the same cement content and aggregate gradation. This comparison demonstrates the superiority of the proposed method. The detailed technical route is shown in Figure 1.

2. Research Significance

The particular significance of this paper lies in the proposal of a statistical method that can reliably evaluate the mean value of the total mean specific surface area by the volume of aggregates with a small sample size. Based on the research results, the refined design of the mix proportion of ecological concrete is achieved.

3. Development of Mesoscopic Aggregate Models

The Revopoint Mini2 3D scanner (made in Xi’an, China and hereinafter referred to as the 3D scanner), with a precision of 0.02 mm and an efficient scanning algorithm, is an ideal device for the high-precision modeling of small objects, being particularly suitable for professional applications with stringent detail requirements.

The 3D scanner supports a minimum scan size of 10 mm and a maximum of 500 mm. The appropriate particle size range for ecological concrete aggregates is 10–30 mm, which falls well within the optimal scanning range of the device. Additionally, the scanner supports a “dark mode” scanning function for aggregates, enabling the construction of high-resolution models without the need for marker points. Therefore, this scanner was selected to develop high-resolution mesoscopic models of ecological concrete aggregates. The levels of detail and visual quality of the resulting mesoscopic models are shown in Figure 2. The point cloud density can exceed 2000 points per square millimeter. As illustrated in Figure 2, the reconstructed aggregate models under high-density point clouds effectively reflect the mesoscopic structural characteristics of the aggregates. This ensures the reliability of the aggregate’s mesoscopic specific surface area analysis and provides a solid foundation for accurate specific surface area calculations.

3.1. Introduction to the Central Limit Theorem

According to the central limit theorem in mathematical statistics, for any population with an unknown distribution, unknown mean, and unknown standard deviation, if simple random sampling with replacement is conducted, and the sample size is sufficiently large (n ≥ 30), the sample means will follow a normal distribution and serve as an unbiased estimator of the population mean [24,25]. To be more specific, the distribution of the mean of a random sample has the following characteristics:

(1)

The sampling distribution of the sample mean will be very close to a normal distribution;

(2)

The mean of the sampling distribution of the sample mean is equal to the population mean;

(3)

The standard deviation (σ) of the sampling distribution of the sample mean is equal to the population standard deviation (SE) divided by the $\sqrt{n}$.

In simple random sampling with replacement, the sample mean is an unbiased estimator of the population mean, meaning that the mathematical expectation of the sample mean is equal to the population mean. The formula is as in Equation (1).

$$\mathrm{E}\left({\mathrm{x}}_{\mathrm{i}}\right)=\mathrm{u}$$

(1)

In Equation (1), μ is the population mean to be estimated and $\mathrm{E}\left({\mathrm{x}}_{\mathrm{i}}\right)$ is the sample mean.

According to the research of Royston [26], when the sample size is 100, the error between the p-value of the Shapiro–Wilk test statistic and the exact simulation result is less than 0.01. During the sampling experiment, setting the sample size as 100 [24], the distribution of the sample mean ($\overline{\mathrm{X}}$) approaches a normal distribution. The normal distribution is shown in Equation (2).

$$\overline{\mathrm{X}}~\mathrm{N}\left(\mathrm{u},{\displaystyle \frac{\mathrm{S}\mathrm{E}}{\surd \mathrm{n}}}\right)$$

(2)

In Equation (2), SE is the sample standard error and n is the sample size.

3.2. Selection of Statistical Indicators

Referring to the aggregate requirements for a single specimen with a 26.5–31.5 mm gradation as provided by Xu Jie [19], mesoscopic mesh models of aggregates from four specimens were constructed using a 3D scanner. The average specific surface area based on volume and mass was then analyzed using CloudCompare software (version 2.14).

Since the units of volume-based and mass-based specific surface area are inconsistent, their dispersion was analyzed using the coefficient of variation. The coefficients of variation for the four groups of specific surface area (SSA) data were calculated using Origin software (version 2022), as shown in

Table 1. The mean SSA discrete analysis with 26.5–31.5 mm particle size.

Group Number	Mean SSA by Mass mm2/g	CV of SSA by Mass	Mean SSA by Volume mm2/mm3	CV of SSA by Volume
A	85.336	0.022	0.226	0.017
B	86.018		0.223
C	87.461		0.225
D	89.695		0.218

. Mean SSA discrete analysis was conducted with 26.5–31.5 mm particle size. For each specimen, the coefficient of variation (CV) of the volume-based specific surface area was smaller than that of the mass-based specific surface area. Therefore, the volume-based average specific surface area of aggregates was selected as the primary statistical indicator to minimize dispersion bias in subsequent statistical analyses.

3.3. Determination of the Sampling Population

The aggregates of the given gradation were cumulatively evaluated in a random, non-sequential manner to calculate the mean volume-based specific surface area. The calculation results are shown in Figure 3. The results indicate that when the number of aggregates included in the cumulative calculation is sufficiently large (no fewer than 150), the mean volume-based specific surface area reaches a stable value. This outcome reflects the fundamental principle of the law of large numbers in statistics. Once the number of aggregates reaches a certain scale (i.e., at least 150), the calculated mean stabilizes near the true population mean. Further increases in the number of aggregates lead to only minor fluctuations around this stable value. Therefore, selecting a total aggregate count exceeding 150 ensures that the overall sample mean reliably represents the population mean. To further ensure the reliability of the sampling results, a total of 300 aggregates (twice the minimum requirement of 150) was selected as the sampling population.

3.4. Implementation of Simple Random Sampling with Replacement

Only samples obtained through random selection can be regarded as representatives of the population and used to infer the population [27,28]. Simple random sampling with replacement satisfies this principle, so the experiment adopted simple random sampling with replacement to meet the requirement of sample representativeness.

For the established population of 300 aggregates in the 26.5–31.5 mm size group, simple random sampling with replacement was conducted by generating random numbers. A total of 100 sample groups were drawn (satisfying the condition n ≥ 30), with each group consisting of 20 aggregates. This provided the sample data for the statistical analysis of the mean volume-based specific surface area. In Figure 4, photographs and mesh models of the first 20 aggregates (by serial number) in the population are displayed.

3.5. Statistical Analysis of Sample Means in Single Extraction

A Shapiro–Wilk normality test was performed on the mean specific surface area data obtained from the single extraction of different sample sizes. Statistical analysis using Origin software showed that, at the 0.05 significance level, the sample means from all four sample sizes were significantly consistent with a normal distribution. The results of the Shapiro–Wilk test are presented in

Table 2. The mean specific surface area Shapiro–Wilk normality test with 26.5–31.5 mm particle size.

Quantity of Coarse Aggregate in a Sample (Pieces)	Sample Size (n)	W-Value	p-Value	Conclusion
15	100	0.99157	0.78927	The data are significantly drawn from a normal distribution at the 0.05 significance level.
20		0.98476	0.70109
25		0.99064	0.71649
30		0.98771	0.48733

. And the normal distribution curve of the sample means is illustrated in Figure 5.

Based on the results of the Shapiro–Wilk test in single extractions, the mean volume-based specific surface area values of the 100 randomly sampled aggregate groups were fitted to a normal distribution well. The fitting parameters and 95% confidence interval for the Mean Volumetric Specific Surface Area (MVSSA) are shown in

Table 3. Parameters for normal distribution fitting of MVSS in single experiment with 26.5–31.5 mm particle size.

Quantity of Coarse Aggre-Gate in a Sample (Pieces)	Sample Size (n)	MVSSA (μ) (mm2/mm3)	Standard Error (SE) of the Sample Mean	95% Confidence Interval for the Mean Volumetric Specific Surface Area [μ − 1.96SE, μ + 1.96SE]
15	100	0.22492	0.000674	[0.22360, 0.22624]
20		0.22423	0.000532	[0.22316, 0.22527]
25		0.22457	0.000420	[0.22375, 0.22539]
30		0.22509	0.000343	[0.22442, 0.22576]

.

According to the results of the large-sample single-population mean test, any single instance of simple random sampling with replacement falls within the interval [μ − 1.96SE, μ + 1.96SE] with a 95% probability. As shown by the statistical distribution in Figure 5, the distribution pattern follows a normal distribution trend, further supporting the rationality of fitting the data using a normal distribution.

3.6. Statistical Analysis of Sample Means in 20 Extraction Tests

The results of a single sampling indicate that each sample containing no fewer than 15 aggregates can effectively reflect the mean specific surface area of aggregates in the population. However, a single sampling result cannot demonstrate the reliability of the number of sampled aggregates; thus, it is necessary to combine multiple sampling results to assess the rationality of the number of aggregates in a single sample.

Considering the convenience of experimental sampling and the reliability of results, the sample sizes of 15 and 20 were further sampled 20 times by Matlab simulation, and the statistical results of the Shapiro–Wilk test are shown in

Table 4. Parameters for the Shapiro–Wilk test of MVSS in single experiment with 26.5–31.5 mm particle size.

Parameters of Extractions	Number of Sampled Aggregates (Pieces)	20			15
	Parameters for Normal Distribution Fitting	H	P	W	H	P	W
Number of experiments	1	0	0.492903	0.957379	0	0.308014	0.949705
	2	0	0.100821	0.920392	0	0.48382	0.956894
	3	0	0.386568	0.951257	1	0.045589	0.902318
	4	0	0.157344	0.933353	0	0.330602	0.947491
	5	0	0.989482	0.986555	0	0.433988	0.954121
	6	1	0.03992	0.899246	0	0.369747	0.954234
	7	0	0.490983	0.957277	0	0.454824	0.955306
	8	0	0.987392	0.986084	0	0.310186	0.945983
	9	0	0.06619	0.911913	0	0.889765	0.976999
	10	0	0.734239	0.969029	0	0.337003	0.951925
	11	0	0.07293	0.914361	1	0.036404	0.896467
	12	0	0.763278	0.970399	0	0.499853	0.957746
	13	0	0.289009	0.948141	0	0.611307	0.963283
	14	0	0.542089	0.959912	0	0.756248	0.970065
	15	0	0.677395	0.966382	0	0.410265	0.956857
	16	0	0.542009	0.959908	0	0.773703	0.970897
	17	0	0.527622	0.959181	0	0.592891	0.966667
	18	0	0.975596	0.984107	0	0.230445	0.939089
	19	0	0.386789	0.951271	0	0.383632	0.95107
	20	0	0.253951	0.944983	0	0.523395	0.958965

Note: ‘H = 0’—‘Do not reject the null hypothesis at significance level ALPHA’; ‘H = 1’—‘Reject the null hypothesis at significance level ALPHA’; ‘W’—the test statistic (non-normalized)’; P—the p-value, or the probability of observing the given result by chance given that the null hypothesis is true.

and Figure 6.

In Figure 6, P-15 means the p-value of the Shapiro–Wilk test using the average specific surface area of 15 aggregates as the statistical value, which is consistent with “the number of sampled aggregates” and “Parameter-P”.

Based on the statistical analysis of the sample means shown in Table 4 and Figure 6, among the results of 20 samplings, the experimental results with 15 aggregates in the sample passed the Shapiro–Wilk test and accepted the normal distribution assumption in 18 cases; the test results with 20 aggregates in the sample passed the Shapiro–Wilk test and accepted the normal distribution assumption in 19 cases. To ensure the reliability of the normal distribution of the sampled data, it is reasonable to use 20 aggregates as a sample when evaluating the mean value of the specific surface area of the overall aggregates.

For any single instance of simple random sampling with replacement, the mean volume-based specific surface area of the 20 selected aggregates falls within the interval [0.22316, 0.22527] with a probability greater than 95%, indicating that the sample mean reliably approximates the population mean. Therefore, in subsequent research, the population-specific surface area of aggregates will be reliably estimated using the average of 20 aggregate models. For convenience in mix proportion calculations, the average volume-based specific surface area of the 26.5–31.5 mm size group can be represented by the sample mean of any one random sampling. In this study, a randomly selected sample with an average volume-based specific surface area of 0.22422 mm²/mm³ was used for calculations, and the mean volume-specific surface area was rounded to 0.225 mm²/mm³. It is apparent from Figure 3 that this result was consistent with the stable result derived from the cumulative calculation of the total volume-specific surface area.

3.7. Statistical Method Verification with 19–26.5 mm Gradation

The same statistical method is applied to aggregates with a gradation of 19–26.5 mm. The following is a brief description of the experimental process and its results. Figure 7 shows that a sampling population of no more 140 pieces is sufficient. To maintain consistency with the analysis of aggregates with a gradation of 26.5–31.5 mm, the total size of the sampling population for aggregates with a gradation of 19–26.5 mm is still randomly set at 300 aggregates.

Given the time cost of grouped statistics, the discrete characteristics of both volume-specific surface area and mass-specific surface area for aggregates in the 19–26.5 mm gradation will not be further subjected to comparative analysis, but the consistency verification of sampling statistical laws is essential.

Table 5. The mean specific surface area Shapiro–Wilk normality test with 19–26.5 mm gradation.

Quantity of Coarse Aggregate in a Sample (Pieces)	Sample Size (n)	W-Value	p-Value	Conclusion
15	100	0.99193	0.81593	The data are significantly drawn from a normal distribution at the 0.05 significance level.
20		0.99263	0.86472
25		0.98195	0.18819
30		0.99339	0.91018

and

Table 6. Parameters for normal distribution fitting of MVSS in single experiment with 19–26.5 mm gradation.

Quantity of Coarse Aggregate in a Sample (Pieces)	Sample Size (n)	MVSS (mm2/mm3)	Standard Error (SE) of the Sample Mean	95% Confidence Interval for the Mean Volumetric Specific Surface Area [μ − 1.96SE, μ + 1.96SE]
15	100	0.29344	0.00113	[0.29115, 0.29566]
20		0.29401	0.0009193	[0.29228, 0.29581]
25		0.29159	0.000836	[0.28851, 0.29323]
30		0.2936	0.000794	[0.29204, 0.29516]

and Figure 8 show the statistical results of single extraction tests.

Table 7. Parameters for the Shapiro–Wilk test of MVSS in single experiment with 19–26.5 mm gradation.

Parameters of Extractions	Number of Sampled Aggregates (Pieces)	20			15
	Parameters for Normal Distribution Fitting	H	P	W	H	P	W
Number of experiments	1	0	0.78283	0.99149	0	0.30976	0.98587
	2	0	0.07856	0.97708	1	0.04939	0.97467
	3	1	0.02715	0.9709	0	0.07706	0.9774
	4	0	0.55921	0.98867	0	0.20997	0.98347
	5	0	0.26311	0.98486	1	0.01652	0.96769
	6	0	0.53867	0.98841	0	0.10472	0.97867
	7	0	0.59732	0.98916	0	0.1778	0.98163
	8	0	0.79692	0.99167	0	0.33871	0.98539
	9	0	0.45618	0.98727	0	0.87665	0.99281
	10	0	0.58748	0.98904	0	0.39858	0.9864
	11	0	0.71516	0.99063	0	0.1831	0.98264
	12	0	0.4913	0.98777	0	0.4483	0.98826
	13	0	0.58997	0.99021	0	0.39829	0.9864
	14	0	0.64797	0.99093	0	0.40037	0.98643
	15	0	0.20975	0.98257	0	0.50534	0.98796
	16	0	0.51283	0.98919	0	0.43985	0.98703
	17	0	0.05825	0.97569	0	0.63929	0.98969
	18	0	0.06913	0.97674	0	0.84126	0.99228
	19	0	0.12716	0.97975	1	0.03964	0.97328
	20	0	0.55869	0.98867	0	0.25475	0.98466

and Figure 9 show the statistical analysis results of sample means in 20 extraction tests.

Based on the statistical analysis of the sample means shown in Table 7, the experimental results with 15 aggregates in the sample passed the Shapiro–Wilk test and accepted the normal distribution assumption in 17 cases; the test results with 20 aggregates in the sample passed the Shapiro–Wilk test and accepted the normal distribution assumption in 19 cases. It is more reliable to still use 20 aggregates as the sample size for the 19–26.5 mm gradation aggregates.

For convenience in mix proportion calculations, the average volume-based specific surface area of the 19–26.5 mm size group can be represented by the 95% confidence interval for the Mean Volumetric Specific Surface Area [0.29228, 0.29581]. In the present paper, the average volume-specific surface area employed for computational purposes is 0.29405 mm²/mm³, which, upon rounding, is taken as 0.295 mm²/mm³. As is evident from Figure 7, this result aligns with the stable outcome obtained through cumulative calculations of the total volume-specific surface area in Figure 7.

3.8. Method for Evaluating the Volume-Specific Surface Area of Overall Aggregate Population

A comparative analysis of the single-sampling and 20 replicate samplings results for two types of graded aggregates reveals that the mean value of volume-specific surface area measured from a single sample of 20 aggregates, when used as a metric to characterize the mean volume-specific surface area of the overall aggregate population, exhibits high reliability.

4. Determination of the Optimal Water–Cement Ratio

Sand-free ecological concrete can be considered a composite material composed of aggregates, cement paste coating the aggregates, and internal pores. The specific-surface-area-based design method using “paste thickness” is the fundamental approach for determining the optimal water–cement ratio, which involves coating saturated surface-dry (SSD) aggregates with cement paste. According to the paste-thickness testing methods proposed by Yang Lixiang and Liu Anyao [4,29], the paste thickness during specimen formation is closely related to the water absorption of dry aggregates. The water absorbed by SSD aggregates is treated as a uniform water film adsorbed on the aggregate surface, which facilitates the calculation of water absorption in practical production. A schematic diagram illustrating aggregate water absorption and paste coating during specimen formation is shown in Figure 10.

In paste coating tests, the water–cement ratio is a key factor influencing the thickness of the paste on aggregates. Given the impracticality of preparing aggregates to a true SSD condition during actual production, the amount of water absorbed by the aggregates during coating is directly added to the mix during the experiment. Therefore, both the water adsorption characteristics of SSD aggregates and the coating thickness of the paste must be considered when designing the mix proportions.

4.1. Determination of the Water Film Thickness Adsorbed by Aggregates

The water absorption of dry aggregates has a direct impact on the paste coating process. To account for the water absorption behavior of dry aggregates during the mixing and forming of specimens, it is necessary to test and evaluate their water absorption capacity. The procedure for determining the thickness of the water film adsorbed on the aggregate surface is illustrated in Figure 11.

During the experiment, three different gradations groups of aggregates (labeled 19.0–26.5 mm, 26.5–31.5 mm, and 19.0–40.0 mm) were subjected to immersion tests. The experimental procedure is illustrated in Figure 12. The water absorption of the aggregates in a saturated surface-dry (SSD) condition was measured after immersion durations of 15, 90, 180, and 270 min. These measurements were then converted into the thickness of the water film adsorbed on the aggregate surface, as shown in Figure 13.

The results indicate that the differences in water film thickness among the three groups were minimal. More than 90% of the total water absorption occurred within the first 15 min. The aggregates showed no visible cracks and had a uniform lithology. Therefore, the water absorption was considered stable, and the average water film thickness across the three groups—calculated as 44.795 μm—was used for further analysis. For simplification in calculations, the water film thickness required for dry aggregates to reach the SSD condition (h_w) was taken as 45 μm.

4.2. Measurement of Paste Coating Thickness

The paste coating thickness was measured using the “paste coating method.” To ensure the reliability of the experiment, the amount of aggregate used should not be too small; therefore, approximately 500 g of dry aggregates (by weight) were uniformly measured for the paste coating test. Based on the average specific surface area (SSA), the aggregate surface area (S) and the mass of adsorbed water (m_wa) were calculated. A fixed cement mass of 500 g was used, and the water–binder ratios (W/B) of 0.31, 0.34, and 0.37 were applied to calculate the amount of mixing water required for the cement paste (m_wb). Taking into account the water absorption of the aggregates, the total water added during paste mixing (m_w) was adjusted accordingly.

The paste coating thickness measured under the three water–binder ratios is shown in

Table 8. Test results of mortar wrapping thickness with 26.5–31.5 mm particle size.

Group	W/B	Quality of Aggregates Before Slurry Coating (g)	Total Surface Area of Aggregates (mm2)	The Water Absorption Quality of Aggregates (g)	Cement Slurry Density (g/cm3)	The Quality After Sizing (g)	Coating Quality (g)	Coating Thickness (mm)	Average Coating Thickness (mm)
1-1	0.31	515	44,913	2.02	2.07	636	121	1.30	1.16
1-2		507	44,215	1.99		621	114	1.25
1-3		511	44,564	2.01		597	86	0.93
2-1	0.34	515	44,913	2.02	2.02	701	186	2.05	1.93
2-2		497	43,343	1.95		683	186	2.12
2-3		501	43,692	1.97		644	143	1.62
3-1	0.37	510	44,477	2.00	1.98	646	136	1.54	1.29
3-2		502	43,779	1.97		605	103	1.19
3-3		509	44,390	2.00		608	99	1.13

. The maximum paste coating thickness (h_c) of 1.93 mm was observed at a water–binder ratio of 0.34 with a 26.5–31.5 mm particle size.

In Table 8, cement slurry density (${\mathrm{\rho }}_{\mathrm{c}\mathrm{s}}$) is calculated by Equation (3). The vibration and post-slurry-coating mass weighing for the slurry coating thickness tests of Groups 1–2 are shown in Figure 14.

$${\mathrm{\rho }}_{\mathrm{c}\mathrm{s}}={\displaystyle \frac{1+\mathrm{W}/\mathrm{B}}{\frac{1}{{\mathrm{\rho }}_{\mathrm{c}}}+\frac{\mathrm{W}/\mathrm{B}}{{\mathrm{\rho }}_{\mathrm{w}}}}}$$

(3)

In Equation (3): ${\mathrm{\rho }}_{\mathrm{c}}$ refers to 3.1 g/cm³, W/B refers to the water–binder ratio, and ${\mathrm{\rho }}_{\mathrm{w}}$ refers to 1.0 g/cm³.

4.3. Aggregate Quality in the Reference Mix Proportion

To calculate the mix proportion of ecological concrete, it is necessary to determine the aggregate dosage per unit volume. The aggregate mass in the reference mix proportion, measured through a dense packing test using standard cubic molds, is shown in

Table 9. Aggregate quality in the reference mix proportion.

Aggregate Grading (mm)	Quality of Naturally Accumulated Aggregate (g)	Average Quality of Naturally Accumulated Aggregate (g)	Natural Bulk Density (kg/m3)	Aggregate Quality in the Reference Mix Proportion (kg/m3)
26.5–31.5	4912.0	5096.3	1510.0	1449.6
	5236.0
	5142.0

Note: Aggregate Quality in the Reference Mix Proportion = Natural Bulk Density × 0.96.

.

4.4. Calculation of the Optimal Water–Cement Ratio

According to the research findings of Yang Lixiang et al. [4], the compressive strength of specimens increases with the amount of paste coating. As shown in Table 4, when the water–cement ratio of the cement paste is 0.34, the paste coating thickness reaches its maximum of 1.93 mm. Based on this water–cement ratio and the corresponding paste thickness, the water–cement ratio calculated is considered the optimal water–cement ratio. The optimal water–cement ratio (W/C) is determined to be 0.354, which is reasonable [12], with the detailed calculation shown in

Table 10. Calculation result of optimal water–cement ratio.

Aggregate Gradation (mm)	Water-Glue Ratio	Quality of Sizing Aggregate (g)	Coating Thickness (mm)	Coating Density (g/cm3)	Total Water Consumption (g)	Cement Quality (g)	W/C
26.5–31.5	0.34	500	1.93	2.02	45.1	127.3	0.354

.

5. Compressive Strength Test

Particle size of aggregates exerts a significant influence on the macroscopic mechanical properties of concrete materials [30]. Given the variations in mesoscopic specific surface area parameters among aggregates with different particle sizes, this paper conducts a mixproportion design for two types of graded aggregates. Furthermore, it compares the strength of the eco-concrete designed in this research with that designed in accordance with the specification, aiming to verify the superiority of the proposed research method.

The material quantities required per cubic meter of concrete were converted according to the dense packing density of the aggregates, as shown in

Table 11. Material usage for each eco-concrete cubic specimen.

Aggregate Gradation (mm)	Method	Aggregate Quantity (g)	Cement Dosage (g)	Water Volume (g)	W/C
26.5–31.5	I	4892.4	1245.6	441.3	0.354
	II	4892.4	1245.6	351.3	0.282

. The specimens used in the test were cubic samples measuring 150 mm × 150 mm × 150 mm.

The cement content plays a decisive role in the strength development of porous eco-concrete [31]. Under the premise of using the same amount of cement, the quantities of cement, water, and coarse aggregates required per cubic meter of ecological concrete were calculated using the method proposed in this study (referred to as Method I) and the empirical method from the Modern Concrete Mix-Proportion Design Manual [32] (referred to as Method II), as shown in Table 11.

The specimens molded according to Methods I and II are shown in Figure 14. The specimens designed using Method I exhibited no slurry segregation and had a uniform paste coating on the aggregate surfaces. In contrast, the specimens designed using Method II were relatively dry due to the lower water–cement ratio, resulting in insufficient paste coating and lower strength, which caused aggregate spalling during the demolding process.

Based on the mix designs in Table 11, three specimens were prepared for both method I and method II, and their compressive strengths after 7 days of natural curing were tested, as shown in

Table 12. Compressive strength test results of two mix proportions under the same cement dosage with 26.5–31.5 mm particle size.

Aggregate Gradation (mm)	Method	Water–Cement Ratio	Specimen Number	Compressive Strength (MPa)	Average Compressive Strength (MPa)
26.5–31.5	I	0.354	1	1.4	1.57
			2	1.6
			3	1.7
	II	0.282	1	1.1	1.17
			2	1.4
			3	1.0

. The loading and failure of the experimental specimens are shown in Figure 15. The results indicate that the mix design calculated using Method I achieved a compressive strength nearly 34% higher than that of Method II.

Using the same method, specimens with 19–26.5 mm aggregates were designed to verify the reliability of the proposed approach. The optimal water–cement ratio was determined to be 0.355 while the manual method designed a water–cement ratio of 0.290. The specimens are shown in Figure 16. The specimens molded by the proposed method exhibited uniform paste coating without slurry segregation whereas the manual method’s lower water–cement ratio resulted in poor paste adhesion on the aggregate surface and lower strength. The compressive strength test results are listed in

Table 13. Compressive strength test results of two mix proportions under the same cement dosage with 19–26.5 mm particle size.

Aggregate Gradation (mm)	Method	Water–Cement Ratio	Specimen Number	Compressive Strength (MPa)	Average Compressive Strength (MPa)
19–26.5	I	0.355	1	2.4	2.4
			2	2.8
			3	2.0
	II	0.290	1	1.1	1.4
			2	1.4
			3	1.7

. The ecological concrete designed by the proposed method with 19–26.5 mm aggregates showed an increase in strength of over 70%. The comparison of strength between the optimal water–cement ratios of the two groups and the manual method is shown in Figure 17.

6. Discussion

Building upon mathematical statistical analyses, this paper has presented a method for estimating the population mean of aggregate properties using the average specific surface area of 20 aggregate particles. According to this approach, the optimal paste thickness of aggregates in the experiments was calculated reliably, and the corresponding mix proportions were determined accordingly. This proposed design methodology has been demonstrated to effectively enhance the compressive strength of eco-concrete by over 30%, thereby providing valuable insights for the refined design of eco-concrete. However, the current research is confined to the exploration of mix proportion design methods, and recycled aggregates were not incorporated into the preparation of eco-concrete—this aspect will constitute a key focus of subsequent investigations. The eco-concrete designed in this research project is mainly used in engineering projects with low strength requirements, such as in slope protection and artificial lake revetment.

7. Conclusions

Based on meso-scale aggregate modeling and statistical methods, a specific-surface-area-based mix-design method for ecological concrete was proposed. The main conclusions are as follows:

• High-precision meso-scale models of test aggregates were established in batches using a 3D scanner, laying the foundation for reliably evaluating the average specific surface area of aggregates. This method demonstrates general applicability, though its modeling efficiency still requires improvement.
• A small-sample estimation method was proposed based on statistical analysis to assess the mean specific surface area of aggregates in a specimen. This approach overcomes the limitations of the traditional wax-coating method and the approximate volume calculation methods such as the “ellipsoid method” and “sphere method.” Additionally, the confidence interval of the calculated mean specific surface area was evaluated.
• Under the condition of equal cement content and same gradation of the aggregate, the ecological concrete specimens designed using the meso-aggregate model exhibited more uniform paste coating and over 30% higher compressive strength compared to those designed using the empirical method in the Handbook of Modern Concrete Mix Design, confirming the scientific validity and reliability of the proposed method.