# Optimizing the Sulfates Content of Cement Using Neural Networks and Uncertainty Analysis

^{*}

## Abstract

**:**

_{3}S, tricalcium aluminate, C

_{3}A). We correlated strength with the ratio %SO

_{3}/CL and the molecular ratios MSO

_{3}/C

_{3}S and MSO

_{3}/C

_{3}A. The data processing stage proved that artificial neural networks (ANNs) fit the results’ distribution better than a parabolic function, providing reliable models. The optimal %SO

_{3}/CL value for 7- and 28-day strength was 2.85 and 3.00, respectively. Concerning the ratios of SO

_{3}at the mineral phases for 28-day strength, the best values were MSO

_{3}/C

_{3}S = 0.132–0.135 and MSO

_{3}/C

_{3}A = 1.55. We implemented some of the ANNs to gain a wide interval of input variables’ values. Thus, the approximations of SO

_{3}optimum using ANNs had a relatively broad application in daily plant quality control, at least as a guide for experimental design. Finally, we investigated the impact of SO

_{3}uncertainty on the 28-day strength variance using the error propagation method.

## 1. Introduction

_{3}) composed mainly of calcium sulfate dihydrate (CASO

_{4}.2H

_{2}O or Cs.2H) and a small content of anhydrite (CaSO

_{4}or Cs). During cement grinding, elevated temperatures partially dehydrate gypsum into hemihydrate form or bassanite (CaSO

_{4}.0.5H

_{2}O or Cs.0.5H) [2]. Copeland et al. [3] and a recent systematic review [4] clarify the solubility of the three phases of CaSO

_{4}.

_{2}or C

_{3}S), dicalcium silicate (2CaO ∙ SiO

_{2}or C

_{2}S), tricalcium aluminate (3CaO ∙ Al

_{2}O

_{3}or C

_{3}A), and tetra calcium aluminoferrite (4CaO ∙ Al

_{2}O

_{3}∙ Fe

_{2}O

_{3}or C

_{4}AF). Bogue established the mathematical formulae for calculating the four clinker compounds as a function of the percentages of the four basic oxides CaO, SiO

_{2}, Al

_{2}O

_{3}, Fe

_{2}O

_{3}, or C, S, A, F, as well as the free lime CaO

_{f}[5] (pp. 245–250).

_{3}S and C

_{2}S, including some small decimal numbers m and n [1].

_{3}A by forming ettringite (C

_{3}A.3Cs.32H), according to Equation (7).

_{3}A hydration, preventing flash set during concrete production, transfer, and placement [6,7,8]. Conversely, adding too much gypsum leads to a drop in strength and detrimental expansion of concrete and mortar, meaning there is a sulfate optimum. Cement standards, such as EN 197-1:2011 [9] and ASTM C150 [10], prohibit sulfates above a certain level per type of cement, though they do not suggest an optimal value because that figure depends on the various physical, chemical, and mechanical properties of cement and clinker produced in the specific conditions of each production unit. Furthermore, the property to be optimized impacts the location of the optimum. Some standards, such as ASTM C563-16 [11], define a method for determining the optimal SO

_{3}, albeit only for a particular property of cement, i.e., 1-day compressive strength.

_{2}emissions in clinker production and clinker consumption per ton of product. Optimizing the composition of raw materials for clinker production can achieve the above goal [12,13]. Optimizing SO

_{3}is essential for reducing clinker incorporation into the cement while retaining or improving product performance. Lerch [14] conducted the first thorough study of sulfate optimization in cement past and mortar. According to the researcher, the optimal sulfate content in cement is closely related to hydration heat, length changes of mortar specimens cured in water, alkalis, C

_{3}A content, and cement fineness. Fincan [15] mentions that Lerch was a pioneer and inspired many studies on sulfate optimization in cement and cementitious systems.

_{3}on the hydration of the clinker mineral phases [16,17,18,19,20,21,22,23,24,25]. Bentur [16] found that gypsum influenced the quantity and quality of the hydrated products by accelerating the hydration process and, at the same time, lowering the intrinsic strength of the gel. He interpreted the effect of gypsum on strength in terms of its influence on the extent of hydration and the chemical composition of the gel. Soroka et al. [18] concluded that gypsum accelerates the rate of hydration when its addition is below the optimum SO

_{3}content. Nevertheless, if the gypsum added exceeds the optimal level, considerable obstruction occurs. The authors did not observe any effect of the SO

_{3}content on the density of the hydration products and pore-size distribution. They pointed out that further study is required to explain the impact of SO

_{3}on strength. On the contrary, Sersale et al. [19] found that gypsum addition modified the microstructure of Portland cement mortars. They correlated these modifications with the mechanical behavior of cement mortars by examining Portland cement with sulfate content ranging between 1.5 and 4.5%. A SO

_{3}content of 2–3.5% promotes a shifting in the pore size distribution to lower values, ranging between 100 and 1000 Å, and a variation in total porosity. They concluded that this issue is likely the main factor governing the influence of SO

_{3}on the compressive strength of Portland cement. Gunay [22], in his thesis, studied the influence of aluminates hydration in the presence of calcium sulfate on C

_{3}S hydration and its consequences on cement optimum sulfate. He observed the SO

_{3}optimum when the hydration of C

_{3}S, during the accelerated period, takes place simultaneously or slightly before the exothermic peak due to the dissolution of C

_{3}A and the precipitation of mono-substituted Al

_{2}O

_{3}and Fe

_{2}O

_{3}(AFm). He concluded that the presence of AFm during the accelerated period of C

_{3}S hydration would be the cause of the observed modification of the microstructure of the cement paste: the porosity increases with the addition of calcium sulfate, though the assembly of hydrates is denser. This effect of the sulfate level is the source of the optimal compressive strength observed by Gunay. Zunino et al. [24] investigated the influence of sulfate addition on hydration kinetics and the C-S-H morphology of C

_{3}S and C

_{3}S/C

_{3}A systems at an early age. Adding gypsum changed the needle length of C-S-H and increased the nucleation density. In C

_{3}S/C

_{3}A systems, they did not observe any difference in C-S-H morphology before and after the aluminate peak. Andrade Neto et al. [25] studied the hydration and interactions between pure and doped C

_{3}S and C

_{3}A in the presence of different calcium sulfates. Except for sulfates of gypsum, Miller et al. [26], Taylor [27], and Horkoss et al. [28] investigated the importance of the amount and phases of SO

_{3}incorporated into the clinker. Mohammed et al. [29] optimized the SO

_{3}content of a CEM I cement via grinding clinker with a ball mill until a fineness of 3270 cm

^{2}/g was achieved, before preparing mixes with ground gypsum ranging from 0 to 9%. According to their results, the water demanded for normal consistency, setting times, compressive strength, the heat of hydration, swelling, drying shrinkage, and hydration degree are adversely affected by gypsum addition above or below the optimal. For this cement, which was composed only of clinker and gypsum, they found that the optimal sulfate for 2-, 7-, and 28-day strength was 3%.

_{2}emissions being imperative, composite types of cement constitute the bulk of the products of the cement industry, rendering the optimization of sulfates in such cement more relevant. During the last few decades, numerous researchers conducted deep research investigating the optimal sulfate content of cement containing supplementary cementitious materials (SCM) [30,31,32,33,34,35,36]. Yamashita et al. [30] investigated the influence of limestone powder (LSP) on the optimum SO

_{3}for Portland cement samples with different Al

_{2}O

_{3}contents, which was not negligible. Analyses showed that if SO

_{3}is less than optimal, an increase in sulfate promotes hydration in C

_{3}A and increases compressive strength. In the presence of higher SO

_{3}content, excess formation of expansive ettringite introduced more pores, and compressive strength decreased. After adding LSP, a lower sulfate content was adequate to obtain the maximum compressive strength. Liu et al. [32] examined the effect of gypsum content on cementitious mixtures containing limestone, fly ash, and slag by studying several properties: initial and final setting time, past fluidity, water demand, and strength. Adu-Amankwah et al. [33] conducted detailed research into the consequence of sulfate additions on hydration and the performance of ternary slag–limestone composite cement using complementary techniques. Their results showed that the presence of sulfate influenced the early-age reaction kinetics of the clinker phases and supplementary cementitious materials. These changes impacted the total porosity and cement strength in opposing ways: porosity was reduced with increasing ettringite fraction, while the lower water content of the C-S-H reduced the space-filling capacity of the C-S-H. Han et al. [34] investigated the effect of gypsum on the properties of composite binders containing high-volume slag and iron tailing powder using multiple measuring techniques. They concluded that although incorporating gypsum promoted early hydration of cement and slag, it limited their further hydration at later ages. Added gypsum formed a large amount of ettringite and densified early-age pore structure, though it coarsened later-age pore structure. Fiscan [15] optimized sulfate in cement–slag blends based on calorimetry and early strength results, investigating the influence of fineness, C

_{3}A, C

_{4}AF of cement, and the Al

_{2}O

_{3}content of slag on the optimum SO

_{3}. Niemuth [7] examined the effect of fly ash on the optimum sulfate content in Portland cement, providing experimental data on strength development and heat release during early hydration for cement–fly ash systems with different SO

_{3}levels. He demonstrated that some fly ash samples achieve their sulfate demand. When a cement contains these samples, there is effectively an increase in the optimal SO

_{3}level compared to the corresponding CEM I Portland cement. In his research into the optimal SO

_{3}content of Portland and pozzolanic cement types, Tsamatsoulis et al. [35] reached the same conclusion.

_{3}, i.e., Cs.0.5H is not preferable when the objective of ΤIPA use is to enhance 28-day strength. Andrade Neto et al. [40] recently conducted a detailed review of the effect of SO

_{3}on cement hydration, noticing that despite many years of research, questions regarding sulfate optimization remain. Further investigation into the influence of clinker and CaSO

_{4}characteristics using different supplementary cementitious materials and chemical admixtures is needed.

_{3}with the cement’s chemical and physical characteristics. For achieving maximum 28-day strength with C

_{3}A, K

_{2}O, Na

_{2}O, and fineness of cement, Schade et al. [41] presented three equations (Haskell, Jawed and Skalny, and Ost). They utilized the third equation to perform a design of experiments (DoE) to model the sulfate amounts in ultra-finely ground and fast-hardening clinker. Kurdowski [42] reported four equations for approximating the optimal SO

_{3}content (Ost, Lerch, Jirku, and Haskell), before noticing that these empirical equations have limited accuracy, even if they include the main factors affecting the optimum gypsum addition. He considered that using experimentation to determine the most convenient sulfate addition is the best approach. Andrade Neto et al. [40] summarized the laboratory methods for determining the correct SO

_{3}content in conjunction with their advantages and disadvantages. From an industrial perspective, strength measurement is the most utilized method, since compressive strength is a key performance criterion for producers and customers. ASTM C563-16 [11] describes the determination of approximate optimum SO

_{3}for maximum compressive strength by measuring the change in this property of cement mortar as a result of substituting calcium sulfate for a portion of the cement. The standard suggests a parabolic equation between strength and sulfate that optimally fits the experimental points when assuming a symmetric distribution. The optimal SO

_{3}approximation corresponds to the value providing the vertex of the parabola. The standard clearly states that in cases of a skewed function of the strength versus sulfate to the right or left of the peak, an asymmetric distribution function may provide a better fit. Niemuth [7], Tsamatsoulis [35], and Fincan [15] applied the parabolic formula in their attempts to find an adequate approximation of SO

_{3}optimal content. Tsamatsoulis [35], when trying to determine a unified function for several cement types with variable clinker content, used the ratio of sulfate amount to clinker percent (%SO

_{3}/CL) as an independent variable.

_{3}content of cement using the maximization of compressive strength as a criterion for cement produced in industrial mills. The experimental design includes tests on four types of cement containing up to three main components, except gypsum, and belonging to three strength classes. We developed several relationships correlating the 7- and 28-day strengths of the sulfate and clinker content of the cement and the clinker mineral composition. We normalized the results to obtain unique functions for all experimental data, using an approach similar to the method presented in [35]. This study proved that a shallow artificial neural network [43,44] fits the data distribution better than a parabolic function. Finally, we focused on the impact of SO

_{3}uncertainty on the 28-day strength variance using the error propagation method. The structure of the paper is as follows: Section 2 includes the sampling procedure, experimental methods, and test results; Section 3 describes the implemented algorithms for data processing, including a detailed discussion of each set of results; and, finally, Section 4 summarizes the main conclusions of this research.

## 2. Materials and Methods

_{3}limit, which was 4% for CEM I 52.5 N and 3.5% for the other three CEM types. We observed that the research covered a wide range of Portland (CEM I, CEM II) and pozzolanic (CEM IV) types. The study was, therefore, general for the cement products of this specific cement plant. The optimal value of cement sulfate depended on various factors, which were summarized by Andrade Neto et al. [40] as follows: (a) clinker mineralogy (C

_{3}S, C

_{3}A) and alkali content, as well as cement and clinker fineness; (b) form of SO

_{3}carriers (Cs.2H, Cs.0.5H, Cs or alkali sulfates), as well as mineral or chemical gypsum; (c) intergrinding or separate grinding of clinker and gypsum; (d) content and type of SCM; (e) grinding aid/strength improver type; (f) hydration age; (g) water/binder ratio; and (h) curing conditions. Using the standard EN 196-1 [45] to create mortars, we attempted to find the SO

_{3}contents for maximum 7- and 28-day strength and examined the optimum position as a function of hydration age, fulfilling the conditions (f)–(h).

_{3}. For each CEM type, by operating the mill in automatic mode and after stabilizing the circuit around the operating set points and the desired fineness, we suddenly decreased the gypsum to 2%. There was no change in the percentage of clinker, while another material increased in proportion (limestone or pozzolan). Sampling of around 20 kg of the final product followed after 1.5 h. The second step was an increase in gypsum to 7.5% and a proportional decrease in limestone or pozzolan to maintain the %clinker constant. Further sampling took place 1.5 h after this change. During this process, with an operation kept as constant as possible, the laboratory sampled clinker at the mill inlet. In this way, the lab created two samples of industrial cement with low and high gypsum for each CEM type. Mixing them in proper proportions yielded samples with suitable SO

_{3}values usable to correlate strength and sulfate. The industrial tests of the four CEM types were realized within two months to allow the clinker composition to incorporate the actual production variances.

#### 2.1. Clinker and Raw Materials Analysis

_{3}and alkalies, i.e., MSO

_{3}.

_{3}, C

_{3}S, and MSO

_{3}. The corresponding annual values for 2022 of the Halyps clinker were as follows: SO

_{3}= 0.86 ± 0.25, C

_{3}S = 64.7 ± 2.5, and MSO

_{3}= 0.88 ± 0.24; these results mean that the variance in the four clinkers’ properties covers a significant amount of the actual variability in the plant clinker in terms of quality. The mineral phases measured via XRD differ from those computed with Bogue formulae, though the C

_{3}S values fit relatively well. XRD did not detect anhydrite and langbeinite because the MSO

_{3}< 1. Table 3 presents the average analysis of the three raw materials. Assuming that CaO, MgO, SO

_{3}, and LOI exist in gypsum in the form of calcium and magnesium carbonates, gypsum dihydrate, and anhydrite, we computed these four components by solving the corresponding linear system. The results show that gypsum is mainly dihydrate, containing a small percentage of anhydrite.

#### 2.2. Cement Analyses

_{CEM}) for each of the eight tests. Although the detection of gypsum phases via XRD could be approximated, the main conclusion was that hemihydrate was null or negligible. This issue arose because the cement temperature at the mill outlet never reached 100 °C, as dihydrate starts to dehydrate to Cs.0.5H [49]. Krause et al. [50] experimentally found that the transformation of gypsum into hemihydrate could take place at a temperature of 50 °C. However, our industrial results do not verify these findings.

_{3 X}, LOI

_{X}, and InsRes

_{X}, we denoted the sulfate, loss on ignition, and insoluble residue of the material X (X = CL, G, Lim, Pz, CEM). Solving the system of (9)–(11) produced the composition of CEM I 52.5 N and CEM II A-L 42.5 N. The calculation of the Pz requires the solution of the system (9)–(12) for CEM II B-M(P-L) 32.5 N. For the CEM IV(P) 32.5 N with high pozzolan content, our algorithm used three additional equations that expressed the mass balances of SiO

_{2}, Al

_{2}O

_{3}, and CaO. The composition results from the error minimization between the actual and calculated chemical analysis of cement were realized using the Generalized Reduced Gradient non-linear regression method. The lab conducted a chemical analysis, except for the eight samples, of the residues at 40, 32, and 20 microns of each of them. By determining the composition of the total sample and the material retained in each sieve, we calculated the %constituents of the material that passes through that sieve. The above approach made it feasible to calculate the composition in each fraction and investigate the grindability of the materials during co-grinding. Table 6 depicts the %components of the samples and passings through the three mentioned sieves and the residues of each material in the three sieves. Equations (13) and (14) provide these values.

_{j}is the residue of a sample at the sieve j, and RM

_{k,j}is the residue of material k at the sieve j. CompSample

_{k}is the percentage of the component k within the sample, while CompRes

_{k,j}and CompPass

_{k,j}are the percentages of the component k in the residue and passing in the sieve j.

_{3}content using industrially produced cement in a closed-circuit mill was crucial. It was difficult to obtain the same particle size per constituent and the distribution of the grinding aid via grinding in an open circuit lab apparatus or interblending the previously ground materials. Tang et al. [51] concluded that when testing to determine the optimum sulfate content, it was significant to co-grind the calcium sulfate with clinker because co-grinding resulted in lower sulfate demand than interblending. Thus, the results obtained via the latter method would not be representative.

#### 2.3. Cement Samples Mixing and Related Tests

_{3}values. Complete chemical analysis of these samples, composition calculation—as described in Section 2.2—mortars preparation, and measuring the 7- and 28-day compressive strengths followed.

_{3}and %Clinker values, the %ratio of sulfates to clinker (%SO

_{3}/CL), and the 7- and 28-day compressive strengths of each mix for the four CEM types investigated. SO

_{3}values stem from XRF measurement, while %Clinker results stem from composition calculation.

## 3. Data Processing and Results

#### 3.1. Correlation of Strength with Sulfates and Clinker Content

_{3}values. Cement fineness, the clinker’s mineral composition, supplementary cementitious materials, and the kind and dosage of strength enhancer also impact compressive strength. The results shown in Table 5 show that the fineness at 40 microns was similar for samples of 2 and 7.5% gypsum for each CEM type. Therefore, the same method would apply to the prepared mixes. Furthermore, we used the same type and dosage of strength improver for all CEM types, which enhances the 28-day strength. As the SO

_{3}value used to obtain maximum strength depends on the clinker content, we used the ratio %SO

_{3}/CL–SO

_{3}/CL as an independent variable. Our algorithm used the following dimensionless 7- and 28-day strengths to normalize the results for all CEM types.

_{3}/CL, MaxStrX is the maximum strength of the cement type, RelStrX is the relative strength for a value of SO

_{3}/CL, and X = 7 or 28. Figure 1a,b depict the relative 7- and 28-day strengths as a function of SO

_{3}/CL, proving that a single distribution can describe the results of the four CEM types.

_{3}/CL: SO

_{3}/CL was the independent variable for the first equation, resulting in a symmetric curve around the maximum, and ln(SO

_{3}/CL) was the independent variable for the second equation, taking into account the asymmetry of these experimental data. Equations (19) and (20) calculate the optimum SO

_{3}/CL of (17) and (18). For the optimal value, RelStrX(SO

_{3}/CL

_{opt}) shall be equal to 1. Constraint (21) guarantees the above values.

_{Res}, and coefficient of determination, R

^{2}, respectively. Our algorithm used the error, s

_{Opt}, which is given in Equation (24), to assess whether a model satisfactorily approximates the sulfate optimum.

_{Calc}(I) are the corresponding computed values for each model. s

_{Exp}is the standard deviation of the population of RelStrX, and k represents the degrees of freedom of the applied equation. The number of variables in (18) and (19) is three; however, k = 2 since both models are subject to the constraint (22). N

_{opt}is the count of RelStrX that is greater or equal to 0.99. Table 8 demonstrates the optimal coefficients A

_{i}, B

_{i}, with i = 0, 1, or 2, as well as the SO

_{3}/CL

_{Opt}, s

_{Res}, s

_{Exp}, s

_{Opt}, and R

^{2}for each model. The algorithm permits up to 1 outlier out of the 33 experimental points when the absolute difference between actual and computed values is higher than 2s

_{Res}.

_{3}/CL as an independent variable covers a part of the data asymmetry. In both the 7- and 28-day strengths, the logarithmic model has a higher R

^{2}than the symmetric parabolic equation and more effectively approximates the maximum strength values that show smaller s

_{Opt}values. In Figure 1a,b, black dashes indicate calculated values using the asymmetric equation, and red points depict outliers. For 7-day strength, optimum SO

_{3}/CL = 2.67 seems reasonable, though it is probably found to the left of the closest approximation. In Figure 1b, all points with RelStr28 ≥ 0.99 are found to the right of the best approximation. Therefore, there is an underestimation of the optimal value, and a better model is necessary. We implemented shallow neural networks (ANN) with one hidden layer to overcome this discrepancy.

**X**contains the SO

_{3}/CL values, and the output vectors

**RelStrX**are the 7- and 28-day relative strengths. The dimension of the vectors is M. Equation (25) provides the respecting normalized variable

**XN**, where X

_{MIN}and X

_{MAX}are the minimum and maximum values of the SO

_{3}/CL. The output vectors are already normalized because their values continuously belong to the interval [0, 1]. The hidden layer has N

_{N}nodes using the sigmoid equation as an activation function, which were provided in Equation (26). Equation (27) gives the input

**Z**to each node, where W

_{J}_{0J}and W

_{1J}are the biases and the synaptic weights between the input and the hidden layer, respectively. Finally, Equation (28) calculates the values of

**RelStrX**, where V

_{Calc}_{J}are the synaptic weights between the hidden layer and the output.

_{ResJ}, for all experimental data. The minimum s

_{Res}occurs when the number of nodes in the hidden layer N

_{N}= 2. Therefore, the continuous ANN function results in the Formulae (29), where Xn is a continuous variable belonging to the interval [X

_{MIN}, X

_{MAX}]. The numerical optimization problem contains two constraints for the optimal Xn: the value of the ANN function for this point should be close to 1, and the derivative should be close to 0. Equation (30) expresses these two constraints. The freedom degrees in Equation (22) are k = 6 − 2 = 4 due to these constraints.

_{Res}, 2.6% for R

^{2}, and 91.2% for s

_{Opt}, i.e., the ANNs more effectively approximate both the shape of the experimental curve and the optimum SO

_{3}. The same conclusion emerges from Figure 2a,b. The optimal SO

_{3}/CL increases with the age of the mortar (SO

_{3}/CL

_{28-day}> SO

_{3}/CL

_{7-day}), which confirms the previous studies’ findings [15,35,40,41].

_{train}. To estimate the test error (s

_{Tes}

_{t}), as well as due to the limited number of data (M = 33), we applied the following procedure for the 28-day ANN: (a) the training set contains M-1 datasets, and the remaining set is used for testing; (b) the non-linear regression technique calculates the optimal parameters of the training set and its s

_{Res}; (c) we compute the value of the test set using these synaptic weights, the input value, and Equation (29); (d) the difference between the experimental and the calculated test value is the error, s

_{T}, of the test; (e) the algorithm considers all possible combinations of training and test sets, resulting in M datasets of both types; and (f) Equation (31) provides the training and test errors, taking into account the error of each dataset.

_{Train}= 0.0167, and s

_{Test}= 0.0172. The coefficient of determination, R

^{2}, of the test set is 0.813. By applying the same procedure to the logarithmic model, the R

^{2}of the test set is 0.793, proving that the implementation of ANNs causes a significant improvement in all investigated statistics.

#### 3.2. Correlation of Strength with Sulfates and Clinker Mineral Phases

_{3}optimum and the C

_{3}S and C

_{3}A of clinker [14,15,16,22,24,25,33,40]. Figure 3a,b shows the relationship between the 28-day relative strength and the ratio of either (a) moles SO

_{3}/moles C

_{3}A (MSO

_{3}/C

_{3}A) or (b) moles SO

_{3}/moles C

_{3}S (MSO

_{3}/C

_{3}S), in a similar way to the method shown in Figure 1a,b. The correlation includes the mineral phases of the clinker contained in the cement, which is initially computed via the Bogue formula and shown in Table 2. Equations (32) calculate the two mentioned molecular ratios, where MW

_{SO}

_{3}= 80, MW

_{C}

_{3A}= 270, and MW

_{C}

_{3S}= 228 are the molecular weights of SO

_{3}, C

_{3}A, and C

_{3}S, respectively.

_{3}/C

_{3}A or MSO

_{3}/C

_{3}S, while Xn

_{opt}corresponds to the optimal position of each variable. Table 10 shows the synaptic weights and statistics of the two ANNs correlating RelStr28, with the mineral phases of the clinker calculated using XRF analysis and the Bogue formulae.

_{3}/CL. Figure 4a,b depicts the fitting of the ANNs results to the experimental data. F

_{ANN}(MSO

_{3}/C

_{3}A) presents a low R

^{2}, and the model is unreliable. In contrast, F

_{ANN}(MSO

_{3}/C

_{3}S) has satisfactory statistics close to those of F

_{ANN}(SO

_{3}/CL). Furthermore, the two outliers (red points) of Figure 4b are at a distance of ~3s

_{Res}from the calculated points.

_{3}A

_{XRD}and C

_{3}S

_{XRD}, as well as M

_{1}C

_{3}S

_{XRD}, which is more reactive, according to the literature [40,52]. Table 2 shows the values of these phases measured via XRD, and Equation (33) calculates the corresponding three molecular ratios.

_{ANN}(MSO

_{3}/C

_{3}A

_{XRD}) and F

_{ANN}(MSO

_{3}/C

_{3}A), we observed that the R

^{2}of the former model is much higher than that of the latter model, before concluding that determining C

_{3}A via XRD results in adequate data fitting and reliable approximation of the optimal sulfates. F

_{AA}(MSO

_{3}/C

_{3}S

_{XRD}) and F

_{AA}(MSO

_{3}/C

_{3}S) fit both similarly and satisfactorily to experimental data, and their optimal molecular ratio of SO

_{3}to mineral phase is roughly the same. The R

^{2}of F

_{ANN}(MSO

_{3}/M

_{1}C

_{3}S

_{XRD}) is relatively low, though the s

_{Opt}is sufficiently reasonable. Therefore, the corresponding optimal molecular ratio could be an initial approximation of the optimum SO

_{3}.

#### 3.3. Correlation of Strengh with Sulfates and Clinker Content under Selected Size

_{3}/CL, considering the total clinker content of each cement sample. We tried to evaluate whether some finer part of the clinker improved the relationship between the RelStr28 and the variable SO

_{3}/CL because it contributed substantially to the 28-day strength, thus applying the subsequent procedure.

- (a)
- We considered the %clinker content to be passing a size (40 μ, 32 μ), CLPass
_{j}, as given in Equation (34). The symbols j, CompPass_{1,j}, and R_{j}are the same in Formulas (13) and (34). The upper part of Table 5 shows the values of CompPass_{1,j}for the samples of low and high gypsum and the four CEM types.

- (b)
- CLL
_{j}and CLH_{j}denote the %clinker passing the sieve j for the low and high gypsum samples. We used the linear combination (35) to calculate the %clinker passing, CLP_{j}, for each mix of percentages P and 100-P of low and high gypsum samples. Table 7 provides these proportions for all CEM types.

- (c)
- We attempted two correlations using CLP
_{1}and CLP_{2}at 40 μ and 32 μ. Equation (36) provides the two independent variables of the two ANN functions.

_{3}/CL, resulting in higher uncertainty.

- (a)
- The total clinker content of the cement is the %clinker in a sieve where the passing is 100%, e.g., 90 microns, as well as SO
_{3}/CL = SO_{3}/CLP_{90}. Thus, the three ratios, i.e., SO_{3}/CL_{90}, SO_{3}/CLP_{40}, and SO_{3}/CLP_{32}, express the clinker content of cement under some sieve. - (b)
- There is an approximate linear increase in the optimal ratio SO
_{3}/CLP_{S}when the sieve size, S, decreases from 90 μ to 32 μ, as shown in Figure 6c. - (c)
- F
_{AA}(SO_{3}/CL) and F_{AA}(SO_{3}/CLP_{40}) have a similarly good fit to the experimental data, though the second ANN has a higher R^{2}, making it probably preferable. - (d)
- F
_{AA}(SO_{3}/CLP_{32}) seems to be the worst model. Its s_{Opt}is higher than those the other two models, and despite accepting two outliers, the R^{2}is the same as that of F_{AA}(SO_{3}/CL). - (e)
- Based on this model, one could conclude that cement hydration in 28 days proceeds to create clinker grains coarser that are than 32 microns, meaning that F
_{AA}(SO_{3}/CL) and F_{AA}(SO_{3}/CLP_{40}) provide a more accurate fitting to the data and a closer approximation to the optimal SO_{3}. - (f)
- The fraction between 3 and 32 microns is critical in achieving maximum 28-day strength, according to Celic [53]. Table 5 results show that P32-R3 ranges between 59.4 and 66.9%. The literature [54] states that the optimum fraction is 70%. If the CEM types studied had such P32–R3, the relationship between RelStr28 and CLP
_{32}might improve.

#### 3.4. Summary of Results

^{2}≥ 0.82. As s

_{Res}is around 0.016 for all ANNs, as well as to achieve an estimation of the confidence interval of each optimum, we considered the ratios to the left and right of the optimal value, where ResStr28 = 0.98. A second check concerned the compatibility of the optimal sulfates calculated from the six ANNs for each CEM type. The algorithm used Table 2, Table 6 and Table 7 and Equation (35) to determine the average C

_{3}S, C

_{3}S

_{XRD}, and C

_{3}A

_{XRD}and the average percentages of Clinker, CLP

_{1}, and CLP

_{2}for each CEM type. Next, it calculated the optimal SO

_{3}content per model and CEM type. Table 13 shows these results.

_{3}optimum, providing roughly the same value per CEM type. Therefore, the ANNs are reliable for the studied clinker mineralogy and cement fineness.

#### 3.5. Uncertainty Analysis

_{3}content affects compressive strength. Approximating the optimal location is critical to cement plant quality control. In this context, the SO

_{3}variance can impact the strength variance. An analytical error propagation model can facilitate a quantitative correlation between these variances. The variance of y, σ

_{y}

^{2}, of an analytical function y = f(x

_{1}, x

_{2}, …, x

_{n}) can be determined using those of independent variables x

_{i}, σ

_{i}

^{2}when applying Equation (37).

_{ANN}(SO

_{3}/CL) and RelStr28 = F

_{ANN}(MSO

_{3}/C

_{3}S). The independent variables of the first model were the SO

_{3}content, the clinker fraction CL = %Clinker/100, and laboratory reproducibility in 28-day strength measurement, σ

_{R}. The second model used the C

_{3}S content as an additional input. Equation (38) gives the derivatives of (29).

_{3}/CL, the partial derivatives of SO

_{3}and CL are:

_{3}/C

_{3}S, the derivatives of the three variables are:

^{2}

_{Str}

_{28}, of the two models.

_{3}S = 64.7%, while also assuming the following values for the input standard deviations: σ

_{CL}= 0.02, σ

_{C}

_{3S}= 2, σ

_{R}= 0.02, SO

_{3}of cement from 2% to 3% with a step of 0.1%, and σ

_{SO}

_{3}from 0.05% to 0.3%, increasing in steps of 0.05%. Figure 7a,b show the standard deviation σ

_{Str28}as a function of SO

_{3}and σ

_{SO}

_{3}for the two models investigated.

- (a)
- The shape of the functions is similar for both models, making the joint conclusions more reliable.
- (b)
- If the location of the SO
_{3}target is near the optimum value and realized independently of the SO_{3}standard deviation, the 28-day strength variance remains low. Therefore, an adequate approximation of the optimal sulfate achieves two goals: maximum strength and reduced strength variance, regardless of SO_{3}variation. - (c)
- As the SO
_{3}value differs from the optimum value, an increase in SO_{3}variance leads to a deterioration of strength variability. For SO_{3}= 2.9%, an increase in σ_{SO}_{3}from 0.05 to 0.2 causes an increase in σ_{Str}_{28}from 1.2 to 1.5 for the first model and from 1.3 to 1.6 for the second model. - (d)
- The above remarks necessitate an automatic controller that regulates SO
_{3}using gypsum dosing to achieve the SO_{3}target with minimum variance, like the model presented in the literature [46]. - (e)
- In Figure 7a,b, the surfaces are not symmetrical around the optimal line σ
_{SO}_{3}= f(SO_{3opt}), where SO_{3opt}denotes the optimal sulfate. The left side slope is steeper than the right, meaning that if SO_{3}< SO_{3opt}, the deterioration in σ_{Str}_{28}variance is more substantial than in SO_{3}> SO_{3opt}.

## 4. Conclusions

_{3}. After air jet sieving, XRF analysis provided the chemical composition of the specimens and their residues at each sieve size. The solution of a system of equations provides the %components of the above. The main conclusions of this study are as follows.

- (1)
- A unique curve can express the function between relative compressive strength and the ratio between sulfate and clinker content—SO
_{3}/CL—for all CEM types. This conclusion holds for both the total clinker and the clinker passing the 40 μ and 32 μ sieves of cement. A parabolic equation between relative compressive strength and SO_{3}/CL can fit the experimental data. Using the logarithm of SO_{3}/CL as an independent variable, the equation provides a better R^{2}than the simple parabola because it covers a part of the data asymmetry. However, the logarithmic model underestimates the optimal SO_{3}/CL position for both 7- and 28-day strength. - (2)
- A shallow ANN with one hidden layer and two nodes provides a better R
^{2}in training and test sets, as well as a closer approximation of the optimal SO_{3}/CL than the simple second-order models. The numerical algorithm for determining the synaptic weights comprises two constraints for the maximum value and its derivative. The optimal SO_{3}/CL is 2.85 for the 7-day strength and 3.0 for the 28-day strength. - (3)
- The ANN using the clinker passing at 40 μ in the sulfate to clinker ratio, SO
_{3}/CLP_{40}, gives equivalently reliable results to the first fundamental ANN with SO_{3}/CLP_{40Opt}= 3.25. The corresponding ANN using the clinker passing through a 32 μ sieve seems to be the worst among the three ANNs for the particle distributions of cement samples investigated with SO_{3}/CLP_{32Opt}= 3.33. SO_{3}/CL = SO_{3}/CLP_{90}because all cement specimens were finer than 90 μ. The three optimal sulfate-to-clinker ratios are in good linear agreement with the sieves’ grid size. - (4)
- The clinker content of the four CEM types varies from 58 to 90%, covering most of modern cement production. The above result means that the implemented ANNs and their approximation to optimal SO
_{3}can have a relatively broad application, at least as a guide for an experimental design. - (5)
- Our research investigated the well-known functions between SO
_{3}, C_{3}S, and C_{3}A computed with Bogue formulae and direct measurement via XRD. We expressed the ratio of SO_{3}using each mineral phase as a molecular ratio, and we developed the corresponding ANN models between 28-day compressive strength and these variables. - (6)
- Determination of C
_{3}A via XRD results in adequate data fitting and reliable approximation of the optimal sulfates. The optimal MSO_{3}/C_{3}A_{XRD}is 1.55, though the C_{3}A range is short for the series of clinkers examined. - (7)
- The ANNs using MSO
_{3}/C_{3}S_{XRD}and MSO_{3}/C_{3}S fit both similarly and satisfactorily to experimental data, and their optimal molecular ratio of SO_{3}to mineral phase is roughly the same, being 0.135 and 0.132, respectively. The clinkers’ C_{3}S ranged from 61 to 68, covering the majority of good reactivity clinkers manufactured nowadays. The above result means that the developed ANNs and the corresponding approximation of optimum sulfate could have a relatively general implementation. The attempt to correlate 28-day strength with M_{1}C_{3}S, which is measured via XRD, did not perform as well as previous models. - (8)
- Particular focus was also given to the impact of SO
_{3}uncertainty on the 28-day strength variance using the error propagation method. One of the main conclusions is that an adequate approximation to the optimal sulfate achieves two goals: maximum strength and reduced strength variance. An automatic controller regulating SO_{3}using gypsum dosing to achieve the SO_{3}target with minimum variance is also necessary. Therefore, if a cement plant operates the cement mills, with the SO_{3}target being close to optimum, and regulates sulfates with an automatic controller, it can gain significant improvement in the products’ quality.

_{3}optimum on an industrial scale, the tools to obtain it, and optimal ratios of sulfate to several characteristics of cement and clinker. We suggest the exclusive use of industrial cement samples of several CEM types with low and high gypsum and their convenient mixing. Using relative strength and the ratio of SO

_{3}to a quality parameter, e.g., %clinker, C

_{3}S, etc., normalizes the results, allowing them to appear in a single curve. The developed algorithm successfully used neural networks to fit and reliably predict the results. According to the authors’ knowledge, it is hard to find application of ANNs for SO

_{3}optimization in the literature. We provide approximations of optimum SO

_{3}as a function of clinker content, %clinker passing through some sieves, C

_{3}S, and C

_{3}A. We calculated some of these functions for a wide interval of input variables’ values. Thus, these approximations could have a relatively broad application in daily plant quality control, at least as a guide for experimental design.

_{3}and some main cement characteristics could continue for the relationship between optimal sulfate and cement fineness, using samples of products manufactured in industrial closed-circuit mills.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Materials**- Clinker and cement: Halyps Building Materials, S. A., Aspropyrgos, Greece.
- Raw materials suppliers
- Pozzolan: Aegean Perlites S. A., Nissiros, Greece.
- Gypsum: George Zervakis S.A.—Gypsum Mines of Eastern Crete, Sitia, Greece.
- Limestone: Aragonitis Quarry, Aspropyrgos, Greece.
- Grinding Aid: GCP Applied Technologies, Milan, Italy.

**Measuring instruments**- XRF analyzer: Malvern-Panalytical, Almelo, the Netherlands.
- XRD analyzer: Malvern-Panalytical, Almelo, the Netherlands.
- Press, mixer, jolting apparatus, molds: Toni Technik, Berlin, Germany.
- Air jet sieving apparatus and sieves: Hosokawa Alpine, Augsburg, Germany.
- Laser particle size analyzer: Cilas, Orléans, France.

**Labs performing the tests**- The laboratory of Devnya Cement AD (Devnya, Bulgaria) performed the XRD analyses. The laboratory of Halyps Building Materials S.A. (Aspropyrgos, Greece) performed the sampling and all remaining physical, chemical, and mechanical tests.

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**Figure 5.**Neural network functions between following models: (

**a**) MSO

_{3}/C

_{3}A

_{XRD}; (

**b**) MSO

_{3}/C

_{3}S

_{XRD}; (

**c**) MSO

_{3}/M

_{1}C

_{3}S

_{XRD}and RelStr28.

**Figure 6.**Neural network functions between (

**a**) SO

_{3}/CLP

_{40}, (

**b**) SO

_{3}/CLP

_{32}, and RelStr28. (

**c**) Function between SO

_{3}/PCL

_{S}and sieve opening size S.

**Figure 7.**σ

_{Str28}as a function of SO

_{3}and σSO

_{3}for models (

**a**) F

_{ANN}(SO

_{3}/CL) and (

**b**) F

_{ANN}(MSO

_{3}/C

_{3}S).

CEM | Constituent (%) | 28-Day Strength Limits (MPa) | ||||
---|---|---|---|---|---|---|

Clinker | Limestone | Pozzolan | Minor | Low | High | |

CEM I 52.5 N | 95–100 | 0–5 | 52.5 | |||

CEM II A-L 42.5 N | 80–94 | 6–20 | 0–5 | 42.5 | 62.5 | |

CEM II B-M (P-L) 32.5 N | 65–79 | 21–35 | 0–5 | 32.5 | 52.5 | |

CEM IV B(P) 32.5 N-SR | 45–64 | 36–55 | 0–5 | 32.5 | 52.5 |

XRF Analysis (%) | Phases According to Bogue | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Clinker for CEM | SiO_{2} | Al_{2}O_{3} | Fe_{2}O_{3} | CaO | MgO | SO_{3} | K_{2}O | Na_{2}O | CaO_{f} | C_{3}S | C_{2}S | C_{3}A | C_{4}AF | MSO_{3} |

I 52.5 N | 21.05 | 4.83 | 3.08 | 66.08 | 1.86 | 1.04 | 0.78 | 0.41 | 1.58 | 65.7 | 10.9 | 7.6 | 9.4 | 0.87 |

II A-L 42.5 N | 20.95 | 5.13 | 3.14 | 66.59 | 1.60 | 0.81 | 0.72 | 0.27 | 1.24 | 67.8 | 9.0 | 8.3 | 9.5 | 0.84 |

II B-M (P-L) 32.5 N | 21.06 | 5.45 | 3.08 | 66.46 | 1.48 | 0.70 | 0.66 | 0.31 | 1.52 | 63.2 | 12.8 | 9.2 | 9.4 | 0.73 |

IV B (P) 32.5 N | 21.08 | 4.78 | 2.98 | 66.15 | 2.12 | 0.97 | 0.78 | 0.34 | 2.56 | 62.2 | 13.6 | 7.6 | 9.1 | 0.88 |

Mean value | 0.88 | 64.7 | 0.83 | |||||||||||

Std. Dev. | 0.15 | 2.5 | 0.07 | |||||||||||

XRD Analysis | ||||||||||||||

Clinker for CEM | M_{3} C_{3}S | M_{1} C_{3}S | Total C_{3}S | C_{2}S | C_{4}AF | C_{3}A Cubic | C_{3}A Ortho | Total C_{3}A | ||||||

I 52.5 N | 23.3 | 37.9 | 61.2 | 20.7 | 8.2 | 2.8 | 3.7 | 6.5 | ||||||

II A-L 42.5 N | 29.8 | 37.1 | 66.9 | 15.4 | 8.8 | 2.8 | 3.7 | 6.5 | ||||||

II B-M (P-L) 32.5 N | 25.2 | 40.2 | 65.4 | 16.9 | 8.5 | 2.5 | 4.2 | 6.7 | ||||||

IV B (P) 32.5 N | 25.1 | 35.7 | 60.8 | 21.2 | 7.9 | 2.6 | 4.0 | 6.6 | ||||||

K_{2}SO_{4} | Na_{2}SO_{4} | K_{2}SO_{4}.2CaSO_{4} | 3K_{2}SO_{4}.Na_{2}SO_{4} | CaSO_{4} | ||||||||||

Clinker for CEM | Arcanite | Thenardite | Langbeinite | Aphthitalite | Anhydrite | |||||||||

I 52.5 N | 0.28 | 0 | 0 | 0.52 | 0 | |||||||||

II A-L 42.5 N | 0.44 | 0 | 0 | 0.58 | 0 | |||||||||

II B-M (P-L) 32.5 N | 0.36 | 0 | 0 | 0.58 | 0 | |||||||||

IV B (P) 32.5 N | 0.21 | 0 | 0 | 0.73 | 0 |

XRF Analysis (%) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Materials | SiO_{2} | Al_{2}O_{3} | Fe_{2}O_{3} | CaO | MgO | SO_{3} | K_{2}O | Na_{2}O | LOI (%) | InsRes (%) |

Gypsum | 1.26 | 0.81 | 0.17 | 31.31 | 1.60 | 43.29 | 0.14 | 0.01 | 20.95 | 0 |

Limestone | 1.08 | 0.33 | 0.06 | 54.22 | 1.39 | 0.02 | 0.10 | 0.03 | 42.82 | 0 |

Pozzolan | 73.61 | 12.24 | 1.90 | 0.87 | 0.19 | 0 | 3.61 | 3.90 | 2.84 | 90.5 |

Gypsum | CaSO_{4}.2H_{2}O | CaSO_{4} | CaCO_{3} | MgCO_{3} | ||||||

Compounds (%) | 87.9 | 4.1 | 1.8 | 3.4 |

XRF Analysis (%) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

CEM Type | Gypsum (%) | SiO_{2} | Al_{2}O_{3} | Fe_{2}O_{3} | CaO | MgO | SO_{3} | K_{2}O | Na_{2}O | LOI (%) | InsRes (%) |

I 52.5 N | 2 | 18.98 | 4.55 | 2.82 | 63.89 | 1.78 | 2.14 | 0.65 | 0.33 | 4.08 | 0.25 |

I 52.5 N | 7.5 | 19.00 | 4.55 | 2.83 | 62.90 | 1.79 | 4.19 | 0.63 | 0.34 | 3.00 | 0.28 |

II A-L 42.5 N | 2 | 16.82 | 3.85 | 2.30 | 63.12 | 1.47 | 1.42 | 0.67 | 0.22 | 9.58 | 0.29 |

II A-L 42.5 N | 7.5 | 16.68 | 3.89 | 2.30 | 62.23 | 1.49 | 3.59 | 0.60 | 0.23 | 8.43 | 0.24 |

II B-M (P-L) 32.5 N | 2 | 19.78 | 4.26 | 2.08 | 58.09 | 1.30 | 1.27 | 0.80 | 0.47 | 11.44 | 6.74 |

II B-M (P-L) 32.5 N | 7.5 | 19.64 | 4.19 | 2.08 | 56.83 | 1.30 | 3.52 | 0.81 | 0.58 | 10.53 | 6.68 |

IV B (P) 32.5 N | 2 | 38.75 | 6.87 | 2.49 | 41.23 | 1.44 | 1.55 | 1.87 | 1.49 | 3.54 | 30.78 |

IV B (P) 32.5 N | 7.5 | 35.16 | 6.33 | 2.43 | 43.54 | 1.52 | 3.26 | 1.67 | 1.29 | 4.06 | 32.37 |

Gypsum phases (%) | |||||||||||

CEM Type | Gypsum (%) | CaSO_{4}.2H_{2}O | CaSO_{4}.0.5H_{2}O | CaSO_{4} | T_{CEM} (°C) | ||||||

I 52.5 N | 2 | 1.52 | 0.01 | 0.21 | 80.3 | ||||||

I 52.5 N | 7.5 | 4.39 | 0.1 | 0.8 | 82.2 | ||||||

II A-L 42.5 N | 2 | 1.56 | 0 | 0.23 | 67.0 | ||||||

II A-L 42.5 N | 7.5 | 5.15 | 0 | 1 | 74.6 | ||||||

II B-M (P-L) 32.5 N | 2 | 1.61 | 0 | 0 | 66.3 | ||||||

II B-M (P-L) 32.5 N | 7.5 | 4.31 | 0.05 | 0.55 | 77.1 | ||||||

IV B (P) 32.5 N | 2 | 1.8 | 0 | 0.26 | 70.6 | ||||||

IV B (P) 32.5 N | 7.5 | 5.5 | 0 | 0.69 | 75.7 |

CEM Type | Gypsum (%) | R40 (%) | R32 (%) | R20 (%) | Blaine (cm^{2}/g) | D50 (μ) | D90 (μ) | D10 (μ) | P32-R3 (%) |
---|---|---|---|---|---|---|---|---|---|

I 52.5 N | 2 | 7.5 | 10.2 | 33.6 | 3350 | 13.5 | 39.3 | 1.5 | 63.4 |

I 52.5 N | 7.5 | 6.5 | 7.0 | 28.7 | 3520 | 12.3 | 34.5 | 1.4 | 66.9 |

II A-L 42.5 N | 2 | 5.5 | 6.7 | 28.5 | 3830 | 11.0 | 34.1 | 1.3 | 65.0 |

II A-L 42.5 N | 7.5 | 6.1 | 6.9 | 27 | 4010 | 10.7 | 33.4 | 1.1 | 65.1 |

II B-M (P-L) 32.5 N | 2 | 8.1 | 10.7 | 33.2 | 3920 | 11.5 | 38.1 | 1.3 | 61.6 |

II B-M (P-L) 32.5 N | 7.5 | 6.8 | 8.8 | 30.8 | 3980 | 11.0 | 35.8 | 1.1 | 62.8 |

IV B (P) 32.5 N | 2 | 8.5 | 17.4 | 39.5 | 3450 | 15.3 | 45.7 | 1.5 | 59.4 |

IV B (P) 32.5 N | 7.5 | 8 | 16.2 | 38 | 3570 | 13.6 | 41.6 | 1.3 | 60.8 |

Gypsum (%) | Compositions of Samples and Passings | Gypsum (%) | Compositions of Samples and Passings | |||||||
---|---|---|---|---|---|---|---|---|---|---|

CEM Type | Sieve | CL ^{1} (%) | G ^{2} (%) | Lim ^{3} (%) | Pz ^{4} (%) | Sieve | CL (%) | G (%) | Lim (%) | Pz (%) |

I 52.5 N | 2, sample | 89.7 | 2.8 | 7.5 | 0 | 7.5, sample | 89.8 | 7.5 | 2.7 | 0 |

2, 40 μ | 89.1 | 3.0 | 7.9 | 0 | 7.5, 40 μ | 89.3 | 7.9 | 2.8 | 0 | |

2, 32 μ | 88.9 | 3.1 | 8.1 | 0 | 7.5, 32 μ | 89.3 | 7.9 | 2.8 | 0 | |

2, 20 μ | 86.3 | 4.0 | 9.8 | 0 | 7.5, 20 μ | 86.9 | 9.7 | 3.4 | 0 | |

II A-L 42.5 N | 2, sample | 77.2 | 1.8 | 21.0 | 0 | 7.5, sample | 77.4 | 6.8 | 15.8 | 0 |

2, 40 μ | 76.4 | 1.9 | 21.7 | 0 | 7.5, 40 μ | 76.3 | 7.2 | 16.6 | 0 | |

2, 32 μ | 76.1 | 2.0 | 21.9 | 0 | 7.5, 32 μ | 76.2 | 7.2 | 16.6 | 0 | |

2, 20 μ | 71.2 | 2.5 | 26.4 | 0 | 7.5, 20 μ | 71.8 | 8.7 | 19.5 | 0 | |

II B-M (P-L) 32.5 N | 2, sample | 66.0 | 1.9 | 24.9 | 7.2 | 7.5, sample | 65.6 | 7.1 | 20.2 | 7.2 |

2, 40 μ | 64.9 | 2.0 | 26.3 | 6.8 | 7.5, 40 μ | 64.7 | 7.4 | 21.2 | 6.7 | |

2, 32 μ | 64.3 | 2.0 | 26.8 | 6.8 | 7.5, 32 μ | 64.3 | 7.5 | 21.6 | 6.7 | |

2, 20 μ | 57.7 | 2.6 | 32.5 | 7.2 | 7.5, 20 μ | 59.1 | 8.9 | 25.9 | 6.2 | |

IV B (P) 32.5 N | 2, sample | 59.2 | 2.3 | 4.3 | 34.2 | 7.5, sample | 57.2 | 6.4 | 3.6 | 32.9 |

2, 40 μ | 59.7 | 2.5 | 4.6 | 33.2 | 7.5, 40 μ | 57.1 | 6.8 | 3.8 | 32.3 | |

2, 32 μ | 60.3 | 2.7 | 5.0 | 32.0 | 7.5, 32 μ | 56.9 | 7.3 | 4.1 | 31.7 | |

2, 20 μ | 59.1 | 3.3 | 6.0 | 31.5 | 7.5, 20 μ | 53.9 | 8.8 | 4.8 | 32.5 | |

Gypsum (%) | Residues (%) | Gypsum (%) | Residues (%) | |||||||

Sieve | CL | G | Lim | Pz | Sieve | CL | G | Lim | Pz | |

I 52.5 N | 2, 40 μ | 8.1 | 1.1 | 2.6 | 7.5, 40 μ | 7.0 | 1.7 | 2.4 | ||

2, 32 μ | 11.0 | 1.6 | 3.8 | 7.5, 32 μ | 7.6 | 1.9 | 2.5 | |||

2, 20 μ | 36.1 | 5.7 | 13.5 | 7.5, 20 μ | 31.0 | 7.6 | 10.8 | |||

II A-L 42.5 N | 2, 40 μ | 6.6 | 0.5 | 2.0 | 7.5, 40 μ | 7.4 | 1.7 | 1.6 | ||

2, 32 μ | 8.0 | 0.6 | 2.3 | 7.5, 32 μ | 8.3 | 1.9 | 2.2 | |||

2, 20 μ | 34.1 | 2.6 | 10.1 | 7.5, 20 μ | 32.3 | 7.1 | 10.0 | |||

II B-M (P-L) 32.5 N | 2, 40 μ | 9.7 | 1.8 | 2.9 | 13.0 | 7.5, 40 μ | 8.1 | 2.4 | 2.3 | 12.3 |

2, 32 μ | 13.0 | 2.3 | 3.9 | 15.7 | 7.5, 32 μ | 10.6 | 3.0 | 2.7 | 15.0 | |

2, 20 μ | 41.6 | 6.6 | 12.7 | 33.8 | 7.5, 20 μ | 37.6 | 13.2 | 11.3 | 40.5 | |

IV B (P) 32.5 N | 2, 40 μ | 7.6 | 1.4 | 1.5 | 11.3 | 7.5, 40 μ | 8.1 | 1.9 | 2.1 | 9.7 |

2, 32 μ | 15.8 | 2.8 | 3.3 | 22.9 | 7.5, 32 μ | 16.6 | 3.8 | 4.9 | 19.2 | |

2, 20 μ | 39.5 | 12.1 | 15.6 | 44.3 | 7.5, 20 μ | 41.6 | 14.2 | 16.8 | 38.7 |

^{1}CL = Clinker,

^{2}G = Gypsum,

^{3}Lim = Limestone,

^{4}Pz = Pozzolan.

Sample with Gypsum | Compressive Strength | ||||||
---|---|---|---|---|---|---|---|

CEM Type | Low (%) | High (%) | SO_{3} (%) | Clinker (%) | %SO_{3}/CL | 7-Day (MPa) | 28-Day (MPa) |

I 52.5 N | 100 | 0 | 2.14 | 89.7 | 2.39 | 47.2 | 60.8 |

80 | 20 | 2.46 | 89.8 | 2.74 | 45.6 | 62.1 | |

70 | 30 | 2.67 | 89.4 | 2.99 | 46.4 | 63.2 | |

60 | 40 | 2.87 | 89.7 | 3.20 | 48.6 | 61.5 | |

50 | 50 | 3.09 | 89.5 | 3.45 | 44.5 | 61.8 | |

40 | 60 | 3.30 | 89.7 | 3.68 | 45.5 | 61.4 | |

30 | 70 | 3.51 | 89.9 | 3.90 | 46.9 | 63.5 | |

20 | 80 | 3.70 | 89.5 | 4.13 | 43.9 | 57.6 | |

II A-L 42.5 N | 100 | 0 | 1.42 | 77.2 | 1.84 | 41.4 | 51.6 |

80 | 20 | 1.87 | 77.5 | 2.41 | 44.2 | 53.6 | |

60 | 40 | 2.28 | 77.5 | 2.94 | 45.6 | 54.0 | |

60 | 40 | 2.29 | 77.5 | 2.95 | 44.3 | 53.4 | |

50 | 50 | 2.49 | 77.6 | 3.21 | 44.5 | 52.3 | |

40 | 60 | 2.71 | 77.5 | 3.50 | 44.0 | 52.3 | |

30 | 70 | 2.98 | 77.3 | 3.86 | 43.3 | 52.6 | |

20 | 80 | 3.15 | 77.5 | 4.07 | 42.7 | 50.5 | |

20 | 80 | 3.17 | 77.5 | 4.09 | 40.7 | 51.3 | |

0 | 100 | 3.59 | 77.4 | 4.64 | 40.0 | 49.2 | |

II B-M (P-L) 32.5 N | 100 | 0 | 1.27 | 66.0 | 1.92 | 34.2 | 43.5 |

80 | 20 | 1.74 | 66.2 | 2.63 | 36.3 | 44.4 | |

65 | 35 | 2.08 | 65.2 | 3.19 | 34.1 | 45.0 | |

60 | 40 | 2.21 | 66.1 | 3.34 | 35.1 | 45.3 | |

50 | 50 | 2.43 | 66.0 | 3.68 | 34.5 | 44.7 | |

40 | 60 | 2.67 | 65.8 | 4.06 | 32.3 | 41.2 | |

40 | 60 | 2.64 | 66.2 | 3.99 | 32.7 | 41.2 | |

20 | 80 | 3.09 | 65.7 | 4.70 | 29.8 | 40.4 | |

0 | 100 | 3.52 | 65.6 | 5.37 | 27.2 | 38.0 | |

IV B (P) 32.5 N | 85 | 15 | 1.77 | 57.4 | 3.08 | 24.9 | 37 |

80 | 20 | 1.84 | 57.3 | 3.21 | 23.5 | 36.9 | |

68 | 32 | 2.04 | 58.3 | 3.50 | 23 | 35.9 | |

55 | 45 | 2.28 | 58.2 | 3.92 | 23.1 | 34.1 | |

40 | 60 | 2.53 | 58.9 | 4.30 | 21.8 | 33.9 | |

20 | 80 | 2.88 | 58.9 | 4.89 | 20.7 | 33.5 |

ResStrX = F(SO_{3}/CL) | ResStrX = F(ln(SO_{3}/CL)) | ||||
---|---|---|---|---|---|

ResStr7 | ResStr28 | ResStr7 | ResStr28 | ||

A_{0} | 0.9269 | 0.9271 | B_{0} | 0.5139 | 0.6918 |

A_{1} | 0.0754 | 0.0673 | B_{1} | 0.9888 | 0.6315 |

A_{2} | −0.0197 | −0.0157 | B_{2} | −0.5039 | −0.3246 |

s_{Res} | 0.0252 | 0.0208 | s_{Res} | 0.0234 | 0.0177 |

s_{Exp} | 0.0552 | 0.0400 | s_{Opt} | 0.0552 | 0.0400 |

s_{Opt} | 0.0250 | 0.0154 | s_{Exp} | 0.0110 | 0.0101 |

R^{2} | 0.792 | 0.726 | R^{2} | 0.821 | 0.802 |

SO_{3}/CL_{Opt} | 1.91 | 2.14 | SO_{3}/CL_{Opt} | 2.67 | 2.65 |

ResStr7 = F_{ANN}(Xn) | ResStr28 = F_{ANN}(Xn) | |
---|---|---|

V_{1} | 0.1572 | 0.1685 |

W_{01} | −2.724 | −2.986 |

W_{11} | 17.33 | 14.82 |

V_{2} | 1.016 | 4242 |

W_{02} | 2.211 | −8.399 |

W_{12} | −1.834 | −0.3446 |

s_{Res} | 0.0227 | 0.0167 |

s_{Exp} | 0.0552 | 0.0400 |

s_{Opt} | 0.0095 | 0.0053 |

R^{2} | 0.831 | 0.823 |

SO_{3}/CL_{Opt} | 2.85 | 3.00 |

**Table 10.**Synaptic weights and statistics of ANNs using C

_{3}A and C

_{3}S as inputs (Bogue formulae).

ResStr28 = F_{ANN}(MSO_{3}/C_{3}A) | ResStr28 = F_{ANN}(MSO_{3}/C_{3}S) | |
---|---|---|

V_{1} | 0.1889 | 0.1916 |

W_{01} | −2.960 | −2.762 |

W_{11} | 12.72 | 13.62 |

V_{2} | 4230 | 4248 |

W_{02} | −8.393 | −8.400 |

W_{12} | −0.3604 | −0.4023 |

s_{Res} | 0.0241 | 0.0161 |

s_{Exp} | 0.0400 | 0.0400 |

s_{Opt} | 0.0065 | 0.0087 |

R^{2} | 0.633 | 0.835 |

Xn_{Opt}(MSO _{3}/C_{3}A_{Opt} or MSO_{3}/C_{3}S_{Opt}) | 1.25 | 0.132 |

**Table 11.**Synaptic weights and statistics of ANNs using C

_{3}A

_{XRD}, C3S

_{XRD}, and M

_{1}C

_{3}S

_{XRD}.

RelStr28 = F_{ANN}(X) | |||
---|---|---|---|

F_{ANN}(MSO_{3}/C_{3}A_{XRD}) | F_{ANN}(MSO_{3}/C_{3}S_{XRD}) | F_{ANN}(MSO_{3}/M_{1}C_{3}S_{XRD}) | |

V_{1} | 0.1785 | 0.2260 | 0.2174 |

W_{01} | −2.721 | −2.537 | −2.574 |

W_{11} | 13.44 | 11.59 | 11.67 |

V_{2} | 4241 | 4241 | 4147 |

W_{02} | −8.404 | −8.405 | −8.384 |

W_{12} | −0.3444 | −0.4396 | −0.4111 |

s_{Res} | 0.0162 | 0.0164 | 0.0203 |

s_{Exp} | 0.0400 | 0.0400 | 0.0400 |

s_{Opt} | 0.004 | 0.0064 | 0.0056 |

R^{2} | 0.833 | 0.830 | 0.738 |

Xn_{Opt}(MSO _{3}/C_{3}A_{XRDOpt} or MSO_{3}/C_{3}S_{XRDOpt} orMSO _{3}/M_{1}C_{3}S_{XRDOpt} | 1.55 | 0.135 | 0.230 |

ResStr28 = F_{ANN}(SO_{3}/CLP_{40}) | ResStr28 = F_{ANN}(SO_{3}/CLP_{32}) | |
---|---|---|

V_{1} | 0.1832 | 0.1853 |

W_{01} | −2.844 | −2.679 |

W_{11} | 13.94 | 13.32 |

V_{2} | 43.87 | 83.92 |

W_{02} | −3.806 | −4.468 |

W_{12} | −0.3836 | −0.3751 |

s_{Res} | 0.0162 | 0.0166 |

s_{Exp} | 0.0400 | 0.0400 |

s_{Opt} | 0.0067 | 0.0083 |

R^{2} | 0.835 | 0.826 |

SO_{3}/CLP_{Opt} | 3.25 | 3.33 |

ANN Model Using | ||||||
---|---|---|---|---|---|---|

SO_{3}/CL | SO_{3}/CL_{40} | SO_{3}/CL_{32} | MSO_{3}/C_{3}S | MSO_{3}/C_{3}S_{XRD} | MSO_{3}/C_{3}A_{XRD} | |

Optimal SO_{3} ratio | 3.00 | 3.25 | 3.33 | 0.132 | 0.135 | 1.55 |

Low ratio for ResStr28 = 0.98 | 2.60 | 2.82 | 2.86 | 0.113 | 0.115 | 1.34 |

High ratio for ResStr28 = 0.98 | 3.46 | 3.76 | 3.89 | 0.153 | 0.156 | 1.79 |

Optimal SO_{3} (%) for | ||||||

I 52.5 N | 2.7 | 2.7 | 2.8 | 2.7 | 2.7 | 2.7 |

II A-L 42.5 N | 2.3 | 2.3 | 2.4 | 2.3 | 2.3 | 2.3 |

II B-M (P-L) 32.5 N | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 |

IV B (P) 32.5 N | 1.7 | 1.7 | 1.8 | 1.7 | 1.8 | 1.8 |

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## Share and Cite

**MDPI and ACS Style**

Tsamatsoulis, D.C.; Korologos, C.A.; Tsiftsoglou, D.V.
Optimizing the Sulfates Content of Cement Using Neural Networks and Uncertainty Analysis. *ChemEngineering* **2023**, *7*, 58.
https://doi.org/10.3390/chemengineering7040058

**AMA Style**

Tsamatsoulis DC, Korologos CA, Tsiftsoglou DV.
Optimizing the Sulfates Content of Cement Using Neural Networks and Uncertainty Analysis. *ChemEngineering*. 2023; 7(4):58.
https://doi.org/10.3390/chemengineering7040058

**Chicago/Turabian Style**

Tsamatsoulis, Dimitris C., Christos A. Korologos, and Dimitris V. Tsiftsoglou.
2023. "Optimizing the Sulfates Content of Cement Using Neural Networks and Uncertainty Analysis" *ChemEngineering* 7, no. 4: 58.
https://doi.org/10.3390/chemengineering7040058