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Optimizing the Cement Compressive Strength Prediction by Applying Coupled Linear Models DIMITRIS TSAMATSOULIS Halyps Building Materials S.A., Italcementi Group 17th Klm Nat. Rd. Athens – Korinth GREECE [email protected] http://www.halyps.gr Abstract: - This study is aiming to search the optimum settings of two kinds of dynamic linear models predicting cement typical strength and afterwards to couple them to achieve optimality. The modeling is based on physical and chemical characteristics and on the early strength of the Portland cement types studied. More than 3000 sets of industrial results were used for this purpose. Models parameters are calculated using a moving past period of length TD and are tested in a future period of length TF. The moving period characterizing the dynamical models attempts to assure that changes in the process are taken into account during the parameters calculation. Among the models the coupled one is noticeably superior as regards the mean square residual error. The implementation of these methods in the daily quality control is an

essential factor of quality improvement by maintaining a low variance of typical strength. Key-Words: - Strength, Model, Cement, Dynamic, Prediction, Quality Control Osbaeck et al. [5] analyzed the effects of particle size distribution and surface area upon Portland cement strength. Tsivilis et al. [6] and GarciaCassilas et al. [7] developed regression models to predict the early and typical strength of Portland cement, pointing out the importance of chemical, mineralogical and fineness factors on the prediction of cement strength. Kheder et al. [8] developed a similar multiple linear regression model, where an accelerated testing of compressive strength has also been utilized. In case the linear model was not sufficient, logarithms of the independent variables were introduced. Very similar modelling has also been performed by Abd et al. [9]. Tepecik et al. [10] presented a multiple linear regression model to predict the strength of 2, 7 and 28 days of CEM I 42.5 cement. Tsamatsoulis et al. [11] using multivariable modelling and uncertainty analysis optimized the SO3 content of cement using maximization of compressive strength as a criterion. Meschling et al. [12] applied a power law model to estimate 28 days compressive strength of CEM I cement using as parameters a coefficient k and an exponent b. The mathematical treatment of the results made it possible to connect parameter k to the C3S rates of the clinker. Popovics [13] introduced a mathematical form for the prediction of concrete strengths obtained at various curing temperatures from the properties of the cement used. He considered the hydrations of C3S and C2S as first

1 Introduction Predicting cement typical compressive strength from earlier analyses results remains a challenging research field, despite the huge number of attempts and the straightforward or more sophisticated techniques chosen. As typical cement strength is defined the one measured 28 days after mortar preparation. This characteristic is thought as the main indicator of the product quality therefore worldwide the cement standards apply specifications with regard to the low and high limit of typical strength. The stability of the cement quality is mainly described by the variance of 28 days strength. Numerous methods and techniques have been developed in estimating the 28-day cement strength, selecting as inputs some of the subsequent characteristics of cement and clinker: clinker mineral compounds, content of main oxides, cement fineness, early strength, composition and chemical analysis. Modeling is mainly based on linear or nonlinear regression algorithmd, fuzzy logic or neural networks. Lee [1], in his historical book “The Chemistry of Cement and Concrete”, referred past attempts to correlate cement strength with clinker mineral composition, cement composition and fineness. Odler in 1991 [2] presented a detailed review of correlations between cement strength and basic factors related with physical and chemical properties of clinker and cement. Zhang et al. [3], Celik [4] and

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dynamic linear model as concerns the prediction of future 28 days strength, by taking into account the information becoming from the second one. This latter information is filtered and added to the result of the first model. The filtering parameters are also optimized. The above constitutes the coupled linear model of 28 days strength prediction. The study is restricted to Portland cement.

order reactions. Tsamatsoulis [14] presented a similar kinetic model of strength development using different cement and aggregate types. De Siqueira Tango [15] presented an extrapolation method for compressive strength prediction of cement products considering that the typical 28 days strength is a function only of the two earlier ages’ of 2 and 7 days strengths. Tsamatsoulis [16] elaborated a multivariable polynomial model by utilizing the 1 and 7 days early strength, physical and chemical data to predict the typical strength. The main feature of the two last contributions [15,16] is the introduction of the early strength as an independent variable, despite that this strength is a function of the physical and chemical structure of cement. The main feature of the described models is that they are static: (a) A set of data is used to estimate the model parameters; (b) the future strength is computed based on these set of parameters. Relis et al. [17] developed a linear regression model to predict the strength of Portland cement by introducing a time-sequence dynamic correction procedure to enhance the model accuracy. Their final model includes constant coefficients of the linear model and the addition of the dynamical correction. Tsamatsoulis [18] performed an initial comparison of the static polynomial equations referred in [16] and movable time horizon models based on linear regression methods. The latter models incorporate the uncertainty due to the time variability of non involved factors during the modelling procedure and they can be characterized as dynamic. In [19] the particularities of these two classes of models have been investigated in detail. This study is initially based on the analysis of Tsamatsoulis in [18, 19] where dynamic linear models have been utilized to predict the typical cement strength. The models are characterized as dynamic because the parameters were estimated from a moving set of data belonging to a predefined past time interval. Two classes of equations developed in these earlier studies: Except physical and chemical data, the first one utilized the one day strength results while the second one the results of seven days strength too. The first model is broadly utilized in Halyps cement plant to regulate the cement composition according to the 28 days strength estimation. The second model is much more accurate from the first one but due to its much bigger delay time compared with the first one, generally it cannot be used for direct control purposes. However, it constitutes additional information for the cement composition adjustment. These two models operate independently. The main objective of this study is to optimize the first class of

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2 Experimental Two Portland cement types produced according to EN 197-1:2011 were studied: CEM II A-L 42.5 N and CEM II B-M (P-L) 32.5 N. The first type, except clinker and gypsum, contains also limestone, while the second type pozzolane and limestone, as main components. The modeling is based on the results of the daily average samples of cement produced in two cement mills (CM) of Halyps plant. The analyses made on these samples were the following: (i) Residue at 40 μm sieve, R40 (%), measured with air sieving . (ii) Specific surface, Sb (cm2/gr), measured according to EN 196-6. (iii) Loss on ignition, LOI (%), and insoluble residue, Ins_Res (%) of the cement measured according to EN 196-2. (iv) SO3 (%) measured with X-ray fluorescence. (v) Compressive strength at 1, 7 and 28 days (MPa). The preparation, curing and measurement of the specimens were made according to the standard EN 196-1. The modeling predicting the 28-day strength was based on more than 3400 data sets of cement fineness, chemical analyses, 1, 7 and 28 days strength.

3 Mathematical Models Predicting Strength The common independent variables in all models are: LOI, SO3, Ins_Res, R40, Sb. The reason to utilize the chemical analysis of cement instead of the cement composition used in the earlier modeling presented in [18], is to generalize as much as possible the derived equations: The direct usage of chemical analysis do not need prior knowledge of raw materials and clinker composition. Two basic and independent equations are initially implemented to predict the 28 days strength. (i) The one where the one day strength - Str_1- constitutes model variable, except the set of physical, chemical data. This model consists of 6 independent variables and is named Str_28_1. (ii) The second one where the seven days strength variable –Str_7- is also included and the

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model, named Str_28_7, is a 7 variables linear equation. (iii) The third model contains all the independent variables of Str_28_1 and an additional correction computed from the moving difference of Str_28_7 and the actual 28 days strength measured in the date the model is to be applied. The model is named Str_28_EW. The linear function describing the three models is given by equation (1).

without new parameters estimation. Consequently TF is ≥ 1. (vii) According to step (vi) for the dates belonging to the interval [t, t+TF-1], the future strength of the cement produced in the time intervals [t-2, t+TF-3], [t-8, t+TF-9] is computed according to the equation of step (v). Otherwise if the date is greater than t+TF-1, new parameters estimation is performed starting from step (i). (viii) As the time span remains TD, when the results of TF days are completed, then, the time interval is moved on by TF days. Thus the future 28 days strengths are calculated using models applied to data sets of movable time span TD and in steps of length TF. (ix) Parameters TD, TF shall be optimized considering the following two criteria: (a) minimum sPast during modeling and (b) minimum error sFutur during the future application of the models. (x) For each TD and TF and for each past and future time interval, a set (AI, sPast, sFutur) is computed from the samples belonging to this interval. Depending on TD and TF values, the number of the consecutive sets (AI, sPast, sFutur) is KTD, the number of data sets in each past interval I is NTD(I) and in each future interval J is NTF(J). The mean square residual errors, sPast, during modeling and sFutur during future prediction are calculated by equations (3) and (4) respectively.

For all the models the parameters AI, I=(0..N) were estimated from a moving set of data belonging to a past time interval of predefined size TD in days and the model is applied for a determined time interval TF before the parameters to be recalculated. The latest date of the time interval TD has two characteristics: (i) the 28-day strength has been measured; (ii) its distance from the first date of prediction is minimal. The parameters AI, I=(0..N) are computed via regression by minimizing the objective function given by equation 2.

where Yact = actual 28 days strength, Ycalc = the calculated one from the model, M = number of data sets, p = number of independent variables. 3.1 Parameters Estimation Algorithm The parameters of models Str_28_1, Str_28_7 were computed by the following algorithm: (i) At date t a new 28-day strength result appears. The specimen was prepared 28 days ago. The production date is in distance t-29 days from the current date t. (ii) A time interval of TD days and the samples belonging to the period [t-29-TD, t-29] are presumed. The dynamic data set consists of this population of samples. (iii) Using multiple regression the model parameters AI (I=0 .. N) and sres are computed. (iv) At day t, the chemical and physical results of the cement produced in the previous day, the 1 day strength of the cement produced 2 days ago and the 7 days strength of cement produced 8 days ago have been measured. (v) With the set of parameters computed in step (iii) the 28 days strength of cement produced at t-2 and t8 days are estimated, by applying the models Str28_1 and Str28_7 respectively. (vi) The steps (iv), (v) are repeated up to the date t+TF-1, where TF is a predetermined time interval

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The described procedure is a generalization of the respecting procedure presented in [18] and constitutes a step forward as to the method optimization. 3.2 Combination of Linear Models and Dynamical Correction The third model – Str_28_EW – constitutes an extension of Str_28_1 and first of all needs the definition and implementation of the moving average filter. The exponentially weighted moving average (EWMA) technique was used for this purpose. As analyzed in [16], for a variable X and discrete time I, the EWMA variable Y is defined by the procedure: (i) For time I=0 the initial moving average Y(0) is expressed by the relation (5):

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(ii)

Table 1. TD, TF, KTD, NTD, NTF values TD NTD TF KTD 90 99 1 2191 120 131 2 1476 180 195 5 634 240 258 10 311 20 153 30 103 60 51 90 34

For a parameter λ, where 0 < λ ≤ 1, the statistic Y(I) is computed by the recursive formula (6):

If λ=1, the moving average values become equal to the current ones. As long as smaller λ value is, the rate of change becomes lower and trends of higher duration can be revealed. It is supposed that at time I the actual 28 days strength is Str_28(I) and the computed one from Str_28_7 model is Str_28_7(I). The difference Diff(I) is defined by formula (7). (iii)

Afterwards the following procedure is applied: (i) The moving average of Diff(I), EW_Diff(I), is calculated by applying equation (6) for a predefined value of λ. (ii) The corrected Str_28_1(J) value, named Str_28_EW(J), is computed from equation (8), where J≥I. The last date I is attributed to date J.

(iii)

NTF 2 4 7 12 22 33 64 96

Figure 1. MSRE as function of TD for Str_28_1 and Str_28_EW models

The parameters k and λ of the coupled linear models, constitute model parameters and need optimization as concerns the mean square residual error (MSRE) given by equations (3), (4).

4 Analysis of Results and Discussion 4.1 Analysis of the modeling mean square errors The modeling and future strength MSRE, were computed for ranges of TD and TF shown in Table 1. The above ranges are applied for all the three models. The average values of KTD, NTD NTF are also presented. The simulation based on 3441 sets of industrial data shows that the modeling MSRE is independent of TF but strong function of TD. The modeling errors of Str_28_1 and Str_28_EW models as function of TD are shown in Figure 1. The respecting errors of Str_28_7 model are demonstrated in Figure 2. The MSRE of Str_28_EW model is derived after an optimization of k and λ parameters. An analysis of this issue follows in the next paragraphs.

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Figure 2. MSRE as function of TD for Str_28_7 model As shown in the earlier works [18, 19] the residual errors of Str_8_7 models are noticeably lower than that of Str_28_1 model. The significant point observed from Figure 1 is that the Str_28_EW model derives better MSRE than Str_28_1 model. The improvement is higher as TD decreases. To investigate closer the improvement of MSRE achieved with Str_28_EW model, the frequency distribution of the sRes of both models for TD=90 and TF=1 and for all the population of KTD points is created and shown in Figure 3.

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Separate (k, λ) are found for each CEM type: The developed software classifies the data set to CEM A-L 42.5 N or CEM II B-M 32.5 N according to LOI and Ins_Res values (v) Therefore, four optimum values are searched, (kAL, λAL, kBM, λBM). (vi) The steps (i)-(iii) are implemented for each (TD, TF) and the optimum (kAL, λAL, kBM, λBM), producing the minimum MSRE of test sets, is determined. The optimization technique used needs further research but the designed one provides the actual optimum despite it makes calculations to all the lattice points. Because MSRE is a function of four parameters cannot be shown in a single drawing. In Figures 4 and 5, the MSRE as a function of kBM, λBM is demonstrated for (kAL, λAL) equal to (1, 0.1) and (0.7, 0.5) respectively, for TD=180 and TF=30. (iv)

Figure 3. Frequency distributions of sRes for TD=90, TF=1 From Figure 3 it is observed that the distributions are bimodal as two relative peaks appear in each one. The above can be attributed to process changes as the period under consideration is more than nine years. The distribution of Str_28_EW model is clearly moved to the left related with the one of Str_28_1 in all the range of sRes. The above explains the noticeable improvement of MSRE. 4.2 Optimization of parameters k and λ Using neural network terminology, the set of data during modelling corresponding to a time period equal to TD, is the “training set” as concerns the parameters of each model. On the other hand the future data set, corresponding to a period equal to TF, is the “test set”. The MSRE of the Str_28_EW model during modelling is lower that the respecting of Str_28_1 for the same TD values. This improvement is due to the optimization of parameters of equations 6 to 8. Two difficulties raised during this problem solving: (a) Not any value of the k, λ, reduces the MSRE but an optimization technique is needed; (b) a non-linear regression technique converges usually to local minimum values, thus it cannot guaranty the optimum values. Thus the following steps were implemented: (i) Minimum and maximum values for k, λ, kMIN, kMAX, λMIN, λMAX are selected. (ii) Steps of k, λ change, dk, dλ are also chosen. (iii) All the rectangular range defined by the vertices kMIN, kMAX, λMIN, λMAX is scanned with steps dk, dλ and the MSRE during training and test provided by equations (3) and (4) are determined but as optimization criterion the minimum MSRE of test set is selected. The optimum k, λ correspond to the minimum MSRE of test set.

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Figure 4. MSRE of test sets for (kAL, λAL)=(1, 0.1)

Figure 5. MSRE of test sets for (kAL, λAL)=(0.5, 0.7) Figures 4, 5 verify that model Str_28_EW efficiency depends on the optimization of the k, λ parameters. Generally the optimum is found at low values of λ and high values of k, in the intervals (0.1, 0.2) and (0.9, 1.0) correspondingly. From these

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results it is concluded that a slow EWMA, e.g. small λ parameter, provides a more effective correction and amelioration of the prediction. 4.3 Analysis of the mean square errors of test sets The combination of the Str_28_1 model with the Str_28_7 one, via the dynamical correction and the optimized parameters k, λ, provides a coupled model of predicting cement 28 days strength. This new linear model, named Str_28_EW contributes to a noticeable improvement of the predicting ability of the linear models. The MSRE of Str_28_1 and Str_28_EW for TD=90, 120, 180, 240 are demonstrated in Figures 6, 7, while the MSRE of Str_28_7 is shown in 8 for all TD values.

Figure 8. MSRE of Str_28_7 as function of TF for TD=90, 120, 180, 240 days.

Figure 6. MSRE as function of TF for TD=90, 120 days

Table 8. Ratio function of TD, TF

MSREStr_28_EW/MSREStr_28_1

as

The results show that: For each TD value as TF decreases the MSRE of all the three models decrease too. - The Str_28_EW model results in a severe reduction of the error compared with the St_28_1. Especially from Figure 8 it is concluded that MSRE of Str_28_EW remains continuously lower than the respecting of Str_28_1. The average reduction is 6%, reaching up to 10%. - The minimum MSRE is achieved with Str_28_EW and TD=240, TF=1, e.g. these are the optimum time parameters -

Figure 7. MSRE as function of TF for TD=180, 240 days

5 Conclusions

The ratio MSREStr_28_EW/MSREStr_28_1 is computed for each (TD, TF) and shown in Figure 9.

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Predicting the 28-day strength of cement using linear models constitutes a methodology easily applied in the daily quality control of the cement production. The essential is to increase the reliability

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[5] Osbaeck, B. and Johansen, V., Particle Size Distribution and Rate of Strength Development of Portland Cement, Journal of the American Ceramic Society, Vol. 72, 2005, pp. 197-201. [6] Tsivilis, S. and Parissakis, G., A Mathematical Model for the Prediction of Cement Strength, Cement and Concrete Research, Vol. 25, 1995, pp. 9-14. [7] García-Casillas, P. E., Martinez, C. A., Montes, H. and García-Luna, A., Prediction of Portland Cement Strength Using Statistical Methods, Materials and Manufacturing Processes, Vol. 22, 2007, pp. 333-336. [8] Kheder, G. F., Gabban, A.M. and Abid, S.M., Mathematical model for the prediction of cement compressive strength at the ages of 7 and 28 days within 24 hours, Materials and Structures, Vol. 36, 2003, pp. 693-701. [9] Abd SM, Zain MFM, Abdul Hamid R. Modeling the Prediction of Compressive Strength for Cement and Foam Concrete, Proc. of International Conference on Construction and Building Technology, Kuala Lumpur, 2008, pp. 343 – 354. [10] Tepecik, A., Altin, Z. and Erturan, S., Modeling, Compressive Strength of Standard CEM I 42.5 Cement Produced in Turkey with Stepwise Regression Method, Journal of Chemical Society of Pakistan, Vol. 31, 2009, pp. 213-220. [11] Tsamatsoulis, D. and Nikolakakos, N., Optimizing the Sulphates Content of Cement Using Multivariable Modeling and Uncertainty Analysis, Chemical and Biochemical Engineering Quarterly, Vol. 27, 2013, pp. 133144. [12] Mechling, J. M., Lecomte, A. and Diliberto, C., Relation between Cement Composition and Compressive Strength of Pure Pastes, Cement and Concrete Composites, Vol. 31, 2009, pp. 255-262. [13] Popovics, S., Model for the Quantitative Description of the Kinetics of Hardening of Portland Cements, Cement and Concrete Research, Vol. 17, 1987, pp. 821-838. [14] Tsamatsoulis, D., Kinetics of Cement Strength Development Using Different Types of Cement and Aggregates, WSEAS Transactions on Systems, 2009, Vol. 8, pp. 1166 - 1176. [15] De Siqueira Tango C.E, An extrapolation method for compressive strength prediction of hydraulic cement products, Cement and Concrete Research, Vol. 28, 1998, pp. 969-983. [16] Tsamatsoulis, D., Control Charts and Models Predicting Cement Strength: A Strong Tool

of these models. For this reason dynamic models have been elaborated, correlating the 28 days strength with physical and chemical features and early strength: The first one named Str_28_1 which takes into account the strength measured at one day, and the second one, the Str_28_7, including also the seven days strength. A third improved model takes into account the moving average difference between the Str_28_7 values and the actual ones to correct the Str_8_1 model. In this way a coupling of the two independent models is succeeded resulting in a combined linear model. The software developed optimizes two parameters per cement type, to determine the minimum MSQE of the test sets. The models are applied to a moving data set belonging to a past time interval of predefined size TD in days, to calculate the parameters and then to a future time interval TF. The first set is the training set, while the second one is the test set. The main conclusions of the presented analysis are the following: (a) for each TD value, as TF decreases the MSRE of the test set decrease too; (b) the Str_28_EW model results in a severe reduction of the error compared with the St_28_1. The MSRE of Str_28_EW remains continuously lower than the respecting of Str_28_1. The average reduction is 6%, reaching up to 10%. (c) The minimum MSRE is achieved with Str_28_EW and TD=240, TF=1, e.g. these are the optimum time parameters. The further improvement of these techniques can follow the next directions. - Combination of the dynamical models with neural network techniques to investigate possible synergies. - Exploitation of the dynamic models to develop robust controllers based on Model Predictive Control (MPD) techniques or other advanced methods.

References: [1] Lee, F.M., The Chemistry of Cement and Concrete, 3rd ed. Chemical Publishing Company, New York, 1971. [2] Odler, I., Cement Strength, Materials and Structures, Vol. 24, 1991, pp. 143-157. [3] Zhang, Y.M. and Napier-Munn, T.J., Effects of Particle Size Distribution, Surface Area and Chemical Composition on Portland Cement Strength, Powder Technology, Vol. 83, 1995, pp. 245-252. [4] Celik, I.B., The Effects of Particle Size Distribution and Surface Area upon Cement Strength Development, Powder Technology, Vol. 188, 2009, pp. 272-276.

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Improving Quality Control of Cement Production, 16th WSEAS International Conference on SYSTEMS, Recent Researches in Circuits and Systems, Kos, Greece, 2012, pp. 136-145. [17] Relis, M., Ledbetter W.B. and Harris, P., Prediction of Mortar-Cube Strength from Cement Characteristics, Cement and Concrete Research, Vol. 18, 1988, pp.674-686. [18] Tsamatsoulis, D., Prediction of cement strength: analysis and implementation in process quality control, Journal of Mechanical Behavior of Materials, Vol. 21, 2012, pp. 81-93. [19] D. Tsamatsoulis, Application of the static and dynamic models in predicting the future strength of pozzolanic cements, 18th International Conference on Circuits, Systems, Communications and Computers, Santorini, Greece, 17-21 July 2014, Latest trends in Systems vol. I, pp. 138-146.

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