Modeling and Optimization of Cement Raw Materials Blending Process

This paper focuses on modelling and solving the ingredient ratio optimization problem in cement raw material blending process. A general nonlinear time-varying G-NLTV model is established for cement raw material blending process via considering chemical composition, feed flow fluctuation, and various craft and production constraints. Different objective functions are presented to acquire optimal ingredient ratios under various production requirements. The ingredient ratio optimization problem is transformed into discrete-time single objective ormultiple objectives rolling nonlinear constraint optimization problem. A framework of grid interior point method is presented to solve the rolling nonlinear constraint optimization problem. Based on MATLAB-GUI platform, the corresponding ingredient ratio software is devised to obtain optimal ingredient ratio. Finally, several numerical examples are presented to study and solve ingredient ratio optimization problems.


Introduction
Cement is a widely used construction material in the world. Cement production will experience several procedures which include raw materials blending process and burning process, cement clinker grinding process, and packaging process. Cement raw material and cement clinkers mainly contain four oxides: calcium oxide or lime CaO , silica SiO 2 , alumina Al 2 O 3 , and iron oxide Fe 2 O 3 . The cement clinkers quality is evaluated by the above four oxides. Hence, ingredient ratio of cement raw material will affect the quality and property of cement clinker significantly. Optimal ingredient ratio will promote and stabilize cement quality and production craft. Therefore, cement raw materials should be reasonably mixed. Hence, it is a significant problem to obtain optimal ingredient ratio.
Outing raw material of ball mill Automatic sampler Finished raw material Raw material homogenizer Xr a y a n a l y z e r Blending controller and blending control system Into cement kiln Artificial sample preparation · · · · · · · · · · · · Ball mill Feeding stuff Back stuff Sand stone Figure 1: The cement raw material blending process and its control system. into certain sizes. The classifier selects suitable size of original cement material which is transported to the cement kiln for burning.
The quality of cement raw material and cement clinkers are evaluated by the cement lime saturation factor LSF , silicate ratio SR , and aluminum-oxide ratio AOR . LSF, SR, and AOR are directly determined by the lime, silica, alumina, and iron oxide which are contained in cement raw material. The LSF, SR, and AOR are critical cement craft parameters, thus ingredient ratio determines critical cement crafts parameters. Likewise, critical cement craft parameters are also used to assess the blending process. In cement production, the LSF, SR, and AOR must be controlled or stabilized in reasonable range. Critical cement craft parameters are not stabilized, so it cannot produce high qualified cement. The X-ray analyzer in Figure 1 is used to analyze chemical compositions of the original cement material or raw material, then X-ray analyzer can feedback LSF, SR, and AOR in fixed sample time. The LSF, SR, and AOR can be affected by many uncertain factors such as composition fluctuation, and material feeding flow. Table 1 shows the chemical composition of original cement materials. Chemical composition is the time-varying function. The symbols μ j μ j t , η j η j t , . . ., ω j ω j t , and ϕ j ϕ j t represent chemical composition of original cement material-j. In Table 1, R 2 O represents total chemical composition of sodium oxide Na 2 O and potassium Why chemical composition is the time-varying function? Original cement materials are obtained from nature mine, thus chemical composition is time-varying function. Composition fluctuation is inevitable and it may contain randomness. With economic development, resource consumption is expanding and the resources are consuming. Therefore, original cement materials with stable chemical composition become more and more difficult to find. From the perspective of protecting environment, cement production needs to use parts of waste and sludge, therefore original cement materials composition fluctuation will be enlarged in the long run.
To some extent, modelling and optimization of the cement raw material blending process becomes more important and challenge. Because of different original cement material 4 Mathematical Problems in Engineering  type, different chemical composition, and different requirements on critical cement craft parameters, ingredient ratio should be more scientific and reasonable in blending process. Therefore, ingredient ratio should adapt to the chemical composition fluctuation and guarantee critical cement craft parameters in permissible scope.

3.11
where δ τ0 , δ r0 , δ s0 , δ λ0 , and δ π0 are the permissible maximum mass percentage of MgO, R 2 O, SO 3 , TiO 2 , and Cl in cement raw material, respectively, and τ, r, s, λ, and π are composition mass percentage vector of MgO, R 2 O, SO 3 , TiO 2 , and Cl, respectively. In cement production, the cement kiln can be divided into wet kiln and dry kiln. In 30-33 , it shows that the cement raw material with high sulphur-alkali ratio SAR will cause some problems in dry kiln. Therefore, it is necessary to control the SAR for preventing cement kiln plug and crust. The cement raw material with small SAR will increase the flammability and improve the cement clinkers quality. Some formulas are presented to calculate the SAR for cement raw material. The world famous cement manufacturers such as KHD Humboldt Company, F.L.Smidth Company, and F.C.B Company in 31 propose their formulas to calculate SAR; in practice, any of the following formulas can be used to compute SAR: The SAR is limited in permissible scope, which will reduce the environmental pollution. Strictly speaking, the blending process does not include the cement ball mill grinding process. Before cement raw materials are transported into the cement burning kiln, cement raw material blending process is considered as a whole process, thus the grinding process could be seen as part of blending process. For integrity and generality, we consider that the cement raw material blending process includes ball mill grinding process. Then, the mass balance equation of active ingredients SiO 2 in ball mill could be obtained as follows: μ j x j , k μ,j k ν μ,j , j 1, 2, . . . , n ,

3.13
where Q is original cement material output flow in ball mill, Q input is original cement material feed flow, F input is SiO 2 mass in feed flow, F output is SiO 2 mass in output flow, and k μ,j is the SiO 2 ouput mass coefficient of original cement material-j. In 3.13 , it assumes that output mass is proportional to the material flow in ball mill and mass composition percentage. Likewise, the A1 2 O 3 , Fe 2 O 3 , and CaO mass balance equation of active ingredients in ball mill will be obtained as follows: k γ,j γ j x j , k η,j k ν γ,j , j 1, 2, . . . , n .

3.15
Mathematical Problems in Engineering

3.17
In order to obtain the general nonlinear time-varying dynamic optimization model, we needs to select suitable optimization objective function. In practice, many factors should be considered such as original cement material cost, grind ability, and the error between the actual critical craft and desired critical craft. To reduce the cement cost, an optimal ingredient ratio should be pursued to reduce the original cement material cost. Thus, original cement material cost function is acquired as where C j /ton is the cost of original cement material-j, and J 1 is the cost function. To improve the grind-ability, it can pursue an optimal ingredient ratio to reduce the electrical power consumption. Thus, the power consumption function is acquired as where P j Kwh/ton is bond grinding power index of original cement material-j, and J 2 is power consumption function. P j represents the grind ability of original cement material-j and also can reflect the ball mill power consumption. To reduce critical cement craft error, it can pursue an optimal ingredient ratio to reduce LSF, SR, and AOR error. Hence, the critical cement craft error function J 3 is obtained as follows:

3.20
where w j j 1, 2, 3 is the weight of LSF error, SR error, and AOR error, Δα, Δβ, and ΔΩ are the error of LSF, SR, and AOR, and α d0 , β d0 , and Ω d0 are the expected LSF, SR, and AOR. Based Table 2: G-NLTV dynamic optimization models of the cement raw material blending process.

Optimization models Optimization objective functions Constraints
Singleobjective optimization on the cement production requirements, various objective functions are obtained. Finally, G-NLTV dynamic optimization models of cement raw material blending process are obtained as

3.21
where Ψ 1 , Ψ 2 , and Ψ 3 are the function weight. The G-NLTV dynamic optimization model includes the single objective and multiple objectives optimization model. All the optimization models contain algebraic constraints and dynamic constraints.

Analysis of Ingredient Ratio Optimization Problem and Grid Interior Point Framework
The object functions J 1 , J 2 , and J 3 in dynamic optimization models are the convex functions.
are also the convex functions. As known, the convex optimization problems have good convergent properties. The optimization problems are the convex optimization problem which is determined by their objective function and constraints. We need to check the constraints of optimization problems shown in where F θ and F α,β,Ω are the feasible regions constructed by constraints 3.12 and 3.8 , respectively. SAR θ is equivalently expressed as

4.2
Then, feasible region F θ can be equivalently written as

4.3
Likewise, critical cement craft parameters α, β, and Ω can be equivalently expressed as

Mathematical Problems in Engineering
Then, feasible region F α,β,Ω can be equivalently written as In the previous section, we know that the m s , m π , m r , m γ , m η , m ρ , and m μ are the linear functions of the ingredient ratio original cement materials mass percentage vector x x 1 , x 2 , . . . , x n T . Therefore, feasible region F θ and F α,β,Ω are the convex or semiconvex region.
Constraints 3.8 and 3.12 are nonlinear algebraic constraints, but their feasible regions are also convex or semiconvex region. Hence, feasible regions constructed by constraints 3.1 -3.12 are obtained as where F is the feasible region constructed by constraints 3.1 -3.12 , F o.e is the feasible region constructed by constraints 3.1 -3.10 , and Π is the convex and semiconvex regions set. The constraints 3.1 -3.10 are the linear algebraic constraints. Hence, the feasible region F o.e constructed by constraints 3.1 -3.10 is the convex or semiconvex. Therefore, feasible regions constructed by constraints 3.1 -3.12 belong to convex or semiconvex region. The constraints 3.13 -3.17 are the time-varying differential equation constraints in dynamic model. The constraints 3.13 -3.17 can be equally written as the following vector form:

4.7
The constraints 3.13 -3.17 in the dynamic optimization model reveal that fluctuations of the cement material flow and chemical composition will have important effects on cement raw material ingredient ratio. The derivative of ingredient ratio is affected by the chemical composition and cement material flow. In practical cement production, chemical composition is analyzed and updated by the X-ray analyzer in fixed sampling period which may be quarter hour, half hour, one hour, and even longer. Therefore, it is hard to accurately solve dynamic optimization problem 3.21 because chemical composition and cement material flow could not be continuously and accurately obtained. To simplify the dynamic model, it is assumed that the derivative of feed flow Q and ingredient ratio x are minor, and they can be ignored. Then, the constraint 4.7 can be equivalently expressed as

4.8
The original cement materials chemical composition will fluctuate with time. To solve the optimization problem 3.21 , dynamic optimization models should be transformed into discrete form. Thus, dynamic constraint 4.8 in optimization model can be transformed into the following discrete forms:

4.9
It is noted μ k μ kT s , . . . , ϕ k ϕ kT s , and x x k in 4.9 , and T s is the sampling period. Differential equation is transformed into difference equation. Constraints 3.1 -3.12 in dynamic optimization model are transformed into the following discrete forms: where h · and g · are the discrete equality and inequality constraint vectors, respectively. Hence, the continuous time dynamic model is transformed into the following discrete time form: Model.1: min J 1 min C T x Model.2: min J 2 min P T x s.t: 4.9 -4.10 .

4.11
It should be noted that i the continuous time dynamic optimization model is transformed into discrete time rolling optimization model; ii chemical composition and cement material flow cannot be obtained in a continuous and accurate way, thus it is necessary to transform the continuous model into the discrete model; iii it is difficult and complex to directly solve the continuous-time dynamic ingredient ratio model; iv the dynamic model of discrete time form is equivalent to a static optimization problem in a specific sampling time. Without losing the generality, the discrete time model can be expressed as the general form in a specific sampling period as follows: where f x : R n → R, h x : R n → R m , and g x : R n → R q are the smooth and differentiable functions, x is the decision variable, and n, m, and q denote the number of the decision variables, equality constraints, and inequality constraints, respectively. The discrete model is seen as a general linear or nonlinear static optimization problem in certain sampling period.
In recent years, various optimization algorithms and optimization toolboxes or softwares in 35-53 are developed to solve optimization problems. The optimization methods in 35-53 such as the Newton methods, conjugate gradient methods, steepest descent methods, interior point methods, trust region methods, quadratic programming QP methods, successive linear programming SLP methods, sequential quadratic programming SQP methods, genetic algorithms, and particle swarm algorithms are well established to solve constraint optimization problems. Based on interior point methods in 43-51 , a framework of grid interior point method is presented for dynamic cement ingredient ratio optimization problem. The optimization problem 4.12 could be transformed into following form: where υ > 0 is the barrier parameter, the slack vector δ δ 1 , δ 2 , . . . , δ q T > 0 is set to be positive, and g ε x is an expanded inequality constraint. It introduces the Lagrange multipliers y and z for barrier problem 4.13 as follows: where L x, y, z, δ is Lagrange function, y y 1 , y 2 , . . . , y q T and z z 1 , z 2 , . . . , z m T are Lagrange multipliers for constraints g ε x δ and h x , respectively. Based on Karush-Kuhn-Tucker KKT optimality conditions 41-43 , optimality conditions for optimization problem 4.13 can be expressed as where S δ diag δ 1 , δ 2 , . . . , δ q is the diagonal matrix and its elements are the components of the vector δ, e is a vector of all ones, ∇h x and ∇g ε x are the Jacobian matrices of the vectors h x and g ε x , respectively, ∇f x is the grand of function f x , and Y diag y 1 , y 2 , . . . , y q is a diagonal matrix and its elements are the components of vector y. The system 4.15 is KKT optimal condition of optimization problem 4.13 . When in the search procedure, δ and y should remain positive δ > 0, y > 0 . To obtain the iteration direction, we can make the point x Δx, δ Δδ, z Δz, y Δy satisfy the KKT conditions 4.15 , then the following system will be obtained: 16

4.17
The system 4.17 is obtained by ignoring the higher order incremental system 4.16 , and replacing nonlinear terms with linear approximation, system 4.17 is written in the following matrix form: (critical cement craft parameters )

Original cement material chemical composition
Select optimization objective function (single objective or multiple objectives ) Select optimization method (grid interior point method, Newton method, and so on )

Optimal ingredient ratio
Does the ingredient ratio satisfy production requirements?

Adjust objective No
End the kth batch cement blending process Yes Update the (k + 1)th batch cement production requirements and cement Choose the initial iteration point Construct the current iteration Calculate the iteration direction (∆x, ∆δ, ∆z, ∆y) Update the iteration point Output optimal ingredient ratio or iteration end conditions? (checking end conditions )

No
Yes The kth batch cement production requirements  where H x, y, z is the Hessian matrix in system. Finally, the new iterate direction is obtained via solving the system 4.18 , which is the essential process of the interior point method. Thus, the new iteration point can be obtained in the following iteration: x, δ, z, y ←− x, δ, z, y ζ 1 Δx, Δδ, Δz, Δy , 4.20 where ζ 1 is the step size. Choosing the step size ζ 1 holds the δ, y > 0 in search process. In this paper, a framework of grid interior point method is presented for optimization problem of ingredient ratio in raw material blending process. The grid interior point method framework is depicted as follows.

Grid Interior Point Method Framework
The following steps are considered.  Step Step 2. For i 1: N,each small feasible region will do the following steps.
Step 4. Constructing current iterate, we have the current iterate value x k,i , δ k,i , z k,i and y k,i of the primal variable x, the slack variable δ, and the multipliers y and z, respectively. Step 5. Calculate the Hessian matrix H x, y, z of the Lagrange system L x, y, z, δ , and the Jacobian matrix ∇h x and ∇g x are of the vectors h x and g x in the current iterate x k,i , δ k,i , z k,i , y k,i .
Step 6. Solve the linear system 4.18 and construct the iterate direction Δx, Δδ, Δz, Δy . Solve the linear matrix equation 4.18 , and then we can obtain the primal solution Δx, multipliers solution Δz, Δy, and also the slack variable solution Δδ.
Step 7. Choosing the step size ζ 1 holds the δ, y > 0 in the search process, ζ 1 ∈ 0, 1 . Update the iterate values: Step 8. Check the ending conditions for region F i . If it is not satisfied, go to Step 5, else the minimum f min,i of feasible region F i is obtained, i ← i 1, go to Step 3.
Compare the minimum f min,i of feasible region F i , output the minimum f min min{f min,i i 1, 2, . . . , N }, end.
Based on the grid interior point method framework, the algorithm structure diagram of cement raw material blending process is shown in Figure 2. In this paper, we develop the ingredient ratio software for cement raw material blending process based on the MATLAB-GUI and grid interior point method. The ingredient ratio software interface is shown in Figures 3 and 4. The ingredient ratio software has strong features which include single objective optimization model, multiple objectives optimization model, and robust ingredient ratio. The software achieves ingredient ratio for four, five, and six types of original cement materials, of course the software can be further improved to achieve ingredient ratio for more types of original cement materials. In practice, it does not exceed eight types of original cement material.

Numerical Results for Blending Process
In production, many field operating engineers will give an ingredient ratio of original cement materials based on critical cement crafts and their experiences. In this paper, a G-NLTV model and ingredient ratio software are shown to provide optimal ingredient ratios for cement raw material blending process under different production requirements. Three numerical examples are shown to depict the proposed method. It does not consider the differential or difference equation constraint because output mass coefficient and flow of original cement materials are unknown. Tables 3-5 in the Appendix display only original cement materials chemical composition in a specific sampling period, wherein the chemical composition in Table 3 1 is used to produce cement raw materials by a cement enterprise in Shan Dong province of China.
There are five types of original cement material in Table 3, and they are the limestone, sandstone, steel slag, shale, and coal ash. The steel slag is the most expensive material, the sandstone is the cheapest material, the limestone has the best grind ability, and the shale has the poorest grind ability. The optimization models discrete time and optimal ingredient ratios under different production requirements are presented in Table 6 and Figure 5. Model.1 has the smallest cost with the optimal ingredient ratio x 1 84.003%, x 2 7.687%, x 3 3.203%, x 4 0.010%, and x 5 5.097%. Model.2 has the smallest power consumption with the optimal ingredient ratio x 1 84.145%, x 2 8.021%, x 3 3.795%, x 4 0.010%, and x 5 4.029%. Model.3 has the smallest critical cement craft deviation with optimal ingredient ratio x 1 84.046%, x 2 7.335%, x 3 3.587%, x 4 0.010%, and x 5 5.021%. Model.4, Model.5, Model.6, and Model.7 are the multiple objectives optimization model which could be equivalently transformed into single objective optimization model via introducing weight Ψ 1 , Ψ 2 , and Ψ 3 . Model.4 makes balance between material cost and power consumption with optimal ingredient ratio x 1 84.658%, x 2 7.349%, x 3 3.122%, x 4 0.010%, and x 5 4.681%. Model.5, Model.6, and Model.7 have the same optimal ingredient ratio with Model.1, Model.2, and Model.4, respectively because the objective function J 3 is far less than the objective function J 1 and J 2 . In addition, the weight of objective function J 3 is not far larger than the weight of objective function J 1 and J 2 , therefore they have the same optimal ingredient ratio.
There are five types of original cement materials in Table 4, and they are the limestone, clay, iron, correction, and coal ash. The iron is the most expensive material, the limestone is the cheapest material, the clay has the best grind ability, and the iron has the poorest grind ability. The optimization models discrete time and optimal ingredient ratios under different production requirements are presented in Table 7 and Figure 6. Model.1 has the smallest material cost with the optimal ingredient ratio x 1 88.257%, x 2 7.503%, x 3 0.010%, x 4 3.731%, and x 5 0.499%. Model.2 has the smallest power consumption with the optimal ingredient ratio x 1 87.565%, x 2 8.480%, x 3 0.040%, x 4 3.905%, and x 5 0.010%. Model.3 has the smallest critical cement craft deviation with optimal ingredient ratio x 1 87.805%, x 2 7.791%, x 3 0.878%, x 4 3.516%, and x 5 0.010%. Model.4 makes balance between material cost and power consumption with optimal ingredient ratio x 1 87.555%, x 2 8.414%, x 3 0.010%, x 4 3.912%, and x 5 0.109%. Model.5, Model.6, and Model.7 have the same optimal ingredient ratio with Model.1, Model.2, and Model.4, respectively. Assuming the cost and bond power index for the cement material in Table 5 Table 3.  Table 4.
There are four types of original cement materials in Table 5, and they are the carbide slag, clay, sulfuric acid residue, and cinder. The sulfuric acid residue is the most expensive material, the cinder is the cheapest material, the carbide slag has the best grind ability, and the sulfuric acid residue has the poorest grind ability. The optimization models discrete time and optimal ingredient ratios under different production requirements are presented in Table 8 and Figure 7. Model.1 has the smallest material cost with the optimal ingredient ration x 1 75.007%, x 2 14.973%, x 3 3.620%, and x 4 6.400%. Model.2 has the smallest power consumption with the optimal ingredient ratio x 1 76.090%, x 2 19.530%, x 3 3.624%, and x 4 0.755%. Model.3 has the smallest critical cement craft deviation with optimal ingredient ratio x 1 75.442%, x 2 19.623%, x 3 4.257%, and x 4 0.678%. Model.4 Table 7: Optimization models and results for cement materials in Table 4.  Table 5.
Overall, numerical examples are presented to demonstrate various optimization problems in the blending process. The dynamic optimal ingredient ratio could be obtained in the blending process and can help to promote the cement quality if raw material chemical composition is updated with time.

Conclusions
This paper focuses on modelling and solving the ingredient ratio optimization problem in cement raw material blending process. A general nonlinear time-varying G-NLTV model of the raw material blending process is established by taking raw material chemical composition fluctuations, feed flow fluctuations, various craft constraints, and various production requirements into account. Various objective functions are presented to obtain optimal ingredient ratios under different cement production requirements. To simplify G-NLTV model and solve the optimization problem with conveniences, the optimal ingredient ratio problem is transformed into discrete time single objective rolling or multiple objectives rolling optimization problem. A framework of grid interior point method is proposed to solve the rolling optimization problem. Based on MATLAB-GUI, the corresponding ingredient ratio software is developed to achieve the optimal ingredient ratio for the cement blending process. Finally, numerical examples are shown to study and solve the ingredient ratio optimization problems in cement raw material blending process.