CEE Oct-Dec 2012

I After the chemical analysis of a blended cement, the analysis gives only the total amounts of oxides inspected. The chemical analysis cannot provide direct information about the percentage of ingredients in a cement sample. This article presents a methodology that estimates the percentages of the ingredients in a blended cement using a mathematical model in combination with the chemical analysis results. In this approach, the chemical compositions of each ingredient and the blended cement are modelled together to estimate the "most likely" amounts of ingredient proportions in the blended cement. This will allow one to determine the amount of any cement ingredient below a certain fineness level after the intergrinding of the blended cement ingredients. There are many cases of mathematical modelling used in estimating desired parameters. Application of the methodology in estimation of ingredient proportions in blended cements is the main contribution of this paper. Mathematical model In theory, the amount of an oxide (i.e. CaO) in the blended cement should be equal to the sum of the amounts contributing from each ingredient, which is the percentage of the oxide in the ingredient times the percentage of the ingredient in the blend. This weighted sum must be equal to the amount in the blended cement, because the ingredients are mixed physically, i.e. no chemical reactions take place between the ingredients. Let's assume there are "m" ingredients in the blended cement and also "n" oxides are inspected in the chemical analysis. A weighted sum equation can be written for each oxide. When the ratio of the "m" ingredients is unknown, the set of "n" weighted sum equations can be solved simultaneously to determine the ratios. The set of weighted sum equations can be written in algebraic form as show in Equation 1. where a;i: weight of oxide "j" in ingredient "i". b 1 : weight of oxide "j" in the blend . x.: unknown ratio of the ingredient "i" in the blend. [1 a] Additionally the per centages of the ingredients have to add up to I and they are greater than zero, as given in the Equation 2 Ill '!\ X= L t: I - I [2a] x,~o [2b] So, the number of equations in this problem adds up to number of oxides analysed , n, plus one. In a linear system of equations, three different cases can be observed based on the number of unknowns and equations: 1. There exists a unique solution, when the number of unknowns is equal to the number of equations. 2. There are multiple solutions, when the number of unknowns is more than the number of equations. 3. There is no solution , when the number of unknowns is less than the number of equations. When there are more equations than unknowns, then there is no solution that can satisfy all equations at the same time, unless some of the equations are linearly dependent. In the case of blended cement, the number of equations of type Equation I (one for each oxide) and Equation 2 result in more equations than unknowns. If the chemical analysis was perfect and the blended cement and ingredients were homogeneous, the equations could be linearly reduced until the number of equations is equal to the number of unknowns. Due to imperfections in the measurements and the nature of the product (i .e. the accuracy of analysis for each oxide, heterogeneous nature of the ingredients and the blended cement, etc.), the equations are not expected to be linearly dependent. Thus, there is no analytical solution that finds the ratio of ingredients in the blended cement directly from the set of equations. When an estimation of the ingredient ratios is put into the Equation I, equality may be valid for some of the oxides but not for all. In Equation I, the left hand side is the oxide amount derived from the weighted sum of the ingredients and the right– hand side is the oxide amount analysed from the sample taken from the blended cement. For any given ratio estimate, in some equations the left– hand side may be equal to the right-hand side. For the rest of the equations, the left hand side will be more or less than the right-hand side. Recognizing this discrepancy, a deviation term S; is 14