Mathematics: A Tool of Quality Control

by Clyde W. Moore

The design and construction of portland cement plants includes provisions for the required raw materials. The number of raw material streams will be determined by an oxide analysis of the local materials and the need for added supplements. The simplest combination available will probably be a limestone, and a clay or shale, from a local quarry. But would that combination need to be supplemented by sand or a high silica waste product from another process to increase the silica content; or iron ore to adjust the C3A content in the finished cement; or a higher alumina-bearing material?

Control of the chemical process will be exercised by controlling the flow rates of the individual raw material streams. The answer to how many streams there will be may rest on what types of portland cement the company management chooses to make at that plant. Provisions for two to four raw materials would likely be included. And a mathematical fact of life emerges from this.

With two raw materials you can control one chemical item. That’s because when the quantity of one raw material is fixed, the quantity of the second is automatically fixed, since the sum of the fractions of the two must equal unity. With three, you can control two items, and with four you can control three items. Referred to as the “degrees of freedom,” the number of items controlled is one less than the number of raw materials used. As the plant gets into operation, a need for other minor constituents may arise. They may be combined with one of the other raw material streams to the raw mill.

Choices must be made
The chemist’s selection of what to control is of fundamental importance. There are a number of choices, but some are better than others. In the early days of cement manufacturing, two raw materials—a limestone and a clay (or shale), carefully selected by their oxide analyses—may have been sufficient. And CaO was chosen to be the item controlled because it was the most abundant oxide in the raw mix, and it was faster and easier to determine than the silicon dioxide.

Logic dictated that if only one item could be controlled, it should be a calculated value composed of several oxides. Michaëlis proposed the hydraulic modulus (H) about 1900. (Ref. 1, pp 75-76) It will work, but is it the best choice?

(1)    H = CaO / (SiO2 + Al2O3 + Fe2O3)

And, in the period 1882 to 1905, Henri Le Chatelier opened the door to use of the potential compound composition as control items (Ref. 2, p 65). In about 1929, Kühl proposed the ratio of aluminum oxide to iron oxide (alumina modulus or A/F ratio), (a)

(2)     a  = Al2O3 / Fe2O3

and the ratio of the silicon dioxide to the sum of the aluminum oxide and the iron oxide. (Ref. 3, p 166) The latter, called the silica modulus, or silica ratio, (R)

(3)    R = SiO2 / (Al2O3 + Fe2O3)

allows control of the “burnability” of raw mix—the ability of the kiln to produce the desired compound composition. As the silica ratio decreases, it becomes easier for solid materials to become liquid in the kiln and tend toward fluxing. As R increases, it becomes more difficult for the potential compounds to form from the components of the raw materials. And the difficulty is increased if the SiO2 is present as quartz rather than in the form of silicates.

The A/F ratio has the property that the ratio of C3A to C4AF is a function of a alone.

(4)    C3A/C4AF = 0.871a – 0.556

Likewise, the fractions of C3A and C4AF in the total aluminoferrites (C3A + C4AF) are also a function of a alone.

(5)    C3A / (C3A + C4AF) = 1.000 –                     [1.148 / (a + 0.510)]
(6)    C4AF / (C3A + C4AF) = 3.043 / (2.650a + 1.351)

So the door was open to the idea of a single control parameter yielding some degree of control over more than one potential compound.

Moore has shown the derivation of a lime factor, which was used by a major cement manufacturer in the U.S. for more than 60 years. (Ref. 4, pp 511-515)

(7)    f = [CaO – (1.65Al2O3 + 0.351Fe2O3)] / SiO2

As written, this lime factor may apply to cement raw mix or to clinker without a correction for loss on ignition, since it is a ratio of oxides. The numerator represents the CaO available to react with the SiO2 in the denominator.  For it to apply to cement containing gypsum, there must be a factor related to the SO3 content included in the numerator within the parentheses. This lime factor has the property that the ratio of C3S to C2S is a function of f alone.
(8)    C3S / C2S = 1.326[(f−1.867) / (2.800 − f)]

Likewise, the reciprocal of the ratio is also a function of f alone, and the fraction of the C3S and C2S in the total silicates (C3S + C2S) is a function of f alone.

(9)    C3S / (C3S + C2S) = (4.071f − 7.600) / (f + 1)           
(10)    C2S / (C3S + C2S) = [11.671/(f +1)] − 3.071

Now, we have a control parameter (a), which relates the ratio of C3A to C4AF; a control parameter (f), which relates to the ratio of C3S to C2S; and a control parameter (R), which relates to burnability and fuel consumption. With one more parameter, the total of the four major oxides, one can write four equations in four unknowns relating oxide analysis to the control parameters. A simple sum of the four major oxides may provide the fourth equation.

(11)    t = CaO + SiO2 + Al2O3
(12)    0 = CaO  – f(SiO2) –                 1.65(Al2O3)–
(13)    0 = −SiO2 + R(Al2O3) +             R(Fe2O3)
(14)    0 = Al2O3 − a(Fe2O3)

Most cement plants use raw materials from reliable sources that limit the variability of both the minor constituents and the major constituents. The result is that the total of the four major oxides varies very little and may be considered constant. Even when there is change in that total, it is reflected in relative amounts of each of the four major compounds. The system will operate within a narrow band of systems of chance causes until a major change in a raw material occurs—a significant increase or decrease of dolomite [CaMg(CO3)2] in the limestone, for instance.

The system of equations above may be solved by algebraic substitution for the individual oxides in terms of t, f, R, and a. Once the oxide analysis is known, it is straightforward to calculate the compound composition in terms of t, f, R, and a using the Bogue Equations as cited in ASTM C 150-07, “Standard Specification for Portland Cement,” appendix A1.

Quantitative relationships with control parameters
With some algebraic manipulation, equations for the four major potential compounds may be written in terms of a, R, f, and t. Those equations are as follows: (Ref. 4, p 513)

(15)    C3S = [R(a + 1)(4.071f – 7.600)t] /                 [R(f + 1) (a + 1) + 2.650a + 1.351]
(16)    C2S = [R(a + 1)(8.600 – 3.071f)t] / [R(f + 1)
(a + 1) + 2.650a + 1.351]
(17)    C3A = [2.650(a − 0.638)t] / [R(f + 1)(a + 1)             +2.650a + 1.351]
(18)    C4AF = (3.043t) / [R(f + 1)(a + 1) + 2.650a +

Notice that in each numerator, the parameter t appears as a multiplier, and the denominators are identical in each case. Therefore, each equation is the decimal fraction of that compound times the total of the four major compounds, which is the same as the total of the four major oxides.

Let’s examine the general mathematical equivalent of this system of equations. Each individual potential compound is a function of four variables, R, a, f, and t.

u = f(R,a,f,t)

Read “u is a function of the variables R, a, f, and t,” and u may represent C3S, C2S, C3A, or C4AF. We have noted that, at most plants, the variable t may be considered a constant. If two additional variables are held constant, and only one variable is allowed to change, then u becomes a function of the single variable that can change in value, and the function can be differentiated in the usual way.

Such a derivative is called a first partial derivative of u with respect to the single variable. We will be interested in three partial derivatives of u with mathematical designations, ∂u/∂R, ∂u/∂a, and ∂u/∂f. These will represent the partial derivatives of each of the potential compounds included in u, with respect to each of the three individual control parameters. They will tell us how much the potential compound will be expected to change in response to a given change in the individual control parameters—with the other items held constant. For chemical control purposes the ∂C3S/∂f and the ∂C3A/∂a will be the most useful.

With the selection of what items to control, one must consider how to exercise control and how to monitor it. The use of quality control charts is a natural choice. Moore has summarized their use as applied to cement plant operations. (Ref. 5, pp 81-111)

Variation in test data is inevitable. Composition of the raw materials is variable; feed rates of belt conveyors are variable; chemical analyses have variations in precision and accuracy; and grinding media wear brings in some variation. But within that framework of variables, the operator can identify and compensate for the items they can control. What remains is a system of chance causes. Even those variations may be reduced further by blending or withdrawing from multiple storage silos simultaneously.

Statistical quality control charts are useful in establishing process capability, so plant standards may be set for control variables, and upper and lower control limits may be applied. On a graph of a control variable in sequence of production, a single point may go out of control limits either because the process average has changed (x-bar), or because the process variability (standard deviation) has increased.

A chart of the moving range of two test results can identify changes in the variability (standard deviation of the data), and a chart of the control variable in time sequence can identify changes in the average value. The charts can tell you when a correction needs to be made, but they give little information on the magnitude of the correction needed. Some say make the smallest change you can make and be sure the control variable moves in the desired direction.

Application of quantitative change
A brief review of six portland cement plants—and the cements they produce—indicates that a high percentage of the cements fall within the following limits of the control parameters and of the potential compounds.

f = 2.25 – 2.73            C3S = 36.4 – 78.9% *
R = 2.14 – 3.40                    C2S =  4.3 – 42.7%
a =  0.638 – 2.80            C3A =  0.0 – 15.6%
t = 91.7 – 99.4  (t-bar = 95.5%)     C4AF = 5.1 – 20.2%

*Note that the range of the alumina modulus (a) has been limited to a > 0.638, and the cements covered do not include Type IV Cement.

A.  Changes in the lime factor
Look at equation 15 for C3S. The first partial derivative of C3S with respect to f, holding R, a, and t constant, is: the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. After collecting terms, that equation is:

∂C3S/∂f = [R2(a+1)2t(11.671) + R(a+1)t(10.78815a + 5.500)] / [R(f+1)(a+1) + 2.650a + 1.351]2

Evaluated at t = 98.00, R = 2.50, f = 2.30, and a = 2.00
(arbitrary choices), yields:

C3S = 41.27%
∂C3S/∂f = 84237 / 986.02 = 85.43 units of C3S per unit of f.

The variable f normally changes in the first and second decimal place. Evaluated at t = 98.00, R = 2.50, a = 2.00, and f =2.40 (increasing f by +0.10 and holding the other variables constant) yields:

C3S = 49.62%
∂C3S/∂f = 84237.4 / 1033.69 = 81.49 units of C3S per unit of f.

Note that the average of the two partial derivatives is (85.43 + 81.49) / 2 = 83.46%, and the change from C3S = 41.27 to C3S = 49.62 is 8.35%—close to 0.10 times the average of the two partial derivatives at the end points of the high and low values of the variable f, as predicted.

If one wanted to lower the C3S to 34% to avoid the heat of hydration performance test on Type IV cement, what change in f would be required? It can be estimated from the C3S level of 41.27% in the example above:

Δf = (34.00 – 41.27) / (85.43) = −0.085

Making that decrease in f (f = 2.3 − 0.085 = 2.215), and reevaluating the C3S and the first derivative of C3S with respect to f at f =2.215, R = 2.50, a = 2.00, and   t = 98.00:
C3S = 33.86%
∂C3S/∂f = 89.01units of C3S per unit of f.

Additional changes in the composition likely would need to be made to meet the ASTM Standard Specification for Type IV Cement.

Consider equation 15 for C3S again, and assign arbitrary values for R=2.80, a=1.70 and t=96.00. The equation simplifies to a function of the single variable, f, and becomes

C3S= (2954.569f−5515.776) / (7.56f+13.416).

The first partial derivative of C3S with respect to f becomes:
∂C3S/∂f = 81337.764 / (7.56f + 13.416)2.

B.  Changes in the alumina modulus
Look at equation 17 for C3A. The first partial derivative of C3A with respect to a, holding f, R, and t constant, is:

∂C3A/∂a =
(2.650(a–0.658)t)(R(f+1)+2.650)] /

This equation is shown in this form so the denominator, numerator, and the first partial derivative of each may be readily identified. When expanded and terms are collected it simplifies to:
∂C3A/∂a = (4.3407R(f+1) + 8.0606)t / [R(f+1)(a+1) + 2.650a + 1.351]2.

Following the example established above for C3S, consider equation 17 for C3A and assign arbitrary (mid-range values for R =2.80, f = 2.50, and t = 96.00.

The equation simplifies to a function of the single variable a as follows:
C3A = (254.4a – 162.3072) / (12.45a + 11.151).

And the first partial derivative becomes: 
∂C3A/∂a = 4857.539 / (12.45a + 11.151)2.

Consider a portland cement plant with four streams of raw materials, and some latitude for chemical control on each of them, in a production run of Type II cement with a maximum C3A of 8%. Control paramters are stable at f=2.50, R=2.30, a= 1.35, and t=96.00%. The compound composition of the clinker averages; C3S = 56.09%, C2S=20.07%, C3A=7.6%, and C4AF=12.25%. Could a change in production to a Type V clinker be made, with a maximum C3A of 5%, and the additional specification requirement that C4AF plus two times the C3A must be less than 25%?

First consider the possibility of making the change by simply reducing the A/F ratio, a, by the equivalent of 3% C3A. Evaluated at f=2.50, R=2.30 and t=96.00, equation 17 for C3A becomes:
C3A = (254.4a – 162.3072) / (10.7a + 9.401).

At a=1.35; ∂C3A/∂a = 7.26 units of C3A per unit of a. The relationship above indicates that at C3A=4.60, the value of a=1.00, and the ∂C3A/∂a=10.22. Reevaluating equations 15 – 18 at R=2.30, f=2.50, t=96.00, and a=1.00 yields a compound composition of: C3S=56.63, C2S=20.27, C3A=4.58, and C4AF=14.53. That satisfies the Type V requirement of C3A<5%, and C4AF plus two times the C3A less than 25%.

Design and construction of a portland cement plant will provide for storage and handling of about two to four raw materials. Mechanisms for conveying and controlling the rate of flow of each raw material provides control of chemical parameters related to the potential compound composition of the clinker.
It can be shown that three control parameters give effective control of four potential compounds. A lime factor, f, controls the ratio of C3S to C2S; the alumina modulus (A over F ratio), a, controls the ratio of C3A to C4AF; and the silica ratio, R, provides control of burning conditions in the kiln.

When fewer than four raw materials are available, the lime factor and silica ratio may be controlled with three raw materials, or the lime factor alone may be selected when only two are available. The remaining potential compounds will vary within a random system of chance causes depending on the uniformity of the raw materials, the uniformity of the weigh feeders, and the speed control of the belt drives.

Quality control charts are the preferred means of monitoring the control parameters. A chart of the moving range will indicate when the variability of the process is out of control. And a chart of the control variable in time sequence of production will identify when the average value of the control parameter (x-bar) is out of control.

The quality control charts tell when a change in the process control is needed, but it gives little information on the size of the change needed in the control parameter. With mathematical equations available to quantitatively relate the compound composition to the control parameters, it is shown that the first partial derivative of C3S with respect to f tells how much the C3S will change in response to a fixed change in f—with the other control parameters each held constant. Likewise, the first partial derivative of C3A with respect to a will tell how much the C3A will change in response to a fixed change in a—when the other control parameters are held constant. Uniformity of the product translates into uniformity of field performance of the product.

1.    Meade, Richard K.  Portland Cement.  3rd ed.  Easton, PA: The         Chemical Publishing Company, 1926.
2.    Bogue, Robert Herman. The Chemistry of Portland Cement. 2nd  ed. New York:  Reinhold Publishing Corporation, 1955.
3.    Lea, F. M.  The Chemistry of Cement and Concrete. 3rd ed. New York: Chemical Publishing Company, Inc., 1971.          
4.    Moore, Clyde W.  “Chemical Control of Portland Cement Clinker”.      The American Ceramic Society Bulletin, Vol. 61, No. 4. pp 511-515, April 1982.
5.    Moore, Clyde W.  Control of Portland Cement Quality.  Skokie, IL:      The Portland Cement Association, EB121, 2007.

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