Environmental Engineering Reference
In-Depth Information
approach, with the variables independent, permits the use of classic factorial
and response surface designs (Box, 1978), but has the undesirable feature that
the experimental region changes depending on how the q mixture components
are reduced to q-1 independent factors.
1.3. Modeling: Classic Mixture Approach
As per the classic mixture approach design, all required properties were
measured for each mixture and modeled as a function of the components
(Myers and Montgomery, 1995). Typically, polynomial functions were used
for modeling, but other functional forms can also be used (Myers and
Montgomery, 1995; Cornell, 1990). In the classic mixture approach, the total
amount (mass or volume) of the product is fixed, and the settings of each of
the
q
component variables are proportions to meet the fixed mass/volume.
Because the total amount is constrained to sum to one, only
q-1
of the factors
(component variables) can be chosen independently. In the present study,
consider the jarosite waste composite brick as a mixture of three components
namely Jarosite waste (
x
1
), clay (
x
2
), and CCRs (
x
3
), where each
x
i
represents
the weight fraction of each component. The weigh fractions of the components
sum to one, and the region defined by this constraint is the regular triangle (or
simplex) as shown in Figure 1. The axis for each component
x
i
extends from
the vertex (
x
i
= 1) to the midpoint of the opposite side of the triangle (
x
i
= 0).
The vertex represents the pure component, where in, the vertex labeled
x
1
is
the jarosite waste mixture with
x
1
= 1,
x
2
= 0, and
x
3
= 0, or (1, 0, 0). The point,
where the three axes intersect, with coordinates (1/3,1/3,1/3), is called the
centroid.
As per the classic mixture approach design, all required properties would
be measured for each mixture and modeled as a function of the components. In
this study, polynomial functions were used for modeling. For three
components, the linear polynomial model for a response ‗
y' be
written as:
y = b
*
0
+ b
*
1
x
1 +
b
*
2
x
2
+ b
*
3
x
3
+ e
(1)
where,
bi
*
are constants
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