25- ((ترجمتها نسرين))
EXPERIMENTAL DESIGNS FOR
PROCESS OPTIMIZATION
(INDEPENDENT VARIABLES)
In this section, we look at methods of obtaining a
mathematical model that can be used for qualitative
predictions of a response over the whole of the
experimental domain. If the model depends on two
factors, the response may be considered a topographical
surface, drawn as contours or in 3D (Fig. 4).
For more factors, we can visualize the surface by taking
‘‘slices’’ at constant values of all but two factors.
These methods allow both process and formulation
optimization.
التصاميم التجريبية لعمليات التحسين (متغيرات مستقلة)
EXPERIMENTAL DESIGNS FOR PROCESS OPTIMIZATION
(INDEPENDENT VARIABLES)
نتطلّع في هذا المقطع إلى طرق للحصول على نموذج رياضي يمكن استخدامه للتنبؤ (التكهّن) الكيفي qualitative للاستجابة في كامل المجال التجريبي. إن كان النموذج معتمداً على عاملين، فقد تكون الاستجابة ممثلة على سطح طبغرافي مرسوم بشكل إطار سطحي أو بشكل ثلاثي الأبعاد (الشكل-4-).
ولمزيد من العوامل يمكن إظهار السطح من خلال "شرائح" بقيم ثابتة لكلا العاملين. تسمح هذه الطرق بتحسين للعملية المستخدمة وللتركيبة أيضاً.
26- (سيترجمها محمد)
Mathematical Models
The design used is a function of the model proposed.
Thus, if it is expected that the important responses
vary relatively little over the domain, a first-order polynomial
will be selected. This will also be the case if the
experimenter wishes to perform rather a few experiments
at first to check initial assumptions. He may
then change to a second-order (quadratic) polynomial
model. Second-order polynomials are those most commonly
used for response surface modeling and process
optimization for up to five variables.
Examples of polynomial models are a first order
model for five factors:
y ¼ b0 þ b1x1 þ b2x2 þ b3x3 þ b4x4
þ b5x5 þ e
and a second-order model for two factors:
y ¼ b0 þ b1x1 þ b2x2 þ b12x1x2 þ b11x1
2
þ b22x2
2 þ e
27-
(سيترجمها محمد)
The coefficients in the models are estimated by multilinear
least-squares regression of the data.
Third-order models are very rarely used in the case
of process studies and, in any case, third-order terms
are only added for those variables where they can be
shown to be necessary (i.e., augmentation of a secondorder
model and the corresponding design). This does
not mean that second-order designs are always sufficient,
and other methods of constructing response
surfaces may sometimes be useful.
28-
Statistical Experimental Designs for
First-Order Models
The design must enable estimation of the first-order
effects, preferably free from interference by the interactions
between factors other variables. It should also
allow testing for the fit of the model and, in particular,
for the existence of curvature of the response surface
(center points). Two-level factorial designs may be
used for this (shown earlier).
29-
Important points to note when using a first-order
model, with or without interactions, are that:
1. Maximum and minimum values of responses
are of necessity predicted at the edge of the
experimental domain;
2. The first-order model should normally be used
only in the absence of curvature of the response
surface. If the experimental values of the center
points are different from the calculate values
(i.e., there is lack of fit), then the response surface
is curved and a second-order design and
model should be used; and
3. The experimenter should test for interaction
terms between two factors in the model. If interactions
seem to be important he should make
sure that they are properly identified.
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ترجمة قسم : Optimization Methods
