Hydrocarbon Processing - January 2021 - 45

Process Optimization
move toward its optimum. We only need
to know the local landscape and to perturb
the process by baby steps. In such a case, a
linear model such as Eq. 3 is adequate:
y = c 0 + c 1x1 + c 2 x2+ ε(3)
where c 0-c 2 are regressed coefficients. If
we are ascending the response surface,
we expect the partial derivatives to decrease if the surface flattens to a maximum. Thus, the direction of steepest
ascent is always given by the negative of
the partial derivatives, which for Eq. 3
becomes Eq. 4:
∂y
∂y
−
= − c1 and −
= − c2
∂x1
∂x 2
(4)

These define a direction vector -
(c 1  c 2 ), which we normalize to a unit
vector via Eq. 5:

(c c )
(5)
c1* c 2* = - 1 2
c12 + c 22
This gives us our step size and direction for the next point in our path of
steepest ascent (FIG. 4). We interrogate
the factor space at this new design center, calculate a new kc1*  c 2* k vector, and
reiterate to make incremental progress
to our goal 3,4 (FIG. 5). With small steps,
a real danger exists that the difference in
response will be small and overwhelmed
by process noise. Replicate measurements subject to all of the normal process uncertainty (so-called genuine
replicates) will increase the informationto-noise ratio, average out the process
noise and allow statistical tests for each
coefficient.
Deriving a steepest ascent. We shall
construct our steepest ascent by reiterating the following sequence:
1.	 Decide on an acceptable
step size for each factor
2.	 Generate candidate coordinates
in factor space
3.	 Collect response data
for each coordinate
4.	 Test for significance.
If the coefficients are statistically significant, we calculate the path of steepest
ascent and go on to collect additional data
at a new design center. If not, we run more
replicates to increase the information-tonoise ratio. Consider an example here.

Example: Calculation of steepest
ascent. Suppose an important process

yield responds to temperature and pressure, that the standard operating parameters (SOP) are 300°C and 500 kPa, and
that incremental variations of each factor are limited to no more than ±5°C and
±10 kPa. To these, we apply the transforms (Eq. 6):
x1 =

T [°C ] − 300[°C ]
,
5[°C ]

(6)
P[ kPa ] − 5[ kPa ]
x2 =
10[ kPa ]
These do nothing more than scale
300 ±5 (°C) and 500 ±10 (kPa) to dimensionless unit ranges of ±1. Scaling
process parameters to a dimensionless
unit range simplifies some statistical calculations, but otherwise has no effect on
the analysis.
These transforms define a unit
square in x1-by-x2 factor space (FIG. 6).
The design center (0,0) is shown in
FIG. 6 as p1 and represents the current
SOP. We now must decide on where to
collect new data.
Logical design candidates include
points at the extremes of our permissible ranges (indicated as points s1 to s4 in
FIG. 6). Such designs are known as factorial designs. For example, in three dimensions, x1 by x2 by x3 would define a unit
cube. In higher dimensions, the space is
known as a unit hypercube. The number

of required design points (nd ) in a factorial design is given by nd = 2nf, where nf is
the number of factors.
Therefore, for the two factors we consider in this example (x1 and x2), nd = 22
= 4 design points (s1 to s4 in FIG. 6) plus
any center points (e.g., p1 ) we choose
to run. Factorial designs are convenient
and often used. However, if we have numerous factors, we may quickly exceed
our resources. For example, a factorial
design in five factors requires 32 points,
not including replicates or center points.
To overcome this limitation, one may
use other designs (e.g., fractional-factorial or simplex designs)5. However, for the
limited number of factors here, a factorial design is sufficient and we shall proceed on this basis.
Since our step size is deliberately
small and our process is noisy, we must
repeat some points to better estimate
the true response value. We can replicate
the entire design multiple times, but in
this case, we will just replicate the center

FIG. 5. Steps on a path of steepest ascent.
Applying the FIG. 4 process in seriatim
constructs a path of steepest ascent
bit by bit (e.g., from p1 to p6 ).

FIG. 4. Method of steepest ascent. Only the
local landscape of the response surface is
needed (large, dotted circle) to determine
the vectors of Eq. 5 and the direction of
steepest ascent (p1 to p2 ).

FIG. 6. Factor space. Unit extensions in x1
by x2 factor space define a unit square.
Hydrocarbon Processing | JANUARY 2021 45
Hydrocarbon Processing - January 2021

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