Chapter 42. The Iterative Approach to Experimentation

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Yes

Stop

experiments



Does the

model fit?

Plot residuals

and confidence

regions



Design

experiment



No

New experiment or

more data



Collect

data



Fit model to

estimate

parameters



FIGURE 42.1 The iterative cycle of experimentation. (From Box, G. E. P. and W. G. Hunter (1965). Technometrics, 7, 23.)



If the dilution rate is sufficiently large, the organisms will be washed out of the reactor faster than they

can grow. If all the organisms are washed out, the effluent concentration will equal the influent concentration, S = S0 . The lowest dilution rate at which washout occurs is called the critical dilution rate (Dc )

which is derived by substituting S = S0 into the substrate model above:



θ1 S0

D c = ---------------θ 2 + S0

When S 0 >> θ 2, which is often the case, D c ≈ θ 1.

Experiments will be performed at several dilution rates (i.e., flow rates), while keeping the influent

substrate concentration constant (S0 = 3000 mg/L). When the reactor attains steady-state at the selected

dilution rate, X and S will be measured and the parameters θ1, θ 2, and θ 3 will be estimated. Because

several weeks may be needed to start a reactor and bring it to steady-state conditions, the experimenter

naturally wants to get as much information as possible from each run. Here is how the iterative approach

can be used to do this.

Assume that the experimenter has only two reactors and can test only two dilution rates simultaneously.

Because two responses (X and S) are measured, the two experimental runs provide four data points (X1

and S1 at D1; X2 and S 2 at D2), and this provides enough information to estimate the three parameters in

the model. The first two runs provide a basis for another two runs, etc., until the model parameters have

been estimated with sufficient precision.

Three iterations of the experimental cycle are shown in Table 42.1. An initial guess of parameter

values is used to start the first iteration. Thereafter, estimates based on experimental data are used. The

initial guesses of parameter values were θ 3 = 0.50, θ 1 = 0.70, and θ 2 = 200. This led to selecting flow

rate D1 = 0.66 for one run and D2 = 0.35 for the other.

The experimental design criterion for choosing efficient experimental settings of D is ignored for now

because our purpose is merely to show the efficiency of iterative experimentation. We will simply say

that it recommends doing two runs, one with the dilution rate set as near the critical value Dc as the

experimenter dares to operate, and the other at about half this value. At any stage in the experimental

cycle, the best current estimate of the critical flow rate is Dc = θ 1. The experimenter must be cautious

in using this advice because operating conditions become unstable as Dc is approached. If the actual

critical dilution rate is exceeded, the experiment fails entirely and the reactor has to be restarted, at a

considerable loss of time. On the other hand, staying too far on the safe side (keeping the dilution rate

too low) will yield poor estimates of the parameters, especially of θ 1. In this initial stage of the experiment

we should not be too bold.

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TABLE 42.1

Three Iterations of the Experiment to Estimate Biokinetic Parameters

Iteration

and

Exp. Run



Best Current

Estimates of

Parameter Values

θ3

θ1

θ2



Iteration 1



0.50



Iteration 2



0.60



Iteration 3



(From iteration 2)



Run 5

Run 6



0.60



Parameter Values

Estimated from

New Data

ˆ

ˆ

ˆ

θ3

θ1

θ2



(From iteration 1)



Run 3

Run 4



Observed

Values

S

X



(Initial guesses)



Run 1

Run 2



Controlled

Dilution

Rate

D = V/Q



0.70



0.55



0.55



200



140



120



0.66

0.35



2800

150



100

1700



0.60



0.55



140



0.52

0.27



1200

80



70

1775



0.60



0.55



120



0.54

0.27



2998

50



2

1770



0.60



0.55



54



Source: Johnson, D. B. and P. M. Berthouex (1975). Biotech. Bioengr., 18, 557–570.



1.00

n= 2

0.75



n= 5



θ 1 0.50

n= 4



0.25

0

0



50



100

θ2



150



200



FIGURE 42.2 Approximate joint 95% confidence regions for θ1 and θ 2 estimated after the first, second, and third experimental iterations. Each iteration consisted of experiments at two dilution rates, giving n = 2 after the first iteration, n = 4

after the second, and n = 6 after the third.



ˆ

ˆ

The parameter values estimated using data from the first pair of experiments were θ 1 = 0.55, θ 2 = 140,

ˆ

and θ 3 = 0.60. These values were used to design two new experiments at D = 0.52 and D = 0.27, and a

second experimental iteration was done. The parameters were estimated using all data from the first and

ˆ

ˆ

ˆ

second iterations. The estimates θ 1 and θ 3 did not change from the first iteration, but θ 2 was reduced

from 140 to 120. Because θ1 and θ 3 seem to be estimated quite well (because they did not change from

one iteration to the next), the third iteration essentially focuses on improving the estimate of θ 2.

In run 5 of the third iteration, we see S = 2998 and X = 2. This means that the dilution rate (D = 0.54)

was too high and washout occurred. This experimental run therefore provides useful information, but

the data must be handled in a special way when the parameters are estimated. (Notice that run 1 had a

higher dilution rate but was able to maintain a low concentration of bacterial solids in the reactor and

remove some substrate.)

At the end of three iterative steps — a total of only six experiments — the experiment was ended.

Figure 42.2 shows how the approximate 95% joint confidence region for θ1 and θ 2 decreased in size

from the first to the second to the third set of experiments. The large unshaded region is the approximate

joint 95% confidence region for the parameters after the first set of n = 2 experiments. Neither θ1 nor

θ 2 was estimated very precisely. At the end of the second iteration, there were n = 4 observations at four

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settings of the dilution rate. The resulting joint confidence region (the lightly shaded area) is horizontal,

but elongated, showing that θ1 was estimated with good precision, but θ 2 was not. The third iteration

invested in data that would more precisely define the value of θ 2. Fitting the model to the n = 5 valid

ˆ

ˆ

ˆ

tests gives the estimates θ 1 = 0.55, θ 2 = 54, and θ 3 = 0.60. The final joint confidence region is small, as

shown in Figure 42.2.

The parameters were estimated using multiresponse methods to fit S and X simultaneously. This

contributes to smaller confidence regions; the method is explained in Chapter 46.



Comments

The iterative experimental approach is very efficient. It is especially useful when measurements are

difficult or expensive. It is recommended in almost all model building situations, whether the model is

linear or nonlinear, simple or complicated.

The example described in this case study was able to obtain precise estimates of the three parameters

in the model with experimental runs at only six experimental conditions. Six runs are not many in this

kind of experiment. The efficiency was the result of selecting experimental conditions (dilution rates)

that produced a lot of information about the parameter values. Chapter 44 will show that making a large

number of runs can yield poorly estimated parameters if the experiments are run at the wrong conditions.

Factorial and fractional factorial experimental designs (Chapters 27 to 29) are especially well suited

to the iterative approach because they can be modified in many ways to suit the experimenter’s need for

additional information.



References

Box, G. E. P. (1991). “Quality Improvement: The New Industrial Revolution,” Tech. Rep. No. 74, Madison,

WI, Center for Quality and Productivity Improvement, University of Wisconsin–Madison.

Box, G. E. P. and W. G. Hunter (1965). “The Experimental Study of Physical Mechanisms,” Technometrics,

7, 23.

Johnson, D. B. and P. M. Berthouex (1975). “Efficient Biokinetic Designs,” Biotech. Bioengr., 18, 557–570.



Exercises

42.1 University Research I. Ask a professor to describe a problem that was studied (and we hope

solved) using the iterative approach to experimentation. This might be a multi-year project

that involved several graduate students.

42.2 University Research II. Ask a Ph.D. student (a graduate or one in-progress) to explain their

research problem. Use Figures 1.1 and 42.1 to structure the discussion. Explain how information gained in the initial steps guided the design of later investigations.

42.3 Consulting Engineer. Interview a consulting engineer who does industrial pollution control

or pollution prevention projects and learn whether the iterative approach to investigation and

design is part of the problem-solving method. Describe the project and the steps taken toward

the final solution.

42.4 Reaction Rates. You are interested in destroying a toxic chemical by oxidation. You hypothesize that the destruction occurs in three steps.

Toxic chemical −> Semi-toxic intermediate −> Nontoxic chemical −> Nontoxic gas

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You want to discover the kinetic mechanisms and the reaction rate coefficients. Explain how

the iterative approach to experimentation could be useful in your investigations.

42.5 Adsorption. You are interested in removing a solvent from contaminated air by activated

carbon adsorption and recovering the solvent by steam or chemical regeneration of the carbon.

You need to learn which type of carbon is most effective for adsorption in terms of percent

contaminant removal and adsorptive capacity, and which regeneration conditions give the best

recovery. Explain how you would use an iterative approach to investigate the problem.



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43

Seeking Optimum Conditions by Response

Surface Methodology



composite design, factorial design, iterative design, optimization, quadratic effects,

response surface methodology, regression, star point, steepest ascent, two-level design.



KEY WORDS



The response of a system or process may depend on many variables such as temperature, organic

carbon concentration, air flow rate, etc. An important problem is to discover settings for the critical

variables that give the best system performance. Response surface analysis is an experimental approach

to optimizing the performance of systems. It is the ultimate application of the iterative approach to

experimentation.

The method was first demonstrated by Box and Wilson (1951) in a paper that Margolin (1985) describes

as follows:

The paper … is one of those rare, truly pioneering papers that completely shatters the existing

paradigm of how to solve a particular problem.… The design strategy…to attain an optimum

operating condition is brilliant in both its logic and its simplicity. Rather than exploring the entire

continuous experimental region in one fell swoop, one explores a sequence of subregions. Two

distinct phases of such a study are discernible. First, in each subregion a classical two-level

fractional factorial design is employed…This first-phase process is iterative and is terminated

when a region of near stationarity is reached. At this point a new phase is begun, one necessitating

radically new designs for the successful culmination of the research effort.



Response Surface Methodology

The strategy is to explore a small part of the experimental space, analyze what has been learned, and

then move to a promising new location where the learning cycle is repeated. Each exploration points

to a new location where conditions are expected to be better. Eventually a set of optimal operating

conditions can be determined. We visualize these as the peak of a hill (or bottom of a valley) that

has been reached after stopping periodically to explore and locate the most locally promising path.

At the start we imagine the shape of the hillside is relatively smooth and we worry mainly about its

steepness. Figure 43.1 sketches the progress of an iterative search for the optimum conditions in a

process that has two active independent variables. The early explorations use two-level factorial

experimental designs, perhaps augmented with a center point. The main effects estimated from these

designs define the path of steepest ascent (descent) toward the optimum. A two-level factorial design

may fail near the optimum because it is located astride the optimum and the main effects appear to

be zero. A quadratic model is needed to describe the optimal region. The experimental design to fit

a quadratic model is a two-level factorial augmented with stars points, as in the optimization stage

design shown in Figure 43.1.



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Dilution Rare (1/h)



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0.01

0.02 0.03

0.04



Optimization

stage

Exploratory

stage

Phenol Concentration (mg/L)



FIGURE 43.1 Two stages of a response surface optimization. The second stage is a two-level factorial augmented to define

quadratic effects.



Case Study: Inhibition of Microbial Growth by Phenol

Wastewater from a coke plant contains phenol, which is known to be biodegradable at low concentrations

and inhibitory at high concentrations. Hobson and Mills (1990) used a laboratory-scale treatment system

to determine how influent phenol concentration and the flow rate affect the phenol oxidation rate and

whether there is an operating condition at which the removal rate is a maximum. This case study is

based on their data, which we used to create a response surface by drawing contours. The data given in

the following sections were interpolated from this surface, and a small experimental error was added.

To some extent, a treatment process operated at a low dilution rate can tolerate high phenol concentrations better than a process operating at a high dilution rate. We need to define “high” and “low” for

a particular wastewater and a particular biological treatment process and find the operating conditions

that give the most rapid phenol oxidation rate (R ). The experiment is arranged so the rate of biological

oxidation of phenol depends on only the concentration of phenol in the reactor and the dilution rate.

Dilution rate is defined as the reactor volume divided by the wastewater flow rate through the reactor.

Other factors, such as temperature, are constant.

The iterative approach of experimentation, as embodied in response surface methodology, will be

illustrated. The steps in each iteration are design, data collection, and data analysis. Here, only design

and data analysis are discussed.



First Iteration

Design — Because each run takes several days to complete, the experiment was performed in small

2

sequential stages. The first was a two-level, two-factor experiment — a 2 factorial design. The two

experimental factors are dilution rate (D) and residual phenol concentration (C ). The response is phenol

oxidation rate (R ). Each factor was investigated at two levels and the observed phenol removal rates are

given in Table 43.1.

Analysis — The data can be analyzed by calculating the effects, as done in Chapter 27, or by linear

regression (Chapter 30); we will use regression. There are four observations so the fitted model cannot

have more than four parameters. Because we expect the surface to be relatively smooth, a reasonable

model is R = b0 + b1C + b2 D + b12CD. This describes a hyperplane. The terms b1C and b2D represent

the main effects of concentration and dilution rate; b12CD is the interaction between the two factors.

The fitted model is R = − 0.022 − 0.018C + 0.2D + 0.3CD. The response as a function of the two

experimental factors is depicted by the contour map in Figure 43.2. The contours are values of R, in

2

units of g/h. The approximation is good only in the neighborhood of the 2 experiment, which is indicated

by the four dots at the corner of the rectangle. The direction toward higher removal rates is clear. The

direction of steepest ascent, indicated by an arrow, is perpendicular to the contour line at the point of

interest. Of course, the experimenter is not compelled to move along the line of steepest ascent.

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TABLE 43.1

Experimental Design and Results for Iteration 1

C (g /L)



D (1/h)



R (g /h)



0.5

0.5

1.0

1.0



0.14

0.16

0.14

0.16



0.018

0.025

0.030

0.040



0.22

Iteration 1

Dilution Rate (1/h)



0.20

Optimum

_

R = 0.040 g/h



0.18

0.16



0.048

0.14

0.036

0.12

0.10



0.012



0.024



1.0

2.0

0.5

1.5

Phenol Concentration (mg/L)

FIGURE 43.2 Response surface computed from the data collected in exploratory stage 1 of the optimizing experiment.



Second Iteration

Design — The first iteration indicates a promising direction but does not tell us how much each setting

should be increased. Making a big step risks going over the peak. Making a timid step and progressing

toward the peak will be slow. How bold — or how timid — should we be? This usually is not a difficult

question because the experimenter has prior experience and special knowledge about the experimental

conditions. We know, for example, that there is a practical upper limit on the dilution rate because at

some level all the bacteria will wash out of the reactor. We also know from previously published results

that phenol becomes inhibitory at some level. We may have a fairly good idea of the concentration at

which this should be observed. In short, the experimenter knows something about limiting conditions

at the start of the experiment (and will quickly learn more). To a large extent, we trust our judgment

about how far to move.

The second factor is that iterative factorial experiments are so extremely efficient that the total number

of experiments will be small regardless of how boldly we proceed. If we make what seems in hindsight

to be a mistake either in direction or distance, this will be discovered and the same experiments that

reveal it will put us onto a better path toward the optimum.

In this case of phenol degradation, published experience indicates that inhibitory effects will probably

become evident with the range of 1 to 2 g/L. This suggests that an increase of 0.5 g/L in concentration

should be a suitable next step, so we decide to try C = 1.0 and C = 1.5 as the low and high settings.

Going roughly along the line of steepest ascent, this would give dilution rates of 0.16 and 0.18 as the

2

low and high settings of D. This leads to the second-stage experiment, which is the 2 factorial design

shown in Figure 43.2. Notice that we have not moved the experimental region very far. In fact, one

setting (C = 1.0, D = 0.16) is the same in iterations 1 and 2. The observed rates (0.040 and 0.041) give

us information about the experimental error.

Analysis — Table 43.2 gives the measured phenol removal rates at the four experimental settings. The

average performance has improved and two of the response values are larger than the maximum observed

in the first iteration. The fitted model is R = 0.047 − 0.014C + 0.05D. The estimated coefficient for the

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