Chemical Reactor Optimization

Summary

This experiment investigates chemical reactor optimization. Box-Behnken design to optimize yield and purity of a batch reactor process.

The design varies 3 factors: temperature (°C), ranging from 150 to 200, pressure (bar), ranging from 2 to 6, and catalyst (g/L), ranging from 0.5 to 2.0. The goal is to optimize 3 responses: yield (%) (maximize), purity (%) (maximize), and cost (USD) (minimize). Fixed conditions held constant across all runs include reaction time = 60, stirring speed = 300.

A Box-Behnken design was chosen because it efficiently fits quadratic models with 3 continuous factors while avoiding extreme corner combinations — requiring only 15 runs instead of the 8 needed for a full factorial at two levels.

Quadratic response surface models were fitted to capture potential curvature and factor interactions. The RSM contour plots below visualize how pairs of factors jointly affect each response.

Key Findings

For yield, the most influential factors were pressure (49.5%), catalyst (31.1%), temperature (19.4%). The best observed value was 78.52 (at temperature = 175, pressure = 6, catalyst = 0.5).

For purity, the most influential factors were temperature (52.0%), catalyst (32.9%), pressure (15.1%). The best observed value was 97.99 (at temperature = 175, pressure = 6, catalyst = 2).

For cost, the most influential factors were pressure (56.1%), catalyst (30.1%), temperature (13.8%). The best observed value was 34.05 (at temperature = 175, pressure = 4, catalyst = 1.25).

Recommended Next Steps

Run confirmation experiments at the predicted optimal settings to validate the model.
Consider whether any fixed factors should be varied in a future study.

The Scenario

You are optimizing a batch chemical reactor. You control three continuous process parameters and want to maximize product yield and purity while minimizing production cost. Running the reactor at all extreme settings simultaneously is risky, so you need a design that avoids corner points.

ℹ

Why Box-Behnken?

With 3 continuous factors, Box-Behnken gives you 15 runs (vs. 18 for CCD), and every run has at most two factors at their extremes — the third stays at center. This avoids the dangerous corner condition of max temperature + max pressure + max catalyst simultaneously.

Experimental Setup

Factors

Factor	Low	High	Unit
`temperature`	150	200	°C
`pressure`	2	6	bar
`catalyst`	0.5	2.0	g/L

Fixed: reaction_time = 60 min, stirring_speed = 300 rpm

Responses

Response	Direction	Unit
`yield`	↑ maximize	%
`purity`	↑ maximize	%
`cost`	↓ minimize	USD

Configuration

use_cases/01_reactor_optimization/config.json
{
  "metadata": {
    "name": "Chemical Reactor Optimization",
    "description": "Box-Behnken design to optimize yield and purity"
  },
  "factors": [
    {"name": "temperature", "levels": ["150", "200"], "type": "continuous", "unit": "°C"},
    {"name": "pressure",    "levels": ["2", "6"],       "type": "continuous", "unit": "bar"},
    {"name": "catalyst",    "levels": ["0.5", "2.0"], "type": "continuous", "unit": "g/L"}
  ],
  "fixed_factors": {"reaction_time": "60", "stirring_speed": "300"},
  "responses": [
    {"name": "yield",  "optimize": "maximize", "unit": "%"},
    {"name": "purity", "optimize": "maximize", "unit": "%"},
    {"name": "cost",   "optimize": "minimize", "unit": "USD"}
  ],
  "settings": {
    "operation": "box_behnken",
    "test_script": "use_cases/01_reactor_optimization/sim.sh"
  }
}

Experimental Matrix

The Box-Behnken Design produces 15 runs. Each row is one experiment with specific factor settings.

Run	`temperature`	`pressure`	`catalyst`
1	175	2	0.5
2	175	4	1.25
3	200	4	2
4	200	4	0.5
5	175	4	1.25
6	175	4	1.25
7	150	4	2
8	200	2	1.25
9	175	2	2
10	200	6	1.25
11	150	4	0.5
12	175	6	2
13	150	2	1.25
14	150	6	1.25
15	175	6	0.5

Step-by-Step Workflow

1

Preview the design

Terminal

$ doe info --config use_cases/01_reactor_optimization/config.json

Plan      : Chemical Reactor Optimization
Operation : box_behnken
Factors   : temperature, pressure, catalyst
Base runs : 15
Blocks    : 1
Total runs: 15
Responses : yield [maximize] (%), purity [maximize] (%), cost [minimize] (USD)
Fixed     : reaction_time=60, stirring_speed=300

2

Generate the runner script

Terminal

$ doe generate --config use_cases/01_reactor_optimization/config.json \
    --output use_cases/01_reactor_optimization/results/run.sh --seed 42

3

Execute the experiments

Terminal

$ bash use_cases/01_reactor_optimization/results/run.sh

4

Analyze results

Terminal

$ doe analyze --config use_cases/01_reactor_optimization/config.json

5

Get optimization recommendations

Terminal

$ doe optimize --config use_cases/01_reactor_optimization/config.json

ℹ

Multi-Objective Optimization

This experiment has conflicting objectives (yield (↑), purity (↑), cost (↓)). Use --multi to find the best compromise using desirability functions:

Terminal

$ doe optimize --config use_cases/01_reactor_optimization/config.json --multi

6

Generate the HTML report

Terminal

$ doe report --config use_cases/01_reactor_optimization/config.json \
    --output use_cases/01_reactor_optimization/results/report.html

Real-World Lab Workflow

ℹ

Physical Experiment? Use the Manual Workflow

The automated workflow above uses a simulation script for demonstration. In a real chemistry lab, you'd physically run each reactor experiment and record the results. The DOE Helper Tool fully supports this with record, status, and export-worksheet commands.

Here's how a research chemist would actually run this experiment over several days in the lab:

1

Design the experiment & print a lab worksheet

Generate the design and export a printable worksheet to take to the lab bench.

Terminal

$ doe generate --config use_cases/01_reactor_optimization/config.json --seed 42

$ doe export-worksheet --config use_cases/01_reactor_optimization/config.json \
    --format csv --output reactor_worksheet.csv

$ doe export-worksheet --config use_cases/01_reactor_optimization/config.json \
    --format markdown

# Experiment Worksheet: Chemical Reactor Optimization
Design: Box-Behnken | 15 runs | 3 factors | 3 responses

| Run | temperature (°C) | pressure (bar) | catalyst (g/L) | yield (%) | purity (%) | cost (USD) | Notes |
| --- | ------------------- | -------------- | -------------- | --------- | ---------- | ---------- | ----- |
| 1   | 200                 | 4              | 0.5            |           |            |            |       |
| 2   | 150                 | 6              | 1.25           |           |            |            |       |
| ... | ...                 | ...            | ...            |           |            |            |       |

Print the CSV and tape it into your lab notebook, or open it in Excel/Google Sheets. The empty response columns are where you'll record measurements.

2

Check what to run next

Before heading to the lab each day, check which runs are pending.

Terminal

$ doe status --config use_cases/01_reactor_optimization/config.json

Experiment: Chemical Reactor Optimization
Design: box_behnken | 15 runs | 3 factors | 3 responses

Progress: 3/15 complete  [####................ ]  20%

Completed runs:
  Run 1: temperature=200, pressure=4, catalyst=0.5
  Run 2: temperature=150, pressure=6, catalyst=1.25
  Run 3: temperature=200, pressure=6, catalyst=1.25

Pending runs:
  Run 4: temperature=175, pressure=2, catalyst=0.5
  Run 5: temperature=150, pressure=4, catalyst=0.5
  ...

Next run to complete: Run 4
  temperature = 175 °C
  pressure    = 2 bar
  catalyst    = 0.5 g/L

Record results with: doe record --config <config> --run 4

3

Run the experiment & record results

After completing a reactor run, enter the measured values. The tool validates input and saves to JSON.

Terminal

$ doe record --config use_cases/01_reactor_optimization/config.json --run 4

Run 4 / 15 (Block 1)
  temperature = 175 °C
  pressure    = 2 bar
  catalyst    = 0.5 g/L

Enter value for 'yield' (%): 62.4
Enter value for 'purity' (%): 91.8
Enter value for 'cost' (USD): 38.50

Saved → use_cases/01_reactor_optimization/results/run_4.json

You can also record all remaining runs in sequence:

Terminal

$ doe record --config use_cases/01_reactor_optimization/config.json --run all

4

Peek at partial results mid-experiment

Don't wait until all 15 runs are done. Use --partial to get early insights while the experiment is still in progress.

Terminal

$ doe analyze --config use_cases/01_reactor_optimization/config.json --partial

Partial mode: analyzing 8/15 completed runs (missing: 9, 10, 11, 12, 13, 14, 15)

=== Main Effects: yield ===
Factor            Effect    % Contribution
-------------------------------------------
catalyst           12.4           45.2%
temperature         9.8           35.7%
pressure            5.2           19.0%

This early analysis already reveals that catalyst loading is the dominant factor for yield — valuable insight for planning the remaining runs.

5

Complete analysis & generate report

Once all 15 runs are recorded, run the full analysis pipeline.

Terminal

$ doe analyze --config use_cases/01_reactor_optimization/config.json
$ doe optimize --config use_cases/01_reactor_optimization/config.json
$ doe report --config use_cases/01_reactor_optimization/config.json \
    --output reactor_report.html

✔

Why This Workflow Works

The analysis pipeline doesn't care how results were produced. Whether you ran a simulation, measured products from a real reactor, or collected data from a manufacturing line — as long as the response values end up in run_N.json files, the analysis, optimization, and reporting work identically. The record command handles the JSON creation so you never have to edit files manually.

Interpreting the Results

Key Findings

Yield: Temperature and catalyst are the dominant factors (~42% and ~32% contribution). Pressure has a moderate effect (~27%).
Purity: Catalyst concentration is the key driver (~44%), followed by pressure (~47%). Temperature has minimal impact (~9%).
Cost: All three factors contribute roughly equally, with pressure and catalyst slightly ahead.

Trade-offs

⚠

Multi-Objective Conflict

Higher temperature increases yield but also increases cost. Higher catalyst improves purity but adds cost. There is no single setting that maximizes yield AND purity while minimizing cost — the experimenter must decide which response matters most, or find a compromise.

Next Steps

Fit a quadratic RSM model to each response
Construct a desirability function that weights the three responses
Run confirmation experiments at the predicted optimum
Narrow the factor ranges and run another Box-Behnken for fine-tuning

Features Exercised

Feature	Value
Design type	`box_behnken`
Factor types	`continuous` (all 3)
Arg style	`double-dash`
Responses	3 (yield ↑, purity ↑, cost ↓)
Total runs	15
Key feature	Multi-response RSM, avoids corner points

Analysis Results

Generated from actual experiment runs using the DOE Helper Tool.

Response: yield

Pressure (46.5%) and catalyst (42.8%) dominate yield, with temperature playing a minor role.

Pareto Chart

Main Effects Plot

Response: purity

Temperature (40.7%) and catalyst (32.2%) dominate purity, making them the key levers for product quality.

Pareto Chart

Main Effects Plot

Response: cost

Pressure (41.9%) and catalyst (40.9%) dominate cost, suggesting these factors must be balanced against yield gains.

Pareto Chart

Main Effects Plot

Response Surface Plots

3D surfaces fitted with quadratic RSM. Red dots are observed data points.

📊

How to Read These Surfaces

Each plot shows predicted response (vertical axis) across two factors while other factors are held at center. Red dots are actual experimental observations.

Flat surface — these two factors have little effect on the response.
Tilted plane — strong linear effect; moving along one axis consistently changes the response.
Curved/domed surface — quadratic curvature; there is an optimum somewhere in the middle.
Saddle shape — significant interaction; the best setting of one factor depends on the other.
Red dots far from surface — poor model fit in that region; be cautious about predictions there.

yield (%) — R² = 0.693, Adj R² = 0.140
Moderate fit — surface shows general trends but some noise remains.
Curvature detected in catalyst, pressure — look for a peak or valley in the surface.
Strongest linear driver: temperature (decreases yield).
Notable interaction: temperature × pressure — the effect of one depends on the level of the other. Look for a twisted surface.

purity (%) — R² = 0.843, Adj R² = 0.561
Moderate fit — surface shows general trends but some noise remains.
Curvature detected in pressure, temperature — look for a peak or valley in the surface.
Strongest linear driver: pressure (increases purity).
Notable interaction: temperature × catalyst — the effect of one depends on the level of the other. Look for a twisted surface.

cost (USD) — R² = 0.642, Adj R² = -0.002
Moderate fit — surface shows general trends but some noise remains.
Curvature detected in catalyst, temperature — look for a peak or valley in the surface.
Strongest linear driver: temperature (decreases cost).
Notable interaction: pressure × catalyst — the effect of one depends on the level of the other. Look for a twisted surface.

cost: pressure vs catalyst

cost: temperature vs catalyst

cost: temperature vs pressure

purity: pressure vs catalyst

purity: temperature vs catalyst

purity: temperature vs pressure

yield: pressure vs catalyst

yield: temperature vs catalyst

yield: temperature vs pressure

Full Analysis Output

doe analyze
=== Main Effects: yield ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
temperature              9.1475       2.0585            40.5%
catalyst                 7.4375       2.0585            33.0%
pressure                 5.9764       2.0585            26.5%

=== Summary Statistics: yield ===

temperature:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  150                 4    64.6450     5.1577    60.6600    71.6600
  175                 7    65.5600     9.3573    52.9300    78.5200
  200                 4    73.7925     4.7382    67.5500    77.8400

pressure:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  2                   4    68.4550     7.2435    60.8300    77.8400
  4                   7    69.3414     9.1916    52.9300    78.5200
  6                   4    63.3650     6.5765    55.4200    69.8300

catalyst:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  0.5                 4    70.9900     1.4384    69.7700    72.7000
  1.25                7    67.7857     9.2044    52.9300    78.5200
  2                   4    63.5525     9.3752    55.4200    77.0800

=== Main Effects: purity ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
catalyst                 6.3261       1.3850            43.8%
pressure                 4.1586       1.3850            28.8%
temperature              3.9521       1.3850            27.4%

=== Summary Statistics: purity ===

temperature:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  150                 4    90.0375     3.3817    85.3000    92.5500
  175                 7    87.3429     6.2959    78.9100    97.6500
  200                 4    91.2950     5.3626    85.4600    97.9900

pressure:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  2                   4    89.5800     4.5572    83.0000    92.7400
  4                   7    90.4586     3.9188    85.4600    97.6500
  6                   4    86.3000     8.2292    78.9100    97.9900

catalyst:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  0.5                 4    87.1000     3.4632    83.0000    90.0300
  1.25                7    92.1486     4.7238    85.3000    97.9900
  2                   4    85.8225     6.0253    78.9100    92.3900

=== Main Effects: cost ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
temperature             15.4050       2.8932            54.4%
pressure                 9.1025       2.8932            32.1%
catalyst                 3.8125       2.8932            13.5%

=== Summary Statistics: cost ===

temperature:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  150                 4    44.4625     7.4242    36.6400    53.1400
  175                 7    48.4843    10.9213    34.0500    65.5200
  200                 4    59.8675    10.8260    47.7600    73.4600

pressure:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  2                   4    53.1675     6.8669    45.5900    62.2200
  4                   7    52.5400    13.7655    34.0500    73.4600
  6                   4    44.0650     9.2419    36.4300    55.4300

catalyst:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  0.5                 4    52.7475     3.8037    47.8100    56.0300
  1.25                7    49.9971    11.8208    34.0500    65.5200
  2                   4    48.9350    16.7759    36.4300    73.4600

Optimization Recommendations

doe optimize
=== Optimization: yield ===
Direction: maximize

Best observed run: #3
  temperature = 175
  pressure = 6
  catalyst = 0.5
  Value: 78.52

RSM Model (linear, R² = 0.06):
  Coefficients:
    intercept:  +67.5113
    temperature:  -0.4100
    pressure:  +2.1775
    catalyst:  +1.3750
  Predicted optimum:
    temperature = 175
    pressure = 6
    catalyst = 2
    Predicted value: 71.0638

Factor importance:
  1. temperature  (effect: 9.9, contribution: 48.8%)
  2. pressure  (effect: 7.6, contribution: 37.6%)
  3. catalyst  (effect: 2.8, contribution: 13.6%)

=== Optimization: purity ===
Direction: maximize

Best observed run: #9
  temperature = 175
  pressure = 4
  catalyst = 1.25
  Value: 97.99

RSM Model (linear, R² = 0.16):
  Coefficients:
    intercept:  +89.1153
    temperature:  +1.0962
    pressure:  +2.2000
    catalyst:  +1.3638
  Predicted optimum:
    temperature = 175
    pressure = 6
    catalyst = 2
    Predicted value: 92.6791

Factor importance:
  1. pressure  (effect: 6.9, contribution: 48.6%)
  2. temperature  (effect: 4.6, contribution: 32.2%)
  3. catalyst  (effect: 2.7, contribution: 19.2%)

=== Optimization: cost ===
Direction: minimize

Best observed run: #13
  temperature = 200
  pressure = 2
  catalyst = 1.25
  Value: 34.05

RSM Model (linear, R² = 0.01):
  Coefficients:
    intercept:  +50.4473
    temperature:  -0.4200
    pressure:  +1.4512
    catalyst:  +0.0688
  Predicted optimum:
    temperature = 150
    pressure = 6
    catalyst = 1.25
    Predicted value: 52.3186

Factor importance:
  1. temperature  (effect: 9.4, contribution: 52.0%)
  2. pressure  (effect: 7.4, contribution: 41.0%)
  3. catalyst  (effect: 1.3, contribution: 7.1%)

Multi-Objective Optimization

When responses compete, Derringer–Suich desirability finds the best compromise. Each response is scaled to a 0–1 desirability, then combined via a weighted geometric mean.

Overall Desirability

D = 0.7328

Per-Response Desirability

Response	Weight	Desirability	Predicted	Dir
`yield`	1.5	0.5648	67.55 0.5648 67.55 %	↑
`purity`	2.0	0.9545	97.99 0.9545 97.99 %	↑
`cost`	1.0	0.6383	47.76 0.6383 47.76 USD	↓

Recommended Settings

Factor	Value
`temperature`	150 °C
`pressure`	4 bar
`catalyst`	0.5 g/L

Source: from observed run #9

Trade-off Summary

Sacrifice = how much worse than single-objective best.

Response	Predicted	Best Observed	Sacrifice
`purity`	97.99	97.99	+0.00
`cost`	47.76	34.05	+13.71

Top 3 Runs by Desirability

Run	D	Factor Settings
#3	0.6895	temperature=175, pressure=4, catalyst=1.25
#6	0.6436	temperature=150, pressure=4, catalyst=2

Model Quality

Response	R²	Type
`purity`	0.1730	linear
`cost`	0.0211	linear

Full Multi-Objective Output

doe optimize --multi
============================================================
MULTI-OBJECTIVE OPTIMIZATION
Method: Derringer-Suich Desirability Function
============================================================

Overall desirability: D = 0.7328

Response                  Weight Desirability    Predicted  Direction
---------------------------------------------------------------------
yield                        1.5       0.5648       67.55 %   ↑
purity                       2.0       0.9545       97.99 %   ↑
cost                         1.0       0.6383       47.76 USD   ↓

Recommended settings:
  temperature = 150 °C
  pressure = 4 bar
  catalyst = 0.5 g/L
  (from observed run #9)

Trade-off summary:
  yield: 67.55 (best observed: 78.52, sacrifice: +10.97)
  purity: 97.99 (best observed: 97.99, sacrifice: +0.00)
  cost: 47.76 (best observed: 34.05, sacrifice: +13.71)

Model quality:
  yield: R² = 0.1193 (linear)
  purity: R² = 0.1730 (linear)
  cost: R² = 0.0211 (linear)

Top 3 observed runs by overall desirability:
  1. Run #9 (D=0.7328): temperature=150, pressure=4, catalyst=0.5
  2. Run #3 (D=0.6895): temperature=175, pressure=4, catalyst=1.25
  3. Run #6 (D=0.6436): temperature=150, pressure=4, catalyst=2