Mixture–Process Variable Experiment: Rocket Propellant Case Study

Introduction to Mixture–Process Designs

When engineers develop formulations — whether that’s rocket fuels, adhesives, or even food products — they often face the same challenge: the ingredients must be combined in specific proportions that sum to a whole.

A well-designed laboratory mixture experiment helps engineers understand how ingredient proportions and processing conditions interact.

This is what makes a mixture experiment different from a typical factorial design.

In a mixture, the sum of all ingredients must equal one:

$\sum_{i=1}^{q} x_i = 1$

where each $x_i$ represents the proportion of an ingredient. The classic Scheffé mixture model describes how the response, $y$ , depends on these ingredient proportions:

$y = \sum_{i=1}^{q} \beta_i x_i + \sum_{i<j} \beta_{ij} x_i x_j + \sum_{i<j<k} \beta_{ijk} x_i x_j x_k + \cdots$

But in real-world manufacturing, it’s rarely enough to look at the recipe alone. Process conditions — such as screw speed, mixing time, or temperature — can all influence performance. To account for this, engineers use what’s known as a mixture–process variable experiment, which extends the model to include the main effect of the process condition and its interaction with the ingredients:

$y = \text{Mixture terms} + \beta_{z} z + \sum_{i=1}^{q} \beta_{iz} x_i z + \cdots$

Here, $z$ is the coded process variable, like a setting for mixing speed. This lets engineers see how changes in processing shift the response surface for the mixture.

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The Kowalski–Cornell–Vining Approach

One of the challenges of adding process variables to a mixture design is that the number of required runs can quickly become impractical. The Kowalski–Cornell–Vining (KCV) method, published in 1991, remains a practical solution for this.

The KCV strategy augments a base mixture design with carefully chosen combinations of the process factor levels. This allows engineers to estimate the main effects and the crucial interactions without having to test every possible combination. The general form of the KCV model is:

$y = \beta_0 + \sum_{i=1}^{q} \beta_i x_i + \sum_{i<j} \beta_{ij} x_i x_j + \beta_{z} z + \sum_{i=1}^{q} \beta_{iz} x_i z + \cdots$

The benefit of this approach is clear: you get the insight you need to develop a robust product or process, but without the cost and time burden of a full factorial test plan.

How Gallant et al. Applied It

A good practical example of a mixture–process variable experiment is the study by Gallant, Prickett, Cesarec and Bruck, published in 2008. Their work focused on composite rocket propellants. They wanted to understand how the burning rate of the propellant changed when they varied not only the mixture of ingredients but also the processing conditions — in this case, the speed of a screw extruder that blends the propellant.

They looked at three key ingredients:
Coarse ammonium perchlorate, fine ammonium perchlorate, and a polymer binder. These ingredients make up the propellant mixture, and their combined proportions always add up to one. The screw extruder speed, coded as $z$ , was studied at low and high levels (–1 and +1) to see how mixing intensity affected the final burning rate.

Using the KCV approach, they designed a 15-run experiment that could estimate the main ingredient effects, the process effect, and the interactions between the mixture and the process variable. This reduced-run approach gave them statistically reliable results without the burden of a much larger test matrix.

The Experiment’s Data

Their dataset is available publicly and is used in R’s mixexp package as an example. It includes the three mixture components, the process variable, and the measured burning rate at a test pressure of 500 psig. Here is an extract from the data, originally published as Table 4 in the paper:

Run	Coarse	Fine	Binder	Screw Speed (z)	Burning Rate (in/s)
1	0.403	0.166	0.431	-1	0.371
2	0.403	0.166	0.431	+1	0.356
3	0.704	0.166	0.130	-1	0.368
…	…	…	…	…	…
15	0.554	0.289	0.157	0	0.370

This design made it possible to model how the burning rate depends on both the recipe and the screw speed.

The Final Predictive Model

By fitting the data to the KCV model, the team derived a predictive equation for burning rate:

$y = 0.365 + 0.027 x_1 - 0.006 x_2 + 0.004 x_3 + 0.018 x_1 x_2 - 0.015 x_1 x_3 + 0.002 x_2 x_3 + 0.003 z - 0.005 x_1 z + 0.001 x_2 z + 0.004 x_3 z$

This equation shows that coarse ammonium perchlorate, $x_1$ , has the largest positive effect on burning rate, but the effect depends on the screw extruder speed, $z$ . This means that the optimal blend of ingredients shifts depending on how fast the extruder runs.

Without including the process variable in the design, they would have missed this critical interaction — potentially leading to an inconsistent product when production conditions changed.

Visualising the Results

To help engineers see these shifts, Gallant et al. used simplex contour plots. You can recreate these plots using the same dataset in R with the mixexp package. Here’s how:

rCopyEditlibrary(mixexp)
data(Burn)

burn_model <- MixModel(Burn, resp = "y",
                       mixcomps = c("Course", "Fine", "Binder"),
                       model = 6,
                       procvars = "z")

ModelEff(nfac = 3, mod = 6, nproc = 1, dir = 1,
         ufunc = burn_model,
         pvslice = c(-1, 1),
         lc = c(0.403, 0.166, 0.130),
         uc = c(0.704, 0.412, 0.210))

This script produces two simplex plots showing how the predicted burning rate surface changes at low and high screw extruder speeds. It’s a powerful way to visualise how both the recipe and processing must be optimised together.

Effect plot generated in R illustrates how each ingredient in the rocket propellant mixture influences burning rate, demonstrating the interactions captured in a mixture–process variable design.

Why This Matters

This case study is a clear reminder of how powerful mixture–process variable designs can be. Without the process factor, you risk developing a formulation that works in the lab but fails under real production conditions. By combining the right design of experiments tools, you can find robust formulations that deliver consistent performance.

Final Thoughts

The Gallant et al. experiment remains a practical example for anyone working with complex formulations where processing conditions matter. If you want to get more from your experiments, save time and material, and make better decisions, a well-planned mixture–process variable experiment is an investment that pays off.

If you’d like support designing, analysing, or scaling up your own experiments, Product Development Engineers Ltd is here to help.

For more information on R software see: The R Project for Statistical Computing

For more information about our DoE and related services see: Development Services

References:
Gallant AJ, Prickett TA, Cesarec T, Bruck HA (2008). Chemometrics and Intelligent Laboratory Systems, 90(2), 156–163.
Kowalski SM, Cornell JA, Vining GG (1991). Journal of Quality Technology.

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Using Mixture–Process Variable Experiments: A Rocket Propellant Case Study

Introduction to Mixture–Process Designs

The Kowalski–Cornell–Vining Approach

How Gallant et al. Applied It

The Experiment’s Data

The Final Predictive Model

Visualising the Results

Why This Matters

Final Thoughts

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