Mathematical Modelling as a tool to inform packaging design

Eli Gray-Stuart, Massey University, School of Engineering and Advanced Technology

Twitter: @Eli_Gray-Stuart


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Developing mathematical models is a great way to further our understanding of different systems and processes. At Massey University we use modelling to gain a better insight into packaging systems and how they can be improved. Mathematical modelling offers great flexibility and the potential to explore various permutations of a problem. This reduces the amount of time-intensive and costly experimental analysis required. Furthermore, a model allows us to visualise a range of parameters e.g. a temperature or moisture distribution in a food or packaging material, where experimentally it may only be feasible to take a limited number of physical measurements.

Our modelling work has covered many industrially relevant packaging applications; from the humble corrugated fibreboard box, which has been a main stay of packaging for decades, to work on “active packaging” which is still an emerging technology.


Modelling the thermal performance of corrugated fibreboard

Finite element model simulation for heat transfer through corrugated board, (left) temperature profile and (right) heat fluxes, the arrow size represents relative heat flux

With corrugated fibreboard, we are interested in how its thermal performance can be optimised. A board with high thermal resistance can increase cooling and freezing times. On the flip side, this can be advantageous by slowing product warming during breakdowns in the cold chain.

We developed a finite element model to predict the thermal resistance of corrugated board based on its geometry and the properties of the liners and medium paper. From this we were able to show how the majority of heat is transferred through the fluted medium. Such a model is useful from a design perspective as the thermal performance of new boards can be optimised to suit the application and give us a better indication of how a box will perform in the cold chain.

This heat transfer model has evolved and been adapted to examine moisture transfer in corrugated board to tackle another issue. In the supply chain the cyclical nature of the environment means the fibres in paper are in a continual state of flux as they absorb and desorb moisture. Over time this compromises the structural integrity of the box which can lead to failure and product losses. Modelling this mass transfer process is helping to increase our understanding of the phenomenon and is allowing us to explore ways to improve box performance through improvements to design and paper materials.


Active packaging is an area of increasing focus globally. Of particular interest is the use of packaging systems which release natural volatile compounds into the product head space. The efficacy of this hinges on the knowledge of the minimum inhibitory concentration (MIC) of the volatiles required to prevent microbial growth and quantification of the interactions of the active agents with food and packaging components. Through modelling we can design active packaging systems that match these different rates to maintain the MIC of the volatiles in the package head space through the distribution chain. Limitations of modelling As a concluding remark, it is important that we are aware of the limitations of our models. George Box famously said “all models are wrong, some are useful”.


In the model development process, we endeavour to make sound assumptions and identify the variables which account for most of the variation in what we want to predict. By complementing modelling with comprehensive experimental validation we can advance our understanding and aid the improvement and development of packaging material and technologies.



Gray-Stuart E. M., Bronlund J. E., Robertson T.R., Navaranjan, N. Modelling of Heat Transfer through Corrugated Cardboard Packaging, Conference of Food Engineering, Omaha, NE, 7-9th April 2014.

Abdul Rahman, M. (2015) Measurement of Minimum inhibitory concentration (MIC) of individual and combinations of essential oil volatiles in food, PhD thesis, Massey University.

Utto W. (2008) Mathematical modelling of active packaging systems for horticultural products. PhD thesis, Massey University.