By Lina Sorg
Powdered milk is a manufactured dairy product formed by dehydrating regular milk to remove nearly all of the water content. Doing so preserves the product, greatly extends its shelf life, and eliminates the need for refrigeration. Milk powder consequently has comparatively low shipping and transportation costs and is used regularly in developing countries, by hikers and campers who need to pack light, or even for everyday cooking
As complex bulk material, milk powder is subject to a series of complicated chemical reactions, all of which are influenced by oxygen, temperature, and humidity. This is especially true during shipment, which exposes the product to a variety of extreme environmental conditions for extended periods of time. Unfortunately, even small alterations can result in significant changes to the powder’s taste, color, and smell. While these changes do not necessarily compromise consumption safety, they typically cause consumers to reject the product.
The mathematics behind the transport and shelf life of milk powder are both multifaceted and highly nonlinear. “Formation of off-flavor compounds is result of a slew of really complex reactions,” Steve Taylor of the University of Auckland said. During a scientific session at the 9th International Congress on Industrial and Applied Mathematics, currently taking place in Valencia, Spain, Taylor described how mathematical modeling can be used to preserve the flavor of milk powder in fluctuating environments. He and his team—which consists of Richard Clarke (University of Auckland), Luke Fullard and Valerie Chopovda (Massey University), and Andrew Fowler (Oxford University and the University of Limerick)—are working with Fonterra, a dairy company in New Zealand that ships its powder in bags lined with a plastic layer to prevent oxygen and moisture from passing through.
Lactose crystallization is another manifestation of prolonged exposure to extreme temperatures and moisture levels. It triggers caking—which affects solubility in addition to texture—and is preceded by a rubbery state that forms at the glass transition temperature. This temperature marks the point at which an amorphous material transitions to a viscous form. Once the associated inter-particle bridging occurs, the rates of other reactions rapidly increase. “You need to keep milk powder from reaching the glass transition temperature,” Taylor said. “Once you’re at this temperature and beyond, that’s when things go wrong very quickly.”
Taylor began with a fairly simple model to investigate the time and spatial dependence of the relevant conditions. He modeled the relative humidity of moisture in the bag as a set of diffusion equations. Milk powder is comprised of small pellets, with gas occupying the free space in between the pellets. Thus, diffusion equations can also appropriately capture the moisture level in each pellet. After estimating the diffusion constants, Taylor calculated a diffusion equation for the bag itself and realized that relative humidity is approximately uniform throughout each milk pellet. However, an extra term is present to account for the pellets’ absorption of moisture. Using data provided by Fonterra about moisture diffusion in the bag, Taylor then solved Poisson’s equation to estimate flux through the bag. The result is a linear ordinary differential equation (ODE) model that can be solved analytically.
The glass transition temperature depends on the amount of water present in the bag; in fact, it is almost linearly dependent on water content. Because chemical reactions occur swiftly once milk powder reaches this temperature, Taylor wanted to model the powder through various environments, calculate the water activity, and ensure that the contents remained under the straight line marking the transition temperature on a graph. He developed numerical calculations for a more detailed ODE model using information about the last time the contents reached the glass transition temperature. By plotting data based on relative humidity, he proved that the product reaches the glass transition temperature more quickly when bags are exposed to high temperatures and relative humidity levels. “Just running that simple ODE model, which you can run using a simple ODE solver, we can create periods like that,” he said.
A single calculation indicated that milk powder has a shelf life of 4.1 months under saturated vapor pressure and when exposed to a temperature of 40 degrees Celsius. Of course, shelf life varies based on the shipment’s location and travel. For example, the trajectory of milk powder properties hit the glass transition line after 36.5 days for in Africa, which is known for its extreme temperatures.
Companies like Fonterra can limit initial water content and use appropriate plastic linings for their shipment bags to avoid reaching the glass transition temperature and unfavorably altering the milk powder’s flavor, appearance, and/or consistency. Fonterra also reduces the oxygen level in the bag by pumping a mix of carbon dioxide and nitrogen through the powder. Thoroughly understanding the specifics of milk powder’s exposure to extreme temperatures and humidity levels during shipping will ultimately result in a higher-quality product with a longer shelf life.