# Determining Shelf Life by Two Criteria

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Consider a food or pharmaceutical product whose shelf life is determined by the loss of either of two nutrients or active components; call them markers. Then the product's shelf life depends on the storage temperature history and the degradation kinetic parameters for the two markers, showing which marker reaches its threshold concentration first. This principle is shown in simulation with markers whose degradation follows fixed-order kinetics, with rate constants following a simple exponential temperature dependence. This can be used interchangeably with the traditional Arrhenius equation. The temperature can be constant, as in some shelf-life studies, or fluctuating, as encountered in many real-life situations.

Contributed by: Mark D. Normand and Micha Peleg (July 2015)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Snapshot 1: temperature profile , where the threshold concentration ratio is reached before

Snapshot 2: temperature profile , where the threshold concentration ratio is reached before

Snapshot 3: isothermal temperature profile, where the threshold concentration ratio is reached before

Snapshot 4: isothermal temperature profile, where and the threshold concentration ratio is reached before

When a compound's degradation follows fixed-order kinetics, its concentration is described by the rate equation , where is the concentration in the chosen units (e.g., ), is the rate constant at the momentary temperature in °C, and is the reaction's kinetic order. The initial condition is , with concentration in the chosen units. For simplicity, assign . The temperature dependence of the rate constant commonly follows the exponential model [1, 2, 3]; that is, , where is a reference temperature and is a characteristic constant having units .

The isothermal solution of the rate equation (i.e. and ) is . For , , and for and , . Where and , the concentration ratio becomes negative and is therefore set equal to zero. For and , becomes a complex number and hence is also set to zero.

Under nonisothermal conditions, the rate equation has an analytical solution only for simple profiles, such as linearly rising or falling temperature. For more elaborate temperature profiles, such as those we have chosen for this Demonstration, the rate equation requires a numerical solution using *Mathematica*'s built-in function NDSolve.

A product reaches the end of its shelf life at time when the concentration ratio is . This time is the analytical solution of the rate equation in the isothermal case and the expression defined by NDSolve in the dynamic case, that is, where is not constant. In those cases, is the threshold concentration ratio for the particular marker.

In this Demonstration you can enter with sliders the two markers' degradation kinetic orders and , their and , their characteristic constants and , threshold concentration ratios and , and the plots' axes limits. The program then plots the temperature profile and the two corresponding degradation curves. It also displays, graphically and numerically, the times and when the degradation curves cross their corresponding threshold concentration ratios. By moving the sliders, you can see how different parameter settings can result in one or the other marker compound reaching its threshold level first, thus terminating the product's shelf life.

Since the calculated times and are obtained by *Mathematica*'s built-in function FindRoot, a solution is not always guaranteed with the default settings. Therefore, the program allows you to enter closer initial guesses of their values with the sliders and .

References

[1] M. Peleg, M. D. Normand, and M. G. Corradini, "The Arrhenius Equation Revisited," *Critical Reviews in Food Science and Nutrition,* 52(9), 2012 pp. 830–851. doi:10.1080/10408398.2012.667460.

[2] M. Peleg, M. D. Normand, and A. D. Kim, "Estimating Thermal Degradation Kinetics Parameters from the Endpoints of Non-isothermal Heat Processes or Storage," *Food Research International*, 66, 2014 pp. 313–324. doi:10.1016/j.foodres.2014.10.003.

[3] M. Peleg, A. D. Kim, and M. D. Normand, "Predicting Anthocyanins' Isothermal and Non-isothermal Degradation with the Endpoints Method," *Food Chemistry*, 187, 2015 pp. 537–544. doi:10.1016/j.foodchem.2015.04.091.

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