## A Mathematica* 9 file for Shelf Life Determination by Two Criteria from Digitized Temperature Data

This web page offers a program developed as part of a project on vitamin loss kinetics in space foods supported by NASA under project SA - 14 - 042.

This program allows the user to estimate the times, tc1 & tc2, at which the degradation curves of two nutrients will cross their respective threshold concentration ratios Concc1 & Concc2, given that the storage temperature profile is entered as a digitized time-temperature data file. The assumptions are that in the pertinent temperature range the degradation of the two nutrients follows fixed order kinetics, n≥0 , and that the temperature-dependence of the corresponding rate constant k(T) follows the exponential model [1, 2], i.e., t(T(t)) = kTref*exp(c*(T(t) - Tref)). Thus, the other entered values are the assumed kinetic orders, n1 & n2, the reference temperatures, Tref1 & Tref2, the rate constants at the corresponding reference temperatures, kTref1 & kTref2, the constants, c1 & c2, and the threshold concentration ratios, Concc1 & Concc2.

To assure a numerical solution with the FindRoot function, the reader can move the t01 & t02 sliders close to the intersection points which will be used as initial guesses of the sought times.

The storage time tmax and temperature range, Tmin & Tmax, can also be set with sliders.

The Manipulate panel display includes the temperature data in the form of an interpolated function plot (top), the calculated numerical values of the two threshold crossing times, tc1 & tc2, (middle) and plots of the two degradation curves with their corresponding threshold levels shown as dashed lines (bottom). The intersection points are plotted as colored dots and the chosen initial guesses in slightly paler colors of the same hue.

For comparison, the program can also be used for isothermal storage by clicking on the isothermal checkbox and then setting the temperature with the T slider below the checkbox.

WARNING: Note that not all possible entry combinations necessarily have a solution within the specified time range.

References

 Peleg, M., Normand, M. D. and Kim, A. D. 2014. Estimating Nutrients' Thermal Degradation Kinetic Parameters with the Endpoints Method. Food Research International 66:313-324.

 Peleg M., Normand, M. D. and Corradini, M. G. 2012. The Arrhenius equation revisited. Critical Reviews in Food Science and Nutrition 52:830-851.

The program is written in Mathematica* 9. It is presented below as a Mathematica* notebook (.nb) file, which a user having the installed Mathematica* software can open, view, modify, print and interact with. It is also presented as a Computable Document Format (.cdf) file which can be opened, viewed, printed and interacted with (but not modified) using the free Wolfram CDF Player* application. It is also presented as a Portable Document Format (.pdf) file which can be opened, viewed and printed (but not modified or interacted with) using the free Adobe Acrobat Reader DC† application.

Micha Peleg and Mark D. Normand
Department of Food Science
University of Massachusetts
Amherst, MA 01003

A notebook (.nb) file may be opened, viewed, printed, modified and interacted with if you have Mathematica* 9 or newer.

A Computable Document Format (.cdf) file may be opened, viewed, interacted with and printed (but not modified) using the free Wolfram CDF Player* application.

A Portable Document Format (.pdf) file may be opened, viewed and printed (but not modified or interacted with) using the free Adobe Acrobat Reader DC† application.

* Mathematica® and Wolfram CDF Player® are registered trademarks of Wolfram Research, Inc.
You can get more information about Mathematica by visiting the web site of Wolfram Research at: http://www.wolfram.com/