## Mathematica* 10 files Calculating Weibullian Degradation Parameters in Dynamic and Isothermal Storage and Heat Processing by the Two Endpoints Method

This webpage offers two programs developed with the support of NASA (Under Grant NNX14AP32G) to extract the Weibullian kinetic degradation parameters of nutrients in stored and heat processed foods (e.g., vitamins in space-foods), which could be used to predict their retention.

The two programs are based on the following assumptions:

1. The degradation follows the Weibullian (stretched exponential) kinetics with a fixed shape factor (exponent), m, which is known a-priori or can be assumed.
2. The food's two past temperature histories can be isothermal, i.e., T1 = constant1 and T2 (t) = constant2, or dynamic, i.e., T1(t) ≠ constant and T2 (t) ≠ constant.
3. The degradation's rate constant temperature dependence follows the exponential model, i.e., b(T(t)) = bTref exp(c(T(t))-Tref)) where b(T(t)) is the momentary degradation rate, bTref is the rate constant at a reference temperature Tref and c a constant having temperature reciprocal units. (It can and has been shown that this simple model can replace the more complicated Arrhenius equation without sacrificing the fit.)

Where these conditions are satisfied, one can show that, in principle at least, two concentration ratio measurements after two temperature histories are sufficient to estimate the magnitudes of bTref and c by numerically solving two simultaneous differential rate equations. In principle, once bTref and c have been estimated in this way, they are used to reconstruct the entire degradation curves for the two particular temperature histories and predict the degradation curves of other temperature histories.

The two programs, labeled Version A and Version B, are written in Mathematica* 10. They are presented below as Mathematica* notebook (.nb) files, which a user having the installed Mathematica* software can open, view, modify, print and interact with. They are also presented as Computable Document Format (.cdf) files which can be opened, viewed, printed and interacted with (but not modified) using the free Wolfram CDF Player* application. They are also presented as Portable Document Format (.pdf) files which can be opened, viewed and printed (but not modified or interacted with) using the free Adobe Acrobat Reader** application.

Program A: The user has two options: "Isothermal" for two constant temperatures in the range of 0 - 100°C, and "Dynamic" where the two temperature profiles, in a temperature range encountered in food storage or accelerated storage, are entered as two digitized time-temperature records.

Program B: The same as the dynamic option of Program A, except that the two entered digitized time-temperature records are in a temperature range encountered in foods pasteurization or sterilization.

In the dynamic version of Program A and in Program B, the two entered digital time-temperature records are converted into continuous Interpolation functions that can be used by Mathematica* to extract the values of bTref and c.

In both programs the user enters, with sliders, the two experimentally determined final concentration ratios and corresponding times. Also entered with sliders are the assumed shape factor m and chosen reference temperature Tref. The upper plot shows the two temperature profiles. Initially, the bottom plot will show the two entered endpoints and two degradation curves calculated with the program's default values of bTref and c. By moving the bTref and c sliders, the user should try to match the generated reconstructed degradation curves with the two entered endpoints. Once matched, as shown, these two sliders' positions mark the sought values of bTref and c, which are also displayed above the reconstructed degradation curves in the bottom graph.

To predict isothermal degradation curves choose the temperatures and rerun the program with the previously calculated bTref and c parameters.

To predict dynamic degradation curves paste their digitized time-temperature data and rerun the program with previously calculated bTref and c parameters.

Failure to match the endpoints with their corresponding reconstructed curve, or to predict correctly degradation curves at temperature histories not used in the parameters calculation, would indicate that the model and its underlying assumptions (notably the assumed m value or its constancy) might be inappropriate and/or that there might be a substantial error in one of or both the entered experimental concentration ratios.

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

### Download two Mathematica* 10 notebook (.nb) files (programs A and B).

The notebook (.nb) files may be opened, viewed, printed, modified and interacted with if you have Mathematica* 10 or newer. If you already have Mathematica* or Wolfram CDF Player installed the file will open in a plug-in in your web browser's window. To save the file choose "Save As..." from your browser's File menu and save it as a Mathematica notebook (.nb) document.

### Download two Mathematica* 10 Computable Document Format (.cdf) files (programs A and B).

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

The Portable Document Format (.pdf) files may be opened, viewed and printed (but not modified or interacted with) using the free Adobe Acrobat Reader** application.

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