# Estimating Equilibrium Moisture Content

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

The equilibrium moisture sorption isotherms of different foods display an important aspect of their chemical and biological stability, processing and preservation protocols, and packaging. Equilibration of a food sample at a given water activity by reaching constant weight in a controlled humidity environment may take a day or two, which is logistically inconvenient. This Demonstration presents a method to estimate the equilibrium moisture content at a given water activity (or relative humidity) from the initial conditions and only two other moisture content determinations after much shorter exposure times, on the order of a few hours, using an empirical dynamic sorption model. You can also estimate the equilibrium moisture content in desorption using an inverted version of the same model.

Contributed by: Mark D. Normand, Viridiana Tejada-Ortigoza and Micha Peleg (July 2019)

Open content licensed under CC BY-NC-SA

## Details

Snapshot 1: equilibration of a dry sample by sorption

Snapshot 2: equilibration of a wet sample by sorption

Snapshot 3: equilibration of a dry sample by desorption

Snapshot 4: equilibration of a wet sample by desorption

We assume that dynamic moisture sorption can be described by the empirical model

,

where is the moisture content (on a dry basis) at time ; and are the initial and equilibrium moisture contents, respectively; and is a characteristic time constant [1].

Suppose that we have determined the initial moisture content followed by two successive moisture contents, at time and at time , . We can now write two simultaneous equations:

and

,

whose two unknowns are the desired equilibrium moisture content and the time constant . These two equations have analytical solutions that can be used to calculate the values of these two parameters.

In desorption, the model has the form

,

and likewise, its two parameters, the desired equilibrium moisture content and the time constant , can be calculated from two successive moisture determinations, at time and at time , .

In both the sorption and desorption cases, the times and should not be too close together to avoid a potential error, but they can be considerably shorter than the time needed to approach constant weight, hence the utility of the method.

To estimate the equilibrium moisture content , use the "mode" button to choose "sorption" or "desorption". Then enter the initial moisture , times and , and their corresponding moisture contents and with their sliders. A plot will be displayed depicting the three entered points, the corresponding sorption or desorption curve calculated with the model, and its asymptote as a horizontal dashed red line. The numerical values of and are displayed above the plot.

The position of a movable red point along the sorption or desorption curve can be adjusted with the "movable point" slider. The numerical values of its corresponding and percentage of the predicted equilibrium are also displayed above the plot. The maxima of the plot axes can be adjusted with the "axes maxima" sliders.

Not all possible plotted curves correspond to actual foods. In cases where any entry or the calculated is inconsistent with the model or in violation of physical considerations, a red error message is displayed.

Most probably, the method can be also be used to estimate the equilibrium moisture contents of nonfood hygroscopic materials, especially pharmaceuticals.

Reference

[1] M. Peleg, "An Empirical Model for the Description of Moisture Sorption Curves," *Journal of Food Science*, 53(4), 1988 pp. 1216–1217 and 1219. doi:10.1111/j.1365-2621.1988.tb13565.x.

## Snapshots

## Permanent Citation