Integrative programming for simulation of packaging headspace and shelf life of fresh produce

Abstract

Fresh horticultural products are exposed to different environmental conditions from farm to fork. Barrier properties of packaging and physiological behaviour of produce, namely respiration and transpiration can affect headspace conditions surrounding produce and consequently remaining shelf life. Packaging material also plays a role in heat and mass transfer, such as thermal conduction and permeation of O₂, CO₂ and water vapour. All of these behaviours are integrated together in the form of ordinary differential equations and solved using numerical methods in MATLAB.

• The simulation program is useful for designing the size and number of perforations to achieve equilibrium modified atmosphere alone or in combination with packaging material having a higher water transmission rate or active moisture absorber.

• The simulation program is also useful for predicting the shelf life of fresh produce under the actual supply chain conditions.

• The simulation program provides a flexible system to input predefined supply chain conditions and the properties of fresh produce and packaging material, thus, minimizing the costly and time consuming experimental procedures for selecting the optimum packaging material for fresh produce.

Graphical abstract

Specifications table

Subject Area	Agricultural and Biological Sciences
More specific subject area	Food Engineering Horticultural Technology Chemical and Process Engineering
Method name	Integrative programming for simulation of packaged fresh produce
Name and reference of original method	The code presented in this paper implements integrative mathematical modelling to simulate the effect of environmental conditions including temperature, relative humidity and gas composition on physiological behaviour of packaged fresh horticultural produce and predict headspace conditions, including gas composition, humidity and moisture condensation, mass loss and shelf life. The computational methods are inspired by the literature and primarily: • JALALI, A., LINKE, M., GEYER, M. & MAHAJAN, P. V. 2020. Shelf life prediction model for strawberry based on respiration and transpiration processes. Food Packaging and Shelf Life, 25, 100525. • JALALI, A., RUX, G., LINKE, M., GEYER, M., PANT, A., SAENGERLAUB, S. & MAHAJAN, P. 2019. Application of humidity absorbing trays to fresh produce packaging: Mathematical modelling and experimental validation. Journal of Food Engineering, 244, 115-125. • JALALI, A., SEIIEDLOU, S., LINKE, M. & MAHAJAN, P. 2017. A comprehensive simulation program for modified atmosphere and humidity packaging (MAHP) of fresh fruits and vegetables. Journal of Food Engineering, 206, 88-97. • Mahajan, P., Oliveira, F., Montanez, J. & Frias, J. 2007. Development of user-friendly software for design of modified atmosphere packaging for fresh and fresh-cut produce. Innov. Food Science & Emerging Technologies, 8, 84-92. • Hertog M, Boerrigter H, Van den Boogaard G, Tijskens L, Van Schaik A. 1999. Predicting keeping quality of strawberries (cv. Elsanta') packed under modified atmospheres: an integrated model approach. Postharvest Biology and Technology, 15(1): 1-12.
Resource availability	MATLAB code (www.mathworks.com) was written and a MATLAB code script file has been created.

Method description

This method aims to simulate the interaction between respiration and transpiration of produce, behaviour of packaging material such as permeability to gases and water vapour as well as moisture absorption and the ambient conditions such as temperature, relative humidity (RH) and gas composition in order to predict in package conditions such as gas composition, RH and moisture condensation [2], [5], [14], and finally quality and shelf life of packaged produce.

Fig. 1 shows the produce-packaging system including a respiring fresh horticultural produce such as strawberries in a closed plastic package covered with lidding perforated film. The produce consumes O₂ and produces CO₂ because of respiration, and releases water vapour to the surrounding area due to the transpiration process. Gases and water vapour can permeate in/out through the packaging film. Respiration also releases heat from the fruit surface to the packaging headspace. Packaging film and tray also exchange heat with the ambient atmosphere when there is a thermal gradient due to ongoing heat and mass transfer processes. There is also a potential of gas transmission across the film and its micro-perforations and moisture absorption by absorbing material embedded in packaging tray or sachets inside the tray. These assumptions and relevant mathematical formulation are discussed in more detail in previous publications [5], [6], [8], [13]. Further details on modelling the effect of respiration and transpiration on shelf life of fresh produce are discussed in Jalali et al. [4].

Fig. 1.

Packaging-ambient system including fresh produce packaged and stored under given ambient conditions.

Numerical simulation

Input variables to the mathematical modelling program cover produce properties, package properties and environmental conditions. These inputs provide initial and fixed values for the mathematical program to solve time dependent ordinary differential equations (ODEs) defining the physiological behaviour of fresh produce and heat and mass transfer phenomena which occur within the packaging and surrounding atmosphere. All these variables and ODEs were integrated into a computer simulation program based on MATLAB software (MATLAB R2010b, MathWorks®, USA). ODEs with the general form $dy/dt=f\left(t,y\right)$ were solved numerically using Euler's method which is a first order Runge-Kutta method for numerical integration of ODEs, to predict output variables. In this method, given (initial) value ($y_i$) at time $t_i$, the value of the equation after one time step ($dt$) is approximated as $y_{i+1}=y_i+f\left(t_i,y_i\right)dt$ (Fig. 2). The simulation program was divided into 5 different modules namely gas concentration, humidity and condensation, fruit mass loss, microbial deterioration and shelf life and temperature, which were solved in that sequence with an iterative step of 1 second.

Fig. 2.

Steps to follow integration and simulation of package headspace and shelf life. The symbols t, i and n stand for the time, the current time step number and the final time step number, respectively.

Table 1 shows the MATLAB script including the modelling duration and time interval, produce and packaging properties, and environmental conditions, defining the initial values to solve ODEs over the modelling duration. This script can be directly used in MATLAB or modified with user defined input parameters.

Table 2.

Code implementation for module 1: Packaging gas concentration.

Table 3.

Code implementation for module 2: Packaging humidity and moisture condensation.

Table 4.

Code implementation for module 3: Fruit mass loss.

Table 5.

Code implementation for module 4: Microbial deterioration and shelf life.

Table 6.

Code implementation for module 5: Temperature.

Table 1.

Basic inputs to integrative mathematical model.

Module 1: Packaging gas concentration

The respiration rate of produce and permeation rate of packaging film O₂ and CO₂ are included in mass balance equations for headspace gases, which determine the equilibrium gas composition as expressed in Eqs. (1) and (2) respectively.

$$\frac{d{\gamma }_{O_2}^{in}}{dt}=\frac{\frac{d{V}_{O_2}^p}{dt}+\frac{d{V}_{O_2}^r}{dt}}{{10}^6{V}_f}=\frac{\left[K_{O_2}A+\frac{{10}^6{N}_p\pi R_p^2{D}_{O_2}}{e+R_p}\right]\left({\gamma }_{O_2}^{out}-{\gamma }_{O_2}^{in}\right)-\frac{W_p{R}_{O_2}}{{\rho }_{O_2}}}{{10}^6{V}_f}$$ (1)

$$\frac{d{\gamma }_{C{O}_2}^{in}}{dt}=\frac{\frac{d{V}_{C{O}_2}^p}{dt}+\frac{d{V}_{C{O}_2}^r}{dt}}{{10}^6{V}_f}=\frac{\left[K_{C{O}_2}A+\frac{{10}^6{N}_p\pi R_p^2{D}_{C{O}_2}}{e+R_p}\right]\left({\gamma }_{C{O}_2}^{out}-{\gamma }_{C{O}_2}^{in}\right)+\frac{W_p{R}_{C{O}_2}}{{\rho }_{C{O}_2}}}{{10}^6{V}_f}$$ (2)

Where ${\mathrm{R}}_{\mathrm{i}}$ is the respiration rate of O₂ consumption and CO₂ production (ml kg⁻¹h⁻¹) respectively, which are functions of temperature and atmosphere composition as presented in Eqs. (3) and (4) reported by Hertog et al. [3] for strawberries.

$$R_{O_2}=\frac{{\gamma }_{O_2}^{in}}{K_{m{O}_2}+{\gamma }_{O_2}^{in}}V{m}_{O_2}^{ref}{e}^{\frac{E_{a{O}_2}}{R}\left[\frac{1}{T_{ref}}-\frac{1}{T}\right]}$$ (3)

$$R_{C{O}_2}=R{Q}_{ox}{R}_{O_2}+\frac{1}{\left(1+{\gamma }_{O_2}^{in}/K_{mC{O}_2\left(f\right)}\right)+1}V{m}_{C{O}_2\left(f\right)}^{ref}{e}^{\frac{E_{aC{O}_2\left(f\right)}}{R}\left[\frac{1}{T_{ref}}-\frac{1}{T_i}\right]}$$ (4)

This module consists of basic aspects of physiological parameters that are also needed in all subsequent modules. This was because the interaction between different modules is a complex system. For example, the respiration will affect the surface temperature of fruit and thereby affects the transpiration and so on. Fig. 3 shows the simulation results for modified atmosphere packaging (MAP) of strawberries with one perforation of 1 mm, containing 400 g fruit stored in two different ambient temperatures of 10°C and 20°C. The results show the effect of temperature on equilibrium gas composition of MAP headspace, so that increasing temperature provides a bigger difference of gas concentration compared to normal air, due to a higher respiration rate.

Fig. 3.

Equilibrium gas composition of headspace for a strawberry package with one perforation of 1 mm diameter in the film under 10°C (left) and 20°C (right) storage temperature.

Module 2: Packaging humidity and condensation

The mass balance of total moisture in the package is presented in Eq. (5) [4], where the rate of total moisture change ($d{M}_t/dt$) depends on the rate of moisture transpiration by produce ($d{M}_{tr}/dt$), permeation through packaging film ($d{M}_f/dt$) and perforations and absorption by packaging tray ($d{M}_a/dt$).

$$\frac{d{M}_t}{dt}=\frac{d{M}_{tr}}{dt}-\frac{d{M}_f}{dt}-\frac{d{M}_{ab}}{dt}$$ (5)

Moisture loss rate by produce depends on the transpiration rate, as a function of vapour pressure deficit (VPD) between the produce surface and headspace air, as well as the mass transfer coefficient for produce skin ($K_s$) and thin air layer ($K_a$) (mg m⁻²h⁻¹Pa⁻¹). VPD itself is generated when there is either difference between the water activity of produce and headspace RH, or a temperature difference between the surface and headspace air (Eq. (6)).

$$\frac{d{M}_{tr}}{dt}=A_s\frac{d_{tr}}{dt}=A_s\frac{VPD}{\left(\frac{1}{K_s}+\frac{1}{K_a}\right)}=A_s\frac{P_s^{sat}{a}_w-P_{in}^{sat}R{H}_{in}/100}{\left(\frac{1}{K_s}+\frac{1}{K_a}\right)}$$ (6)

The total moisture permeation rate through packaging film and perforations is a function of film permeability to water vapour, the size and number of perforations as well as the vapour pressure gradient between headspace and ambient air as presented in Eq. (7).

$$\frac{d{M}_f}{dt}={\rho }_{wv}\left[K_{H_{2}O}A+\frac{{10}^6{N}_p\pi R_p^2{D}_{H_{2}O}}{e+R_p}\right]\left(\frac{R{H}_{out}-R{H}_{in}}{100}\right)\frac{P_{in}^{sat}}{R\left(T+273.15\right)}$$ (7)

Eq. (8) represents a Weibull type moisture absorption rate for humidity packaging trays reported by Rux et al. [11] and Jalali et al. [5].

$$\frac{d{M}_{ab}}{dt}={10}^3{M}_{eq}{e}^{\left(\frac{-t}{24\beta }\right)}$$ (8)

In Eq. (8), the moisture absorption capacity at equilibrium, $M_{eq}$ is a function of RH in packaging headspace [5], [13]. Integration of Eq. (9) against time gives the cumulative moisture within the package. The difference between the cumulative moisture and maximum moisture holding capacity of air at each temperature determined the total condensation ($M_c=M_t-M_{a.max}$), while the total moisture present in headspace air is the difference between cumulative moisture and cumulative condensation ($M_a=M_t-M_c$), so that until there is no water condensed, the air moisture would be equal to cumulative moisture. The RH of headspace is then calculated as a function of headspace air moisture based on psychometric rules [1] as expressed in Eq. (9).

$$R{H}_{in}=100\times \frac{P_{atm}\times \frac{M_a\times {10}^{-3}}{W_{air}}}{P_{sat}\left(0.622+\frac{M_a\times {10}^{-3}}{W_{air}}\right)}$$ (9)

Fig. 4 shows the evolution of different moisture components over the simulation time for a normal plastic tray and a moisture absorber tray under 20°C storage temperature. Both packages had a perforation of 3 mm diameter in packaging film. Results showed that most of the transpired moisture from fruit was condensed in the normal plastic package. Moisture absorber in a package could prevent condensation by absorbing the excess water, however, imposed a higher total moisture transpiration compared to the normal plastic package. This was due to moisture absorption decreases vapor pressure in packaging headspace, which is compensated by more transpiration consequently. Fig. 5 shows the headspace RH of both packaging types. RH in the moisture absorbing tray was only slightly less than the normal plastic tray. This was also due to the higher transpiration of fruit in the moisture absorbing tray, which prevented the decrease in RH.

Fig. 4.

Moisture distribution for strawberry packaged in the normal plastic tray (left) and moisture absorber tray (right) under 20°C storage temperature.

Fig. 5.

Headspace RH for strawberry packaged in the normal plastic tray (left) and moisture absorber tray (right) under 20°C storage temperature.

Module 3: Fruit mass loss

Fruit mass loss (as a percentage compared to initial mass) calculated as one of the quality indices was used to predict the shelf life of fresh produce during the supply chain. Mass loss based shelf life is the time it takes to reach the maximum acceptable mass loss (MAM) as a percent of initial fruit mass and was determined 4.3% for strawberries [4]. It is separated into two different parts. The first part was moisture loss due to fruit transpiration and the second part was due to oxidative mass loss resulting from fruit respiration. The overall equation for the oxidation of glucose as the substrate for aerobic respiration is ${\mathrm{C}}_6{\mathrm{H}}_{12}{\mathrm{O}}_6+6{\mathrm{O}}_2\to 6\mathrm{C}{\mathrm{O}}_2+6{\mathrm{H}}_2\mathrm{O}+\text{energy}$. Then for respiratory oxidation of 180 g (1 mole) of glucose, 192 g (6 moles) of O₂ is consumed which diffuses into the tissue from the surrounding atmosphere, while 264g (6 moles) of CO₂ diffuses out. The 108 g (6 moles) of H₂O produced is simply incorporated into the aqueous solution of the cell [7]. Hence the total mass loss ($\mathrm{d}{\mathrm{L}}_{\mathrm{t}}/\text{dt}$) was estimated as the sum of mass loss from respiratory oxidation of the substrate ($\mathrm{d}{\mathrm{L}}_{\text{ox}}/\text{dt}$) [6] and moisture loss due to transpiration (Eq. (5)).

$$\frac{d{L}_{ox}}{dt}=W_p{R}_{C{O}_2}\left(180-108\right)/264$$ (10)

$$\frac{d{L}_t}{dt}=\frac{d{M}_{ox}}{dt}+\frac{d{M}_{tr}}{dt}$$ (11)

Fig. 6 shows the simulation results of fruit mass loss and its components for strawberry packages with the same conditions as the previous section. Oxidative mass loss is almost the same for both packages as it changes with storage temperature. Total mass loss in moisture absorber package was higher than the normal plastic package, since higher transpiration rate imposed by moisture absorption.

Fig. 6.

Percentage mass loss components for strawberry packaged in the normal plastic tray (left) and moisture absorber tray (right) under 20°C storage temperature.

Module 4: Microbial deterioration and shelf life

Microbial deterioration was another quality index to predict the shelf life of fresh produce. Deterioration based shelf life is the time it takes the produce to reach the maximum acceptable deterioration (MAD) as a percentage. Matar et al. [10] provided a mathematical model to predict fungal decay for packaged strawberries Eq. (12) and ((13)), where they defined 13% of visible deterioration as MAD, using a global visual method for measuring the deterioration of strawberries in MAP [9].

$$\frac{\mathit{dN}}{\mathit{dt}}=K_{\mathrm{d}}N\frac{N_{\max }-N}{N_{\max }}C{O}_{2_{\mathit{rel}}}$$ (12)

$$C{O}_{2_{\mathit{rel}}}=1-\frac{y_{C{O}_2}}{y_{C{O}_2}^{\max }}$$ (13)

Where deterioration constant, $K_d$ is a function of temperature and metabolic rate, $C{O}_{2_{\mathit{rel}}}$ is a deterioration-inhibiting factor due to CO₂ evolution.

The effect of two different packaging types, including non-MAP (19.5% O₂ & 1.2% CO₂ at equilibrium) made by 20 perforations of 1 mm diameter in packaging film and MAP (3.5% O₂ & 16.1% CO₂ at equilibrium) made by one perforation of 1 mm diameter in the film is showed in Fig. 7. Lower respiration of fruit under MAP conditions led to a lower deterioration rate, thereby, increased the shelf life to 214 h compared to 105 h under non-MAP conditions.

Fig. 7.

Microbial deterioration for strawberry package in (left) non-MAP (19.5% O₂ & 1.2% CO₂) and (right) MAP (3.5% O₂ & 16.1% CO₂) stored under 20°C.

Module 5: Temperature

Thermal equilibrium equations were used to calculate steady-state heat and mass transfer to predict the temperature changes on produce surface, headspace air, internal and external tray and film surface according to changes in ambient temperature. Thermal equilibrium between produce and headspace air is a balance between rates of internal heat production due to respiration, convective heat transferred from headspace air to produce surface. Eq. (14) is based on the thermal equilibrium between produce surface and headspace air to calculate the rate of surface temperature change [12].

$$\frac{d{T}_s}{dt}=\frac{6.21R_{C{O}_2}{W}_p-\frac{d{M}_{tr}}{dt}\lambda +h_s{A}_s\left(T_i-T_s\right)}{W_p{C}_p}$$ (14)

In a similar way, the rate of temperature change on the internal surface of the packaging tray was calculated from the thermal equilibrium between headspace air and the inner surface of the tray and film (Eq. (15)).

$$\frac{d{T}_t}{dt}=\frac{\frac{K_t{A}_t\left(T_o-T_t\right)}{x}-h_t{A}_t\left(T_i-T_t\right)+\left(\frac{d{M}_{ab}}{dt}+\frac{d{M}_c}{dt}-\frac{d{M}_f}{dt}\right)\lambda }{W_t{C}_t}$$ (15)

In addition, the overall thermal equilibrium inside the package used to calculate the rate of temperature change in headspace air.

$$\frac{d{T}_i}{dt}=\frac{h_t{A}_t\left(T_t-T_i\right)-h_s{A}_s\left(T_i-T_s\right)+\frac{d{M}_t}{dt}\lambda }{W_a{C}_a}$$ (16)

Fig. 8 shows temperature evolution for packaging tray wall, headspace and fruit surface for normal plastic and moisture absorber tray packaging under 20°C. As the moisture absorbing tray led to a higher transpiration rate of fruit, the cooling effect of moisture evaporation from the fruit surface led to a lower fruit surface and consequently headspace air temperature in this package compared to normal plastic tray packaging. Internal tray wall temperature was very slightly higher than the ambient air temperature, which was not visible, so that tray wall temperature fitted on the ambient temperature curve.

Fig. 8.

Temperature evolution for strawberry packaged in the normal plastic tray (left) moisture absorber tray (right) under 20°C storage temperature.

Method application under realistic supply chain conditions

The ultimate objective of the integrating simulation program was to provide flexibility to adapt to varying supply chain conditions of fresh horticultural produce, therefore, optimization of packaging material. Fig. 9 shows an example of a typical supply chain of harvested strawberries, in which fluctuating conditions fitting different postharvest processes served as simulation input in order to compare keeping quality of fruit under different postharvest chain scenarios including MAP and open tray packaging [4]. Under the identical supply chain conditions, MAP showed different packaging headspace temperature and RH compared to the open tray packaging where the fruits are exposed directly to the ambient conditions of the supply chain (Fig. 9).

Fig. 9.

Supply chain profile for postharvest handling of fresh strawberries under MAP and open tray packaging conditions (total time subdivisions: Grower (G), Cooling (C), Packaging (P), Wholesale (W), Transport (Tr), Supermarket Warehouse (SW), Supermarket Fridge (SF) and Consumer Fridge (CF)).

Fig. 10 shows the effect of different packaging types on the shelf life of fresh strawberries under the same supply chain conditions. The values for using MAP, it was possible to delay the microbial deterioration and percentage mass loss to meet the maximum acceptable value for about 2.5 d and 1 d, respectively rather than open tray packaging. However, both cases showed that the shelf life was limited by the percentage mass loss, as it reached the maximum acceptable value in a shorter time compared to the microbial deterioration. The integrating simulation program, thus, was a useful tool for the design and optimization of packaging for fresh produce under dynamically changing environmental conditions.

Fig. 10.

The effect of open tray packaging (left) and MAP conditions (right) on different quality indices (percentage mass loss and percentage microbial deterioration under realistic supply chain conditions).

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.