Mathematical modelling of diffusion channel length to maintain steady-state oxygen concentration for controlled atmosphere storage of tomato

ABSTRACT

In this study, potential of the air-diffusion channels to maintain the desired controlled atmosphere condition in the storage chambers was evaluated. The diffusion channels of different combination in length (60, 120, 180, 240 mm) and diameter (3, 6, 9, 12 mm) were installed in the storage chambers of 2 L capacity. Tomato samples (0.805 kg) were taken in the chambers and stored at 10°C. Oxygen concentration in the chambers was measured periodically until the end of storage period. The tomatoes were stored upto 40 days under the channel length of 180 mm and diameter of 9 mm with harvest-fresh appearance, better texture and colour. The experimental results showed that the oxygen concentration ranging from 7% to 15% was maintained depending on the channel length and its tendency was found directly proportional to the channel diameter and inversely related to the channel length. Simultaneously, respiration rate of tomato was determined by closed respiration system to facilitate working with diffusion channel model. A model was developed based on Fick’s first law of gas diffusion to predict the channel length. The model was fitted to the experimental data and channel lengths were calculated theoretically in the ranges between 30 and 320 mm. The predicted channel lengths maintained the oxygen concentration between 4% and 16% irrespective of the channel diameters. The predicted results were in accordance with the experimental results. The model was very useful in predicting the diffusion channel length and establishing the controlled atmosphere condition for storage of tomatoes.

KEYWORDS:

• Tomato
• Diffusion channel
• Modelling
• O₂ concentration
• Respiration rate

Introduction

Tomato (Solanum lycopersicum) is one of the most consumed vegetables worldwide. It is a climacteric vegetable, so it is highly perishable in nature that encounters several problems in its transportation, storage, and marketing.^[1] It has been estimated that the postharvest losses of tomato range from 20% to 50% in tropical countries.^[2] Therefore, an increase in postharvest life of tomato is really desirable to reduce the losses during supply chain. One of the major physiological activities in the postharvest life of tomato is respiration. Fresh tomatoes generally have higher respiration rates that indicate more active metabolism and subsequently a faster deterioration rate in the normal course of time.^[3] Respiration involves the consumption of oxygen (O₂) for oxidative breakdown of organic components into simple molecules such as carbon dioxide (CO₂) and water (H₂O) with concurrent release of energy and other intermediates which can be used by the cell for synthetic reactions.^[4] Respiration rates and deterioration rates can be minimized by storing the produce in controlled/modified atmospheric condition at low temperature. Modified atmosphere can be achieved by maintaining the natural interplay between respiration rate of the commodity and transfer of gases through the storage that lead to an atmosphere richer in CO₂ and poorer in O₂.^[5] Controlled atmosphere storage (CAS) has been defined as a form of storage where the concentration of gas is initially modified according to optimal conditions and then maintained during the period of storage.^[6] Over ripening, senescence, ethylene production rates, and some physiological alterations such as mould growth, rotting, decay, butt discolouration can be minimized greatly in CAS condition.^[7,8] The levels of both O₂ and CO₂ concentrations during storage affect the quality of the final product. Absence of O₂ results in development of off-flavours, and presence of high CO₂ concentration (more than 5% for tomato) may cause physiological injures.^[9]

There are different techniques such as O₂ control system, CO₂ control system, hypobaric system, membrane system, and diffusion channel system to provide the desired gas composition in the storage environment depending on the degree of control of the gases.^[6,10] The final O₂ concentration in the storage chamber depends on the permeability of the membrane and area available for O₂ transfer. Moreover, for those commodities that need a high concentration of O₂ during storage, the use of polymeric membranes has not been successful due to their low O₂ permeability.^[11] Among different techniques, the diffusion channel system is indeed a potential alternative for products that cannot be packed in polymeric films, and is capable of maintaining a specific steady-state gas composition in CAS systems over a longer period of time, relatively insensitive to changes in barometric pressure and fluctuations in the storage room temperature, provides great flexibility in the design of storage chamber, and materials used are considerably less expensive and are structurally simple.^[12] Diffusion channel system works on the principle of Fick’s first law of gas diffusion. Diffusion channel is a hollow tube fitted with an airtight storage chamber in which produce is stored and other end of the tube is exposed to the atmosphere. As a result of consumption of O₂ and liberation of CO₂ during respiratory process of the stored commodity, a great change in concentration of gases inside the storage chamber occurs. With respect to ambient condition, O₂ concentration decreases and that of CO₂ increases at rates depending on respiration rate of the produce. This creates a concentration gradient between inside and outside the chamber. Diffusion of gases takes place through the channel due to concentration gradient of gases between inside and outside the storage chamber, thus modified atmosphere is created inside the chamber which is beneficial for storage. The rate of diffusion of gases is dependent on length and cross-sectional area of the channel and gas concentration gradients between the chamber and ambient air. The gas composition in the storage chambers could easily be altered by varying the dimensions of diffusion channels.^[12–16]

There is not much research work on the concept of diffusion channel system to enhance the shelf life of fruits and vegetables. Some researchers have published their research work on diffusion channel system. Emond et al.^[11] studied the gas exchanges through a perforation (considered as channel of zero length) and developed a model to predict the gas exchanges through the perforation. Baugerod^[12] proved that the diffusion channel system is capable of maintaining the steady-state O₂ concentration from 1.5% to 18% depending on the mass of stored product, and length and cross-sectional area of the diffusion channel. Ratti et al.^[13] showed the influence of diffusion channel length on the concentration of O₂ in the storage chambers but no theoretical interpretation of the experimental data. Ratti et al.^[14] developed mathematical model based on the molecular diffusion theory and this model includes the effect of mass of the produce, length and cross-sectional area of the diffusion channel on the final gas composition in the storage chamber. Different combinations of diffusion channel length (0.6, 3, 7, 12, 18, and 25 cm) and cross-sectional area (0.04, 0.18 and 1.15 cm²) were tested for storage of broccoli at 0°C.^[15] The length and cross-sectional area have a significant effect on the final O₂ steady-state concentration. Fick’s first law of molecular diffusion theory was used to predict the required length of diffusion channel. Spinach was stored up to 30 days at 2°C under diffusion channel length of 0.18 m and cross-sectional area of 4×10⁻⁶ m² with best colour and freshness.^[16] CO₂ produced by the respiring spinach was completely scrubbed using calcium hydroxide, and O₂ was maintained in the range of 1–5%. The O₂ steady-state levels decreased with an increase in length and became more distinct as the cross-sectional area increased. A model was developed based on Fick’s first law of molecular diffusion for predicting the length of diffusion channel. Stewart et al.^[17] confirmed that the Cavendish bananas (also a climacteric fruit) can be stored for 42 days at 15°C under diffusion channel systems (4, 7, and 10 cm channel length). The shortest diffusion channel achieved 5% CO₂ and 3% O₂ in 12–16 days. Karthiayani et al.^[18] stored banana (Musa acuminate) under diffusion channel system (length: 5, 10, 15, and 20 cm and inner diameter: 3, 5, and 7 mm) at room temperature, 24°C and 14°C. They showed that the shelf life of banana can be increased 3–4 -folds as compared with control. They also developed a model using non-linear multiple regression to fit the observed data, and the model was found to be useful in determining the length of the diffusion channel for required steady-state O₂ concentration for banana.

Since the polymeric membranes have low O₂ permeability, maintaining required O₂ concentration in the storage is not feasible. The use of diffusion channel system is theoretically able to handle the whole range of possible O₂ concentrations in the storage. The main objective of this work is to develop mathematical model to predict the optimal length of diffusion channel to maintain the steady-state O₂ concentration in the storage.

Theory of modelling

Fick’s first law of gas diffusion

CAS involves atmospheric gas mixtures and their diffusion characteristics. Gas diffusion describes the spread of particles through random motion from regions of higher concentration to regions of lower concentration. Diffusion occurs in response to a concentration gradient and is expressed as change in concentration due to a change in position. The gas diffusion generally obeys Fick’s first law of gas diffusion, which states that the gas flux moves from region of higher concentration to region of lower concentration, with a magnitude that is proportional to the concentration gradient. The movement of gases is due to the random motion of the individual molecules caused by their kinetic energy. The molecular diffusion is typically described mathematically using Fick’s first law of gas diffusion^[19] as follows:

(1)

where N_A is the rate of gas diffusion per unit area per unit time (mol m⁻² s⁻¹), D_AB is the diffusion coefficient that describes the speed of gas that diffuses from region A to region B (m² s⁻¹), C_A is the concentration of gas in A, ∂x is the distance between where the object started and ended after diffusion, (−) sign indicates that N_A is positive when movement is down the gradient. In steady-state conditions, diffusion rate of a gas is proportional to the concentration difference across the material and the area perpendicular to the diffusion.^[20]

Development of modelling

The molecular diffusion theory on diffusion of O₂ through diffusion path was adopted in this study to develop a mathematical model to predict the optimal length of diffusion channel as described by Ratti et al.^[14] for storage of cauliflower. This model takes into account the functionality between the mass of the produce, final O₂ concentration level in the storage chamber, diffusion channel length, and its cross-sectional area. While modelling process, the following key assumptions have been made. (i) total pressure in the chamber is 1 atm; (ii) the temperature is constant; (iii) the concentration of O₂ in the chamber is uniform at any time; (iv) the diffusion process is binary, since the CO₂ produced by the stored commodity is chemically scrubbed off, thus O₂ and N₂ are diffusing; (v) there is no concentration gradient for nitrogen (N₂) between the outside and the inside, thus O₂ will be diffusing in a stagnant N₂ gas; (vi) the diffusion coefficient of O₂ in N₂ is constant; and (vii) diffusion of gases in the chamber is one-dimensional.

Figure 1 shows the diffusion of O₂ into a storage chamber through the diffusion path. The mass balance is written for a strip of the diffusion channel of length Δz. For a binary gas mixture, the general equation that describes the flux of the diffusing substance (O₂) into a stagnant gas (N₂) based on molecular diffusion can be stated as follows:

(2)

Figure 1. Diffusion of O₂ into a storage chamber through the diffusion path.

where is the O₂ molar flux (mol cm⁻² h⁻¹), c is the total gas molar concentration (mol cm⁻³), is the diffusion coefficient of O₂ gas through diffusion channels (cm² h⁻¹), is the O₂ mole fraction in sealed storage chamber (dimensionless), z is the spatial coordinate (cm). Under steady-state condition, the amount of O₂ utilized in the respiration process is equal to that diffusing into the chamber. Applying boundary conditions to the diffusion channel system

The mass flux of O₂ can be obtained by rearranging Eq. (2) and integrating with proper limits of boundary conditions.

(3)

(4)

where L is the length of the diffusion channel (cm), is the O₂ mole fraction in air (dimensionless), is the O₂ mole fraction in storage chambers installed in diffusion channel. Under steady-state conditions, the O₂ consumed by the commodity is equal to the O₂ diffusing through the channel into the chamber. The O₂ consumed in the respiration process can be expressed numerically as follows:

(5)

where is the mass of O₂ consumed in the respiration process (mg O₂), is the respiration rate as a function of O₂ (mg O₂ kg⁻¹ h⁻¹), M_s is the mass of the stored products (kg), t is the time (h). Under steady-state condition (z = 0), the mass flux of O₂ can be written as

(6)

where A_C is the cross-sectional area of the diffusion channel (cm²), is the O₂ molecular weight (kg mol⁻¹), t is the time (h). Equating Eqs. (5) and (6) and simplifying, we get

(7)

The length of diffusion channel required to maintain a desired steady-state O₂ concentration inside the storage chamber can be obtained by equating Eqs. (4) and (7)

(8)

The following model was adopted to predict the respiration rate of tomato as a function of O₂ and CO₂ concentration:^[21]

(9)

where r is the respiration rate (mg O₂ kg⁻¹ h⁻¹ or mg CO₂ kg⁻¹ h⁻¹), is the O₂ concentration (mol cm⁻³), is the CO₂ concentration (mol cm⁻³), K₁ is the constant (mg kg⁻¹ h⁻¹), K₂ is the constant (mol O₂ mol total⁻¹), K₃ is the inhibition constant (dimensionless). The inhibition constant (K₃) for CO₂ concentration was tended to zero as the rate of CO₂ evolution during the respiration process is not same as the rate of O₂ consumption. The constant K₃ was negligible in either case, which nullifies the effect of changing CO₂ concentration on the respiration rate.^[21] Hence, the respiration rate was assumed only as a function of O₂ concentration as follows:

(10)

To facilitate working with the diffusion channel model, the concentration of O₂ is expressed as O₂ mole fraction and stated as follows:

(11)

At steady-state condition, concentration of O₂ can be expressed as follows:

(12)

Substituting Eqs. (11) and (12) in Eq. (8) can be written as:

(13)

Diffusion coefficient of oxygen

The diffusion coefficient () of O₂ into N₂ can be determined with following key assumptions: (i) length of diffusion channel is unity, (ii) Fick’s law defines this transport process clearly, (iii) O₂ concentration in the chamber is uniform at any time (steady state). Based on these assumptions, the diffusion coefficient of O₂ through diffusion channels can be calculated by following equation:^[22]

(14)

where Q_m is the mass flow rate of O₂ through diffusion channel (mg h⁻¹), Δl is the unit length of diffusion channel (cm), Δ_c is the difference in gas concentration between inlet and outlet end of the diffusion channel (mg cm⁻³).

(15)

(16)

where a₁ and a₂ are constants.

Model for length of diffusion channel

The model to predict the length of diffusion channel can be obtained by substituting Eqs. (15) and (16) in Eq. (13) and simplifying

(17)

Materials and methods

Sample preparation

The popular tomato variety Roma in the West Bengal (India) was selected for this study. The matured tomatoes at breaker stage of ripeness (definite break in colour from green to tannish-yellow, pink, or less than 10% red coloration^[23]) and uniform size were procured from a farmer field. The tomatoes were transported to the lab immediately after harvest. The harvested tomatoes were graded manually and washed in chlorinated water with the concentration of 100 mg L⁻¹ to remove adhering dirt on their surface.^[24] The tomato samples of 0.805 kg were chosen since the selected storage chamber’s capacity was 2 L and two-third of the headspace in the chambers was kept as free volume.

Experimental design

The experiments were designed with four levels of diffusion channel length (60, 120, 180, and 240 mm) and four levels of channel diameter (3, 6, 9, and 12 mm) as independent variables and gas concentration as dependent variable. The diffusion channels (fibre glass tubes) of different geometries were purchased from Hindusthan Industrial Syndicate, Kolkata, India. The geometries of diffusion channel were selected based on the information available in the previous studies.^[14–17] Different symbols (Table 1) were used to denote different treatments with three replicates. Factorial completely randomized design using AGRESS software package (P ≤ 0.05) was used for analysis of the effect of different lengths and diameters of the diffusion channel on gas concentration in the storage chambers. The storage temperature of 10°C was selected as recommended by the previous research (fresh tomatoes stored at around 10°C were more favourable as compared with high temperature (24–30°C) for prolonged shelf life and to retain fresh tomato quality,^[25] and postharvest recommendations indicate that tomatoes, including cherry and grape tomatoes, should be stored at 10°C or higher to avoid chilling injury^[26]).

Table 1. Treatment details of the experiments.

Treatments (T)	Diffusion channel dimensions
	Diameter of channel, D (mm)	Length of channel, L (mm)	Cross-sectional area of channel, AC (cm^2)
T1	3	60	0.071
T2	3	120	0.071
T3	3	180	0.071
T4	3	240	0.071
T5	6	60	0.283
T6	6	120	0.283
T7	6	180	0.283
T8	6	240	0.283
T9	9	60	0.636
T10	9	120	0.636
T11	9	180	0.636
T12	9	240	0.636
T13	12	60	1.131
T14	12	120	1.131
T15	12	180	1.131
T16	12	240	1.131

Fabrication of experimental setup

The storage chambers of 2 L capacity made up of polyethylene tetrachloride (PET) were used for conducting the experiments. The experimental setup (Fig. 2) was fabricated in such a way that four holes were provided on the lid. A silicone septum was fitted in the first hole to facilitate the withdrawal of gas sample from the chamber for analysis. Brass nipples with rubber gaskets were connected in second and third holes and tightened with nuts. Rubber tubes of length 200 mm were connected to these nipples and closed by pinch clips after purging gas. In the fourth hole, a conical-shaped hallow rubber cork was fitted. The diffusion channel of required size was rigidly fixed in the rubber cork. The joints were crammed by melted paraffin wax to secure air tightness. The absence of air bubble ensured the air tightness of diffusion chambers.

Figure 2. Experimental setup of diffusion channel system for storage of tomato.

Experimental procedure

The experimental chambers were washed with chlorinated water of concentration 500 mg L⁻¹ and wiped completely by cotton cloth. Calcium hydroxide (50 g kg⁻¹) with tomato wrapped in a tissue paper was placed at the bottom of the each storage chamber to prevent any accumulation of CO₂ and absorb condensed moisture within the storage chambers.^[14–16] The tomato samples (M_s = 0.805 kg) were taken in the storage chambers. About two-third of headspace was left in the chambers as free volume. The chambers were provided perfect air tightness by top lid and wrapped with Teflon tape. To achieve a fast pull-down O₂ level inside the chamber, the storage chamber with tomato sample was flushed with N₂ until O₂ level is brought down to required level (6%). The selected gas mixture was flushed from one sampling port, and the other sampling port was opened as gas outlet to atmosphere. The gas was flushed for about 3–5 min. During flushing, diffusion channels were closed with a thin wax plate. At the end of flushing, both inlet and outlet ports were closed with pinch clips. The storage chambers were transferred to the cold room where the temperature of 10°C was maintained throughout the storage period. The chambers were kept in vertical position until end of the storage period.

Measurement of respiratory gases

The storage chambers were allowed to sit for about 30 min for their temperature to come to equilibrium with room temperature. The first gas samples were drawn without breaking the wax cover of channel opening. The readings indicated that gas concentrations in the chamber were uniform. The wax plate was then removed from the diffusion channels and the gases were allowed to diffuse. Gas samples of 0.5 mL were drawn periodically from each chamber through the silicone septum using syringes. The gas samples were analysed for CO₂ and O₂ concentration (%) using gas chromatograph (SRI 8610A model, SRI Instruments, USA) equipped with thermal conductivity detector and operating at an oven temperature of 45°C and detector temperature of 100°C using helium as the carrier gas. Calculation of concentration of the gases was made with the Peak Simple software. The standard gas mixture (purchased from Span Gas Equipments Ltd., Mumbai) was analysed, and calibration curves were developed for determining the amount of components in experimental gas mixture quantitatively.

Determination of respiration rate

Respiration rate of tomato was measured using the closed respiration system, which involves monitoring the O₂ and CO₂ concentrations as a function of time inside a closed chamber containing the tomatoes. The respiration rate can be measured by observing the concentration of O₂ consumption or CO₂ evolution per unit time per unit weight of the produce^[27,28] (it can be expressed by mg O₂ kg⁻¹ h⁻¹ or mg CO₂ kg⁻¹ h⁻¹). The experimental respiration rate was calculated periodically using the data such as difference in gas concentration per time, weight of the stored produce, and free volume of the chamber. The mass concentration of O₂ and CO₂ inside the closed experimental chamber were determined and plotted as a function of time. Curve fitting was done using second-order polynomial regression equation.^[29] The fitted polynomial was differentiated to determine the change of rate of gas concentrations. The first derivative of the regression functions were used to obtain the experimental respiration rates as a function of O₂ consumption () and CO₂ evolution ()^[30]

(18)

(19)

where [O₂] is the oxygen concentration (%), [CO₂] is the carbon dioxide concentration (%), d[O₂] is the change in concentration of O₂ with time, d[CO₂] is the change in concentration of CO₂ with time, dt is the difference in time between two gas measurements (h), P is the gas pressure (1 atm), M_a is the mean molecular mass of air (g mol⁻¹), V_f is the free volume of the chamber (L), R is the gas constant in the ideal gas equation (0.08206 L atm mol⁻¹ K⁻¹), T is the temperature of the gas (K), W_p is the weight of the stored tomatoes (kg). The negative sign in Eq. (18) signifies that the O₂ concentration in the chamber decreases with time.

Results and discussion

The respiration rate of tomato was obtained through the closed respiration system experiments at 10°C. Kandasamy et al.^[21] reported detailed experiment as well as calculation procedures. The experimental respiration data were used to estimate the respiration rate model constants in Eq. (9). The respiration rates were plotted as a function of O₂ and CO₂ concentration (Fig. 3). The model was then fitted to the experimental data. The respiration rate model constants K₁, K₂, and K₃ in Eq. (9) were determined by non-linear regression with experimental data using Gauss–Newton procedure for non-linear least squares. The regression described that the experimental data, that is, O₂ concentration was very well (R² = 0.970), whereas CO₂ concentration was fair (R² = 0.812) for tomato. The results of fitting for K₁, K₂, and K₃ were 208.552, 526.810, and 0.007, respectively. Constant K₃ is quite close to zero and nullifies the influence of CO₂ concentration on respiration rate. Therefore, the respiration rate of tomato was considered mainly as a function of O₂ concentration.^[21] The inhibition parameter K₃, calculated as the rate of CO₂ evolution, was inversely related to temperature, whereas the one calculated as the rate of O₂ consumption was directly proportional to the temperature. This is because the rate of CO₂ evolution during the respiration process is not same as the rate of O₂ consumption. The parameter K₃ was negligible in either case which nullifies the effect of changing CO₂ concentration on the respiration rate.^[21] As the inhibition constant K₃ for CO₂ concentration in Eq. (9) was negligible, the model was modified mainly as a function of O₂ concentration (Eq. 10).

Figure 3. Respiration rate of tomato: (a) function of O₂, (b) function of CO₂ concentration at 10°C.

To facilitate working with the diffusion channel model, the concentration of O₂ was expressed as O₂ mole fraction (Eq. 11). The respiration rate of tomato in sealed storage chambers was calculated as a function of O₂ molar fraction. The respiration rate was plotted against the O₂ molar fraction. Curve fitting was done using non-linear regression equation as shown in Fig. 4. Constants K₁ and K₂ in Eq. (11) were obtained from experimental data through the curve fittings. Results of fitting for K₁ and K₂ were 13.764 mg kg⁻¹ h⁻¹ and 14.328 mol O₂ mol total⁻¹, respectively. The predictions of the respiration rate model [Eq. (11) with fitted constants] are shown in Fig. 4 together with experimental results. The linear tendency shown by the model in this range of O₂ concentrations is in accordance with the predictions of enzymatic reaction theory.

Figure 4. Respiration rate of tomato as a function of O₂ mole fraction at 10°C.

The diffusion coefficient of O₂ () through the diffusion channel was determined through Eq. (14) by fitting the experimental data, and the results obtained are presented in Table 2. The diffusion coefficient of O₂ depends on length and cross-sectional area of the diffusion channel, mass flow rate of O₂ through diffusion channels, and difference in gas concentration between inlet and outlet end of the diffusion channel.^[22] It is clearly noticed from Table 2 that the decreasing trend in diffusion coefficient of O₂ was observed when increasing the channel length and diameter.

Table 2. Different parameters for predicting the model for diffusion channel length.

Treatments	Ac	Ms		a1	a2
T1	0.071	0.805	1.824	0.273	4.135	0.025	32	0.189	0.106
T2	0.071	0.805	0.943	0.187	4.135	0.025	32	0.189	0.093
T3	0.071	0.805	0.801	0.076	4.135	0.025	32	0.189	0.073
T4	0.071	0.805	0.466	0.067	4.135	0.025	32	0.189	0.068
T5	0.283	0.805	1.355	0.089	4.135	0.025	32	0.189	0.113
T6	0.283	0.805	0.242	0.061	4.135	0.025	32	0.189	0.101
T7	0.283	0.805	0.114	0.029	4.135	0.025	32	0.189	0.081
T8	0.283	0.805	0.082	0.021	4.135	0.025	32	0.189	0.073
T9	0.636	0.805	0.943	0.061	4.135	0.025	32	0.189	0.130
T10	0.636	0.805	0.152	0.038	4.135	0.025	32	0.189	0.113
T11	0.636	0.805	0.067	0.017	4.135	0.025	32	0.189	0.086
T12	0.636	0.805	0.048	0.012	4.135	0.025	32	0.189	0.077
T13	1.131	0.805	0.478	0.045	4.135	0.025	32	0.189	0.141
T14	1.131	0.805	0.099	0.025	4.135	0.025	32	0.189	0.122
T15	1.131	0.805	0.049	0.013	4.135	0.025	32	0.189	0.094
T16	1.131	0.805	0.038	0.011	4.135	0.025	32	0.189	0.086

The constants a₁ and a₂ were obtained through Eqs. (15) and (16), respectively, by fitting the experimental data such as diffusion coefficient of O₂ through the channel (Table 2), constants (K₁ and K₂), and total gas molar concentration, that is, 3.466 mol cm⁻³. Results of fitting for a₁ and a₂ are presented in Table 2. The constant a₁ was found to be varied because the diffusion coefficient of O₂ varied depending on the channel length and cross-sectional area. O₂ molecular weight (), O₂ mole fraction in sealed chamber (), and O₂ mole fraction in air () were 32 kg mol⁻¹, 0.025, and 0.189, respectively. Moreover, O₂ mole fraction in storage chambers equipped with diffusion channel () was determined using experimental data and the results obtained are presented in Table 2.

The diffusion channel length model (Eq. 17) was fitted to the experimental data (Table 2) and the results obtained were found negative. Since the length cannot be negative, the model (Eq. 17) was rewritten as follows:

(20)

Then the corrected model (Eq. 20) was fitted to the experimental data, and channel lengths were obtained. Predicted length of diffusion channel for maintaining the steady-state O₂ concentration is presented in Fig. 5. The channel lengths, respectively, were predicted in the range of 120–320 mm, 80–170 mm, 30–80 mm, and 30–60 mm for maintaining the O₂ concentration of 4%, 8%, 12%, and 16% irrespective of the channel diameter (3, 6, 9, and 12 mm). It is clearly noticed that the longer diffusion channel maintained lower range of O₂ steady-state concentration, whereas shorter diffusion channel maintained higher range of O₂ steady-state concentration irrespective of the channel diameter. These results were confirmed with the results reported by Ratti et al.^[14] for cauliflower, Chimphango^[16] for spinach, and Karthiayani et al.^[18] for banana. The model (Eq. 20) developed for predicting the length of diffusion channel fits the experimental data well when individual channel diameters were considered, and is more appropriate for establishing the controlled atmosphere condition for storage of tomato. The model suggests that the desired O₂ level in the storage chambers for any length can be precisely obtained if the diameter of the channel is known. To get same O₂ level as in a chamber installed with a shorter diffusion channel length of a small diameter, one can use a longer diffusion channel of a larger diameter. A parallel trend was observed when these results (predicted) were compared with the experimental results.

Figure 5. Predicted length of diffusion channel for maintaining the steady-state O₂ concentration at 10°C (error with 5% value).

The steady-state O₂ concentration in experimental storage chambers equipped with diffusion channel at 10°C is presented in Fig. 6. From the figure, it is noticed that the longer diffusion channel (240 mm) maintained lower O₂ concentration (7–9%), whereas shorter diffusion channel (60 mm) maintained higher O₂ concentration (11–15%) depending on the channel diameter. On the other hand, higher channel diameter (12 mm) maintained higher O₂ concentration (9–15%), whereas lower channel diameter (3 mm) maintained lower O₂ concentration (7–12%) depending on the channel length. It is evident from the results that the steady-state O₂ concentration was inversely proportional to the length of channel and directly proportional to the channel diameter. The statistical analysis showed that the effect of different length and diameter of the diffusion channel on concentration of O₂ was found to be significant at 5% level. Similar trends in the steady-state O₂ concentration under diffusion channel system were observed by Chimphango^[16] and Karthiayani et al.^[18] for modified atmosphere packaging of spinach and banana, respectively. Moreover, the tomatoes were stored up to 40 days at 10°C under diffusion channel length of 180 mm and diameter of 9 mm (Treatment 11) with harvest-fresh appearance, better texture, good colour, retained nutritional values, and good marketability conditions.

Figure 6. Steady-state O₂ concentration in experimental storage chambers with diffusion channel at 10°C (error with 5% value).

Conclusion

The experimental results showed that the diffusion channel system can well maintain the O₂ concentration in the storage chamber. The model for predicting length of diffusion channel fits the experimental data well when individual channel diameters were considered and was more appropriate for establishing the controlled atmosphere condition for storage of tomato. The fitted model closely represented the experimental results and can be used to predict the length of diffusion channel for the same commodity. The model suggests that the desired O₂ level in the storage chambers for any length can be precisely obtained if the diameter of the channel is known. The model was very useful for predicting the diffusion channel length and more appropriate for establishing the controlled atmosphere condition for storage of tomato.