Introduction

Sweet lime (Citrus limetta), a member of the Rutaceae family, is a tropical fruit valued for its distinctive balance of sweet and tangy flavor as well as its appealing aesthetic qualities (Mondal et al., 2019). It is known for its rich nutritional profile and therapeutic benefits, which include a high concentration of vitamin C, copper, zinc, iron, calcium, potassium, antioxidants, and folic acid (Shariq & Sohail, 2019; Younis et al., 2015). Additionally, the juice derived from sweet lime is recognized for its ability to alleviate fatigue and possesses anti-carcinogenic and antibacterial properties. Given its versatility and substantial economic potential, the cultivation of sweet lime varieties plays a significant role in commercial agriculture (Bhaumik et al., 2018).

As a perishable fruit, sweet limes require a critical number of postharvest procedures, including sorting, washing, waxing, packing, storage, and transportation (Kumari et al., 2020). However, a lack of understanding of the physical, thermal, mechanical, and chemical properties of sweet limes often results in considerable fruit wastage during these postharvest processes (Azman et al., 2020; Jahanbakhshi et al., 2019; Tewari et al., 2019). Among these properties, the physical characteristics of fruits significantly influence the design of equipment and procedures for storage, transportation, sorting, and packing (Birania et al., 2022; Pathak et al., 2019; Shelke et al., 2018). Similarly, the thermal properties of fruits play a vital role in thermal processing operations, such as drying, pasteurization, sterilization, and controlling storage conditions (Barbhuiya et al., 2020; Mukama et al., 2020; Tewari et al., 2019). The textural characteristics of fruits, including compression, penetration, cutting, and crushing forces, are crucial for optimizing packaging, transportation, and processing operations (Murakonda et al., 2022; Panda et al., 2022). Chemical properties, such as total solids, pH, moisture content, and ascorbic acid, significantly influence fruit quality and, consequently, consumer satisfaction (Barbhuiya et al., 2020; Shelke, Kad, Pandiselvam, et al., 2023). Postharvest losses of fruits have a substantial impact on both the economy and nutrition (Roumani et al., 2024). Optimizing equipment selection and process design can reduce losses, product failures, and equipment breakdowns (Roumani et al., 2024; Sanganamoni et al., 2024). Therefore, understanding the engineering properties of fruits is essential for the design and development of processes and equipment that optimize the efficiency of postharvest operations.

An analysis of consumer preferences reveals that customers typically prefer fruits that exhibit consistency in both weight and shape (Bibwe et al., 2022). Therefore, fruit grading is an essential process based on factors such as shape, size, color, texture, and weight. Size grading is a time-consuming and costly process, making it impractical for fruits with irregular shapes. However, the grading process becomes problematic when fruits have similar shapes but differ in weight (Mahawar et al., 2019). Furthermore, mass grading surpasses size grading in efficiency, cost-effectiveness, and accuracy due to its automated fruit sorting methodology. Mass-based grading can be performed by directly measuring the weight of the fruit or using models based on its dimensional characteristics. Understanding the correlation between mass and the physical properties of fruits can enhance the efficiency and precision of grading systems (Pathak et al., 2020; Shelke et al., 2020). Hence, it is crucial to comprehend the correlation between mass and the major, minor, and intermediate dimensions, as well as the projected areas and volume. Regression analysis is commonly used to develop models that predict the statistical correlation between one or more independent variables and the response factor. Previous studies have extensively used various regression models, including linear, S-curve, power models, and quadratic (Azman et al., 2020; Shousha et al., 2024). Several studies have investigated mass modeling for various fruits, including sohiong (Vivek et al., 2018), chebula fruit (Pathak et al., 2019), cantaloupe (Jahanbakhshi et al., 2019), gooseberry (Tewari et al., 2019), sweet orange (Mahawar et al., 2020), belleric myrobalan (Pathak et al., 2020), guava (Bibwe et al., 2022), strawberry (Birania et al., 2022), kadamb (Panda et al., 2022), amla (Tomar & Pradhan, 2022), wood apple (Murakonda et al., 2022), date fruit (Roumani et al., 2024), and oil palm (Sanganamoni et al., 2024). Currently, there is a lack of published research on the engineering properties and mass modeling of sweet lime fruit. Therefore, to develop suitable processing methods, a comprehensive investigation of the engineering properties of sweet lime is essential. Accordingly, this study aims to evaluate the physical, chemical, thermal, and mechanical properties of sweet lime fruit and to identify an appropriate model for predicting its mass.

Materials and Methods

Fresh sweet lime fruit samples were obtained from the Department of Horticulture at Mahatma Phule Agricultural University in Maharashtra, India. The harvested fruits were packed into boxes and delivered to the Process and Food Engineering laboratory. Subsequently, the fruits were washed with water and stored at 10 ± 1 °C and 90 ± 3 % relative humidity to minimize environmental damage. The moisture content of the fruits was measured using a standardized methodology (Pathak et al., 2020). This involved drying six sliced fruits in an oven (Tempo Instruments Pvt. Ltd., Mumbai) at 80°C for 72 hours, after which the initial moisture content was determined. The remaining samples were then used for the experiments.

Morphological properties of sweet lime

The weight of the sweet lime fruits was measured using a digital weighing balance (Indosaw, with an accuracy of 0.001 g, Haryana, India). Equations 1 to 4 were used to calculate the percentages of pulp, peel, seeds, and the pulp-to-peel ratio (Abhishek et al., 2017; Patil et al., 2017).

Physical properties of sweet lime

The dimensions of the fruits were measured using a vernier caliper (Mitutoyo Corp, Japan) with an accuracy of 0.01 mm. A total of 100 sweet lime fruits were selected for physical analysis. The dimensions of each fruit were measured along three linear axes: the major axis (LA), intermediate axis (IA), and minor axis (TA) (Figure 1).

Figure 1. Pictorial view of sweet lime fruit.

Meanwhile, using three linear axes, the remaining physical attributes were determined: arithmetic mean diameter (D_a, mm), geometric mean diameter (D_g, mm), equivalent mean diameter (D_e, mm), P_LA is projected area perpendicular to major axis (mm²). While P_IA refers to the projected area perpendicular to the intermediate axis (mm²), whereas P_TA is the projected area perpendicular to the minor axis; (mm²). Additionally, C_PA represents the critical projected area (mm²). The aspect ratio (R_a), sphericity (∅), surface area (S_a, mm²), ellipsoid volume (V_ellip, mm³), prolate spheroid volume (V_pro, mm³), and oblate spheroid volume (V_osp, mm³) were determined using established formulas (Bibwe et al., 2022; Khoshnam et al., 2007; Mahawar et al., 2019; Murakonda et al., 2022; Pathak et al., 2020), as shown in Equations 5 and 17.

The bulk density of sweet lime (ρ_b, kg/m³) was determined using established methodologies (Bibwe et al., 2022), while the true density (ρ_t, kg/m³) was measured using the toluene displacement technique (Murakonda et al., 2022). Porosity (ε, %) was calculated using conventional formulas (Tomar & Pradhan, 2022), as shown in Equations 18 and 20.

where, M_f is the mass of fruits (kg), V_b is the volume of the container (m³), V_f is the volume of fruit (m³), which represents the volume of toluene displaced.

Frictional characteristics of sweet lime

The frictional characteristics of the fruit, such as the static coefficient of friction (μ) and the angle of repose (α), were assessed using the methodologies proposed by Pathak et al. (2020) and Murakonda et al. (2022), respectively. The static coefficient of friction was determined by conducting experimental trials on various surfaces, including rubber, plywood, stainless steel, and aluminum sheets. The static coefficient of friction was calculated using Equation 21, while the angle of repose was determined using Equation 22.

μ = tan α (21)

where, D refers to the plate diameter (mm), H refers to the heap height(mm).

Thermal characteristics of sweet lime

The specific heat, thermal diffusivity, thermal conductivity, and latent heat of fusion of sweet lime fruit were determined using established mathematical models suggested by Barbhuiya et al. (2020) and Vivek et al. (2018), as shown in Equations 23 and 26.

C_P = 1.667 + 0.025M (23)

K = 0.148 + 0.00493M (25)

λ = 335 W (26)

Where, C_p is specific heat (KJ/kg °C), M is moisture content (%), K is thermal conductivity (J/m s ºC), α is the thermal diffusivity (× 10⁻⁷ m²/s), ρ is true density (kg/m³), λ is the latent heat of fusion (J/kg). These models were developed based on moisture content of fruit. To use these mathematical models, the fruit moisture content must be greater than 60 %.

Mechanical characteristics of sweet lime

The texture properties of the fruits, such as penetration, compression, and cutting, were evaluated using a universal testing machine (AG-X, Shimadzu Analytical, India) equipped with various stainless-steel probes and a 1000 N load cell. The experiments were conducted using established methodologies previously utilized by multiple researchers (Barbhuiya et al., 2020). The penetration test was performed using a cylindrical probe with a 6 mm diameter. The fruit sample was initially placed on a lower horizontal surface, and the probe was then moved at a speed of 1 mm/s to a depth of 10 mm. Each fruit was pierced from all four directions. For the compression test, a stainless-steel probe with a circular flat shape and a diameter of 75 mm was used to assess the compressive force. The fruit was subjected to compression at a speed of 1 mm/s, resulting in a maximum deformation of 20%. For the cutting test, a knife probe with a fixed blade was employed. The probe was operated at a speed of 1 mm/s, and the resistance to the depth of the incision was continuously measured. A force-displacement curve was generated for each test, with the force being recorded at every incremental displacement.

Chemical characteristics of sweet lime

Protein content was quantified using the Kjeldahl method, with a correction factor of 6.25 applied (Yıldız et al., 2015). The ash content was measured by exposing the sample to a muffle furnace set at a temperature of 550°C (Barbhuiya et al., 2020). Fat content was determined using the solvent extraction technique with petroleum ether as the solvent (Hacıseferogulları et al., 2007). Carbohydrate content was calculated by subtracting the combined mass of protein, ash, fat, and other discernible constituents from the total mass (Hacıseferogulları et al., 2007). The vitamin C content in the fruit sample was determined using the AOAC method 967.21. The sample was homogenized, extracted with metaphosphoric-acetic acid, filtered, and titrated with 2,6-dichloroindophenol (DCPIP) until a pink color appeared. The vitamin C content was calculated based on the volume of DCPIP used and expressed in mg per 100 g of the sample (Barbhuiya et al., 2020). Fiber content was analyzed according to Hacıseferogulları et al. (2007). The pH and total soluble solids in the fruit juice were measured using a digital pH meter (ELICO Ltd., Hyderabad, India) and a refractometer (Erma, Tokyo, Japan), respectively (Shelke, Kad, Yenge et al., 2023).

Color characteristics of sweet lime

A color-scanning apparatus (Premium Color Scan, Japan) was used to measure the color of the sweet lime fruit. The hue angle and chroma were calculated using the formulas provided by Vivek et al., (2018) and Barbhuiya et al., (2020), as shown in Equation 27 and 28.

Where, C is chrome, α is hue angle, a* is red/green coordinate and b* is yellow/blue coordinate.

Model selection for mass prediction

The following models were presumptive when estimating the mass models for sweet lime fruit.

• The study employed single variable regression analysis to examine the correlation between the mass of sweet lime fruit and its LA, IA, and TA dimensions.

• Single or multiple variable regression analysis was employed to correlate the mass of sweet lime fruit mass with its projected areas, including P_LA, P_IA, P_TA, and C_PA.

• The mass of the sweet lime fruit is modeled as a function of the fruit’s volume, assuming three different shapes: ellipsoid (V_ellip), oblate spheroid (V_osp), and prolate spheroid (V_pro).

Based on earlier studies, four regression models were used for the mass modeling of sweet lime fruit: linear, S-curve, quadratic, and power models, as shown in Equation 29 and 32.

M = b₀ + b₁X (29)

M = b₀ + b₁X + b₂X² (30)

Whereas, M is the mass of the fruit in grams, X is the independent variable (dimensions, volume, or projected area), and b₀, b₁, and b₂ are the parameters used for curve fitting.

Statistical analysis

The statistical measures of mean, minimum, maximum, and standard deviation for each fruit property were computed using Microsoft Excel 2007 (Microsoft Corporation, Redmond, Washington, USA) and SPSS Statistics 20 (IBM Corporation, Armonk, New York, USA). Accurate models were then constructed based on the physical property data to determine the mass of the fruits. The model with the highest regression coefficient (R²) was selected as the optimal mass model.

Result and Discussion

Morphological properties of sweet lime

The pulp, peel, and seed content of sweet lime fruit were extracted, and their percentages were calculated using Equations (1)–(3), while the pulp-to-peel ratio was determined through Equation (4). The mean results with standard deviation (SD) are shown in Table 1. The average percentages of pulp, peel, and seed were found to be 44.25%, 54.45%, and 1.31%, respectively, while the average pulp-to-peel ratio was 0.83. Additionally, the average peel thickness of sweet lime fruit was found to be 4.26 mm. The pulp, peel, seed content, and peel thickness of sweet lime fruit may be useful for the design and development of peeling, shredding, and crushing equipment.

Physical properties of sweet lime

Table 2 displays the physical characteristics of the sweet lime fruit. The study revealed that the mean measurements for the major axis (LA), intermediate axis (IA), and transverse axis (TA) were 78.71 mm, 77.07 mm, and 74.96 mm, respectively. Meanwhile, the average values for the geometric mean D_g, D_e, and D_a were 76.92 mm, 76.89 mm, and 76.9 mm, respectively. These diameters are essential for calculating the projected area and for designing engineering machinery used in washing, grading, sorting, and packaging (Moradi et al., 2020; Pathak et al., 2019 ).

P_LA, P_IA, P_TA, and C_PA were determined by computing their projected areas using Equations (8) to (11). The results indicated that the average values for P_LA, P_IA, P_TA, and C_PA were 4663.01, 4567.45, 4444.17, and 4558.21 mm², respectively. Meanwhile, the surface area (S_a), another important physical characteristic of sweet lime fruit, was calculated using Equation (12), with an average value of 18694.41 mm². These fruit areas are essential for assessing the ripeness index, transpiration rate, gas permeability, simulating mass and heat transfer, and measuring water loss (Jahanbakhshi et al., 2019). The shape and flowability characteristics of sweet lime fruit, such as aspect ratio and sphericity, were calculated using Equations (13) and (14), respectively. The mean values of sphericity and aspect ratio were found to be 0.98, suggesting that the sweet lime fruit has a nearly spherical shape. Mahawar et al. (2020) made similar observations regarding kinnow fruits. The aspect ratio and sphericity properties significantly influence the fruit’s flowability. The volumes of sweet lime fruit, including V_ellip, V_pro, and V_osp, were calculated using Equations (15), (16), and (17), respectively. The average volumes for V_ellip, V_pro, and V_osp were 242.95, 254.97, and 249.79 cm³, respectively. These volumes are crucial in various postharvest processes, such as determining heat transfer rates in refrigeration and drying operations, as well as estimating fruit density and packaging coefficients during packaging and storage (Murakonda et al., 2022; Roumani et al., 2024). The average bulk density (ρ_b) and true density (ρ_t) were found to be 449.75 and 931.41 kg/m³, respectively. The porosity of sweet lime fruit was determined using Equation (20), yielding a value of 51.68%. Measuring the bulk density (ρ_b) and true density (ρ_t) of the fruit can aid in the development of washing and sorting machinery, as well as the design of storage structures (Mahawar et al., 2020). The porosity of the fruit is also essential for evaluating airflow and heat transfer characteristics (Pathak et al., 2019).

Frictional properties of sweet lime

The frictional characteristics of fruit are crucial factors in the design of fruit processing equipment and operations (Murakonda et al., 2022; Pathak et al., 2020). These properties specifically influence the design of feed hoppers and mechanical conveyors (Panda et al., 2022). They also significantly affect the cohesion and flow of the fruit, making them essential for the design of storage structures.

The study found that sweet lime fruit exhibits good flowability, with a mean angle of repose of 6.83° (Table 3). The average coefficient of static friction for rubber, plywood, stainless steel, and aluminum sheets was 0.629, 0.523, 0.336, and 0.448, respectively (Table 3). Stainless steel showed the lowest coefficient of static friction, making it the most suitable material for constructing conveying equipment. Similar findings were reported by Motie et al. (2014) concerning lime fruit.

Thermal properties of sweet lime

The thermal characteristics of fruits play a vital role in the design of various processes, such as drying, cooling, freezing, sterilization, evaporation, boiling, and crystallization (Vivek et al., 2018). The thermal properties of different fruits are typically determined using moisture-based regression models. In this study, the thermal properties of sweet lime fruit pulp were assessed at a moisture content of 69.74% (wet basis, w.b). Table 4 displays the thermal characteristics of sweet lime fruit. The average specific heat (Cp) value was 3.861 kJ/kg°C, which is higher than the specific heat values found in previous studies on coffee plums and Sohiong fruits (Barbhuiya et al., 2020; Vivek et al., 2018). Fruits with a higher Cp require more heat to be cooled or heated. Similarly, the mean value for thermal conductivity (K) was 0.581 J/m•s•°C. Thermal conductivity is crucial for determining the rate of heat transfer within food.

On the other hand, α and λ were found to be 0.00016 m²/s and 78.52 kJ/kg, respectively, which are essential for calculating freezing and drying times. The results of this investigation are consistent with those published by Barbhuiya et al., (2020) and Vivek et al., (2018).

Mechanical properties of sweet lime

The mechanical characteristics of fruits are essential for designing machinery used in various processes such as peeling, shredding, handling, packaging, and cutting. These properties are also instrumental in predicting fruit quality parameters (Moradi et al., 2020; Singh & Reddy, 2020). Table 4 presents the mechanical characteristics of sweet lime fruit. The analysis revealed that the average maximum penetration force was 9.44 N. Similarly, the average peak values for compression and cutting forces were 311.08 N and 148.68 N, respectively. The results are consistent with data reported by Singh & Reddy (2020) for orange fruit.

Chemical properties of sweet lime

The chemical characteristics of fruits provide a reliable method for accurately evaluating fruit quality (Barbhuiya et al., 2020). The chemical composition of sweet lime fruit was analyzed using well-established techniques. The results, including the mean values and standard deviations (SD), are presented in Table 5. The fruit exhibited a mean moisture content of 86.40% (w.b.). For protein, fat, carbohydrates, fiber, and ash, the average values were 0.87%, 0.35%, 10.35%, 0.55%, and 0.14%, respectively. The pH level was 5.44, and the total soluble solids (TSS) had a Brix value of 11.68. The ascorbic acid (vitamin C) content was 54.50 mg/100g, highlighting the fruit’s rich vitamin C concentration. These findings are consistent with the results reported by Bhaumik et al. (2018). The results provide a comprehensive understanding of sweet lime fruit quality, offering valuable insights for future research and practical applications in the food industry.

Color properties of sweet lime

The color of fruit significantly influences consumer preferences and serves as an indicator of its ripeness (Murakonda et al., 2022). Table 5 presents the color values of the sweet lime fruit. The study found that sweet lime fruit has a greenish-yellow color, with an L* value of 58.07, an a* value of -13.15, and a b* value of 51.38. The hue angle reflects the extent of light purity, while the chroma value indicates the level of color saturation and its relationship to intensity (Barbhuiya et al., 2020; Vivek et al., 2018). The analysis revealed that the hue angle and chroma values were 105.06 and 53.32, respectively.

Model selection for mass prediction of sweet lime

The study employed regression models, including linear, S-curve, quadratic, and power models, to predict the mass of sweet lime fruit. The dimensions, volume, and projected area were used as independent variables, while the fruit’s weight was the dependent variable. Table 6 presents the best-fitted mass models obtained. Among the physical dimensions, the intermediate axis (IA) was selected as the most suitable dimension for mass modeling of sweet lime fruit, with the quadratic model showing the highest R² value (0.93). In contrast, the geometric mean diameter (D_g), which had the highest R² value (0.95) for the quadratic model, was identified as the most suitable parameter among all diameters. For the projected area, P_LA and P_IA demonstrated the highest R² value (0.92) for the linear model. Based on their respective volumes, the ellipsoid (V_ellip) and oblate spheroid (V_osp) volumes were found to be the most appropriate for estimating the mass of sweet lime fruit, resulting in the highest R² value (0.93).

Finally, models based on the intermediate axis (IA), geometric mean diameter (D_g), and ellipsoid volume (V_ellip) were found to be the most accurate methods for estimating the mass of sweet lime fruit. Additionally, the following regression models can be applied to determine the mass of sweet lime fruit.

Best dimension-intermediate axis (IA)

M = 0.2424 – 0.0078IA + 0.0001IA²,R² = 0.93.

Best diameter-geometric mean diameter (D_g).

M = 0.0358 – 0.0028D_g+ 0.00008D_g²,R² = 0.95

Best projected area – (P_LA) and (P_IA)

M = –0.067+ 0.00007P_LA,R² = 0.92

M = –0.0594 + 0.00007P_IA,R² = 0.92

Best volume – Ellipsoid volume (V_ellip) and Oblate spheroid volume (V_obl)

M = 0.0509 + 0.0009V_ellip,R² = 0.936

M = 0.0488 + 0.0009V_obl,R² = 0.933

Conclusions

This study aimed to evaluate the engineering properties of sweet lime fruit and develop a mass model based on its physical characteristics to reduce postharvest losses. The findings include the following: The mean proportions of pulp, peel, and seeds were 44.25%, 54.45%, and 1.31%, respectively. The average values of the major axis (LA), intermediate axis (IA), transverse axis (TA), arithmetic mean diameter (D_a), geometric mean diameter (D_g), and equivalent diameter (D_e) were 78.71 mm, 77.07 mm, 74.96 mm, 76.92 mm, 76.89 mm, and 76.9 mm, respectively. The mean values for projected areas (P_LA, P_IA, P_TA, C_PA) and surface area (S_a) were 4663.01 mm², 4567.45 mm², 4444.17 mm², 4558.21 mm², and 18694.41 mm², respectively. The average volumes for sweet lime (V_ellip, V_pro, V_osp) were 242.95 cm³, 254.97 cm³, and 249.79 cm³, respectively. The aspect ratio and sphericity were 0.94 and 0.95, respectively. The average bulk density and true density were 449.75 kg/m³ and 931.41 kg/m³. Stainless steel exhibited the lowest coefficient of static friction. Thermal properties (specific heat - Cp, thermal conductivity - K, thermal diffusivity - α, latent heat of fusion - λ) were measured as 3.861 kJ/kg°C, 0.581 J/m•s°C, 0.00016 m²/s, and 78.524 kJ/kg, respectively. The average peak values for penetration, compression, and cutting forces were 9.44 N, 311.08 N, and 148.68 N, respectively. The chemical analysis indicated that sweet lime fruits contain a significant concentration of ascorbic acid, making them suitable as dietary supplements. The average L*, a*, and b* values for color were 58.07, −13.15, and 51.38, respectively. The mass modeling analysis determined that the linear and quadratic models were the most appropriate, with the highest R² values. Mass models based on the intermediate axis (IA), geometric mean diameter (D_g), and ellipsoid volume (V_ellip) achieved R² values ranging from 0.93 to 0.95. However, the study acknowledges that the effectiveness of these models depends on the unique attributes of each fruit, which may vary among different fruit varieties. Understanding the physical, thermal, mechanical, and chemical engineering properties of fruits can enhance industrial handling, postharvest processing, storage, grading, and sorting operations. Additionally, mass modeling improves the efficiency and accuracy of industrial fruit sorting.