Mathematical modeling and dimensional analysis of hot-air tray drying kinetics of Khoa

Introduction

Drying has remained a popular method for preserving dairy and food products for many years. The drying process involves reducing the water content of the product to an acceptable level for marketing, storage or processing. Drying is usually done by vaporizing water from the product. Thus, the latent heat of evaporation must be supplied. Air flow is also necessary to remove the vapor from the product. The lower the humidity of the hot air supplied to the drying chamber, the better the drying speed, as the less humid air can transport more moisture from the product surface than the more humid air. In general, increasing temperature and the speed of air reduces drying time. However, high temperatures in heatsensitive products such as dairy and food reduce the quality of products. In this case, low temperature and humidity drying is required to maintain the fresh colour of the product (Sato et al., 1997) [17] . The dryers that are commonly used are tray, tunnel, drum, fluidized bed, spray, flash, rotary, belt, vacuum and freeze dryers. Tray dryer is the widely used because of its simple and economic. Normally, the moisture content is determined by using an electronic balance to get the difference between the final and initial mass of the product. (Bakhshipour et al., 2012) [3] . Khoa, a traditional dairy product in South Asian cuisine, is a heat-coagulated milk solid prized for its granular texture and rich, milky flavor. Despite its cultural and economic importance as a primary ingredient in numerous sweet and savory dishes, khoa is highly perishable. Its high moisture content makes it an ideal medium for microbial spoilage and chemical deterioration, limiting its shelf life to only a few days under ambient conditions (Meena et al., 2021) [11] .

To address these stability issues and facilitate large-scale distribution, the dairy industry frequently employs dehydration to transform fresh khoa into a shelf-stable khoa powder. Drying it into a powder or shelf-stable concentrate is challenging due to its high fat (20-30%) and protein content. These components create a complex food matrix that resists moisture migration. Among various industrial dehydration technologies, tray drying remains the most extensively utilized method due to its simplistic design, economic viability, and operational flexibility (Chaudhari et al., 2022) [4] . Drying kinetics study the rate of moisture removal as a function of time and drying conditions. The efficiency of this process is governed by drying kinetics. During the initial constant rate period, evaporation occurs freely from the saturated surface. However, as the process transitions into the falling rate period, internal mass transfer becomes the limiting factor (Sarkar et al., 2023) [16] . To optimize dryer performance must develop robust mathematical models. However, the interplay of variables including air density, viscosity, thermal conductivity, mass diffusivity and characteristic length presents a high degree of complexity. However dimensionless number comes in it groups variables like air velocity, temperature and moisture diffusivity into ratios that remain valid regardless of the dryer's size. By reducing the physical problem to universal parameters such as the Biot and Fourier number of mass transfer of researchers can generalize drying behavior across different scales and equipment geometries. This research aims to apply the principles of dimensional analysis to the tray drying of khoa to better understand the underlying transport phenomena. This study seeks to model the relationship between processing parameters (temperature, air velocity) and drying kinetics.

Material and methods 2.1 Raw material and Equipment

Tray dryer has a heating element of 1.0 kW capacity for air heating was used to drying of khoa to make khoa powder in this study. The tray was made up of stainless steel (AISI-316). The balanced draught fan (forced and induced fans) were used to proper speed regulator. During the experiment blower speed was controlled by a regulator to regulate the air flow rate. The tray dryer was connected with the energy meter to measure energy consumption. The inlet, inside and exit air temperatures of tray dryers were measured using a calibrated copper constantan thermocouple digital temperature indicator (accuracy:±0.1%). The inlet and outlet air velocity of the tray dryer was measured with an electronic digital anemometer (least count of 0.1 m/s with an accuracy of ±2% on the full-scale range of 0.2 m/s to 30.0 m/s).

Methodology 2.2.1 Khoa preparation

Khoa was prepared by traditional khoa making process in manner as described in the flow chart given below (Aneja et al., 2002) [1] .

Working principle of khoa drying

The khoa was dried with the help of hot air circulation in a tray dryer. Forced convection heating took place to remove moisture from the khoa. Khoa was taken in clean stainless steel (AISI-316) trays and uniformly spread in a layer about 0.5 cm thick layer. The tray dryer was turned on and temperature and time in the tray dryer were recorded. Inlet and exit fan (blower) knobs were adjusted to control the air flow rate. After reaching the set temperature, trays were loaded into the tray dryer and mass of partially dried khoa lumps were manually mixed and re-spread at every 30minute interval. The weight of khoa mass (kg) was recorded until the equilibrium relative humidity achieved. The air temperature was established at 55, 60, and 65°C with velocity of air drying was established at 2.0 m/s (Prasad et al., 2024) [14] . The tests were replicated 3 times for each experimental, and the average values were reported. Khoa were uniformly positioned on the tray (40 x 30 cm) in a thin layer with a loading rate of 2.0 kg/m 2 .

Drying kinetics

The drying kinetics of khoa were described as a decrease in moisture ratio (MR) versus drying time (t). Moisture Ratio (MR) is a dimensionless number that represents the amount of removable moisture remaining in the product at any given time relative to the initial moisture content. The MR was analysed using using equation (1) (Jahanbakhshi et al, 2020) [8] .

where, MR: Moisture ratio in dimensionless value M t: Moisture content at any time t (% wb) M i: The initial moisture content (% wb) Variations of the drying rate of khoa (kg moisture.h -1 .m -2) at different temperatures with drying time were evaluated (Sekar et al., 2023b) [18] . The drying rate for khoa was determined using Equation ( 2 where, D r: Drying rate of khoa, kg.h -1 .m -2 M w: Mass of moisture evaporated, kg A: Area of the tray for the spread of khoa for drying, m 2 t: Time taken for drying of khoa, h.

Mathematical modelling of drying kinetics

Table 1 shows the equations used to determine the best model for explaining the khoa drying curve. Two semiempirical drying equations were operated to describe the khoa drying kinetics and calculate the Drying Rate Constant (k) in this investigation (Darvishi et al., 2013) [5] .

Result and discussion

Moisture evaporation rate of khoa at different drying temperatures

The moisture evaporation rate (kg moisture.m -2. h -1 ) of khoa at three different temperatures (55°C, 60°C and 65°C) was estimated, having the khoa drying area 0.12 m 2 and fixed mass of khoa (0.625 kg) were fixed. The temperature and velocity (2 m/s) of air were fixed during the period. During drying, the process of changing of mass of khoa (g) over time was recorded. The mass of khoa was taken at different temperatures, with an interval of half hour (Prasad et al., 2024) [14] . Drying ends at 6, 5.5 and 5 hours for drying temperatures of 55, 60 and 65 °C, respectively. Drying of khoa was continued up to equilibrium conditions and recorded mass change of full cream based khoa (Figure 2). Sanika et al. (2021) [15] reported that nonlinear decrease of moisture with the time progressing of tray-dried peanut milk residue. Author reported that the time taken to reach EMC decreases with increase in temperature; which supports findings of the current investigation for the trends obtained. Korotkiy et al. (2023) [9] reported that the time duration (h) required for complete drying of bananas at 50 °C, 60 °C and 70 °C were 12, 9 and 9 h in tray dryer.

The rate of removal of moisture increased with drying temperature. The time to reach the equilibrium moisture content decreased with the increase in temperature. When khoa is dried at 65ºC, it takes about 5 hours to reach the equilibrium condition. However, at 55 and 60 ºC, thorough drying took 6 and 5.5 hours, respectively. For khoa, the final moisture content (%, dry basis) of khoa powder obtained at 55, 60 and 65 °C were 4.83, 4.00 and 3.17% respectively. Moisture carrying capacity of air increases with temperature rise. The low relative humidity of air in the tray dryer caused drying in less time. At higher air drying temperatures in tray dryer for drying of quinces to make quince powder, the gradient potential was high which led to a reduction in drying time Tzempelikos et al., (2014) [19] . Arulkumar et al.

(2023) [2] reported that the range of equilibrium moisture content (% wb) of tray dried paneer cube samples (8 cm 3 ) at 50, 55 and 60 °C were 15.26 to 16.64, 14.65 to 16.59 and 12.46 to 15.89%, respectively; shows similar trends of current investigation, higher drying temperature recorded lower equilibrium moisture content. According to Sanika et al. (2021) [15] , tray-dried peanut milk residue at 60, 70, and 80°C required 9, 7, and 5 hours, respectively, to reach equilibrium moisture content (EMC); these results are consistent with our observations.

Variation of drying rate of khoa with respect to drying time

Khoa (0.625 kg) was evenly distributed in a tray with a bed thickness of about 0.5 cm and the area of tray used for this study was 0.12 m 2 . The moisture evaporation per unit area per unit time (kgmoisture.h -1 .m -2 ) vs. time (h) relationship for drying rate of full cream milk (FM) khoa is depicted in Figure 4. At the commencement of the khoa drying process, the water content of the khoa was significantly greater, hence the drying rate faster. Increasing the temperature resulted increase in the drying rate and hence water content in the material decreased. As temperature rises, the diffusivity of water increases both on the khoa surface and within the mass, which results in faster drying rate. Figure 4 shows the full cream milk based khoa drying at a temperature of 65 o C gives a higher drying rate than at 60 o C and 55 o C. At the starting (0.5 h) of full milk based khoa (FM-Khoa) drying, drying rates (kg moisture . h -1 .m -2 ) at 55, 60 and 65°C were obtained as 0.58, 0.72 and 0.88 kg moisture.h -1 .m -2 , respectively. Drying rate (kg moisture.h -1 .m -2 ) of FM-Khoa at 55, 60 and 65°C were ceased at 6, 5.5 and 5.0 h, respectively. The average drying rate (kg moisture.h -1 .m -2 ) of FM-khoa during the drying period at 55, 60 and 65°C were at 0.34, 0.38 and 0.42 kg moisture.h -1 .m -2 , respectively. Arulkumar et al. (2023) [2] reported maximum drying rate (g/g.h) for tray drying of paneer cube samples (8 cm 3 ) at 50, 55 and 60°C as 0.2080, 0.3352 and 0.3213 kg moisture.kg - 1 .h -1 , respectively. Drying rate best fitting equation at 55°C, 60 and 65°C (shown in equation 8 to 10) with R 2 values 0.95, 0.99 and 0.97 respectively.

Dr (at 55°C) = -0.052t + 0.6771 ……………………….( 8)

Dr (at 60°C) = -0.0657t + 0.7723 ……………………….( 9)

Dr (at 65°C)= -0.0844t + 0.8964 ……………………….( 10)

Modelling of drying curves

Moisture ratio (MR) is defined as the ratio of moisture percent in the product at a particular time to initial moisture content. The moisture ratio (%) of full cream based khoa (Figure 5) with respect to drying time (h) at three different drying temperatures (55, 60 and 65 °C) in tray dryer. The moisture ratio (%) decreased with time. The FM khoa (FM-KP) at 55°C was the slowest. The final moisture ratio (%) of full cream milk based khoa powder at 55, 60 and 65 °C were 6.67, 5.52 and 4.38 respectively. According to Sanika et al.

(2021) [15] , tray-dried peanut milk residue's moisture ratio dropped over time and showed that it fell inside the falling rate zone, which is consistent with current findings. Table 2 shows the statistical measurements of data goodness of fit to drying equations. Both equations exhibited the high (R 2 ) values than 0.95, could equitably represent the drying behaviour.

Dimensionless Number of khoa drying

Figure 6 shows ln (MR) vs time for a thin layer khoa drying at three different temperature as a semi-infinite medium. Factors that will change your result include temperature of drying, composition of khoa and shrinkage of khoa during drying. Evaluation of effective moisture diffusivity (D eff ) for khoa, according equation 3 to 5. Ndukwu et al. (2017) [13] reported that effective moisture diffusivity (D eff ) value for cocoyam slices found from 3.37 x10 -10 to 1.13 x10 -10 for drying temperature 50-70°C. As the drying temperature was stepped up from 55°C to 65°C, the negative slope of the drying curves accelerated significantly from -0.3958 to -0.5533. Correspondingly, the D eff values increased from 1.11 x 10 -9 m²/s to 1.55 x 10 -9 m²/s. Elevating the drying temperature imparts greater kinetic energy to the water molecules within the khoa matrix. This structural collapse accounts for the slight non-linearities and the constant intercept of -0.21 observed in the regression models.

Biot Number for Mass Transfer (Bim)

The Biot numbers for mass transfer were determines using equation 6. Table 4 shows the Biot numbers for mass transfer during khoa drying at different drying temperature.

Here, Bi m > 40: Internal resistance controls the process. Fick's Second Law is highly accurate here because the surface dries almost instantly, and the core moisture struggles to reach the surface. The calculated mass transfer Biot numbers (Bi m > 9.0x 10 6 ) mathematically confirmed that internal mass transfer resistance vastly dominates over surface convective resistance. Since khoa is dense and sticky, you will likely find Bi m is very high, validating that internal diffusion is indeed the limiting factor. Mondal and Das (2017) [12] reported that mass transfer biot number of frying of Chenna jhilli from 4.21 to 4.286 for frying temperature (125-145°C). [6] reported that mass transfer Fourier number of frying of Gulab jamun from 0.29 to 0.51 for frying temperature (135-155°C). Fourier numbers (Fo m ) crossed the critical threshold of 0.2 within the first 1.0 to 1.5 hours, showing that the drying front rapidly advanced from the surface down to the core of the 0.5 cm khoa layer. A Fourier number approaching or exceeding 1.0 indicates that the mass transfer process is deeply developed and moving toward equilibrium conditions throughout the entire thickness of the material. At 65°C, the Fo m crossed the unity threshold quite rapidly, reaching 1.106 at 5.0 h. In contrast, at 55°C and 60°C, the Fo m values remained below unity even at 5.0 h (0.792 and 0.885, respectively), failing to reach advanced equilibrium states within that time frame.

Conclusion

This study applied dimensional analysis and thin-layer modeling to evaluate the hot-air tray drying kinetics of khoa.

Drying occurred entirely within the falling-rate period.

Increasing the drying temperature from 55°C to 65°C enhanced the effective moisture diffusivity (D eff ) from 1.11 × 10 -9 m 2 /s to 1.55 × 10 -9 m 2 /s. This structural change effectively shortened the total drying time by 16.7% (from 6.0 hours to 5.0 hours) and decreased the final moisture content from 4.83% to 3.17% (dry basis). Among the mathematical models tested, the Wang and Singh equation provided the best fit for predicting moisture ratio (MR) behavior (R 2 = 0.998). The calculated mass transfer Biot numbers (Bi m > 9.0 × 10 6 ) mathematically confirmed that internal mass transfer resistance vastly dominates over surface convective resistance. Fourier numbers (Fo m ) crossed the critical threshold of 0.2 within the first 1.0 to 1.5 hours, showing that the drying front rapidly advanced from the surface down to the core of the 0.5 cm khoa layer.

Ultimately, these scale-independent dimensionless numbers (Bi m and Fo m ) offer universal transport parameters that can be directly applied to optimize industrial equipment design, control energy consumption, and scale up shelf-stable khoa powder manufacturing.

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