International Journal of Thermal Sciences 117 (2017) 266e273 Contents lists available at ScienceDirect International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts Experimental investigation of bound and free water transport process during drying of hygroscopic food material Md Imran H. Khan a, b, R. Mark Wellard a, Szilvia Anett Nagy c, d, M.U.H. Joardder e, M.A. Karim a, * a Science and Engineering Faculty, Queensland University of Technology (QUT), Brisbane, Queensland, Australia Department of Mechanical Engineering, Dhaka University of Engineering & Technology, Gazipur, 1700, Bangladesh P ecs Diagnostics Center, H-7623, P ecs, R et Street 2, Hungary d g Street 20, Hungary MTA - PTE Neurobiology of Stress Research Group, H-7624, P ecs, Ifjúsa e Department of Mechanical Engineering, Rajshahi University of Engineering and Technology, Bangladesh b c a r t i c l e i n f o a b s t r a c t Article history: Received 15 July 2016 Received in revised form 30 March 2017 Accepted 4 April 2017 Hygroscopic food materials contain free (FW) and bound (BW) water in different cellular environments. In-depth understanding of the mechanisms of moisture migration from different cellular environments during drying is crucial for optimising heat and mass transfer as well as for obtaining better quality dried foods. Therefore, the main aim of the present work is to investigate the transportation mechanisms of FW and BW during drying. Experiments were performed on the potato tissue using 1H-NMR T2 relaxometry to uncover the mechanisms involved in FW and BW transportation. The results have confirmed the view that BW migrates after the rupture of the cell membranes. It is interesting to highlight that the cell membranes rupture at different stages of drying rather than collapsing at one time. The membrane collapse depends predominantly on the penetration rate of heat energy and the pressure gradient between intracellular and intercellular environments. All test results suggest that most of the cell membranes rupture at the middle stage of drying where the moisture content is about 2e4 kg/kg (db.). Furthermore, the moisture distribution profile confirmed that some moisture remained around the centre of the dried sample although the surface of the sample became dry. © 2017 Elsevier Masson SAS. All rights reserved. Keywords: Bound water Free water Drying Cell rupture Transport 1. Introduction Drying is an excellent method of food preservation. However, the dehydration of foodstuffs is a very complex process because of its complexity in internal structure and simultaneous heat and mass transfer during drying [1]. Understanding of the actual heat and mass transfer during drying is crucial for optimising energy efficiency of the drying process and preserving the quality attributes of food materials [2]. Many physical and chemical changes take place in food tissue during drying and moisture and temperature distributions significantly contribute to these changes [3]. Therefore, understanding of moisture and temperature distributions and transport mechanism during drying is important. The rate of moisture transport depends on the size and orientation of cells as well as the types of cellular water in the samples [4]. Plant-based * Corresponding author. E-mail address: [email protected] (M.A. Karim). http://dx.doi.org/10.1016/j.ijthermalsci.2017.04.006 1290-0729/© 2017 Elsevier Masson SAS. All rights reserved. foods materials are hygroscopic and porous in nature and contain two types of water, namely free water (FW) and bound water (BW) [5]. FW is present in capillaries or intercellular spaces; whereas, the water in the intracellular space is referred to as BW [6], as shown in Fig. 1. The migration pathways of FW and BW are also different. It is assumed that most of the BW migrate after the collapse of the cells. When cells are collapsed, BW from cells moves to intercellular spaces and shows the characteristics of FW [7]. Due to insufficient knowledge about the migration pathways and characteristics of FW and BW during drying, the current food drying models consider bulk water transport mechanisms in the mathematical modelling [8e12]. Compared to single phase models, multi-phase models considering transport of liquid water, water vapour and air inside the food materials are more realistic [5]. These three phases (water, gas and solid matrix) inside plant-based food structures, as shown in Fig. 2, represent the domain which is commonly used in most multiphase drying models. According to the assumptions presented in Refs. [8e12], there is no water inside the solid matrix. M.I.H. Khan et al. / International Journal of Thermal Sciences 117 (2017) 266e273 Fig. 1. General water distribution inside Plant-based food materials [5]. Fig. 2. Domains considered in existing literature [8e12]. Although this assumption leads to the simplified process of multiphase transport, the real transport process remains unclear [7,13]. It is reported that migration of free water has a minimum effect on the food quality. Migration of BW causes cellular shrinkage, pore formation and the collapse of the cells and pore structure and hence has a major effect on food quality [14,15]. Moreover, the energy requirements for drying of a particular food material depend on its cellular structure and cellular level moisture distributions [16]. Transport of bound water requires more energy compared to transport of free water [17]. In most of the cases, energy and time required for first 90% of water is almost equal to the energy and time required for removing last 10% of water present in a food sample [14]. Therefore, in order to accurately predict the heat and mass transport during drying of food material, better understand of the migration mechanisms of BW water is necessary. 267 There are several techniques available for analysing bound water transport including bioelectric impedance analysis (BIA), differential scanning calorimetry (DSC), differential thermal analysis (DTA), centrifugal settling method (CSM) and nuclear magnetic resonance (NMR) methods. However, not all these methods are suitable for investigating moisture migration mechanisms while drying is in progress. BIA and NMR methods are evaluated as the most appropriate for this type of investigations. BIA is a very simple and established technique for assessing body compositions by measuring the resistance of tissues to the flow of electrical current [6]. The proportion of different contents in a tissue can be calculated as the current flows more easily through the parts of the material that contain more water. This method was used in the study of Halder et al. [4] for calculation of the migration pathways of intracellular water in the plant-based food materials. The temperature influence on the cellular water migration was examined in the same study. It was stated that all of the membranes of the cells would collapse at once after specific temperature is reached. They suggested that below 50 C the cells remain intact and therefore the conversion of the intracellular water to free water remains unchanged during the drying. As cells remain intact, the intracellular water (BW) moves to intercellular space only through microcapillaries, denoting this moisture transport as the slow one. However, their argument is not justified because cell collapse depends on internal thermal stress [15] that first develops near the surface and gradually penetrates to the centre of the sample during convective drying. In other words, entire food sample does not reach ‘cell rupture temperature’ at a time. Therefore, it is logical that the cells may collapse progressively from the surface to centre [14,15]. Moreover, BIA is mainly used for analysing animal fat composition. It may not be a sufficiently accurate technique for predicting moisture migration pathways in plant tissue as BIA cannot detect the position of different types of water (FW and BW). Nuclear magnetic resonance (NMR) is a widely used method for investigating the distribution of different types of water in various locations inside plant-based food material [18]. Proton nuclear magnetic resonance (1H-NMR) relaxometry study has been proven to be a viable method in the study of plants and plant-based food materials submitted to stress reflecting anatomical details of the entire tissue and the water status in particular [19,20]. 1H-NMR signals, which are an average over the whole sample, provide information about the water content of the plant tissue since the proton signal is dominated by water protons and the proton NMR signal intensity is directly proportional to the proton density of the tissue [21,22]. The water proton relaxation behaviour strongly depends on the water mobility in the microscopic environment of the tissue, local magnetic field fluctuations (related to the molecular environment) and the strength of the applied magnetic field. The spin-spin (T2) relaxation is the transverse component of the magnetization vector, which exponentially decays towards its equilibrium value after excitation by radio frequency energy. Some researchers have tried to investigate the changes in water compartmentation during drying of plant tissue [23,24]. These studies found a strong relationship between different water T2 relaxation times and the percentage of moisture loss during drying. However, they did not investigate the migration mechanisms of bound water and free water separately. Moreover, it is not clear when the cell exactly collapses and what are the consequence for the migration of bound water. Therefore, the primary aim of this study is to investigate the free and bound water migration mechanism in plant-based food material during drying. 268 M.I.H. Khan et al. / International Journal of Thermal Sciences 117 (2017) 266e273 2. Material and methods 2.1. Sample preparation and drying Experiments were performed on the potato (Eureka) samples collected from a local market in Brisbane, Australia. Samples were stored in a refrigerator at 4 C until the start of drying experiments, which were performed in a cabinet dryer. The dryer was started about 30 min before drying experiments to achieve steady state conditions before each drying run. At the start of each experiment, the materials were washed and cut into cylindrical slices of 30 mm thickness and 20 mm diameter. Experiments were performed at a temperature of 60 C and a constant air velocity of 0.7 m/s. The relative humidity of the inlet air was in the range of 60%e65%. During the drying process, the tray was taken out at 30-min intervals and weighted using a digital electronic balance (model BB3000; Mettler-Toledo AG, Grefensee, Switzerland). The measurement range of the balance was 0e100 g with an accuracy of 0.01 g. The temperature was measured by a thermal imaging camera Flir-i7. The temperature range of the thermal camera was 20 C to 250 C with 140 140 pixels resolution. In order to better interpret and analyse the results, a simulated 3D temperature distribution profile was developed (Fig. 7) using a validated heat transfer model. The details of the theoretical model, from where simulation results were generated, can be found in authors' previous publications [25,26]. 2.2. NMR measurements At different stages of the drying process, the samples were taken out of the dryer and immediately placed in a 25 mm diameter NMR tube. To protect the sample from oxidation, it was immersed in Fomblin PFPE (Grade 06/6, USA) oil to provide a barrier for mass transfer between samples and surroundings and to assist with sample shimming. The tube was sealed with a standard NMR tube cap and incubated at 22 C for 5 min to reach thermal equilibrium. Measurements were made with a Bruker DRX wide-bore spectrometer (Bruker Bio-spin, Karlsruhe, Germany) operating at 300 MHz for hydrogen and fitted with a micro-imaging (micro 120) gradient set and a birdcage coil. Data were collected and processed using Paravision 4 software (Bruker). Regional T2 relaxation times were measured from relaxation maps acquired with a multi-slicemulti-echo (MSME) sequence using the following acquisition parameters: 64 averages, 1000 echoes with 10 ms echo time and 5.0 s repetition time. The slice thickness and the matrix size were 3 mm and 64 64, respectively. The spatial resolution of the scans was 468 mm. 2.3. Mathematical analysis 2.3.1. Theory The water mobility in potato tissue was investigated quantitatively using multicomponent analysis of T2 relaxation decay curves. A nonlinear least-squares method was applied for data analysis [27]. Generally, the free induction decay of the proton relaxation follows an exponential decay. For multiple environments, there will be corresponding T2 relaxation time constants. A multi-exponential equation can describe these functions. Each tissue compartment corresponds to a different environment and will have a distinct relaxation time constant (T2). It was assumed that these compartments were not inter-reliant at the time of the measurements such that the multi-exponential nature of the T2 decay curve relates to the different water compartments in the tissue, and the water molecules do not undergo rapid exchange between compartments on the NMR timescale. However, when water proton relaxation follows a pattern of mono-exponential decay, there is a fast exchange of protons between tissue water and macromolecules, showing that the water compartments are symbiotic [28e30]. 2.3.2. Data analysis For each sample, two regions-of-interest (ROI) were defined manually on a given slice of the mid transverse section of the MSME images [6]. The mean value of these ROIs was computed for each sample. The accuracy of the multi-exponential parameters considerably depends on the noise level; therefore, a third ROI was assessed outside the sample for determining the signal-to-noise ratio (SNR, average signal intensity over the standard deviation of the noise). Noisy T2-signals were eliminated from the original signal, as described in authors' previous publications [6,31], and data with appropriate SNR (SNR>5) were used for parameter fitting. In order to achieve SNR >5, a constant cut-off of TE < 600 ms was used for all datasets [32]. The mean signal of the two ROIs at each echo time were measured and plotted as a function of time on a semi-log scale where curvature was determined [33]. The number of inflection points was used to determine the number of T2 components. The different T2 relaxation times comprising the signal from each sample was determined by bi-exponentially fitting the mean signal intensity using the following equation. Y ¼ A1 expt =T2 þ A2 expt =T2 1 2 (1) where, Y is the function of T2 relaxation time constant, A1 and A2 represent the relative contributions of the two proton environments, and T21 , T22 are the relaxation time constants of the different components. For T2 measurements, odd echoes were excluded to minimize error due to the influence of stimulated echoes [6,34,35]. T2 relaxation time data processing was carried out with a nonnegative least squares algorithm using self-written program code in the curve fitting toolbox of Matlab® software (The MathWorks, Inc., Natick, MA). The experiments were replicated three times for each sample, and the average of these measurements was used for analysis. The details of the experimental procedure can be found in the authors' previous publication [6]. 2.4. Statistical analysis Data are expressed as mean ± SD of the mean. For statistical analysis least-squares linear regression analysis was used. A 95% confidence level was regarded as significant, as shown in Table 1. 3. Result and discussions The water migration mechanisms at different stages of drying were investigated using 1H-NMR relaxometry. The T2 relaxation decay curves obtained from samples at different drying times are presented in Fig. 3. Each curve was fitted using the multiexponential decay equation (1). After fitting the bi-exponential decay curve with different T2 relaxation intensity data, two different proton components of water relaxation (long and short) were determined, as shown in Table 1, with their corresponding standard deviation and statistical confidence level. Depending on the water mobility and pore size, two signal components namely, long and short were categorised as intracellular environmental water (BW) and intercellular environmental water (FW) [6]. 3.1. Collapse of the cell membrane during drying Fig. 4 shows the change in the proportion of BW over time during the drying process, calculated on the assumption that the total moisture present at that time is considered 100%. The curve is 30.65 32.92 24.62 74.81 41.85 48.72 37.28 50.21 66.50 33.86 79.87 44.50 102.30 53.97 20.08 ± 4.2 18.02 ± 3.5 22.75 ± 3.8 49.8 ± 5.6 23.78 ± 3.5 14.51 ± 2.8 15.42 ± 2.7 49.28 ± 6.2 17.97 ± 2.9 58.23 ± 6.8 53.78 ± 5.8 52.95 ± 4.9 92.98 ± 8.2 99.9 ± 10.2 97.02 82.50 90.84 104.70 97.23 106.20 137.00 124.30 148.20 101.70 139.80 105.80 76.30 103.20 98.92 ± 10.2 100.75 ± 11.2 92.32 ± 9.5 118.21 ± 10.8 98.87 ± 10.3 109.92 ± 12.2 138.87 ± 14.4 127.02 ± 10.8 150.84 ± 15.3 104.78 ± 11.3 149.92 ± 13.2 107.74 ± 10.4 332.75 ± 16.2 532.70 ± 19.4 Fresh Potato 30 60 90 120 150 180 210 240 270 300 360 420 450 79.92 ± 6.8 81.98 ± 8.2 77.25 ± 7.3 50.2 ± 6.4 76.22 ± 5.5 85.49 ± 7.5 84.58 ± 7.2 50.72 ± 5.3 82.03 ± 6.8 41.77 ± 4.8 46.22 ± 5.7 47.05 ± 5.3 7.02 ± 3.5 0.1 ± 2.4 (%) T2 (ms) 100.80 87.01 93.80 131.70 100.50 113.70 140.70 129.70 153.40 107.90 160.00 109.60 318.70 134.30 26.47 24.25 29.25 65.72 38.87 37.99 32.83 48.69 60.80 32.57 74.87 43.50 95.42 98.21 ± ± ± ± ± ± ± ± ± ± ± ± ± ± 6.2 5.8 3.9 9.9 7.5 6.8 6.4 5.9 8.9 7.7 11.2 8.5 10.2 12.4 22.29 28.07 18.47 56.66 35.90 27.27 28.40 47.16 55.09 31.27 69.88 42.50 86.15 43.71 % T2 (ms) T2 values Contribution (A) 95% CI T2 values 95% CI Short component Long component Drying time (min) Table 1 The different T2 component at different drying times, each measurement is the mean and SD of 3 samples. 95% CI Contribution (A) 95% CI 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998 0.9997 0.9994 0.998 0.9997 0.9998 0.9998 0.9996 0.9995 Goodness of fit (R2) M.I.H. Khan et al. / International Journal of Thermal Sciences 117 (2017) 266e273 269 divided into four zones according to the cell membrane collapse phenomenon. In Fig. 5, these zones are presented separately for clearly showing the cell membrane rupture points. For each zone, the peak of the BW curves is represented as cell membrane collapse point. It can be seen in the figure that the duration of the Zone - 1 is approximately 100 min. In this zone, BW is about 80e85% and this proportion remains constant up to 55 min of drying, as shown in Fig. 5a. This is because only free water migrates at the initial stage of drying. After 55 min of drying, the curve drops sharply to a BW level of 50e55%. In order to interpret the zone, an average surface temperature profile and 3D temperature distribution profile have been presented in Figs. 6 and 7 respectively. A 3D coupled heat and mass transfer model for convective drying of food materials was developed earlier by the present authors [25,26]. COMSOL Multiphysics, finite element-based engineering simulation software was used to simulate the model in 2D and 3D and to solve the coupled heat and mass transfer equations. That model was validated using comprehensive experimental data. The details of the model can be found in authors' previous publications [25,26]. Fig. 7 presents simulated temperature profile of the sample, which was developed using above validated heat transfer model. From Fig. 7a, it can be observed that after 55 min of drying the temperature at the surface rose to 53e54 C, and the heat energy penetrated slowly towards the centre of the sample. It has been reported that cell membranes start to collapse when the drying temperatures are at or above 50 C [4]. It has also been stated that the cell membrane collapses due to the thermal stress which are induced by the temperature and pressure gradients [15]. As mentioned earlier, it can be assumed that when the temperature reaches 52e53 C, the cell membranes close to the surface of the samples will start to rupture first. As a result, the exposed BW (intracellular water) is able to readjust rapidly to become part of the intercellular space (i.e. FW) which is registered as a rapid BW drop (Fig. 5a). Within the next drying Zone, the FW is gradually decreasing due to its migration up to the surface. According to Vasi c et al. [36], this internal mass transport is controlled by several drying mechanisms. Consequently, the relative proportion of BW (compared to FW) increases during this period. The total water at any point (instantaneous moisture) is assumed as 100%, although the average moisture content is decreased gradually (Fig. 4, zone 2). This process continues up to 170 min of drying. Simultaneously, with the previously mentioned process, heat is continuously propagated towards the centre of the drying sample (Fig. 7). At that time the surface temperature increased to about 58 C (Fig. 6), whereas inside the sample the temperature was about 56e57 C as shown in Fig. 7c. The sample surface temperature increased up to 58 C at the end of the Zone 2, while the temperature inside the sample was 2 C lower (see Figs. 7c and 6). As a result of this temperature difference, the cell membrane ruptured again and the BW curve dropped rapidly as it was converted to FW (Fig. 5b). Within the next drying Zone (Zone 3) the internal moisture transport of the FW to the surface is continued. In other words, FW is gradually decreasing (see Fig. 4). Consequently, the relative proportion of BW (compared to FW) increases during this period. The majority of cell membranes are damaged at the end of the Zone 3 (after 180 min) when the temperature of the entire sample has reached 59.5 C (above cell collapse temperature). This point is registered at Fig. 5c as the cell membrane collapse point. From this point up to the end of the Zone 3 gradual decrease of BW is registered (see Fig. 4) as BW is transferred into FW. The final cell collapse is registered in the Zone 4 (after 350 min), as shown in 270 M.I.H. Khan et al. / International Journal of Thermal Sciences 117 (2017) 266e273 30000000 Fresh 25000000 After drying of 30 min 60 min 90 min Int e n sit y 20000000 120 min 150 min 180 min 15000000 210 min 240 min 270 min 10000000 300 min 360 min 420 min 5000000 450 min 0 15 65 115 165 215 265 315 365 415 465 515 565 615 Echo time (ms) Fig. 3. T2 relaxation at different drying time. Fig. 4. The change in percentage of free and bound water with drying time. Fig. 5d. The sample surface and internal temperature have reached 60 C (see Fig. 7d) in the final collapse point. From this moment up to the end of the Zone 4 progressive decrease of BW is registered (see Fig. 4). Detail analysis of Figs. 4 and 5 showed that the Zone 2 lasts longer than other three Zones. Initially, cell membranes close to the sample surface will be the first to rupture (zone 1). The rest of the tissue (i.e. cells far away from the surface) gains energy slowly prior to developing thermal stresses. As a result, in zone 2, the cell membrane takes more time to collapse. On the other hand, at zone 3, the cell membranes inside the sample have already absorbed sufficient heat energy to develop higher thermal stresses. Consequently, the cell membranes in zone 3 will collapse faster (Fig. 5c). As a final conclusion, it can be stated that the cell membranes in plant-based food material collapse in different stages of drying rather than collapsing altogether. In the case of conventional drying the cell rupture mainly depends on the rate of penetration of heat energy as well as the resulting pressure gradient between intracellular and intercellular spaces for conventional drying. 3.2. Moisture distribution Several NMR images showing the moisture distribution profile at different drying times are presented in Fig. 8. The lighter parts of the NMR images are related to the higher water content while darker parts are associated with the dried sample segments. It is interesting to note that the moisture distribution in fresh potato tissue is uniform throughout the whole sample (see Fig. 8a). At the M.I.H. Khan et al. / International Journal of Thermal Sciences 117 (2017) 266e273 (a) 80 Percentage of water Percentage of bound water(%) (%) %)) PercePnetracgeenotfabgoeuonfdw waatteerr ((% 90 70 60 Cell membrane rupture 50 40 30 Zone- I 20 10 0 0 10 20 30 40 50 60 70 80 90 (b) 90 80 70 60 50 40 30 20 10 0 Cell membrane rupture Zone- II 90 100 110 (c) 80 70 Cell membrane rupture 40 30 Zone- III 20 10 0 Percentage of water Percentage of bound water(%) (%) d awtaetrer(% (%)) PerP ceenrtcaegnetoafgbeooufnw 90 50 50 45 40 35 30 25 20 15 10 5 0 220 230 240 250 150 170 190 210 (d) Ultimate cell membrane rupture Zone- IV 270 210 130 Time (min) Time (min) 60 271 260 270 290 310 330 280 350 370 390 410 430 Time (min) Time (min) Fig. 5. The percentage of bound water at different stages of drying. collapse in each of the remained three Zones. After the membrane collapse is reached in each zone, a fraction of BW is transformed into FW fraction, and removed along with the remained FW fraction. Simultaneously the fraction of the wet surface will gradually decrease until the last “wet patches” are removed from the surface while the drying front will be directed toward the centre of the drying sample (See Fig. 8cee). This observation is consistent with onard and Blacher [39]. the findings of Belton [38], and Le 4. Conclusions Fig. 6. Average surface temperature profile. initial drying stage, only FW is gradually removed causing the surface of the sample to become partially dried (Fig. 8b). After rupturing the cell membranes, migration of BW takes place and consequently the surface of the sample becomes dried as shown by the gray colour in Fig. 8c. After the first cell membrane collapse had taken place, a fraction of BW was rapidly redistributed into the intercellular space and removed along with the FW fraction. This creates “dry patches” on the sample surface which are registered as darker segments on Fig. 8c. As the drying continues, heat penetrates towards the sample centre [37] causing the pressure gradient between intracellular and intercellular spaces to grow. This process will eventually fulfill the requirements necessary for the membrane In this study, the process of free and bound water migration and cell collapse during drying of plant-based food material was investigated using 1H-NMR T2 relaxometry. It was found that the cell membranes rupture at different stages of drying rather than collapsing all at once. The collapse mainly depends on the penetration rate of heat energy and the resulting pressure gradient generated between intracellular and intercellular environments. Test results showed that when cells are collapsed at different stages of drying, BW migrates from intracellular space to the intercellular spaces (hence become available for migration). This study also identified the timing of the ruptures of the cell membranes. Understanding of the periods when cell membranes are ruptured are crucial as this knowledge can be used for designing more efficient drying system. The findings of this study will enhance the understanding of heat and mass transfer during drying of plant-based food material and will also contribute to the development of accurate heat and mass transfer models and the prediction of deformation. 272 M.I.H. Khan et al. / International Journal of Thermal Sciences 117 (2017) 266e273 Fig. 7. Simulated 3D temperature distributions at different drying time (a) at 55 min of drying (b) at 90 min of drying (c) at 200 min of drying (d) at 300 min of drying. Fig. 8. Moisture distribution at different drying time with high intensity (light colour) representing greater water content. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) M.I.H. Khan et al. / International Journal of Thermal Sciences 117 (2017) 266e273 Acknowledgements The authors are sincerely grateful to the Queensland University of Technology, Australia for funding a QUTPRA scholarship and HDR tuition fee sponsorship, which has enabled the conduct of this research. Thanks also to Dr. Chandan Kumar, for helpful discussion and suggestion of the manuscript. Moreover, S.A.N. was supported by the Hungarian Brain Research Program “B” (KTIA_NAP_13-22014-0019). References [1] Esfahani JA, Majdi H, Barati E. Analytical two-dimensional analysis of the transport phenomena occurring during convective drying: apple slices. J Food Eng 2014;123:87e93. [2] Khan MIH, Kumar C, Joardder MUH, Karim MA. Determination of appropriate effective diffusivity for different food materials. Dry Technol 2017;35:335e46. [3] Barati E, Esfahani JA. Mathematical simulation of convective drying: spatially distributed temperature and moisture in carrot slab. Int J Therm Sci 2012;56: 86e94. 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