# Mathematical Modelling of Heat and Mass Transfer during Jackfruit Drying Considering Shrinkage

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Sample Preparation

#### 2.2. Experimental Procedure

## 3. Mathematical Model Development

#### 3.1. Heat and Mass Transfer Governing Equations

#### 3.1.1. Energy Balance Equation

^{3}), c

_{p}is the material specific heat capacity (J/kg·K), and k is the thermal conductivity of the sample (W/m·K). The value of Q is zero due to there being no internal heat generation.

#### 3.1.2. Mass Conservation Equation

^{3}), t is the time (s), and ${D}_{eff}$ is the effective moisture diffusivity (m

^{2}/s).

#### 3.2. Deformation Prediction

^{2}/s), density ratio $\left(\frac{{\rho}_{w}}{{\rho}_{p}}\right)$ is the ratio of water density and particle density, respectively, and L is the half thickness of the sample (m). The considered shrinkage velocity of the solid matrix during drying is calculated using the mass-average approach. This approach considers the change in mass of the entire solid matrix during drying and assumes that the velocity is zero when the mass of the solid matrix does not change.

#### 3.3. Initial Conditions and Input Parameters

#### 3.3.1. Initial and Boundary Conditions

_{(t = 0)}= 28.5 °C and C

_{(t = 0)}= C

_{o}

_{o}represents initial moisture concentration of the sample (mol/m

^{3}).

^{2}. K), ${h}_{m}$ being the mass transfer coefficient (m/s), ${h}_{fg}$ as the latent heat of evaporation (J/kg), ${M}_{w}$ as the molar mass of water (kg/mol), R as the universal gas constant, and ${p}_{v,eq}$ and ${p}_{v,air}$ being the equilibrium vapour pressure (Pa) and vapour pressure of ambient air (Pa), respectively.

#### 3.3.2. Input Parameters

#### Thermophysical Properties of Jackfruit Sample

#### Moisture Effective Diffusivity

^{2}/s), ${L}_{o}$ and ${L}_{t}$ are the sample thickness (m) at initial time and time t, respectively. The thickness ratio is found from the following equation [39,40]:

^{3}).

#### Determination Effective Moisture Diffusivity

^{2}/s), L is the half thickness of the sample (m), and t is the drying time (s).

^{−10}–3.15 × 10

^{−10}m

^{2}/s for the temperature ranges of 50–70 °C, which closely agrees the values seen in the literature [22,23].

#### Determination of Heat and Mass Transfer Coefficient

^{2}/s). The values of Re, Sc, and Pr numbers are calculated by Equations (19)–(21), respectively.

^{3}), $v$ is the drying air velocity (m/s), ${\mu}_{a}$ is the dynamic viscosity of air (Pa·s), ${k}_{a}$ is the thermal conductivity of air (W/mK), and ${C}_{p}$ is the specific heat capacity of air (J/kg·K).

#### 3.4. Simulation Methodology

## 4. Results and Discussion

#### 4.1. Moisture Content and Distribution

#### 4.2. Average Sample Temperature

#### 4.3. Shrinkage Profile

_{o}) with reduced moisture content for jackfruit can be found in Figure 11a.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Wu, B.; Guo, X.; Guo, Y.; Ma, H.; Zhou, C. Enhancing jackfruit infrared drying by combining ultrasound treatments: Effect on drying characteristics, quality properties and microstructure. Food Chem.
**2021**, 358, 129845. [Google Scholar] [CrossRef] - Zhang, Y.; Li, B.; Xu, F.; He, S.; Zhang, Y.; Sun, L.; Zhu, K.; Li, S.; Wu, G.; Tan, L. Jackfruit starch: Composition, structure, functional properties, modifications and applications. Trends Food Sci. Technol.
**2021**, 107, 268–283. [Google Scholar] [CrossRef] - Saxena, A.; Maity, T.; Raju, P.S.; Bawa, A.S. Optimization of pretreatment and evaluation of quality of jackfruit (Artocarpus heterophyllus) bulb crisps developed using combination drying. Food Bioprod. Process.
**2015**, 95, 106–117. [Google Scholar] [CrossRef] - Haque, M.A.; Begum, R.; Shibly, A.Z.; Sultana, M.M.; Khatun, A. Influence of jackfruit pulp on the quality and shelf life of jackfruit cake. J. Environ. Sci. Nat. Resour.
**2015**, 8, 59–64. [Google Scholar] [CrossRef] - Anaya-Esparza, L.M.; González-Aguilar, G.A.; Domínguez-Ávila, J.A.; Olmos-Cornejo, J.E.; Pérez-Larios, A.; Montalvo-González, E. Effects of minimal processing technologies on jackfruit (Artocarpus heterophyllus Lam.) quality parameters. Food Bioprocess Technol.
**2018**, 11, 1761–1774. [Google Scholar] - Swami, S.B.; Thakor, N.J.; Orpe, S.; Kalse, S.B. Development of osmo-tray dried ripe jackfruit bulb. J. Food Res. Technol.
**2014**, 2, 77–86. [Google Scholar] - Prasantha, B.; Amunogoda, P. Moisture adsorption characteristics of solar-dehydrated mango and jackfruit. Food Bioprocess Technol.
**2013**, 6, 1720–1728. [Google Scholar] [CrossRef] - Chowdhury, M.; Bala, B.; Haque, M. Energy and exergy analysis of the solar drying of jackfruit leather. Biosyst. Eng.
**2011**, 110, 222–229. [Google Scholar] [CrossRef] - Taib, M.R.; Muhamad, I.I.; Ngo, C.L.; Ng, P.S. Drying kinetics, rehydration characteristics and sensory evaluation of microwave vacuum and convective hot air dehydrated jackfruit bulbs. J. Teknol.
**2013**, 65, 51–57. [Google Scholar] [CrossRef] - Giraldo-Zuniga, A.D.; Arévalo-Pinedo, A.; Rodrigues, R.M.; Lima, C.S.S.; Feitosa, A.C. Kinetic drying experimental data and mathematical model for jackfruit (Artocarpus integrifolia) slices datos experimentales y modelo matemático de la cinética de secado de rodajas de jaca (Artocarpus integrifolia). CyTA J. Food
**2006**, 5, 89–92. [Google Scholar] [CrossRef] - Saxena, A.; Maity, T.; Raju, P.S.; Bawa, A.S. Degradation kinetics of colour and total carotenoids in jackfruit (Artocarpus heterophyllus) bulb slices during hot air drying. Food Bioprocess Technol.
**2012**, 5, 672–679. [Google Scholar] [CrossRef] - Souza, M.A.; Bonomo, R.C.; Fontan, R.C.; Minim, L.A.; COIMBRA, J.S.D.R. Thermophysical properties of jackfruit pulp affected by changes in moisture content and temperature. J. Food Process Eng.
**2011**, 34, 580–592. [Google Scholar] [CrossRef] - Yi, J.; Wang, P.; Bi, J.; Liu, X.; Wu, X.; Zhong, Y. Developing novel combination drying method for jackfruit bulb chips: Instant controlled pressure drop (DIC)-assisted freeze drying. Food Bioprocess Technol.
**2016**, 9, 452–462. [Google Scholar] [CrossRef] - Nansereko, S.; Muyonga, J.; Byaruhanga, Y.B. Optimization of drying conditions for Jackfruit pulp using Refractance Window Drying technology. Food Sci. Nutr.
**2022**, 10, 1333–1343. [Google Scholar] [CrossRef] - Gan, P.L.; Poh, P.E. Investigation on the effect of shapes on the drying kinetics and sensory evaluation study of dried jackfruit. Int. J. Sci. Eng.
**2014**, 7, 193–198. [Google Scholar] [CrossRef] - Leite, D.D.D.F.; Queiroz, A.J.D.M.; Figueirêdo, R.M.F.D.; Lima, L.S.L. Mathematical drying kinetics modeling of jackfruit seeds (Artocarpus heterophyllus Lam.). Rev. Cienc. Agron.
**2019**, 50, 361–369. [Google Scholar] [CrossRef] - Aprajeeta, J.; Gopirajah, R.; Anandharamakrishnan, C. Shrinkage and porosity effects on heat and mass transfer during potato drying. J. Food Eng.
**2015**, 144, 119–128. [Google Scholar] [CrossRef] - Tuly, S.S.; Mahiuddin, M.; Karim, A. Mathematical modeling of nutritional, color, texture, and microbial activity changes in fruit and vegetables during drying: A critical review. Crit. Rev. Food Sci. Nutr.
**2023**, 63, 1877–1900. [Google Scholar] [CrossRef] - Mahiuddin, M.; Khan, M.I.H.; Kumar, C.; Rahman, M.M.; Karim, M.A. Shrinkage of food materials during drying: Current status and challenges. Compr. Rev. Food Sci. Food Saf.
**2018**, 17, 1113–1126. [Google Scholar] [CrossRef] - Taoukis, P.; Giannakourou, M. Modelling food quality. Food Sci. Technol.
**2018**, 32, 38–43. [Google Scholar] - Bakhara, C.K.; Pal, U.S.; Bal, L.M. Drying characteristic and physico-chemical evaluation of tender jackfruit slices during osmo-convective drying. J. Food Meas. Charact.
**2018**, 12, 564–572. [Google Scholar] [CrossRef] - Saxena, J.; Dash, K. Drying kinetics and moisture diffusivity study of ripe Jackfruit. Int. Food Res. J.
**2015**, 22, 414. [Google Scholar] - Praneetpolkrang, P.; Sathapornprasath, K. Thin-layer drying model of jackfruit using artificial neural network in a far infrared dryer. Eng. Appl. Sci. Res.
**2021**, 48, 181–189. [Google Scholar] - Mabel Queiroz de Oliveira, T.; da Silva, A.F.; Farias, V.S.D.O.; Alves de Medeiros, R.; Nascimento Lima, A.R. Description of drying of jackfruit seed through diffusive models. J. Food Process. Preserv.
**2022**, 46, e16389. [Google Scholar] [CrossRef] - Kaushal, P.; Sharma, H. Osmo-convective dehydration kinetics of jackfruit (Artocarpus heterophyllus). J. Saudi Soc. Agric. Sci.
**2016**, 15, 118–126. [Google Scholar] [CrossRef] - Aswin, G.; Bhasin, A.; Mazumdar, A. Utilization of jackfruit by-products and application in food industry. Pharma Innov. J.
**2022**, 11, 2293–2299, TPS. [Google Scholar] - Nidhina, K.; Abraham, B.; Fontes-Candia, C.; Martínez-Abad, A.; Martínez-Sanz, M.; Nisha, P.; Lopez-Rubio, A. Physicochemical and functional properties of pectin extracted from the edible portions of jackfruit at different stages of maturity. J. Sci. Food Agric.
**2023**, 103, 3194–3204. [Google Scholar] [CrossRef] - Joardder, M.U.; Kumar, C.; Karim, M. Prediction of porosity of food materials during drying: Current challenges and directions. Crit. Rev. Food Sci. Nutr.
**2018**, 58, 2896–2907. [Google Scholar] [CrossRef] - Joardder, M.U.; Kumar, C.; Karim, M. Multiphase transfer model for intermittent microwave-convective drying of food: Considering shrinkage and pore evolution. Int. J. Multiph. Flow
**2017**, 95, 101–119. [Google Scholar] [CrossRef] - Joardder, M.U.; Karim, M. Development of a porosity prediction model based on shrinkage velocity and glass transition temperature. Dry. Technol.
**2019**, 37, 1988–2004. [Google Scholar] [CrossRef] - Pham, N.D.; Khan, M.; Karim, M. A mathematical model for predicting the transport process and quality changes during intermittent microwave convective drying. Food Chem.
**2020**, 325, 126932. [Google Scholar] [CrossRef] [PubMed] - Ratti, C.; Crapiste, G.; Rotstein, E. A new water sorption equilibrium expression for solid foods based on thermodynamic considerations. J. Food Sci.
**1989**, 54, 738–742. [Google Scholar] [CrossRef] - Vega-Mercado, H.; Góngora-Nieto, M.M.; Barbosa-Cánovas, G.V. Advances in dehydration of foods. J. Food Eng.
**2001**, 49, 271–289. [Google Scholar] [CrossRef] - Cengel, Y.A.; Boles, M.A.; Kanoğlu, M. Thermodynamics: An Engineering Approach; McGraw-Hill: New York, NY, USA, 2011; Volume 5. [Google Scholar]
- Cengel, Y.; Heat, T.M. A practical approach. Heat Mass Transf.
**2003**. [Google Scholar] - Sahin, S.; Sumnu, S.G. Physical Properties of Foods; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2006. [Google Scholar]
- Khan, M.I.H.; Kumar, C.; Joardder, M.U.H.; Karim, M.A. Determination of appropriate effective diffusivity for different food materials. Dry. Technol.
**2017**, 35, 335–346. [Google Scholar] [CrossRef] - Kumar, C.; Millar, G.J.; Karim, M. Effective diffusivity and evaporative cooling in convective drying of food material. Dry. Technol.
**2015**, 33, 227–237. [Google Scholar] [CrossRef] - Karim, M.A.; Hawlader, M. Mathematical modelling and experimental investigation of tropical fruits drying. Int. J. Heat Mass Transf.
**2005**, 48, 4914–4925. [Google Scholar] [CrossRef] - Kumar, C.; Karim, A.; Joardder, M.U.H.; Miller, G. Modeling heat and mass transfer process during convection drying of fruit. In Proceedings of the 4th International Conference on Computational Methods, Washington, DC, USA, 17–19 August 2012; Queensland University of Technology: Brisbane, Australia, 2012. [Google Scholar]
- Feng, Y.; Wu, B.; Yu, X.; Yagoub, A.E.A.; Sarpong, F.; Zhou, C. Effect of catalytic infrared dry-blanching on the processing and quality characteristics of garlic slices. Food Chem.
**2018**, 266, 309–316. [Google Scholar] [CrossRef] - Hirt, C.; Amsden, A.; Cook, J. An arbitrary Lagrangian–Eulerian computing method for all flow speeds. J. Comput. Phys.
**1997**, 135, 203–216. [Google Scholar] [CrossRef] - Zhou, J.; Yang, X.; Ye, J.; Zhu, H.; Yuan, J.; Li, X.; Huang, K. Arbitrary Lagrangian-Eulerian method for computation of rotating target during microwave heating. Int. J. Heat Mass Transf.
**2019**, 134, 271–285. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Cylindrical sample. (

**b**) Boundary conditions for shrinkage and heat–mass transfer for the developed model.

**Figure 5.**Moisture profile of jackfruit samples with and without shrinkage phenomena at a drying air temperature of 60 °C.

**Figure 7.**Comparison of moisture ratio variation among model predictions and experimental outcomes over the drying time.

**Figure 11.**Influence of moisture variation on (

**a**) volume ratio and (

**b**) shrinkage coefficient of jackfruit sample.

References | Used Model | Model Type | Remarks | Drawback |
---|---|---|---|---|

[21] | Lewis model $\frac{M-{M}_{eq}}{{M}_{\theta i}-{M}_{eq}}=A\mathrm{exp}(-K\theta )$ Page power model $\frac{M-{M}_{eq}}{{M}_{\theta i}-{M}_{eq}}=\mathrm{exp}(-K{\theta}^{n})$ | Semi-theoretical | - Osmo-convective drying.
- Drying characteristics and evaluation of physico-chemical behaviour was carried out.
- Thin-layer drying was well described by Page model.
| Lacks the physics-based analysis of heat and moisture transfer. |

[22] | Midilli model $MR=a\mathrm{exp}\left(-k{t}^{n}\right)+b$ | Empirical | - Hot air drying.
- Moisture diffusivity and drying kinetics was determined.
- Midilli model fitted well among 14 thin-layer drying models.
| Analysis for shape and size changes of dried samples was neglected. |

[23] | Thin-layer models using Artificial neural network (ANN) | Empirical | - ANN structure of 2-12-1 with Tan-sigmoid transfer functions was found as the optimal one.
- Page model showed the best results.
| No consideration of shrinkage. |

[1] | Midilli model $MR=a\mathrm{exp}\left(-k{t}^{n}\right)+b$ | Semi-theoretical | - Infrared drying with ultrasound Treatments.
- Drying kinetics were properly described by Midilli model.
| Textural and microstructural analyses were conducted, but shrinkage was neglected. |

[24] | Diffusive models | Theoretical | - Drying of jackfruit seed (with and without endocarp) was conducted.
- Influence of endocarp presence and temperature on thermos-physical parameters were investigated applying diffusive models.
| Basic transport process was ignored. |

[11] | Zero and first order kinetic models $\frac{dQ}{dt}=\pm k{Q}^{n}$ | Empirical | - Hot air drying.
- Degradation kinetics of total carotenoids and colour were investigated.
| No analysis for shrinkage of the dried samples. |

[25] | Thin-layer drying models (Newton, Logarithmic, Henderson and Pabis, Magee, Modified Page, Page, Wang and Singh) Wang and Singh model: $MR=1+at+b{t}^{2}$ | Semi-theoretical, empirical | - Osmo-convective drying.
- Wang and Singh model provided the best result for jackfruit dying kinetics.
| Dimensional changes of the dried samples were ignored. |

[15] | Nine thin-layer drying models Midilli–Kucuk model: $MR=a\mathrm{exp}\left(-k{t}^{n}\right)+b\mathrm{t}$ Modified Midilli–Kucuk model: $MR=a\mathrm{exp}\left(-k{t}^{n}\right)+b$ | Semi-theoretical | - Convective drying.
- Influence of shape was evaluated on sensory property and drying kinetics
- Modified Midilli–Kucuk model was found to be the best kinetic model.
| Lacks the fundamental analysis of heat–mass transfer phenomena. |

Parameter | Values | Reference |
---|---|---|

Sample diameter | 0.025 (m) | This study |

Sample thickness | 0.006 (m) | This study |

Molecular weight of water | 0.018 (kg/mol) | [34] |

Initial temperature, T_{o} | 28.5 °C | This study |

Air temperature, T_{air} | 60 °C | This study |

Air velocity, v | 1.1 (m/s) | This study |

Reference effective diffusivity | 2.27 × 10^{−10} (m^{2}/s) | This study |

Latent heat of evaporation, h_{fg} | 2,358,600 (J/kg) | [35] |

Initial moisture content, M_{o} | 5.25 (kg/kg db) | This study |

Initial wet bases moisture content, M_{o, wb} | 0.84 (kg/kg wb) | This study |

Equilibrium moisture content, M_{e} | 0.2 (kg/kg db) | [12] |

Heat transfer coefficient, h_{T} | 17.96 W/(m^{2}·K) | This study |

Mass transfer coefficient, h_{m} | 0.003 (m/s) | This study |

Density of jackfruit, ρ | 1023 (kg/m^{3}) | [12] |

Density of water, ρ_{w} | 1000 (kg/m^{3}) | [35] |

Activation energy, E_{a} | 30,325.65 (J/mol) | This study |

Arrhenius factor, D_{o} | 1.29 × 10^{−5} (m^{2}/s) | This study |

Universal gas constant, R_{g} | 8.3145 J/(mol·K) | [34] |

Nusselt number, Nu | 16.444 | This study |

Sherwood number (Sh) | 37.163 | This study |

Schmidt number (Sc) | 77.526 | This study |

Reynolds number (Re) | 177.36 | This study |

Prandlt number (Pr) | 6.5527 | This study |

Model Parameter | Model Types | Statistical Error | |
---|---|---|---|

MAE | MRE | ||

Average Moisture content | With shrinkage | 0.27 kg/kg db | 0.19 |

Without shrinkage | 0.46 kg/kg db | 0.35 | |

Average temperature | With shrinkage | 2.07 °C | 0.04 |

Without shrinkage | 2.93 °C | 0.06 | |

Shrinkage | With shrinkage | 0.04 | 0.05 |

$Meanabsoluteerror\left(MAE\right)=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}}\left|{Y}_{exp,i}-{Y}_{pre,i}\right|$ | |||

$Meanrelativeerror\left(MRE\right)=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}}\frac{\left|{Y}_{exp,i}-{Y}_{pre,i}\right|}{{Y}_{exp,i}}$ | |||

$\mathrm{where}{Y}_{exp}$$\mathrm{and}{Y}_{pre}$ are the experimental and predicted values, respectively, for either moisture content, temperature, or shrinkage, and N is the total number of observations. |

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## Share and Cite

**MDPI and ACS Style**

Tuly, S.S.; Joardder, M.U.H.; Welsh, Z.G.; Karim, A.
Mathematical Modelling of Heat and Mass Transfer during Jackfruit Drying Considering Shrinkage. *Energies* **2023**, *16*, 4461.
https://doi.org/10.3390/en16114461

**AMA Style**

Tuly SS, Joardder MUH, Welsh ZG, Karim A.
Mathematical Modelling of Heat and Mass Transfer during Jackfruit Drying Considering Shrinkage. *Energies*. 2023; 16(11):4461.
https://doi.org/10.3390/en16114461

**Chicago/Turabian Style**

Tuly, Sumaiya Sadika, Mohammad U. H. Joardder, Zachary G. Welsh, and Azharul Karim.
2023. "Mathematical Modelling of Heat and Mass Transfer during Jackfruit Drying Considering Shrinkage" *Energies* 16, no. 11: 4461.
https://doi.org/10.3390/en16114461