Numerical Solution of Mass Transfer Process during Drying of Apple Slices Using Pseudospectral Method

ماشین های کشاورزی

Article 9, Volume 9, Issue 1 - Serial Number 17, 2019, Pages 113-122   

Document Type: Research Article

DOI: 10.22067/jam.v9i1.63511

Authors

A. Pasban¹; M. Mohebbi^{* 1}; H. Sadrnia²; S. A. Shahidi³

¹Department of Food Science and Technology, Ferdowsi University of Mashhad, Mashhad, Iran

²Department of Biosystems Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

³Department of Food Science and Technology, Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran

Abstract

Introduction
Convective air drying is one of the oldest and most popular drying methods. Designing and controlling the convective air drying needs the mathematical description of the moisture transfer during the drying process, known as drying kinetics. Fick’s second law of diffusion can be used for modelling the moisture distribution inside the moist object during drying process.
Mathematical modeling of drying process is a very important tool, as it contributes to understand better moisture distributions inside the product which helps designing, improving and controlling drying operation in the food industry.
Implementation of the partial differential equations subject to the correspondent initial and boundary conditions is one of the main methods of mathematical modeling to describe the physical phenomena such as moisture transfer during drying. In the recent decades, considerable number of research works have been devoted to numerical solution of mass transfer phenomena during convective drying of food products by using the common numerical solution such as FDMs, FEMs and FVMs.
The spectral collocation (pseudospectral) methods is a powerful tool for the numerical solutions of smooth PDEs like mass transfer equations. Pseudospectral methods are able to achieve the high precision with using a small number of discretization points compared to FDMs and FEMs and with low computational time and computer memory.
The objective of present research is to simulate the mass transfer phenomena in one dimension during convective drying of apple slices. The validation of the presented numerical model was done by comparing experimental drying data taken from Kaya et al. (2007) and Zarein et al. (2013). For more confirming the numerical approach, a numerical example with the exact solution is provided and the related errors were evaluated.
 
Materials and Methods
Estimation of mass transfer coefficients
The convective mass transfer coefficient in the surface of the apple slice was obtained according to the relationship presented by Paitil (1988) and Janjai et al. (2008).
    (1)  
 
Estimation of effective moisture diffusivity coefficient
Fick’s second law of diffusion was applied to obtain the effective moisture diffusivity coefficient of the apple slices. The analytical solution of this equation can be written as follows (Crank, 1975):
         (2)
In this study, we consider the Pseudospectral methods for solving 1D mass transfer equation. In order to develop the model, the following common assumptions are considered: negligible heat changes during drying process, moisture is transferred inside the slices by diffusion, one-dimensional mass transfer in apple slices, non-shrinkage and non-deformation of the slice.
Results and Discussion
In the field of numerical analysis, the main advantage of pseudospectral methods compared to others such as FDMs and FEMs are exponential convergence and sufficient accuracy (Sun et al., 2012). The values of parameters and coefficients of mathematical model are summarized in Table 1. The comparisons between the predicted average moisture content and the experimental data are shown in Fig. 1 & 2. It can be seen, the numerical results are in good agreement with the experimental data. The values of the correlation coefficient and the root mean square error from comparison of numerical result with experimental data taken from Zarein et al. (2013) and Kaya et al. (2007) were 0.9996, 0.0729 and 0.997, 0.1561 respectively. Moreover, the running time for solving 1D mass transfer equations was about 3 seconds. This result is the evident that the presented model is successful for predicting the moisture content history during drying process.
Moreover, by using the considered numerical method the approximate solutions of defined numerical example for different discretizing points was evaluated and the associated error history are shown in Figure 3. It can be seen that the values of errors are very low and about 10-3 and 10-5, that confirms the high accuracy, robustness and efficiency of the suggested numerical approach.
 
Conclusion
Spectral collocation (pseudospectral) method is presented to solve mass transfer equation in one dimensional in during convective drying process approximately. The model was validated by the reported experimental data from convective drying of apple slices. Also, a numerical example, which had an exact solution in a closed form, was provided to illustrate the high accuracy of the proposed method. The results of statistical computations (r and RMSE) and numerical example showed the efficiency, applicability and robustness of the presented approach.

Keywords

Apple drying; Modeling; Numerical solution; Pseudospectral method

References

1. Tavakoli pour, H. 2002. Drying foods, Principles and Methods. Aegee publisher, Tehran.

2. ASHRAE. 2009. ASHRAE Handbook-Fundamentals. American Society of Heating, Refrigeration and Air-Conditioning Engineers Inc., Atlanta.

3. Aregawi, W., T. Defraeye, S. Saneinejad, P. Vontobel, E. Lehmann, J. Carmeliet, P. Verboven, D. Derome, and B. M. Nicolai. 2013. Understanding forced convective drying of apple tissue: Combining neutron radiography and numerical modeling. Innovative Food Science and Emerging Technologies 24: 97-105.

4. Ateeque, M., R. K. Mishra, V. P. Chandra Mohan, and P. Talukdar. 2014. Numerical modeling of convective drying of food with spatially dependent transfer coefficient in a turbulent flow field. International Journal of Thermal Sciences 78: 145-157.

5. Bhrawy, A. H., L. M. Assas, E. Tohidi, and M. A. Alghamdi. 2013. A Legendre-Gauss collocation method for neutral functional-differential equations with proportional delays, Advances in Difference Equations 63: 1-16.

6. Aversa, M., S. Curcio, V. Calabro, and G. Iorio. 2007. An analysis of the transport phenomena occurring during food drying process. Journal of Food Engineering 78 (3): 922-932.

7. Bruce, D. M., and S. A. Giner. 1993. Mathematical modeling of grain drying in counter-flow beds: investigation of crossover of air and grain temperatures, Journal of Agriculture Engineering. 55: 143-161.

8. Canuto, C., M. Y. Hussaini, A. Quarteroni, and T. A. Zang. 1988. Spectral methods in fluid dynamics. Springer, New York.

9. Costa, B. 2004. Spectral Methods for Partial Differential Equations. A Mathematical Journal 6 (4): 1-32.

10. Crank, J. 1975.The Arithmetics of Diffusion, second ed. Oxford University Press, Oxford, UK.

11. Doymaz, I. 2007. Air drying characteristics of tomatoes. Journal of Agriculture Engineering 78: 1291-1297.

12. Graham, N., M. Quandt, and H. Weigel. 2009. Spectral Methods in Quantum Field Theory, Springer-Verlag, Berlin.

13. Hussain, M. M., and I. Dincer. 2003. Numerical simulation of two-dimensional heat and moisture transfer during drying of a rectangular object. Numerical Heat Transfer, Part A: Appl. 43 (8): 867-878.

14. Janjai, S., N. Lamlert, P. Intawee, B. Mahayothee, M. Haewsungcharern, B. K. Bala, and J. Muller. 2008. Finite element simulation of drying of mango. Biosystems Engineering 99 (4): 523-531.

15. Kaya, K., O. Aydın, and C. Demirtas. 2007. Drying Kinetics of Red Delicious Apple. Biosystems Engineering 96 (4): 517-524.

16. Kuo, D. C., J. C. Morales, and K. S. Ball. 1999. Combined natural convection and volumetric radiation in a horizontal annulus: spectral and finite volume predictions. ASME Transaction. Journal of Heat Transfer 121 (3): 610-615.

17. Mujumdar, A. S. 2006. Book Review: Handbook of Industrial Drying, third ed. CRC Press.

18. Lemus-Mondaca, R. A., C. E. Zambra, A. Vega-Galvez, and N. O. Moraga. 2013. Coupled 3D heat and mass transfer model for numerical analysis of drying process in papaya slices. Journal of Food Engineering 116 (1): 109-117.

19. Li, B. W., Y. S. Sun, and D. W. Zhang. 2009. Chebyshev collocation spectral methods for coupled radiation and conduction in a concentric spherical participating medium. ASME Transaction, Journal of Heat Transfer 131 (6).

20. Paitil, N. D. 1988. Evaluation of diffusion equation for simulating moisture movement within an individual grain kernel. Journal of Drying Technology 6 (1): 21-42.

21. Sabarez, H. T. 2014. Mathematical Modeling of the Coupled Transport Phenomena and Color Development: Finish Drying of Trellis-Dried Sultanas. Journal of Drying Technology 32: 578-589.

22. Sadrnia, H., H. Monfared, and M. Khojastehpour. 2015. Comparison of two methods of purple top turnip drying based on energy consumption and quality parameters. Journal of Agricultural Machinery 5 (1):143-153. (In Farsi).

23. Seyedabadi, E., M. Khojastehpour, and M. H. Abbaspour-Fard. 2017. Convective Drying Simulation of Banana Slabs Considering Non-Isotropic Shrinkage Using FEM with Arbitrary Lagrangian- Eulerian Method. International Journal of Food Properties. DOI: 10.1080/10942912.2017.1288134.

24. Shan, X. W., D. Montgomery, and H. D. Chen.1991. Nonlinear magneto hydrodynamics by Galerkin-method computation, Physical Review A. 44 (10): 6800-6818.

25. Srikiatden, J., and J. S. Roberts. 2008. Predicting moisture profiles in potato and carrot during convective hot air drying using isothermally measured effective diffusivity. Journal of Food Engineering. 84: 516-525.

26. Sun, Y., M. Jing, and B. W. Li. 2012. Chebyshev collocation spectral method for three dimensional transient coupled radiative conductive heat transfer. ASME Transaction. Journal of Heat Transfer 134.

27. Tohidi, E., and O. R. N. Samadi. 2013. Optimal control of nonlinear Volterra integral equations via Legendre polynomials, IMA Journal of Mathematical Control and Information 30: 67-83.

28. Tohidi, E., and A. Kilicman. 2014. An Efficient spectral approximation for solving several types of parabolic PDEs with nonlocal boundary conditions. Mathematical Problems in Engineering.

29. Tohidi, E. 2015. Application of chebyshev collocation method for solving two classes of non-classical parabolic PDEs. Ain Shams Engineering Journal 6: 373-379.

30. Tohidi, E., and Saberi Nik, H. 2015. A Bessel collocation method for solving fractional optimal control problems, Applied Mathematical Modeling 39: 455-465.

31. Trefethen LN. Spectral methods in MATLAB. Philadelphia: SIAM; 2000.

32. Tzempelikos, D. A., D. Mitrakos, A. P. Vouros, A. V. Bardakas, A. E. Filios, and D. P. Margaris. 2015. Numerical modeling of heat and mass transfer during convective drying of cylindrical quince slices. Journal of Food Engineering 156: 10-21.

33. Villa-Corrales, L., J. J. Flores-Prieto, J. P. Xaman-Villasenor, and E. Garcia-Hernandez. 2010. Numerical and experimental analysis of heat and moisture transfer during drying of Ataulfo mango. Journal of Food Engineering 98 (2): 198-206.

34. Zare, D., S. Minaei, M. Mohamad Zadeh, and M. H. Khoshtaghaza. 2006. Computer simulation of rough rice drying in a batch dryer. Energy Conversion and Management 47: 3241-3254.

35. Zarein, M., S. H. Samadi, and B. Ghobadian. 2013. Kinetic Drying and Mathematical Modeling of Apple Slices on Dehydration Process. Journal of Food Processing and Technology.

36. Zhou, R. R., and B.W. Li. 2017, Chebyshev collocation spectral method for one-dimensional radiative heat transfer in linearly anisotropic-scattering cylindrical medium. Journal of Quantitative Spectroscopy and Radiative Transfer 189: 206-220.