Fig. 1 Schematic geometry of a date pit
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Horticultural Research Institute, Shahid Bahonar University of Kerman, P.O. Box 76169-133, Kerman, Iran
1.Horticultural Research Institute, Shahid Bahonar University of Kerman, P.O. Box 76169-133, Kerman, Iran
Published: 2010-07 ,
Received: 24 October 2009 ,
Revised: 31 May 2010 ,
Accepted: 28 March 2010
Cite this article
Motahareh Waezi-Zadeh, Ahmad Ghazanfari, Shahin Noorbakhsh. Finite element analysis and modeling of water absorption by date pits during a soaking process. [J]. Journal of Zhejiang University-SCIENCE B (Biomedicine & Biotechnology) 11(7):482-488(2010)
Motahareh Waezi-Zadeh, Ahmad Ghazanfari, Shahin Noorbakhsh. Finite element analysis and modeling of water absorption by date pits during a soaking process. [J]. Journal of Zhejiang University-SCIENCE B (Biomedicine & Biotechnology) 11(7):482-488(2010) DOI: 10.1631/jzus.B0910641.
Horticultural Research Institute, Shahid Bahonar University of Kerman, P.O. Box 76169-133, Kerman, Iran
Date pits for feed preparation or oil extraction are soaked in water to soften before milling or extrusion. Knowledge of water absorption by the date pits helps in better managing the soaking duration. In this research, the process of water absorption by date pits was modeled and analyzed using Fick’s second law of diffusion, finite element approach, and Peleg model. The moisture content of the pits reached to its saturation level of 41.5% (wet basis) after 10 d. The estimated coefficient of diffusion was 9.89×10−12 m2/s. The finite element model with a proposed ellipsoid geometry for a single date pit and the analytical model fitted better to the experimental data with R2 of 0.98. The former model slightly overestimated the moisture content of the pits during the initial stages of the soaking and the latter model generally underestimated this variable through the entire stages of soaking process.
The date tree (Phoenix dactylifera L.) is an important staple food and a strategic plant in many arid regions of the world. The worldwide production of date fruits exceeded 6×106 t in 2007 (FAO,
A major portion of annually produced date fruits is used by confectionaries and beverage production industries. Date pits are the main by-products in these processing plants. On average, the pits from different varieties contain about 9.5% moisture, 6.0% protein, 10.5% fat, and 70% carbohydrates (Hamada et al.,
For either feeding or extracting oil, the date pits are initially soaked in water and then they are ground. Soaking is necessary for softening the hard pit to facilitate grinding. Soaking of date pits like soaking of many other agricultural seeds is a time-consuming process. For instance, corn kernels are soaked for up to 72 h before milling (Ji et al.,
The amount of water absorbed by seeds during soaking is affected by different factors such as the initial moisture content, variety of the seeds, soaking duration, and temperature and acidity level of the water (Hsu et al.,
Modeling the process of water absorption by agricultural seeds helps in understanding the dynamic and kinetic of this process and this knowledge is valuable for proper management of their soaking processes. Mathematical modeling of the agricultural seeds has been approached by three general methods: (1) empirical approach wherein a suggested model is fitted to experimental data (Hung et al.,
The conventional models for describing the moisture transfer phenomenon by a seed, generally predict the total amount of moisture absorbed or desorbed and they cannot estimate the temporal moisture distribution within a seed.
The numerical modeling methods such as finite element and finite difference offer an alternative tool for conventional approaches to analyze various processes (Inc,
Date pits are hard, and the process of water uptake by them takes an extended period of time. In this research, for efficient control of the soaking process, the water absorption by date pits was studied and modeled by three modeling approaches: (1) analytical approach using Fick’s second law of diffusion; (2) finite element approach based on Fick’s law; and (3) Peleg (
Date pits dedicated for the experiments were derived from Mozafati cultivar fruits harvested in summer of 2008 from a date palm grove in Bam City, Kerman, Iran. The pits were cleaned, washed, and spread on a tray at laboratory conditions for a period of two weeks to allow the pits to reach uniform moisture content. Prior to soaking, the moisture content of the pits was determined by oven method at 103 °C for 36 h (ASAE,
For conducting the experiments, 250 g of the pits were placed in a mesh fabric bag and the bag was immersed in 25 °C distilled water. At predetermined time intervals, the pits were removed from water and superficially dried by a paper towel and immediately weighed using an electronic balance with 0.01 g accuracy. This procedure continued until there were no appreciable changes in the mass of the soaked pits. The experiments were performed in three replicates and their averages were used for modeling and analyses.
During a diffusion process at constant temperature, it is assumed that the process follows Fick’s second law of diffusion. For an axisymmetric diffusion, Fick’s three-dimensional (3D) equation is given by: ∂M∂t=D(∂2M∂x2+∂2M∂y2+∂2M∂z2), (1) MR=M−MiMe−Mi=1−(6π2)∞∑i=1(1i2)exp(−Di2π2tr2), (2)
In this research, the experimental MR values at specific time intervals were calculated and used as input to the curve fitting tool box of MATLAB (R2006a) software and the diffusion coefficient of the date pits, D, was estimated. The typical shapes for side and cross section views of a date pit are presented in Fig.
Fig. 1 Schematic geometry of a date pit
(a) Side view; (b) Cross sectional view along the width
Since the general shape of a typical pit is closer to an ellipsoid than to a sphere, the coefficient of diffusion should be adjusted in Eq. ( De=f2e×D, (3)
In a finite element approach, an estimated value for D is supplied to the Eq. (
The geometry of a date pit was considered to be an ellipsoid. For modeling a date pit, one quarter of an ellipse was used in a time-dependent axisymmetric two-dimensional (2D) analysis (Fig.
Fig. 2 Finite element grid for a single date kernel
One of the most frequently used empirical models for estimating the overall moisture content M as a function of time t during a moisture absorption process is the Peleg model defined as: M=Mi+tk1+k2t, (4)
The goodness of fits for the finite element model, analytical solution, and Peleg model was evaluated by calculating the coefficient of determination (R2) and root mean square error (RMSE) by the following formulae: R2=(∑MReMRp)2∑MR2e∑MR2p, (5) RMSE=√∑(MRp−MRe)2n, (6)
The measured physical characteristics of the date pits before and after soaking are presented in Table
Length (mm) | Width (mm) | Thickness (mm) | Sphericity (mm) | GMD (mm) | |
---|---|---|---|---|---|
Before soaking | 23.77±3.61 | 6.6±1.91 | 5.38±1.87 | 0.41±0.03 | 9.44±1.77 |
After soaking | 24.61±3.82 | 8.1±1.23 | 6.51±1.51 | 0.49±0.11 | 9.88±2.02 |
Data are expressed as mean±SD. GMD: geometric mean diameter
The moisture percentages, both on wet and dry bases, absorbed by the pits are presented in Fig.
Fig. 3 Changes in the moisture content of the date pits during the soaking process
The sphericity factor (fe) for an average date pit was calculated to be equal to 0.78. Using Eqs. (
The obtained coefficient of diffusion was supplied to Eq. (
The MR values calculated from the moisture absorption experimental data and their different fitted models are presented in Fig.
Fig. 4 Comparison of moisture ratios determined by the experimental data and three modeling approaches
The visualization of finite element predictions for the moisture content is presented in Fig.
Fig. 5 Typical distribution of moisture in a date pit during soaking as determined by finite element analysis
(a) 1 h; (b) 17 h; (c) 41 h; (d) 240 h. The bar below each shape represents the level of moisture content (%) within a pit region at a given time
Fig. 6 Moisture contents at various points along the cross section from the surface to the center
Using the experimental data for moisture contents of date pits at different time intervals and the curve fitting tool box of MATLAB commercial software, the following model was obtained based on the Peleg formula: M=0.12+t/(37.68+1.685t), R2=0.94. (7)
The Peleg model, as shown in Fig.
The R2 and RMSE values calculated for the finite element model, analytical solution, and Peleg model are shown in Table
Model | R2 | RMSE |
---|---|---|
Finite element | 0.98 | 0.0724 |
Analytical solution | 0.98 | 0.0788 |
Peleg | 0.94 | 0.1097 |
The process of water absorption by date pits was investigated and the process was modeled by three different approaches. The rate of water diffusion was high during the first 2 h of soaking and then it gradually decreased. The date pits reached their equilibrium moisture content of 41.5% on the wet basis after 240 h of soaking in distilled water. The analytical approach based on Fick’s second law of diffusion and the finite element approach based on Fick’s law fit well to the experimental data. However, the RMSE value calculated for the finite element model was slightly lower than that for the analytical model. The finite element model is able to predict the moisture distribution at any given point within the pit as a function of time while the analytical model gives an overall moisture content of the pit at a specified time. The Peleg model is inferior to the analytical and finite element models, and predicts overall moisture content for the pit as a function of time.
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