Finite element analysis and modeling of water absorption by date pits during a soaking process

Finite element analysis and modeling of water absorption by date pits during a soaking process first-author corresp ‡Corresponding Author ‡Corresponding Author †E-mail:ahg212@mail.usask.ca †E-mail:ahg212@mail.usask.ca ahg212@mail.usask.ca Horticultural Research Institute, Shahid Bahonar University of Kerman, P.O. Box 76169-133, Kerman, Iran 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 m 2 /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 R 2 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. Date pits Soaking Water absorption Modeling Introduction 1 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 6×10 6 t in 2007 (FAO, 2009 2009 ). Date fruits are an important food item, with plenty vitamins and minerals. They are eaten fresh or are dried and stored for later consumption. Individual date fruits contain a pit which, depending on the variety, accounts for about 10% to 18% weight of the fruits (Salem et al., 1989 1989 ). 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., 2002 2002 ). The pits are generally used as complementary feed materials (Vandepopuliere et al., 1995 1995 ) or as a conventional soil fertilizer. Portions of the pits are also used for extracting oil for cosmetic and pharmaceutical purposes (Devshony et al., 1992 1992 ). 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., 2004 2004 ). 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., 1983 1983 ; Karapantsios et al., 2002 2002 ; Laria et al., 2005 2005 ). On the other hand, the agricultural seeds are not uniform and consist of three major parts, namely, seed coat, endosperm, and embryo. Since in most seeds the endosperm occupies the major part of the seed, a seed is usually considered as a uniform entity in many moisture transfer studies (Gaston et al., 2004 2004 ; Bakalis et al., 2009 2009 ). 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., 1993 1993 ; Turhan et al., 2002 2002 ; Badau and Jideani, 2005 2005 ; Noorbakhsh et al., 2006 2006 ); (2) theoretical methods which are generally based on principles of moisture diffusion, sometimes coupled with heat or mass transfer equations (Vizcarra Mendoza et al., 2003 2003 ; Gaston et al., 2004 2004 ; Gely and Giner, 2007 2007 ); and (3) semi-theoretical approaches which are generally a modified form of theoretical models (Misra and Brooker, 1980 1980 ). 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, 2005 2005 ). Using these methods, the temporal moisture distribution within a seed is estimated by a suitable discrete model composed of piecewise continuous functions defined over a finite number of internal elements. Examples are found in Igathinathane and Chattopadhyay ( 1999 1999 ), Gaston et al. ( 2004 2004 ), and Bakalis et al. ( 2009 2009 ) who successfully applied the finite element modeling method for water diffusivity in wheat kernels, rice hydration, and drying of parboiled paddy components, respectively. 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 ( 1988 1988 ) model. In this investigation, the date pits were assumed to have a uniform material with uniform initial moisture content. The goodness of fit of each model was evaluated by calculating the coefficient of determination ( R 2 ) and root mean square error (RMSE) between the experimental moisture data and the values obtained in the models. Materials and methods 1 Material preparation and soaking process 2 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, 1997 1997 ). The dimensions of the kernels including length, width, and thickness were measured by a digital caliper. From these measured values the sphericity and geometric mean diameter (GMD) of the seed were determined (Jain and Bal, 1997 1997 ). 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. Analytical approach 2 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: $$\frac{{\partial M}}{{\partial t}} = D\left( {\frac{{{\partial ^2}M}}{{\partial {x^2}}} + \frac{{{\partial ^2}M}}{{\partial {y^2}}} + \frac{{{\partial ^2}M}}{{\partial {z^2}}}} \right)$$ , where M is the instantaneous moisture content at a specified time t and D is the diffusion coefficient. A solution for the above equation for an object with a sphere shape of radius r was presented by Crank ( 1975 1975 ), as $${\text{MR}} = \frac{{M - {M_{\text{i}}}}}{{{M_{\text{e}}} - {M_{\text{i}}}}} = 1 - (\frac{6}{{{\pi ^2}}})\sum\limits_{i = 1}^\infty {(\frac{1}{{{i^2}}})\exp ( - D{i^2}{\pi ^2}\frac{t}{{{r^2}}})}$$ , where MR is the moisture ratio, and M i and M e are, respectively, the initial and the equilibrium moisture contents of the object during a moisture transformation process. In most cases, only a finite number of Eq. ( 2 2 ) is used for estimating MR values. In fact, most researchers use only the first two terms of this equation. 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. 1 1 . (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. ( 1 1 ). Gaston et al. ( 2004 2004 ) presented a procedure to estimate the coefficient of diffusion for an ellipsoid ( D e ) using the following equation: $${D_{\text{e}}} = f_{\text{e}}^2 \times D$$ , where f e is the sphericity factor of ellipsoid (Gaston et al., 2002 2002 ). Then, the coefficient of diffusion D in Eq. ( 2 2 ) should be replaced by the calculated D e . Finite element approach 2 In a finite element approach, an estimated value for D is supplied to the Eq. ( 1 1 ) and the MR values are numerically determined for specified locations at given time intervals. In applying this approach the following assumptions were made for the individual date pits: 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. 2 2 ). The grids for date pit consisted of 243 triangular elements with a total of 546 nodes. The commercial finite element analysis software ANSYS (Rel. 10.0, 2005) was used to evaluate instantaneous moisture distribution at the nodes. The coefficient of diffusion estimated by fitting Eq. ( 2 2 ) to the experimental data was provided to the software and the moisture content of each node was calculated at one-second time steps. The overall moisture content for a date pit was also calculated at every 30 min by averaging the moisture contents of the nodes. Peleg model 2 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: $$M = {M_{\text{i}}} + \frac{t}{{{k_1} + {k_2}t}}$$ , where M i is initial moisture content (decimal, dry basis), k 1 and k 2 are two constants known as the Peleg rate constant (h∙% −1 ) and the Peleg capacity constant (% −1 ), respectively. These two constants must be determined for a particular product under certain moisture absorption processes. We used the Peleg model to estimate the overall moisture content of the date pits during the soaking process. The data for overall moisture contents at specified time intervals were used as input to curve fitting tool box of MATLAB commercial software and the two constants of the Peleg model were determined. Comparison and evaluation of the models 2 The goodness of fits for the finite element model, analytical solution, and Peleg model was evaluated by calculating the coefficient of determination ( R 2 ) and root mean square error (RMSE) by the following formulae: $${R^2} = \frac{{{{\left( {\sum {{\text{M}}{{\text{R}}_{\text{e}}}{\text{M}}{{\text{R}}_{\text{p}}}} } \right)}^{\text{2}}}}}{{\sum {{\text{MR}}_{\text{e}}^{\text{2}}\sum {{\text{MR}}_{\text{p}}^{\text{2}}} } }}$$ , $${\text{RMSE}} = \sqrt {\frac{{\sum {{{({\text{M}}{{\text{R}}_{\text{p}}} - {\text{M}}{{\text{R}}_{\text{e}}})}^2}} }}{n}}$$ , where MR e and MR p are the experimental and the predicted moisture ratios, respectively. A model having a high value of R 2 (close to 1) and a low value of RMSE (close to 0) is considered to be a well-fitted model. Results and discussion 1 The measured physical characteristics of the date pits before and after soaking are presented in Table 1 1 . From the dimensions, the date pit geometry is very close to an ellipsoid and it retains its general shape after soaking. However, due to a relative increase in the thickness and width of the pits as a result of moisture gain, the sphericity of the pits increases and changes from 0.41 to 0.49. The increase in the dimensions is due to the migration of water between and within the cells of the pits which causes an overall swelling of the pits. A detailed discussion on changes of dimensions during soaking is given by Ahromrit et al. ( 2006 2006 ). Data are expressed as mean±SD. GMD: geometric mean diameter 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 The moisture percentages, both on wet and dry bases, absorbed by the pits are presented in Fig. 3 3 . During the first 2 h of soaking, the moisture content increased rapidly from 12% to near 30% on the dry basis. Then, the rate of water absorption gradually decreased until the saturation moisture content, 71% on the dry basis, was reached after 240 h. Thus, under normal environmental conditions, it takes about 10 d for the date pits to reach their saturation moisture level. The sphericity factor ( f e ) for an average date pit was calculated to be equal to 0.78. Using Eqs. ( 2 2 ) and ( 3 3 ), the coefficient of diffusion D e was calculated to be equal to 9.89×10 −12 m 2 /s. The same procedure was used by Gaston et al. ( 2002 2002 ) and they calculated the coefficient of diffusion for wheat kernels within a range of 1.35×10 −11 to 6.88×10 −11 m 2 /s. Bakalis et al. ( 2009 2009 ) estimated this coefficient for rice equal to 7×10 −10 m 2 /s. The lower D e calculated for date pits indicates that the date pits absorb water at a slower rate than wheat and rice kernels. The obtained coefficient of diffusion was supplied to Eq. ( 1 1 ) and this equation was solved by finite element approach to estimate the moisture distribution within the pits at various time intervals. The MR values calculated from the moisture absorption experimental data and their different fitted models are presented in Fig. 4 4 . The predicted MR values calculated by finite element method were close to those of both measured and analytical solutions. As illustrated in Fig. 4 4 , the finite element model provided a good prediction of the MR values at the end of the soaking process, while during the initial and intermediate stages it slightly overestimated the MR values. The analytical model generally underestimated the MR values. The visualization of finite element predictions for the moisture content is presented in Fig. 5 5 at 1, 17, 41, and 240 h after the commencement of the soaking. Fig. 5 5 shows a gradual migration of the water from the surface to the center of a pit. Theoretically, the finite element model, as shown in Fig. 5 5 , indicates that even at the end of the soaking process the moisture content throughout the pit is not quite uniform. In a practical sense, as shown in Fig. 6 6 , the average moisture content reached its saturation level, 71.0% on dry basis, after 240 h. Fig. 6 6 indicates that the moisture gradient within a pit is large during the initial stages of soaking and the gradient decreases with the progress of soaking, and after 240 h the moisture content is practically uniform and reaches a saturation level at all the nodes. The shortcoming of the finite element model, with respect to the analytical model, is that the former relies on a coefficient of diffusion developed by other sources, i.e., analytical modeling or experimental values. (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 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+1.685 t ), R 2 =0.94. The Peleg model, as shown in Fig. 4 4 , underestimated the MR values at the initial and the final stages of the soaking process. Comparing the R 2 value of the three developed models, the Peleg model is inferior to analytical and finite element models for predicting the water absorption by date pits. The R 2 and RMSE values calculated for the finite element model, analytical solution, and Peleg model are shown in Table 2 2 . The R 2 values were 0.98 for finite element and analytical solution. The RMSE value for finite element model is very close to, but slightly lower than, that for the analytical solution. Thus, both analytical and finite element approaches are well suited for modeling this water absorption process. However, the finite element model slightly overestimated the MR values of the pits at the initial and intermediate stages while the analytical approach generally underestimated these values (Fig. 4 4 ). The superiority of the finite element model is due to its power to predict the moisture content changes at specified nodes within a date pit rather than estimating an overall moisture transfer for a pit. Model R 2 RMSE Finite element 0.98 0.0724 Analytical solution 0.98 0.0788 Peleg 0.94 0.1097 Conclusions 1 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. References A Ahromrit , DA Ledward , K Niranjan . High pressure induced water uptake characteristics of Thai glutinous rice . Journal of Food Engineering , 2006 . 72 ( 3 ): 225 - 233 . DOI: 10.1016/j.jfoodeng.2004.11.013 http://doi.org/10.1016/j.jfoodeng.2004.11.013 . High pressure induced water uptake characteristics of Thai glutinous rice ASAE ASAE Standard S352.2: Moisture Measurement–Unground Grain and Seeds , 1997 . : St. Joseph, MI American Society of Agriculture Engineers , . ASAE Standard S352.2: Moisture Measurement–Unground Grain and Seeds MH Badau , IA Jideani . Water absorption characteristics of various pearl millet cultivars and sorghum grown in Northern Nigeria . Journal of Food Process Engineering , 2005 . 28 ( 3 ): 282 - 298 . DOI: 10.1111/j.1745-4530.2005.00392.x http://doi.org/10.1111/j.1745-4530.2005.00392.x . Water absorption characteristics of various pearl millet cultivars and sorghum grown in Northern Nigeria S Bakalis , A Kyritsi , VT Karathanos , S Yanniotis . Modeling of rice hydration using finite elements . Journal of Food Engineering , 2009 . 94 ( 3-4 ): 321 - 325 . DOI: 10.1016/j.jfoodeng.2009.03.023 http://doi.org/10.1016/j.jfoodeng.2009.03.023 . Modeling of rice hydration using finite elements J Crank . The Mathematics of Diffusion , 1975 . Oxford University Press, New York , . The Mathematics of Diffusion S Devshony , E Eteshola , A Shani . Characteristics and some potential application of date palm ( Phoenix dactlifera L.) seeds and seed oil . Journal of the American Oil Chemists Society , 1992 . 69 ( 6 ): 595 - 597 . DOI: 10.1007/BF02636115 http://doi.org/10.1007/BF02636115 . Characteristics and some potential application of date palm (Phoenix dactlifera L.) seeds and seed oil FAO Citing Electronic Sources of Information . 2009 . Food and Agricultural Organization of the United Nations , Available from http://www.fao.org http://www.fao.org [accessed on Aug. 20, 2009] . Citing Electronic Sources of Information AL Gaston , RM Abalone , SA Giner . Wheat drying kinetics. Diffusivities for sphere and ellipsoid by finite elements . Journal of Food Engineering , 2002 . 52 ( 4 ): 313 - 322 . DOI: 10.1016/S0260-8774(01)00121-2 http://doi.org/10.1016/S0260-8774(01)00121-2 . Wheat drying kinetics. Diffusivities for sphere and ellipsoid by finite elements AL Gaston , RM Abalone , SA Giner , DM Bruce . Effect of modeling assumptions on the effective water diffusivity in wheat . Biosystems Engineering , 2004 . 88 ( 2 ): 175 - 185 . DOI: 10.1016/j.biosystemseng.2004.02.009 http://doi.org/10.1016/j.biosystemseng.2004.02.009 . Effect of modeling assumptions on the effective water diffusivity in wheat MC Gely , SA Giner . Diffusion coefficient relationship during drying of soya bean cultivars . Biosystems Engineering , 2007 . 96 ( 2 ): 213 - 222 . DOI: 10.1016/j.biosystemseng.2006.10.015 http://doi.org/10.1016/j.biosystemseng.2006.10.015 . Diffusion coefficient relationship during drying of soya bean cultivars JS Hamada , IB Hashim , FA Sharif . Preliminary analysis and potential uses of date pits in foods . Food Chemistry , 2002 . 76 ( 2 ): 135 - 137 . DOI: 10.1016/S0308-8146(01)00253-9 http://doi.org/10.1016/S0308-8146(01)00253-9 . Preliminary analysis and potential uses of date pits in foods KH Hsu , J Kim , LA Wilson . Factors affecting water uptake of soybeans during soaking . Cereal Chemistry , 1983 . 60 ( 3 ): 208 - 211 . . Factors affecting water uptake of soybeans during soaking TV Hung , LH Liu , RG Black , MA Trewhella . Water absorption in chickpea ( C. arietinum ) and field pea ( P. sativum ) cultivars using the Peleg model . Journal of Food Science , 1993 . 58 ( 4 ): 848 - 852 . DOI: 10.1111/j.1365-2621.1993.tb09374.x http://doi.org/10.1111/j.1365-2621.1993.tb09374.x . Water absorption in chickpea (C. arietinum) and field pea (P. sativum) cultivars using the Peleg model C Igathinathane , PK Chattopadhyay . Moisture diffusion modeling of drying in parboiled paddy components. Part I: starchy endosperm . Journal of Food Engineering , 1999 . 41 ( 2 ): 79 - 88 . DOI: 10.1016/S0260-8774(99)00074-6 http://doi.org/10.1016/S0260-8774(99)00074-6 . Moisture diffusion modeling of drying in parboiled paddy components. Part I: starchy endosperm M Inc . Decomposition method for solving parabolic equations in finite domains . Journal of Zhejiang University-SCIENCE , 2005 . 6A ( 10 ): 1058 - 1064 . DOI: 10.1631/jzus.2005.A1058 http://doi.org/10.1631/jzus.2005.A1058 . Decomposition method for solving parabolic equations in finite domains RK Jain , S Bal . Properties of pearl millet . Journal of Agricultural Engineering Research , 1997 . 66 ( 2 ): 85 - 91 . DOI: 10.1006/jaer.1996.0119 http://doi.org/10.1006/jaer.1996.0119 . Properties of pearl millet Y Ji , K Seetharaman , PJ White . Optimizing a small scale cornstarch extraction method for use in the laboratory . Cereal Chemistry , 2004 . 81 ( 1 ): 55 - 58 . DOI: 10.1094/CCHEM.2004.81.1.55 http://doi.org/10.1094/CCHEM.2004.81.1.55 . Optimizing a small scale cornstarch extraction method for use in the laboratory TD Karapantsios , EP Sakonidou , SN Raphaelides . Water dispersion kinetics during starch gelatinization . Carbohydrate Polymers , 2002 . 49 ( 4 ): 479 - 490 . DOI: 10.1016/S0144-8617(02)00005-X http://doi.org/10.1016/S0144-8617(02)00005-X . Water dispersion kinetics during starch gelatinization J Laria , E Meza , M Mondragon , R Silva , JL Pena . Comparison of overall water uptake by corn kernel with and without calcium hydroxide at room temperature . Journal of Food Engineering , 2005 . 67 ( 4 ): 451 - 456 . DOI: 10.1016/j.jfoodeng.2004.05.013 http://doi.org/10.1016/j.jfoodeng.2004.05.013 . Comparison of overall water uptake by corn kernel with and without calcium hydroxide at room temperature MK Misra , DB Brooker . Thin-layer drying and rewetting equations for shelled yellow corn . Transactions of the ASAE , 1980 . 23 1254 - 1260 . . Thin-layer drying and rewetting equations for shelled yellow corn S Noorbakhsh , T Lope , A Ghazanfari . Analysis and modeling of water absorption by yellow dent corn kernels before and during gelatinization process . Asian Journal of Plant Sciences , 2006 . 5 ( 5 ): 805 - 810 . DOI: 10.3923/ajps.2006.805.810 http://doi.org/10.3923/ajps.2006.805.810 . Analysis and modeling of water absorption by yellow dent corn kernels before and during gelatinization process M Peleg . An empirical model for the description of moisture sorption curves . Journal of Food Science , 1988 . 53 ( 4 ): 1216 - 1217 . DOI: 10.1111/j.1365-2621.1988.tb13565.x http://doi.org/10.1111/j.1365-2621.1988.tb13565.x . An empirical model for the description of moisture sorption curves AE Salem , SS Younis , AA Shabana , MM Misbah . Thermal study of date kernels via TG and DTA . Journal of Thermal Analysis , 1989 . 35 ( 7 ): 2243 - 2254 . DOI: 10.1007/BF01911888 http://doi.org/10.1007/BF01911888 . Thermal study of date kernels via TG and DTA M Turhan , S Sauer , S Gunasekaran . Application of Peleg model to study water absorption in chickpea during soaking . Journal of Food Engineering , 2002 . 53 ( 2 ): 153 - 159 . DOI: 10.1016/S0260-8774(01)00152-2 http://doi.org/10.1016/S0260-8774(01)00152-2 . Application of Peleg model to study water absorption in chickpea during soaking JM Vandepopuliere , Y Al-Yousef , JM Lyons . Dates and date pits as ingredients in broiler starting and coturnix quail breeder diets . Poultry Science , 1995 . 74 ( 7 ): 1134 - 1142 . 7479490 . Dates and date pits as ingredients in broiler starting and coturnix quail breeder diets MG Vizcarra Mendoza , C Martínez Vera , FV Caballero Domínguez . Thermal and moisture diffusion properties of amaranth seeds . Biosystems Engineering , 2003 . 86 ( 4 ): 441 - 446 . DOI: 10.1016/j.biosystemseng.2003.07.003 http://doi.org/10.1016/j.biosystemseng.2003.07.003 . Thermal and moisture diffusion properties of amaranth seeds 10.1631/jzus.B0910641 2009-10-24 2010-05-31 2010-03-28 2022-01-05 Copyright © Zhejiang University and Springer-Verlag Berlin Heidelberg 2010 2010 Journal of Zhejiang University-SCIENCE B (Biomedicine and Biotechnology) 07 11