Mehrdad Negahban

Spatial and temporal variability of soil nitrate and corn yield: Multifractal analysis

1997). Ferguson et al. (2002) found reduction in soil nitrate concentration due to variable rate fertilizer N High levels of residual soil NO3–N can contaminate ground water application in only 3 out of 12 site-years as compared by leaching through the soil. Our objective was to reduce the level and spatial variability of residual soil NO3–N while maintaining optiwith uniform N application. Machado et al. (2000) indimum corn (Zea mays L.) production by variable rate N fertilizer cated that management zones for variable rate fertilizer application. The experiment was located on a 60-ha sprinkler-irrigated and water applications should be based on information corn field in central Nebraska and included four N management pracabout soil elevation, texture, and soil nitrate. Spatial tices: uniform rate, variable rate (VRAT), variable rate at 75% of dependence of soil NO3–N was found to be time depenrecommended amount (VRAT @ 75%), and variable rate plus 10% dent in irrigated salad crops (Bruckler et al., 1997). (VRAT 10%). VRAT @ 75% decreased the amount of residual Fractal analysis can provide insight into the spatial or NO3–N in the soil while maintaining similar grain yield to the other temporal variability of crop or soil parameters. Fractal treatments, indicating over-application of N with treatments receiving analysis has been shown to be useful in a variety of the recommended rate. Increasing the recommended rate by 10% (VRAT 10%) did not increase corn yield or residual soil NO3–N. scientific disciplines. The use of fractals for numerical Based on multifractal spectrum, no consistent pattern of spatial varianalysis of soil and plant parameters is still a relatively ability of soil NO3–N was observed for each treatment across years. new technique. It has been used for characterizing soil Spatial variability in corn grain yield was much lower than that for structure (Eghball et al., 1993b; Perfect and Blevins, soil NO3–N, indicating noneffectiveness of using soil NO3–N spatial 1997), soil chemical and physical parameters (Burrough, distribution for variable rate N application unless some areas in the 1983), root morphology (Eghball et al., 1993a), temporal field are severely N deficient. Variable rate N application did not yield variations (Eghball and Power, 1995; Eghball and reduce variability of residual soil NO3–N or corn grain yield as comVarvel, 1997), and spatial variability of soil and crop pared with uniform N. Multifractal analysis quantitatively characteryield (Eghball et al., 1997, 1999). Fractal analysis was ized the extent and pattern of spatial and temporal variability in corn grain yield and residual soil nitrate. found to be useful in characterizing soil and plant parameters that was not possible or very difficult to do before. Fractal dimension (D) of a curve can have a value between 1 and 2, giving a quantitative indication R developments in agricultural technology have of the function’s shape or roughness. made site-specific fertilizer application a reality. Multifractal analysis has been proposed for determiVariable rate (site-specific) N application should pronation of spatial variability of soil parameters (Folovide the plant with the appropriate amount of N while runso et al., 1994; Kravchenko et al., 1999, 2000). Multireducing the quantity and variability of residual soil fractal parameters were found to reflect many of the NO3–N after harvest. One may also expect to find a more major aspects of variability in soil properties, provided homogeneous yield response across the field following a unique quantitative characterization of the data spatial adoption of variable rate N application. By reducing distribution, and multifractal parameters were useful in variability and quantity of residual soil NO3–N, its leachchoosing an appropriate interpolation procedure for maping and subsequent ground water contamination potenping soil properties (Kravchenko et al., 1999). Multitial should be reduced. Eghball et al. (1999) found that the extent of variability in residual soil NO3–N was sigfractal analysis was used to characterize particle-size nificantly reduced following adoption of variable rate distribution of soils with wide range of particle sizes N application in a continuous corn system under gravity (Posadas et al., 2001). A single fractal dimension might irrigation. The residual soil NO3–N to a depth of 0.9 m not be sufficient to characterize soil spatial variability was high (avg. 6.8 mg kg 1, max. 12.0 and min. 2.4) across because of the heterogeneous nature of soil parameters. the field before initiation of variable rate N application. A set of fractal dimensions, called a multifractal specAfter 1-yr variable rate N application, average residual trum, is referred to as multifractal analysis (Frisch and soil NO3–N was 5.0 mg kg 1 with a maximum of 7.9 and Parisi, 1985). Multifractal analysis needs to be evaluated a minimum of 3.7. In another study where residual soil to determine its usefulness in comparing spatial variabilNO3–N was low (avg. 4.0 mg kg 1, max. 7.8 and min. ity of soils treated with different treatments. The objec1.5), variable rate N application did not significantly tive of this study was to characterize and compare spatial reduce residual soil NO3–N variability (Eghball et al., and temporal variability of residual soil NO3–N and corn grain yield in a variable rate N study using multifractal analysis. B. Eghball, J.S. Schepers, and M.R. Schlemmer, USDA-ARS, 121 Keim Hall, Univ. of Nebraska, Lincoln, NE 68583; and M. Negahban, Dep. of Eng. Mechanics, Univ. of Nebraska, Lincoln, NE 68583. Joint Abbreviations: adiff, the distance between minimum and maximum contribution of the USDA-ARS and the Univ. of Nebraska Agric. a values of each multifractal spectrum; CEC, cation exchange capacity; Res. Div., Lincoln, NE, as paper no. 13618. Received 9 Feb. 2002. VRAT, variable rate; VRAT @ 75%, variable rate at 75% of the *Corresponding author (beghball1@unl.edu). recommended amount; VRAT 10%, variable rate of the recommended amount plus 10%. Published in Agron. J. 95:339–346 (2003).

Heat and Mass Transport during Microwave Heating of Mashed Potato in Domestic Oven—Model Development, Validation, and Sensitivity Analysis

UNLABELLED A 3-dimensional finite-element model coupling electromagnetics and heat and mass transfer was developed to understand the interactions between the microwaves and fresh mashed potato in a 500 mL tray. The model was validated by performing heating of mashed potato from 25 °C on a rotating turntable in a microwave oven, rated at 1200 W, for 3 min. The simulated spatial temperature profiles on the top and bottom layer of the mashed potato showed similar hot and cold spots when compared to the thermal images acquired by an infrared camera. Transient temperature profiles at 6 locations collected by fiber-optic sensors showed good agreement with predicted results, with the root mean square error ranging from 1.6 to 11.7 °C. The predicted total moisture loss matched well with the observed result. Several input parameters, such as the evaporation rate constant, the intrinsic permeability of water and gas, and the diffusion coefficient of water and gas, are not readily available for mashed potato, and they cannot be easily measured experimentally. Reported values for raw potato were used as baseline values. A sensitivity analysis of these input parameters on the temperature profiles and the total moisture loss was evaluated by changing the baseline values to their 10% and 1000%. The sensitivity analysis showed that the gas diffusion coefficient, intrinsic water permeability, and the evaporation rate constant greatly influenced the predicted temperature and total moisture loss, while the intrinsic gas permeability and the water diffusion coefficient had little influence. PRACTICAL APPLICATION This model can be used by the food product developers to understand microwave heating of food products spatially and temporally. This tool will allow food product developers to design food package systems that would heat more uniformly in various microwave ovens. The sensitivity analysis of this study will help us determine the most significant parameters that need to be measured accurately for reliable model prediction.

Modeling the mechanical response of a material undergoing continuous isothermal crystallization

Abstract The gradual transition seen in polymer crystallization is modeled. A constitutive equation is developed to follow the mechanical behavior of a crystallizing polymer before, during, and after the completion of crystallization. The post-crystallization response of the material is studied and shown to be “elastic”. The symmetries of the post-crystallization response are defined and calculated for crystallization under several deformation histories.

Simulation of mechanical response during polymer crystallization around rigid inclusions and voids: homogeneous crystallization

The mechanical effects of homogeneous polymer crystallization around single defects are studied, showing how crystallization can develop residual stresses, and change material moduli. Defects in the form of rigid inclusions or voids are considered, either having spherical or cylindrical geometry. Problems with spherical symmetry are considered in the case of a spherical defect, and plane strain problems with axial symmetry are considered in the case of a cylindrical defect. The predicted response is based on a constitutive model developed by Negahban et al. (1993, Int. J. Eng. Sci. 31(1), 93–113), and shows that large residual stresses develop, which may result in debonding or fracture.

Thermodynamic modeling of the thermomechanical effects of polymer crystallization: A general theoretical structure

A general theoretical structure is developed based on continuum thermodynamics to model the thermomechanical effects of polymer crystallization. This phase transition, seen in many polymers, involves the gradual transformation of the polymers microstructure from an unorganized amorphous structure to that of a much more rigid semi-crystalline structure. This smooth transition is captured by a set of integral models which obtain the response by averaging the apparent response of the amorphous portion and a continuum of different crystals. A commonly used empirical relation between the extent of crystallization and volume change is imposed as a restriction on the material, and the implication of the entropy production inequality in the presence of this constraint is evaluated. General representations are provided.

Modeling the thermomechanical effects of crystallization in natural rubber: I. The theoretical structure

Abstract Using the theoretical framework introduced in a previous work (Negahban, M., 1997. Thermodynamic modeling of the thermomechanical effects of polymer crystalliation: A general theoretical structure. International Journal of Engineering Science 35, 277–298), a model is proposed for capturing the thermomechanical response of natural rubber during and after crystallization. The model is given in a form which will allow the incorporation of both the known mechanical response and the known thermal response observed before, during, and after crystallization in natural rubber. In particular, one can include in this model known experimental results characterizing the stress relaxation due to crystallization, increase in rigidity with crystallization, heat capacity, heat of crystallization, and the melting temperature. In this first article, a basic overview is presented of the model, and the thermal expansion of the amorphous and crystalline phases of natural rubber are incorporated into the model. The specific form of the free energy used to characterize the response of natural rubber is presented in the following articles.

Simulation of mechanical response in polymer crystallization

Abstract The problem of following the mechanical response of a polymer during crystallization is studied using a theory developed by Negahban and Wineman [1]. Elastic modulus, shear modulus, and Poissons ratio are defined in the context of polymer crystallization. For unconstrained crystallization and crystallization under constant uniaxial stretching, the values of elastic modulus, shear modulus, poisons ratio, and residual stretch are evaluated. The proposed model is fit to data available for natural rubber and the predictions of the model are discussed.

Modeling the thermomechanical effects of crystallization in natural rubber: II. Elementary thermodynamic properties

The model proposed in the first part of this series for characterizing the thermomechanical response of natural rubber during crystallization is used in this article to model the elementary thermodynamic properties of natural rubber. In particular, known experimental results for the heat capacity, heat of fusion, fundamental melting temperature, and equilibrium crystallinity are used to calculate specific material functions associated with the proposed model. Based on these material parameters the model is used to evaluate the dependence of equilibrium crystallinity on pressure and temperature. The dependence of the melting temperature on pressure is also evaluated.

A study of thermodynamic restrictions, constraint conditions, and material symmetry in fully strain-space theories of plasticity

A thermodynamic theory is presented for modeling elastic-plastic response at large deformations. A study is conducted on how one may impose different constraint conditions, including that imposed by the yield function, and the implications of the entropy production inequality on the elastic-plastic response in the presence of these constraints. Both single and multiple constraints are considered. Representations are provided for Cauchy stress, heat flux, free energy, entropy, the flow rule, and the hardening rule for an initially isotropic material.

Simulation of mechanical response in crystallizing polymers: crystallization under a constant shear force

SummaryThe mechanical response of crystallizing polymers under shearing controlled by a constant load is simulated by the model developed by Negahban et al. [10]. It is shown that under a constant shear load the shear strain increases with crystallization. The history of this shear strain directly affects the residual shape of the material after unloading. The shear modulus and elastic moduli are calculated as functions of the shear load and the degree of crystallization. Comparison with simulations based on a constant shear strain shows that some material moduli have the same functional relations and others are different. Specific simulations based on parameters selected for natural rubber are presented and compared to the response under constant shear strain.