FINITE ELEMENT SIMULATION OF CORN MOISTURE ADSORPTION

K. Muthukumarappan, S. Gunasekaran

ABSTRACT. Thin-layer moisture adsorption tests were conducted by exposing com kernels to different air relative humidities (RH of 75, 80, and 90%) and temperatures (25, 30, 35, and 40°C). A two-dimensional finite element model was developed to simulate moisture diffusion into a com kernel during adsorption. The finite element model satisfactorily predicted experimental moisture ratio of corn samples and performed better than the analytical model in predicting the adsorption behavior of corn kernels. Moisture profiles predicted with the finite element model showed that the moisture gradient between the center and boundary of a corn kernel exposed to air at 25, 30, and 35°C each at 90% RH was about 4% after one hour and reached a maximum of about 9% after 7.5 h and declined during subsequent adsorption. Keywords, Adsorption, Com, Diffusivity, Moisture, Simulation.

Corn kernel is hygroscopic which adsorbs or desorbs moisture under changing environmental conditions. Moisture transport within grains during desorption (i.e., drying) has been widely

investigated (Irudayaraj et al., 1992; Syarief et al., 1987; Suarez et al., 1981; Steffe and Singh, 1980a, b, c; Misra and Young, 1980). However, moisture transport during adsorption (i.e., wetting) has received only little attention (del Giudice, 1959; Chittenden, 1961; Misra, 1978; Muthukumarappan and Gunasekaran, 1990; Osborn et al., 1991; Lu and Siebenmorgen, 1992). In addition, the pattern of moisture movement inside a corn kernel during adsorption has not been reported. Because of the hysteresis between desorption and adsorption, desorption analyses are not directly applicable to adsorption analyses.

Adsorption of com takes place during grain conditioning, storage, deep-bed drying, and aeration processes. For efficient processing operations, quantitative and predictive models relating the physical properties to transient time-moisture profiles that determine product quality are needed. Collecting moisture content data at various locations inside a com kernel as a function of time requires sophisticated sensors and is cumbersome. Mathematical models, based on physical principles, have the potential to accurately predict the moisture distribution inside the kemel during adsorption. The exact solution of the goveming equations might be difficult to obtain. Therefore, approximate solution techniques have to be sought.

Development of fissures in com kernels is caused by both external and internal stresses. Fissured or stress-

Article was submitted for publication in February 1996; reviewed and approved for publication by the Food and Process Engineering Inst, of ASAE in August 1996.

The authors are K. Muthukumarappan, ASAE Member Engineer, Associate Researcher, and Sundaram Gunasekaran, ASAE Member Engineer, Professor, Biological Systems Engineering Department, University of Wisconsin-Madison, Wis. Corresponding author: K. Muthukumarappan, Biological Systems Engineering Dept., University of Wisconsin-Madison, 460 Henry Mall, Madison, WI 53706; telephone: (608) 262-7794, e-mail: <muthukum@facstaff.wisc.edu>.

cracked kemels are objectionable because they are quite susceptible to breakage during handling and cause problems in storage, shipping, and processing (Gunasekaran and Paulsen, 1985). Moisture and temperature gradients prevalent within the grain cause undue expansion and contraction in the grain leading to the development of internal stresses (Gunasekaran et al., 1985). In general, moisture gradients have a predominant effect on the expansion and shrinkage of grains while the effect of temperature gradients is negligible (Suresh et al., 1975; Muthukumarappan et al., 1992). If the stresses developed within the kernels can be calculated accurately, better processes can be designed to reduce fissure development. However, such an estimation also requires the knowledge of transient time-moisture profiles and moisture gradients prevalent within a corn kernel during adsorption.

Partially coupled heat and mass transfer equations have been solved for an isotropic sphere with constant material properties (Haghighi and Segerlind, 1988) and coupled equations with varying material properties (Haghighi et al., 1990) was used to study the drying of barley, soybean, and com kemels (Irudayaraj et al., 1992). Coupling effects of moisture and temperature, although important for accurately modeling desorption, is not important for adsorption since the adsorption process takes much longer (48 to 50 h) than the desorption process (6 to 10 h). And also when diffusion in a corn kemel takes place at constant temperature, the moisture diffusion equation alone is sufficient for describing moisture movement.

Finite difference and finite element methods (FDM and FEM) have been extensively used to solve problems numerically. Husain et al. (1973) solved simultaneous heat and mass diffusion equations with the aid of altemating direction explicit scheme. They tested the model with rough rice and their prediction agreed well with the experimental data. Fortes et al. (1981) analyzed wheat drying and rewetting by applying a model based on non-equilibrium thermodynamics. Steffe and Singh (1980c) modeled thin-layer drying of rough, brown and white rice using Crank-Nicolson scheme and determined liquid

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diffusivities of starchy endosperm, bran, and hull of rough rice. The FEM is a powerful numerical technique for solving differential equations. The method assumes that any continuous quantity such as moisture can be approximated by a set of discrete piecewise-continuous functions defined over a finite number of subdomains or elements (Segeriind, 1984; Cook et al., 1989).

Zhang et al. (1984) used the FEM to model water diffusion in rice with diffusion coefficient as a function of moisture concentration, and they also were able to account for the increase in the size of rice by swelling during soaking. Chinnan and Bakshi (1984) modeled the moisture transfer in rewetted peas during drying. Their approach was the same as that of Lomauro and Bakshi (1985). Sokhansanj and Gustafson (1980) applied the FEM and solved coupled heat and mass transfer equations for drying cereal grains. Both com and rice kemels were considered. When modeling the rice kemel, they considered endosperm, bran, and hull. But for corn, only the endosperms and germ were considered. The pericarp of the com, which acts as a barrier for moisture diffusion, was neglected.

Haghighi et al. (1990) presented the solution of a set of coupled conductive heat and diffusive moisture transfer equations for grain drying simulation of axisymmetric bodies. The model assumed that the moisture diffuses to outer boundaries of the kernel in liquid form and evaporates at the surface of the grain. Lu and Siebenmorgen (1992) simulated the moisture adsorption of long-grain rough rice using the FEM. They determined the diffusivities of the hull, bran, and endosperm by assuming rice kemel as a composite body with the shape of a prolate spheroid.

Corn kernel comprises of four major components, namely pericarp, germ, and soft and hard endosperms (Pomeranz, 1987). These components fill the kernel in a complex manner than any other grain which makes the diffusion analysis difficult. The moisture diffusivity of corn germ, pericarp and soft and hard endosperms was reported in Muthukumarappan and Gunasekaran (1994a, b and c). The moisture diffusivity of corn germ, pericarp, soft and hard endosperms was determined using a one-dimensional (1-D) analytical method, a 1-D FDM, and a 2-D FEM, respectively. The main objective of this article was to determine how well we can simulate the moisture diffusion in corn kemels using the diffusivity values reported in Muthukumarappan and Gunasekaran (1994a, b and c).

In this study the FEM was used to solve the governing differential equations for the mass transfer during com kernel moisture adsorption. Since the moisture diffusivity of com components was determined using different methods, the moisture diffusion simulated using FEM should be verified against an analytical model. Moreover, the moisture diffusivity of corn germ which was determined using 1-D analytical method was used in the finite element simulation. Therefore it is necessary to compare the moisture diffusion simulated using the FEM with results of the analytical model for com adsorption. The specific objectives of this study were to: (1) develop a finite element model to simulate the moisture adsorption behavior of a whole corn kemel; and (2) verify the model by predicting nodal and average moisture content of the corn and comparing with experimental and analytical results.

THEORETICAL CONSIDERATIONS The following theoretical development assumes that

diffusivity is the dominant factor in the moisture transport process and, therefore, it is treated as a variable. Cartesian coordinate system was used to represent the corn kernel as a two-dimensional body. The two-dimensional diffusion equation which describes the moisture transport process has the form:

3M_ a 3t dx

D aM ax ay

D aM ay

(1)

with initial and boundary conditions:

M = M^ , at t = 0 (2)

and

M3= [ 1 - exp( -Kt ) ] X [ M , - M j + M^ ,

for t > 0 on n (3)

where, Q. constitutes the complete boundary surface for the body. The boundary condition was proposed by Shivhare et al. (1991) and used by Muthukumarappan and Gunasekaran (1994b, c). Explanations for all symbols and notations are provided separately under nomenclature.

The element equations were developed from the governing differential equations by transforming the governing equation by use of the Galerkin's weighted residual approach. After the formulation, the final system of equations, incorporating the known boundary conditions, has the following form:

( [ c ]+At [K] )M,^^ ,= [C]M, + A t F , +At (4)

In developing the final system of equations, for the transient case under consideration, an implicit technique (backward difference scheme) was used. As a preliminary analysis, the central difference and backward difference schemes were tested. Both schemes yielded the same results and because the backward difference scheme is unconditionally stable for smaller time steps, it was used in solving the transient finite element equations.

MATERIALS AND METHODS SAMPLE PREPARATION

Corn kernels of FR27 x MO 17 variety were used in this study. The com was grown on the Agricultural Research Station Farm at the Purdue University, West Lafayette, Indiana and combine-harvested at about 27% moisture content during Fall, 1990. The corn was dried using natural air at a temperature of 23°C and RH of 55%.

Dried samples were hand-cleaned to remove broken kernels. The initial moisture content of the samples was about 9 to 10%, determined by the oven method (ASAE, 1990). The samples were stored in a refrigerator maintained at 5°C and 58% RH until the adsorption tests.

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ADSORPTION TESTS All adsorption tests were conducted in a controlled

environment chamber (2.21 x 0.74 x 1.95 m) available in the Biotron at the University of Wisconsin-Madison. Four air temperatures of 25, 30, 35, and 40°C each at three RH values of 75, 80, and 90% were used. These temperatures and RHs are typical for natural storage and deep-bed drying. Temperature and RH of the air in the chamber could be maintained within 0.1 °C and 1.0%, respectively, and were monitored periodically using a Weston (Model TH65) thermocouple/thermometer coupled with thermocouple psychrometer. Air was circulated at 0.5 m/s continuously during the tests. The air velocity was monitored with a Hastings (Model G-11) hot-wire anemometer.

Before each adsorption test, the samples were removed from the refrigerated storage and left to equilibrate to room temperature. Any stress-cracked kernels present in the samples were removed by visual examination. Round and other shaped kernels were discarded and only flat kernels were used. Fifty-gram samples of corn kernels and 25 g samples of germ were placed in the perforated wire-meshed containers giving a sample depth of about 10 mm. This depth was chosen to obtain thin-layer adsorption data. The tests were conducted for 48 to 72 h during which the sample moisture content reached near equilibrium with the environment. The details of the experimental set-up used are explained in Muthukumarappan (1993).

The cross-section of the corn kernel showing four distinct regions of germ, soft and hard endosperms, and pericarp is shown in figure 1. The cross section presented is through the narrowest dimension of the kernel and was selected based on the average dimensions of two corn kernels. To obtain the dimensions, the two corn kernels were cut and mounted on 10 x 10 mm aluminum cylinder stubs using double-sided sticky tape. Silver paint was then

applied around the sides of the kernel. The mounted samples were sputtered with gold to a thickness of about 270°A using Bio-Rad Polaron Division Gold Coater (Model E5000M SEM Coater). The samples were examined in a scanning electron microscope (Model Hitachi S-570) at an accelerating potential of 10 kV and corresponding dimensions were used.

A finite-element discretization of the kernel is shown in figure 2. Four-noded quadrilateral finite elements were used for discretization of the kernel. The 2-D model in Cartesian coordinates consists of 85 elements. Using the finite element formulations developed by Muthukumarappan (1993), the nodal moisture content of a corn kernel during adsorption was predicted. The average kernel moisture content (as distinguished from the nodal moisture values) was defined as the mass average value. Assuming constant density, the mass average moisture content of a body was determined as defined by Haghighi and Segeriind (1988). The listing of the finite element program is presented in Muthukumarappan (1993).

The diffusivity values (table 1) reported in Muthukumarappan and Gunasekaran (1994a, b, c) were used along with the necessary initial and boundary conditions for the finite element simulation. The validation of the FEM was fully explained in Muthukumarappan and Gunasekaran (1994c). A time step of 0.25 h was used for time marching. The time step was selected after numerous preliminary runs with different time steps and error analysis.

Fick's law of diffusion (Crank, 1975) model considering the geometry of corn as an infinite slab (Muthukumarappan and Gunasekaran, 1990, 1994a) was also used for moisture adsorption simulation. The first 10 terms of the model were considered using the non-linear, least square multivariate secant method (SAS, 1987). The characteristic dimension

11

10

9

7

r

Pericaip

X - Axis, mm

Figure 1-Cross-section of a corn kernel through its narrowest dimension.

X - Axis, mm

Figure 2-Finite-element discretization of the corn kernel (• node).

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Table 1. Moisture diffusivities (rnVh) of pericarp, germ, soft and hard endosperms of a corn*"

Table 2. Mean sum of square deviation (MSSD) values obtained for simulation models

Adsorpl Condit

Temp. (°C)

25

30

35

40

ion on

RH (%)

80 90 75 80 90 75 80 90 75 80

Pericarp xlO-8

0.34 0.30 0.45 0.42 0.42 0.52 0.49 0.47 0.57 0.53

Germ xlO-7

0.17 0.15 0.54 0.20 0.18 0.66 0.33 0.24 1.18 0.40

Moisture Diffusivity

Soft Endosperm

xlO-7

0.733 0.546 1.014 0.997 0.923 1.245 1.142 1.056 1.460 1.221

Hard Endosperm

xlO-7

0.320 0.420 0.652 0.566 0.549 0733 0.680 0.639 0.919 0.687

Composite xlO-7

0.68 0.60 1.01 0.90 078 1.24 1.20 0.88 1.40 1.32

Adsorption

Temp. (°C)

25

30

35

40

Condition

RH (%)

80 90 75 80 90 75 80 90 75 80

MSSD

Finite Element Model

0.002 0.004 0.002 0.003 0.002 0.002 0.001 0.002 0.003 0.001

ratio of corn samples exposed at all air

Analytical Model

0.010 0.019 0.014 0.010 0.010 0.018

0.008 0.011 0.016 0.003

humid conditions * Obtained from Muthukumarappan and Gunasekaran (1994a, b, c).

(half-thickness) of the kernels was determined at the initial moisture content. The thickness of 50 kernels were measured using a micrometer. The average half-thickness value (2.08 mm) was used for the infinite slab model. The composite moisture diffusivity values of com samples (FR27xM017) reported in table 4 of Muthukumarappan and Gunasekaran (1994a) were used in the analytical model to simulate the com moisture adsorption (table 1).

JlESULTS AND DISCUSSION MOISTURE ADSORPTION SIMULATION

The experimental, analytical and FEM simulated moisture ratio of FR27xM017 corn samples exposed to air at 35°C and 75% RH are presented in figure 3. In general, FEM simulated very well the experimental moisture ratio. The analytical model poorly over-predicted than the finite element model in the early stage of adsorption and under-predicted in the final stage of adsorption.

Mean sum of squares deviation (MSSD) was used an indicator to determine the prediction accuracy of the models studied. The MSSD between the experimental and the FEM simulated moisture ratio and MSSD between the experimental and the analytical model simulated moisture

1.0

are presented in table 2. Based on the MSSD values, the FEM predictions were clearly better than the corresponding analytical solutions. This may be because individual component moisture diffusivities of corn for the FEM was considered but a composite moisture diffusivity was considered for the analytical model. Another possible reason may be that the FEM was 2-D and analytical model was 1-D. However, the comparison is valid due to the fact that the composite moisture diffusivity values used in the analytical model and the germ and pericarp diffusivity values used in the FEM were based on a 1-D model.

EFFECT OF DIFFERENT CORN COMPONENTS ON THE

MOISTURE DIFFUSION

The variation of moisture ratio with time for corn germ, corn without pericarp and corn with pericarp exposed to air at 35°C and 75% RH is presented in figure 4. The corn germ attained higher moisture ratio than the corn samples without pericarp. This might be related to the non-homogeneity and interfaces between different components of corn without pericarp. Moreover, the corn without pericarp reached higher moisture ratio at a given time than com with pericarp. This was due to the resistance of the pericarp for moisture movement interacting with the interface between the pericarp and the corn without

Com without Pericarp

Com with Pericarp

0 10 20 30 40 20 30 40 Time, h

Time, h Figure 4-Comparison between moisture ratio of corn germ, corn

Figure 3-Comparison of moisture ratio of corn samples exposed at without pericarp, and corn with pericarp samples exposed at 35°C 35°C and 75% RH with analytical and finite element solutions. and 75% RH.

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pericarp. Ruan et al. (1991) presented 3-D transient moisture profiles of corn kernels during steeping process using magnetic resonance imaging technique. From the images they reported that the steepwater moved first into the corn kernel through the space between the germ and endosperm, and through the cross and tube cells of the pericarp layers. Then it quickly diffused into the germ, and slowly diffused into the endosperm.

From the results of the present study and Ruan et al. (1991) it is clear that the moisture diffusion in a corn kernel is a complex phenomena and more work is needed to better understand this behavior.

TIME-MOISTURE PROFILES The FEM predicted nodal moisture contents were

transformed to contour plots using the Surfer software (Surfer, 1990). The moisture profiles for corn samples after 1 h of exposure to air at 35°C and 90% RH is presented in figure 5. The moisture gradient between the center and surface of corn kernels during simulated moisture adsorption at 25, 30, and 35X each at 90% RH is presented in figure 6. In general, the moisture gradient

9.00 h

1.00

0.00 0.00 1.00 2.00 3.00

Kernel Thickness, mm 4.00

inside a corn kernel during adsorption at 25°C was lower than at 35°C. The temperature effect on moisture gradient was significant during early stage of adsorption (up to 5 h). This is because of different moisture diffusivity values and varying boundary condition assumption used in the simulation model. The moisture gradient between the center and boundary of a corn kernel exposed to air at 25, 30, and 35°C each at 90% RH was about 4% after 1 h, and reached a maximum of about 9% after 7.5 h and declined during subsequent adsorption. These times compare well with Sarwar and Kunze's (1989) experimental observations that the corn samples took about 1 h of exposure for fissures to start developing when exposed to 92% RH at 21°C. Further, they reported that all the kernels exposed to 92% RH at 2 PC fissured within 8 h of adsorption. This shows that the difference in moisture gradient may cause the kernels to fissure. In addition, all the kernels may fissure when the moisture gradient was maximum.

CONCLUSIONS A 2-D FEM based inodel was developed to simulate

moisture diffusion into a com kernel during adsorption. When compared to an 1-D analytical model, the 2-D finite element model better predicted the experimental moisture ratio of corn samples. Moisture profiles predicted with the FEM showed that the moisture gradient between the center and boundary of the com kernel exposed to air at 25, 30, and 35°C each at 90% RH was about 4% after 1 h, and reached a maximum of about 9% after 7.5 h and declined during subsequent adsorption.

ACKNOWLEDGMENT. We thank Prof Stroshine, Department of Agricultural Engineering, Purdue University, West Lafayette, Indiana, for providing the com samples.

c

1 3

Figure 5-Moisture profile (% wb) within a corn kernel after 1 h of exposure to 35°C and 90% RH during adsorption.

Time, h

Figure 6-Moisture gradient (% wb) within a corn kernel with time when exposed to 25,30, and 35°C each at 90% RH air condition.

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NOMENCLATURE [C] = global moisture capacitance matrix D = diffusivity (m^/h) F = varying boundary condition K = surface adsorpdon coefficient (h~^) [K] = global moisture conductance matrix M = moisture content at time t (h) (% wh) MQ = average initial moisture content of the kernel (% wb) Mg = equilibrium moisture content of the kernel (% wb) Mg = surface moisture content of the kernel (% wb) t = adsorption time (h) x,y = cartesian co-ordinates At = time step (h) Q. = boundary surface of the body

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