EFFECTIVE WATER DIFFUSION COEFFICIENT IN FABA BEAN SEEDS DURING DRYING.PART I. DETERMINATION OF VALUES

Ireneusz Białobrzewski, Ryszard Myhan, Romuald Cydzik
Department of Agricultural Process Engineering, University of Warmia and Mazury in Olsztyn, Poland

ABSTRACT

The effective water diffusion coefficient was determined in faba bean seeds during drying. Faba bean seeds with an initial moisture content of 0.240 kg/kg were dried under natural convection conditions at 15, 20, 25, 30 and 35°C. The results obtained in the study show that in the process of convective drying of faba bean seeds the effective water diffusion coefficient is related to moisture content in a temperature range [15, 35]°C, and moisture content range. It was also found that the effective water diffusion coefficient is significantly affected by temperature in a range [15, 35]°C, but not in a range [20, 30]°C. The exponential form of the Arrhenius equation describes, with sufficient accuracy, the effective water diffusion coefficient in faba bean seeds during drying, as a function of material moisture content and temperature.

Key words: faba bean, drying, effective water diffusion coefficient, mathematical models.

NOMENCLATURE

Bi - Biot number ,
D - effective water diffusion coefficient, m² s^-1,
D_AB - moisture diffusivity in the gaseous phase, m² s^-1,
Fo - Fourier number ,
Gr - Grashof number ,
h_m - mass transfer coefficient, m s^-1,
k - drying rate coefficient, s^-1,
M₀ - initial moisture content, kg_water kg_db^-1,
M - moisture content, kg_water kg_db^-1,
M_e - equilibrium moisture content, kg_water kg_db^-1,
MR - reduced moisture content, -,
R_z - equivalent radius of faba bean seeds, m,
j - relative humidity of drying air, %,
j₁₄ - relative humidity of ambient air, %,
p - air-pressure, N m^-2,
p_s14 - partial pressure of water vapor at 14°C, N m^-2,
p_si - partial pressure of water vapor at particular drying temperatures, N m^-2,
j_r - air relative humidity in the drying equilibrium state, %,
Sc - Schmidt number ,
Sh - Sherwood number ,
t - temperature of drying air, °C,
V - volume of 1000 faba bean seeds, m³,
n - air kinematic viscosity, m² s^-1,
Q - drying time, s.

INTRODUCTION

In north eastern Poland faba bean seeds are harvested in the second and third decade of September, and in the first decade of October. Over this period air temperature and relative humidity vary in a range of [9.2, 14.2]°C and [75, 92]%, respectively. It follows that seeds always require preservation or drying. Faba bean seed drying in industrial dryers, where temperature of seed heating exceeds 55°C, results in considerable thermal damage (deteriorating seed quality) and is conducive to fungal infestations. Results of previous research [12] show that seed heating temperature exceeding 35°C causes excessive damage. Thus, the so called low-temperature drying, with unheated air or air heated to 35°C, seems to be a good solution. This method has been universally applied to cereal grain drying and ventilation in silos. However, the practical use of this method in faba bean drying should be preceded by a thorough investigation into the process. The necessary information may be obtained in the course of natural experiments or computer simulations. The latter solution, using models based on partial differential equations, is less expensive and time-consuming. The application of these models must be preceded by determining the physical properties of raw material, including the water diffusion coefficient. In the case of biological material, this coefficient reflects the whole complexity of moisture transport [6, 17, 18], so it can be treated as an effective coefficient.

Due to the shape of faba bean seeds, it seems reasonable to use a water diffusion equation for a spherical body (1) [14]:

(1)

Certain simplifications were made in this equation. It was assumed, among other, that the effective water diffusion coefficient is constant, and that the geometry and dimensions of faba bean seeds do not change. It was also assumed that shrinkage does not occur, because faba bean seeds have low initial moisture content, and seed coat structure minimizes changes in seed geometry during convective drying. Hatamipour and Mowla [9] presented the occurrence and effects of shrinkage on green pea drying, but the initial moisture content of this material is about 3 kg/kg.

Many authors apply dimensionless numbers to estimate the value of the convective mass transfer coefficient [3, 5, 7]. Equation (2) is often applied under free convection conditions [4]:

(2)

During convective drying (gas - motionless fluid), the value of mass transfer coefficient may be estimated assuming that Sh = 2. The humid air diffusion coefficient, D_AB, was calculated using relationship (3) [10]:

(3)

At mean air temperature of 25°C, D_AB = 2.7.10^-5 m² s^-1, and h_m = 0.012 m s^-1. The value of the Biot number for mass transfer enables to determine which factor decides about the rate of this process: diffusion internal resistance or transfer resistance in the boundary layer of the dried particle. Literature data allow estimating the range of variation of the effective water diffusion coefficient. According to Senaderra et al. [15], the value D for green peas is 10^-10 ÷ 10^-9 m² s^-1, whereas according to Simal et al [16] - 10^-6 m² s^-1. The values obtained by Simal et al. [16] are probably by many orders of magnitude higher than those expected for faba bean seeds, since the former were received for a moisture content range of 0.3 to 3 kg/kg and M₀ for faba bean seeds is 0.240 kg/kg. For estimated h_m = 0.012 m s^-1 and the expected range of the effective water diffusion coefficient (10^-10 ÷ 10^-9 m² s^-1), the estimated value of the Biot number amounted to 9.10⁴ and was >> 100; in consequence it was assumed that first kind boundary conditions occur on the surface of the material [7].

An analytical solution of equation (1) at the initial and boundary conditions adopted in the study, i.e. (4) and (5):

(4)

(5)

with respect to mean moisture content as a function of time, takes the form (6):

(6)

Equation (6) shows that computer simulation of the drying process is based on such parameters as dimensions of the product, initial and equilibrium moisture content and effective water diffusion coefficient, affecting the Fourier number. Professional literature on the topic provides numerous models of equilibrium moisture content and water diffusion coefficients of various granular products, together with ranges of temperatures and relative air humidity in which they can be applied. These are, among other, models of water diffusion coefficients for parsley seeds [13], white bean (Phaseolus Vulgaris) seeds [1], and seeds at different ripeness stages [16]. However, there are no models describing the effective water diffusion coefficient for faba bean seeds. The value D is often determined by comparing dMR/dFo and dMR/dQ derivatives, describing theoretical and experimental shrinkage curves, respectively [2, 8, 9, 15]. According to this method, the effective water diffusion coefficient is calculated on the basis of equation (7):

(7)

A graphical interpretation involves comparing slopes of the above shrinkage curves.

The aim of the present study was to determine the effects of temperature and moisture content on the effective water diffusion coefficient in faba bean seeds dried under natural convection conditions, in a temperature range of 15°C (ambient temperature) to 35°C (maximum permissible seed heating temperature).

MATERIAL AND METHODS

The experimental material comprised seeds of faba bean var. "Nadwiślański" with an initial moisture content of 0.240 kg/kg. The material was stored in an exsiccator placed in a Nagema GO-185 cooling chamber, at a temperature of 14°C. Faba bean seeds were dried under natural convection conditions, on a test stand that consisted of a KC100/200 dryer, Medicat 1600C balance, and a computer recording results. A single layer of seeds weighing 80 ± 0.01 g was placed on an openwork pan of the balance placed in the dryer chamber. Dry-bulb temperature and relative air humidity, measured with an Assmann psychrometer, were 14° C and 62% respectively. Drying temperature in successive measurement series was 15, 20, 25, 30 and 35°C. Relative air humidity in the dryer chamber was calculated using the following relationship:

(8)

At the above air temperatures in the dryer chamber, relative air humidity was 58, 42, 31, 23 and 18% respectively. Sample weight losses were recorded every two hours.

In order to determine the equivalent radius R_z, six samples, 1000 seeds each, were separated with a LN-S50 seed counter. The volume V of each of the samples was determined in a measuring cylinder, and the equivalent radius R_z was calculated from dependence (9):

(9)

The equivalent radius of faba bean seeds (mean of R_i) was R_z = 0.0044 ± 6.5.10^-5 m.

The effective water diffusion coefficient, as a parameter of equation (4), was determined by solving an inverse problem [19]. This method consists in calculating the unknown value D on the basis of known experimental values u, so that equation (10) is true:

(10)

where M_exsp is the mean moisture content determined experimentally. It was assumed that the iteration process of searching for the value D is completed when dependence (10) is true at err = 0.001 kg/kg i.e. for moisture content determination accuracy. The computational process was divided into a finite number of time steps. It was assumed that the effective water diffusion coefficient is constant within a given time step, but may change in consecutive time steps. 100 first terms of the series were taken into account in computations. At this value of n the series reached convergence [11]. Simulations were performed in MATLAB software (MathWorks Inc., MA, USA).

The equilibrium moisture content was not reached during a natural experiment. This value, M_e, was calculated based on non-linear estimation of the exponential form of the Lewis model (11) describing changes in the mean moisture content over drying:

(11)

The values M_e and k, treated as model parameters, were estimated for experimental data, M.

The significance of temperature effects on the effective water diffusion coefficient was determined by the Duncan test. A statistical analysis was made in the STATISTICA 6.1 package (StatSoft, Inc., Ok., USA).

RESULTS AND DISCUSSION

Figure 1 presents experimental and simulated (model 11) changes in moisture content at 15, 20, 25, 30, 35°C. The scale of the time axis shows that the model moisture content approaches asymptotically the equilibrium moisture content. Table 1 presents the values of model determination coefficient, equilibrium moisture content, drying rate coefficient and air parameters in the drying equilibrium state. The analysis of these values confirms (for all temperatures R² >0.999) that the accuracy of equilibrium moisture content determination was satisfactory.

Figure 2 illustrates changes in the effective water diffusion coefficient at particular drying temperatures, i.e. 15, 20, 25, 30 and 35°C. The effect of temperature inside faba bean seeds on the effective water diffusion coefficient was estimated on the basis of results of the Duncan test (significance of differences between means). Taking into account seed size and process duration, it was assumed that the seed temperature was equal to the temperature of the drying medium over the entire process. The values included in table 2 show that the effective water diffusion coefficient was significantly (p < 0.05) affected by temperature in a range [15, 35]°C, but not in a range [20, 30]°C. This could be related to the structure of seed coat, with a palisade layer of tightly packed cells underneath the cuticle. Heat transported to seeds during drying at higher temperature increases palisade cell lignifications and causes cuticle shrinkage. The protective coat formed in this way efficiently inhibits heat and moisture transfer.

The non-linear estimation procedures offered by STATISTICA were used for testing the generalized Arrhenius equation, describing the correlations between effective diffusivity, moisture content and temperature. The coefficient in resultant equation (12) is significant at p < 0.01:

(12)

The value of the determination coefficient (0.945) proves that this model provides satisfactory accuracy of effective water diffusion coefficient determination for faba bean seeds dried at various levels of M and t.

CONCLUSIONS

The results obtained in the study show that in the process of convective drying of faba bean seeds the effective water diffusion coefficient is related to moisture content in a temperature range [15, 35]°C, and moisture content range. It was also found that the effective water diffusion coefficient is significantly affected by temperature in a range [15, 35]°C, but not in a range [20, 30]°C. Equation (12) can be used for describing changes in the effective water diffusion coefficient within the examined ranges of material moisture content and temperature.

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