EFFECTIVE WATER DIFFUSION COEFFICIENT IN FABA BEAN SEEDS DURING DRYING.PART II. ANALYSIS OF ESTIMATION ERRORS

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

ABSTRACT

The objective of this study was to determine the effects of measuring errors on the accuracy of diffusion coefficient estimation as dependent on the method used. The estimation was based on a numerical solution of the inverse function, and inverse problem solving. The analysis concerned results of convective drying of faba bean seeds. The seeds, with an initial moisture content M₀=0.240 kg_water/kg_db were dried at 15, 20, 25, 30 and 35°C. The results obtained in the study show that the method based on inverse problem solving is more effective at extreme moisture levels (Θ ≈ 0 and Θ ≈ Θ_k). This method, used for simulating successive replications, enables to acquire information not only on estimated values of the diffusion coefficient and measuring errors, but also on their distribution.

Key words: faba bean, drying, effective water diffusion coefficient, estimation errors.

NOMENCLATURE

D - internal water diffusion coefficient, m² s^-1,
DD - error of effective diffusion coefficient estimation, m² s^-1,
Fo_m - Fourier number,
DFo_m - numerical error of Fourier number estimation,
k - drying rate coefficient, s^-1,
m - sample weight, kg,
Dm - weight measurement error, kg,
M - moisture content, kg_water kg_db^-1,
DM_e - standard error of equilibrium moisture content estimation, kg_water kg_db^-1,
MR - Moisture ratio, -,
R_z - equivalent radius of faba bean seeds, m,
DR_z - error of estimation of the equivalent radius of faba bean seeds, m,

t - temperature of drying air, °C,
Q - drying time, s.

Subscripts:
0 - initial,
e - equilibrium,
exp - experimental,
sym - calculated using a model.

INTRODUCTION

The current state of art allows replacing natural experiments by computer simulations, using models based on partial differential equations. The application of these models must be preceded by determining the physical properties of raw material, including the water diffusion coefficient. Due to the shape of faba bean seeds, it seems reasonable to use a water diffusion equation for a spherical body [4]. Certain simplifications were made, assuming among other that the effective water diffusion coefficient is constant, and that the geometry and dimensions of faba bean seeds do not change. The following:

- initial

(1)

- and boundary

(2)

conditions were also adopted. An analytical solution of this equation takes the form:

(3)

In order to estimate the diffusion coefficient D on the basis of the above equation, we have to know the dimensions of dried material R_z, instantaneous M(Q ), initial M₀ and equilibrium M_e moisture content. Due to the fact that the values of these parameters are determined empirically, each of them can be burdened with certain measuring or estimation errors.

The objective of this study was to determine the effects of measuring errors on the accuracy of effective diffusion coefficient estimation as dependent on the method used.

MATERIAL AND METHODS

The experimental material comprised seeds of faba bean var. `Nadwiślański´ with an initial moisture content of 0.240 kg/kg. The seeds were dried under natural convection conditions, at 15, 20, 25, 30 and 35°C. Sample weight losses were recorded every two hours [1]. In particular experimental series initial sample weight was m₀ = 0.080 kg, and the absolute error of weight measurement - Dm = ±1.0.10^-5 kg. In order to determine the equivalent radius R_z, six samples, 1000 seeds each, were separated and the volume V of each of the samples was determined in a measuring cylinder. Then radiuses R_i and R_z were calculated from dependence (4)

(4)

as means of R_i. The equivalent radius calculated in this way was R_z = 0.0044 m, at the statistical estimation error D R_z = ±6.5.10^-5 m.

The effective water diffusion coefficient D, as a parameter of equation (3), was determined by two methods:

- a numerical solution of the inverse function [2],
- inverse problem solving [5].

The application of both methods must be preceded by calculating the equilibrium moisture content M_e, if it was not estimated experimentally, and by determining estimation accuracy DM_e.

Equilibrium moisture content. Theoretically, it follows from equation (3) that the equilibrium moisture content may be determined if the drying process is carried out long enough (Q ® ¥  ). This approach assumes that after this time the weight of a dried sample does not change. However, in practice the drying process last for a specified period of time Q_k, and each measurement is burdened with a certain error resulting from weight measurement accuracy. This was also observed in the present natural experiment. For the purpose of this study, the equilibrium moisture content M_e was determined at each drying temperature, based on non-linear estimation of the exponential form of the empirical Lewis model:

(5)

A statistical analysis was made in the STATISTICA 6.1 package (StatSoft, Inc., Ok., USA). The parameters k and M_e of equation (5) were estimated by the Lewenberg-Marquardt method [3] at a significance level p < 0.05.

Numerical solution of an inverse function. Using this method, the effective diffusion coefficient was estimated at six consecutive stages, taking into accounts errors of estimation. The final value of this coefficient for particular empirical datasets was determined from equation (6):

(6)

where F(Q) is the Fourier number for a given temperature and drying time. The value of this number was calculated based on a generated tabular form of the function inverse to that described by equation (3):

(7)

Successive values of the Fourier number were generated in the <0, 1> range with the 0.01 step, and reduced moisture content was calculated by summing up 100 elements of a series (n = 100). The value of the Fourier number at the reduced moisture content corresponding to empirical data was obtained by approximation in the appropriate interval of the tabular function from equation 7. A detailed algorithm is given in table 2, and the results obtained are presented in figure 1.

Reverse problem solving. According to this method, the unknown value of the effective diffusion coefficient D is calculated on the basis of known experimental values of moisture content M_exp(Q). The value of the diffusion coefficient was sampled from a gradually narrowing domain of feasible solutions. While solving equation 3, the calculated moisture content M_sym(Q) was determined for each case. 100 first terms of the series were taken into account in computations (as in Numerical solution of an inverse function). The optimization criterion was the estimation error of moisture content:

(8)

It was assumed that the iteration process of searching for the value D is completed when dependence (8) is true at err = 0.001 kg/kg. In order to determine the value and distribution of estimation error of the effective diffusion coefficient for all results of the natural experiment, the simulation experiment was repeated 100 times, each time modifying the parameters of equation 3 by error values sampled from normal distribution: weight measurement error Dm = ±0.00001 kg, estimation error of the equilibrium moisture content DM_e(t) (tab. 1), and estimation error of the equivalent radius DR_z = ±6.5.10^-5 m. Simulations were performed in MATLAB software (MathWorks Inc).

RESULTS

Figure 2 presents the results of diffusion coefficient estimation, obtained using both methods. Figure 3 shows differences in estimation, and figure 4 - estimation errors.

A comparative analysis of diffusion coefficient estimation shows that:

1. Both methods provided similar results for all drying temperatures analyzed, in the range. The absolute estimation error increases with the decrease in moisture content, and the relative estimation error remains at a stable level of about 5%.

2. At the initial stage of drying, at moisture content u(Θ) close to the initial value M₀ = M(0), inverse function solving leads to overestimation of the diffusion coefficient. This is related to underestimation of the calculated moisture content (tab. 2), visible especially at Θ ≈ 0. The absolute value of this underestimation may be reduced by increasing the number of elements in the series of equation 3, but this will not reverse the general tendency.

3. At the final stage of drying M(Θ)→M_e diffusion coefficient estimation with both methods is burdened with a gross error, which results primarily from the error in equlibrium moisture content (ΔM_e) estimation. The inverse function method underestimates the value of the diffusion coefficient in this range, and results are burdened with a greater error than with inverse problem solving (tab. 3). This results from the use of the tabular form of the function from equation 7, and numerical errors of approximation.