A comparison of concentration measurement techniques for the estimation of the apparent mass diffusion coefficient

A COMPARISON OF CONCENTRATION MEASUREMENT TECHNIQUES FOR THE ESTIMATION OF THE APPARENT MASS DIFFUSION COEFFICIENT

L.M.Pereira¹, R. De Souza², H.R.B.Orlande² and R.M.Cotta²

¹UNIVAP, Av. Shishima Hifumi, 2911, 12244-000São José dos Campos - SP, Brazil

²Department of Mechanical Engineering, POLI/COPPE/UFRJ

Cx. P. 68503, Cidade Universitária, 21945-970,

Phone: [phone], Fax: [phone],

Rio de Janeiro - RJ, Brazil

E-Mail: [e-mail]

(Received: April 19, 2001 ; Accepted: July 13, 2001)

Abstract - In this article we compare two different techniques to measure the concentration of saline solutions for the identification of the apparent mass diffusion coefficient in soils saturated with distilled water. They are the radiation measurement technique and the electrical conductivity measurement technique. These techniques are compared in terms of measured quantities, sensitivity coefficients with respect to unknown parameters and the determinant of the information matrix. The apparent mass diffusion coefficient is estimated by utilizing simulated measurements containing random errors. The Levenberg-Marquardt method of minimization of the least-squares norm is used as the parameter estimation procedure. The effects of the volume of saline solution injected into the column devised for the experiments on the accuracy of the estimated parameters are also addressed in this article.

Keywords: radiation measurement technique, electrical conductivity measurement technique.

INTRODUCTION

Accurate knowledge of parameters appearing in mathematical models used in numerical simulations of physical problems is of capital importance for the accuracy of the results obtained. One such parameter, appearing in the analysis of mass diffusion problems, is the so-called apparent mass diffusion coefficient.

In previous works, we addressed the estimation of the apparent mass diffusion coefficient by using the radiation measurement technique (Pereira et al., 1998, 1999). A radiation detector was used to measure the number of counts, which is associated with the salt concentration at several positions along a column filled with sand and saturated with distilled water. These measurements were then used to estimate the apparent diffusion coefficient of the salt in the column by inverse analysis, utilizing the Levenberg-Marquardt Method of minimization of the least-squares norm (Levenberg, 1944; Marquardt, 1963; Beck and Arnold, 1977; Ozisik and Orlande, 2000). This type of measurement technique is especially suitable for diffusion experiments involving radioactive materials found in nature, but it can also be used for the study of the diffusion of several nonradioactive materials, which can be irradiated before the experiments. It also has the advantages of being nonintrusive and nondestructive. On the other hand, it involves the handling of radioactive materials, thus requiring strict safety procedures.

In this work we present a comparison of the radiation measurement technique described above with a technique based on the use of electrical conductivity measurements (Schackelford and Redmond, 1995). The electrical conductivity measured by cells placed inside the column can be associated to the salt concentration by calibration curves established a priori. Despite being an intrusive technique, the use of electrical conductivity cells for problems involving inactive salts is appealing, because they do not need to be irradiated beforehand and, hence, the safety procedures do not have to be as strict as those for radioactive measurement techniques. Also, the use of electrical conductivity cells is more appropriate than radioactive measurements for two- and three-dimensional experiments.

The two measurement techniques discussed above are compared here in terms of measured quantity, sensitivity coefficients with respect to unknown parameters and the determinant of the information matrix. The D-optimum approach is used for the design of the experiment (Beck and Arnold, 1977; Ozisik and Orlande, 2000). We also examine the effects of the volume of the saline solution injected into a column containing the soil under analysis on the accuracy of the estimated parameters. The accuracy of the estimated parameters is examined by using simulated measurements containing random errors. The Levenberg-Marquardt Method (Levenberg, 1944; Marquardt, 1963; Beck and Arnold, 1977; Ozisik and Orlande, 2000) is utilized for the solution of the inverse problem of parameter estimation, as described below.

PHYSICAL PROBLEM AND MATHEMATICAL FORMULATION

The physical problem addressed in this paper consists of a column filled with soil and saturated with distilled water (Pereira et al., 1998, 1999). The salt concentration in the column is zero for t < 0. At t = 0, a saline solution is injected into the bottom of the column through a capillary tube and then the mass diffusion process takes place in the column. This physical problem can be mathematically formulated by considering the column as a semi-infinite medium, since the diffusion process is so slow that the boundary condition at the upper end of the column does not influence the solution during the time range of interest. Due to the symmetry of the column and the no-flux boundary conditions on its lateral surfaces, the diffusion process can be formulated in terms of a one-dimensional problem given by

where D^* is the apparent mass diffusion coefficient, l is the radioactive decay coefficient, C₀ is the initial concentration of the salt injected into the column and a is the initial length of the column filled with the injected saline solution, under the hypothesis that this injected solution displaces the water in the pores perfectly.

The apparent mass diffusion coefficient is defined as

where D is the effective diffusion coefficient, which takes into account the tortuosity of the porous media, and K is the retardation factor.

Problem 1 above is denoted as a direct problem when D^*, C₀, l and a are known. The objective of the direct problem is then to determine the transient concentration field, C(z,t), in the column.

The analytical solution of the direct problem can be obtained by Fourier transform (Ozisik, 1993) as

where erf(.) is the error function (Ozisik, 1993).

The analytical solution given by equation 3 is valid for both radioactive (l ¹ 0) and inactive (l = 0) salts.

INVERSE ANALYSIS

The objective of the present study is to estimate the apparent mass diffusion coefficient D* appearing in problem 1, by using transient concentration measurements taken at different points the column. This kind of problem is known as an inverse problem of parameter estimation (Beck and Arnold, 1977; Alifanov, 1994; Ozisik and Orlande, 2000). The other quantities appearing in the formulation of the direct problem are assumed to be exactly known for the analysis, with the exception of the initial length filled with saline solution (a). This quantity was also regarded as unknown and was left to be estimated as part of the solution procedure because we detected some inconsistencies between different experiments, when using actual experimental data for the estimation of the apparent diffusion coefficient (Pereira et al., 1999).

By assuming the measurement errors to be additive, uncorrelated and normally distributed, with zero mean, as well as known and constant standard deviations, the apparent mass diffusion coefficient can be estimated by the minimization of the least squares norm. For a general case involving N unknown parameters, i.e., P = [P₁, P₂, , P_N], such a norm can be written as

Superscript T above denotes transpose and [Y-C(P)]^T is given by

where , i=1,..., I is a row vector containing the differences between the measured and estimated concentrations at measurement points z_m, m=1,..., M at time t_i, i.e.,

For the minimization of the least squares norm (4), we consider the Levenberg-Marquardt method (Levenberg, 1944; Marquardt, 1963; Beck and Arnold, 1977; Ozisik and Orlande, 2000). The iterative procedure for this method is given by

where superscript k denotes the number of iterations, m^k is the so-called damping parameter and W^k is a diagonal matrix, which can be taken as the identity matrix or as the diagonal of J^TJ. The sensitivity matrix, J, is given by

where for i=1,..., I.

After estimating the unknown parameters by using the iterative procedure of the Levenberg-Marquardt method (equation 6), we can estimate the standard deviations for the parameters with the covariance matrix given by (Beck and Arnold, 1977)

where s is the standard-deviation of the measurement errors.

Different experimental variables, such as the number of measurement locations, frequency of measurements and duration of the experiment, can be determined by the analysis of the sensitivity coefficients, i.e., the elements of the sensitivity matrix, and of the determinant of the information matrix J^TJ. These analyses are also performed in order to compare the two measurement techniques under study. Measurements shall be taken in regions where the sensitivity coefficients for the different parameters attain large magnitudes and are not linearly dependent. Maximization of the determinant of J^TJ is generally aimed in order to obtain estimates with minimum confidence regions, by utilizing the so-called D-optimum experimental design approach, as described by Beck and Arnold (1977).

RESULTS AND DISCUSSIONS

This estimation problem is non-linear, since the sensitivity coefficients depend on the unknown parameters D* and a (Pereira et al., 1998, 1999). As a result, the following analysis of the sensitivity coefficients and of the determinant of J^TJ is local and dependent on the values of D* and a assumed for the simulation. For the results presented below, we assumed D*=1x10^-9 m²/s, a = 0.02 m and l = 5.426x10^-6 s^-1 (corresponding to Br82).

Figures 1.a and 1.b show the time variation of the salt concentration measured at different points in the column for experiments using the radiation measurement technique and the electrical conductivity measurement technique, respectively. Figure 1.a shows that the measured concentration approaches zero for long periods of time. In addition, the concentration is practically zero at measurement points far from the region initially filled with saline solution, such as for z=0.06 m. Therefore, despite the diffusion process taking place inside the column, the radiation measurement technique may not be able to detect the salt concentration as a result of the radioactive decay. This behavior is not observed in Figure 1.b, where electrical conductivity cells are used to measure concentration. In fact, we can observe in Figure 1.b values for the measured concentration at the different points in the column generally higher than those in Figure 1.a.

The effect of radioactive decay can also be clearly observed by comparing Figures 2.a and 2.b, which show the normalized sensitivity coefficients with respect to D* for the radiation measurement technique and the electrical conductivity technique, respectively. The normalized sensitivity coefficients are obtained by multiplying the sensitivity coefficients by the parameters to which they refer. The use of normalized sensitivity coefficients in the analysis is appropriate to detect small magnitudes and linear dependence when parameters of different orders of magnitude come into the picture, as in the present case. Figure 2.a shows that the normalized sensitivity coefficients are small for long periods of time and for measurement points far from the region initially filled with saline solution, as a result of the radioactive decay. In fact, although the behaviors for the normalized sensitivity coefficients are similar in Figures 2.a and 2.b, i.e., the sensitivity coefficients are negative inside the region initially filled with saline solution and positive outside it, the sensitivity coefficients for Figure 2.b (electrical conductivity measurement technique) are generally larger and do not decay for long periods of time. The sensitivity coefficients for D* are negative at the points inside the region initially filled with saline solution because an increase in D* tends to decrease the concentration there. On the other hand, an increase in D* tends to increase the concentration at points not inside the region initially filled with saline solution. A comparison of Figures 1.a to 2.a and 1.b to 2.b reveals that the normalized sensitivity coefficients for D* are of the same order of magnitude of the measured concentrations with both measurement techniques. Hence, the measured concentrations can provide useful information for the estimation of D* by inverse analysis.

Figures 3.a and 3.b show the normalized sensitivity coefficients for the length in the column initially filled saline solution, a, for the radiation measurement technique and the electrical conductivity technique, respectively. As with the sensitivity coefficients for D* shown in Figures 2.a and 2.b, in Figures 3.a and 3.b we observe that the sensitivity coefficients for a do not decay for long periods of time and for measurement positions far from the region initially filled with saline solution when the electrical conductivity measurement technique is utilized in the analysis. Figures 3.a and 3.b show that the relative sensitivity coefficients for a are generally of the same order of magnitude of the measured concentrations (see Figures 1.a and 1.b). Therefore, the measured concentration is as sensitive to variations in the initial length filled with saline solution, a, as to variations in the apparent mass diffusion coefficient, D*, and this conclusion does not depend on the technique utilized to measure concentration. This fact reveals that, since there is some uncertainty in the volume of the solution injected, a needs to be estimated together with the apparent mass diffusion coefficient. A comparison of Figures 2.a and 3.a, as well as of Figures 2.b and 3.b, shows that the sensitivity coefficients for D* and a are not linearly dependent.

The foregoing analysis of the sensitivity coefficients reveals that the conditions are favorable for the simultaneous estimation of a and D* with the two measurement techniques examined. The normalized sensitivity coefficients are not linearly dependent and attain magnitudes of the same order as the measured variable.

Figures 4.a and 4.b show the behavior of the determinant of the information matrix, det(J^TJ), for the radiation measurement technique and the electrical conductivity measurement technique, respectively. For the cases shown in Figures 4.a and 4.b, we considered available for the analysis one measurement per sensor every 4 hours. Figures 4.a and 4.b show the curves for det(J^TJ) obtained with a single sensor (z=0.01 m), two sensors (z=0.01 and 0.02 m), three sensors (z=0.01, 0.02 and 0.03 m), and so on. In Figure 4.a we can observe that, for each number of sensors examined, det(J^TJ) increases very fast for short periods of time. However, for times longer than approximately 100 hours, the rate of increase of det(J^TJ) is quite small. Such is the case because the sensitivity coefficients approach zero for long periods of time due to radioactive decay, as can be seen in Figures 2.a and 3.a. We can also see in Figure 4.a that det(J^TJ) increases when more sensors are used in the analysis, since more information becomes available for the estimation of the parameters. However, the use of six sensors instead of five adds very little useful information for the inverse analysis because the curves for det(J^TJ) for five and six sensors are practically identical. This behavior is due to the fact that the magnitudes of the sensitivity coefficients decrease for sensors far from boundary z=0.0 m. The same behavior with respect to the number of sensors is observed with the electrical conductivity measurement technique, as can be seen in Figure 4.b. On the other hand, Figure 4.b shows that det(J^TJ) still increases for long periods of time, although at a lower rate than for short periods, since this technique is not sensitive to the effects of radioactive decay.

A comparison of Figures 4.a and 4.b reveals that more accurate estimates can be obtained with the concentration measurements based on the electrical conductivity technique, since the values for det(J^TJ) are larger than those for the radiation measurement technique.

Let us now consider the effect of the volume of saline solution injected into the column (i.e., of the length initially filled with saline solution, a) on the accuracy of the estimated parameters. Figure 5 shows the transient variation in the concentration at different points in the column for electrical conductivity measurements for a=0.001 m. A comparison of Figures 1.a and 5 shows a large reduction in the concentration in the column, when a is reduced from 0.02 m to 0.001 m. The normalized sensitivity coefficients for D* and a undergo a similar reduction in their magnitudes, as shown in Figures 6.a and 6.b, respectively. The values of det(J^TJ) obtained for a=0.001 m are shown in Figure 7. We observe that det(J^TJ) for a=0.001 m is smaller than it is for a=0.02 m (see also Figure 4.b); hence, a larger injected volume should be used in order to obtain more accurate estimates for the unknown parameters. The behavior discussed above, regarding the use of small volumes injected, is also observed when the radiation measurement technique is used in the analysis.

After addressing the effects of the measurement technique, as well as of the number of sensors and the duration of the experiment, let us now consider the simultaneous estimation of D* and a by using simulated measurements (Ozisik and Orlande, 2000). Table 1 shows the estimated values for the unknown parameters by using simulated measurements with a standard deviation of 1% of the maximum measured concentration. The exact parameters used to generate the simulated measurements were a=0.02 m and D*=1x10^-9 m²/s. Measurements were considered available from six sensors located at positions z=0.01, 0.02, 0.03, 0.04, 0.05 and 0.06 m. The duration of the experiment was 120 hours and we assumed available one measurement per sensor every four hours. The measurement technique for the results presented in Table 1 was the use of electrical conductivity cells. The parameters were estimated by using the Levenberg-Marquardt method of minimization of the least squares norm (Levenberg, 1944; Marquardt, 1963; Beck and Arnold, 1977; Ozisik and Orlande, 2000).

An analysis of Table 1 reveals that accurate estimates can be obtained for the apparent mass diffusion coefficient and for the length of the column initially filled with saline solution, by using the electrical conductivity measurement technique. The estimated values are independent of the initial guess utilized for the iterative procedure of the Levenberg-Marquardt method. Also, as expected from the analysis of the sensitivity coefficients and of the determinant of the information matrix, det(J^TJ), the parameters estimated with the electrical conductivity technique are more accurate than those obtained with the radiation measurement technique (Pereira et al., 1999).

CONCLUSIONS

This work dealt with a comparison of techniques for the measurement of concentration, aimed at estimating the apparent mass diffusion coefficient of salts in soils saturated with water.

Analyses of the sensitivity coefficients and of the determinant of the information matrix revealed that more accurate estimates can be obtained using electrical conductivity cells to measure concentration. Also, injection of small volumes of solution into the column should be avoided, because this results in small concentration values in the column and in small values for the sensitivity coefficients.

The accuracy of the estimated parameters was examined by using simulated measurements containing random errors. The present solution approach was able to obtain the exact parameters quite accurately when electrical conductivity measurements were used.

ACKNOWLEDGEMENTS

This work was funded by Comissão Nacional de Energia Nuclear, Brazil (COREJ/CNEN), under a contract with Fundação COPPETEC. Dr. Paulo Heilbron Filho and Dr. Nerbe Ruperti Jr. from CNEN contributed significantly to this work through their important technical suggestions.

NOMENCLATURE

a length of the column initially filled with the saline solution C estimated concentration C₀ concentration of saline solution injected into the column D effective mass diffusion coefficient D* apparent mass diffusion coefficient J sensitivity matrix K retardation factor P vector of unknown parameters S(P) least squares norm Y measured concentration Greeks Symbols l coefficient of radioactive decay Superscripts k number of iterations

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Publication Dates

• Publication in this collection
  10 Oct 2001
• Date of issue
  Sept 2001

History

• Accepted
  13 July 2001
• Received
  19 Apr 2001