Model-free estimation of mass fluxes based on concentration profiles. II. Effect of noise on concentration profiles and of time resolution
Introduction
Water diffusivity in food materials has been the object of numerous studies and data compilations (Gekas, 1992, Krokida et al., 2001, Maroulis et al., 2001, Mittal, 1999, Saravacos, 1995, Zogzas et al., 1996). However the reported values vary considerably from one study to another, which limits confidence and re-use. The most commonly given reasons for this variability are (i) the identification of the diffusivity coefficient using a simple diffusion law with constant diffusivity, thus neglecting the multiple driving forces for water transport in food and the physicochemical changes occurring in food as a result of mass transport, and (ii) the lack of knowledge and equipment for accurately characterizing the concentration at the boundary. Clearly, new approaches need to be developed to better characterize the uncertainty arising from the method used for the identification of diffusivity. This can be done by generating a set of concentration data with known state equations and transport property values and applying the estimation method to the set of virtual data. The error on estimating D can then be fully quantified by comparing the estimated and expected values. The error can obviously be affected by many factors that merit investigation, such as the use of an inappropriate transport mechanism or the spatial or temporal resolutions for the set of concentration data.
Such an approach has been applied to a model-free estimation of diffusivity in various case studies (Bardow et al., 2005, Bardow and Marquardt, 2004, Lucas and Bohuon, 2005). Bardow and Marquardt (2004) and Bardow et al. (2005) developed this type of approach for the identification of diffusivities in solutions using experimental concentration data obtained by Raman spectroscopy. Their method gives very good results in the case of high spatial and temporal resolutions, such as those provided by Raman spectroscopy, but is not compatible with solid food systems. In the first part of the present study (Lucas & Bohuon, 2005), we proposed a similar approach for use with diffusion phenomena in solid food systems and tested it on a simple configuration with constant diffusivity and no noise added to the concentration data.
This second part of the study extends the approach to more realistic and complex conditions, still focusing only on diffusive transport. The handling of noisy concentration data requires the calculation of smooth curves passing through the data points before mass fluxes and concentration gradients are calculated (calculation steps using derivatives were expected to be particularly affected). The main reason for choosing a smoothing function rather than another type of function was to avoid implying that the function includes any (transport) properties as is the case when exponential functions are used in diffusion studies. Cubic polynomial spline functions were selected for this purpose. Another advantage of splines is the local character of the approximation. This means that the result at a given point does not depend on the whole dataset, only on data adjacent to the point (de Boor, 1978). Two previous studies of diffusion phenomena have already demonstrated the efficiency of spline functions in “smoothing” concentration profiles as a step preceding other calculation steps in the final estimation of diffusion coefficients (Bardow et al., 2005, Kapoor and Eagar, 1990, Kokkonis and Leute, 1996). Secondly, diffusivity is known to be concentration- and composition-dependent, due to effects of viscosity, thermodynamic behaviour, etc. A linear variation of D with concentration was selected as a case study. Handling data generated with variable D requires separate representations of mass fluxes against concentration gradients at different levels of concentration. Noise on concentration data was expected to jeopardize the differential analysis, so this effect was investigated.
Section snippets
Virtual experimental concentration data
For a plane sheet of thickness L with no mass flux at z = 0, and with the opposite surface (z = L) exposed at constant concentration c∞, the dimensionless concentration was simulated in the region 0 < z < L, as a function of Fourier number (Fo = Dt/L2) and dimensionless coordinate in two cases: with constant diffusion coefficient (D) and with variable diffusion coefficient. The dimensionless concentration was defined as , where c0 = c(z, 0)∀z is the initial uniform concentration
Estimation of mass fluxes
The mass fluxes as function of time t and space z, noted J(z, t), were estimated from the concentration profile by mass balances. The corresponding dimensionless mass flux, noted , was stated (Lucas & Bohuon, 2005) aswithEq. (10) was numerically solved in two steps (Fig. 1). In the first step, was calculated as the direct analytical integral of the cubic smoothing spline function applied on . In the second step, at
Benefits and limits of smoothing procedure in the noise-free-case
The effect of the smoothing steps on the error on D was evaluated against the errors previously obtained without any smoothing steps in the calculation process (Lucas & Bohuon, 2005). In other words, the evaluation was performed in the absence of noise on the concentration data (a = 0) and with a single spatial resolution (Fig. 2). In practical terms, 10 appeared to be the appropriate number of concentration data per profile with uniform distribution: with slice sampling, it corresponds to ten 2
Conclusion
This second part of the study adapts an inverse mass diffusion method to the case of noisy concentration data and evaluates its performance in identifying the diffusion coefficient.
In the case of constant diffusivity, nFo = 10 and are typical values used with the conventional slicing method for determining concentration profiles. Using the proposed method, the error in estimating D with constant diffusivity was about 10% for the highest noise intensities (8%) and could be reduced for lower
Acknowledgement
The authors thank Garth Evans for his assistance with the English.
References (13)
- et al.
Concentration-dependent diffusion coefficients from a single experiment using model-based Raman spectroscopy
Fluid Phase Equilibria
(2005) - et al.
Identification of diffusive transport by means of an incremental approach
Computers & Chemical Engineering
(2004) - et al.
Least squares splines approximation applied to multicomponent diffusion data
Computational Materials Science
(1996) - et al.
Model-free estimation of mass fluxes based on concentration profiles. I. Presentation of the method and of a sensitivity analysis
Journal of Food Engineering
(2005) The mathematics of diffusion
(1986)A practical guide of splines
(1978)