Critical assessment of quasi-steady-state method to determine effective diffusivities in alginate gel membranes

The application of the quasi-steady-state (QSS) method for measuring effective diffusivities (D(e)) using the diffusion-cell technique was evaluated critically. Diffusion experiments with glucose and lactic acid across a gel membrane containing immobilized Lactobacillus rhamnosus were used as illustration cases. The correct estimation of D(e) is based on the assumption that a QSS is satisfied. A theoretical error analysis was performed to quantitate the impact of estimating D(e) when some conditions result in the QSS not being fulfilled. These conditions may be caused by: (1) not measuring/calculating the solute mass in the gel (M(m,t)); (2) errors in measurements of solute concentration; and (3) when the volume of sample withdrawal is not factored in. It was found that the errors in estimating D(e) when M(m,t) and the concentration measurement errors were not taken into account would be over 200%, depending on the prevailing conditions. In addition, the errors associated with the sample withdrawal increased linearly with the relative sampling fraction. In order to apply the QSS method, a procedure based on an accurate mass balance has been proposed to qualitatively, and quantitatively, validate the QSS assumption. This procedure uses a statistical t-test to check whether the solute flux-in equals the flux-out. To estimate D(e) accurately, it is essential that the attainment of the QSS is validated, the accuracy of concentration measurement is established, and the effect of sample withdrawal taken into account.