A Michaelis-Menten rate model for the electrodialysis of concentrated salts

Abstract

• A modified Michaelis-Menten model is presented for electrodialysis (ED). • Kinetics with concentrated salts are different than at low salt concentrations. • Electrodialysis of concentrated salts appear to follow saturation kinetics. • A standard electrical resistance model failed to adequately model the process. • The chemistry of the electrode rinse solution impacts the process kinetics. Volt-Amp profiles are used to describe electrodialysis (ED) dynamics when the concentration of the diluate is equal to the concentration of the concentrate and both are at high concentration. A standard electronic model that treats the ED process as a series of resistors failed to describe the performance of a 10-cell pair, 200 cm 2 per membrane pilot electrodialysis unit treating 0.5–6% NaCl. The standard electronic model resulted in a regression coefficient of R 2 < 0.76 and underestimated the observed system resistance by 0.54 to 1.1 Ohms depending on the conductivity of the test water. However, if the overall process was treated as a saturated chemical reaction, such as a modified Michaelis-Menten function, the regression exceeded R 2 > 0.999. The current was plotted as a function of the conductivity of the salt at a constant conductivity of electrode rinse solution. Each curve was a function of the applied voltage. The value of the saturation coefficient for the system plots as {D sat } ∼ 22.1 mS/cm (2.21 S/m). The kinetic coefficient i MAX plotted as a linear function of the effective voltage at 25 °C (298°K). More importantly, a series of tests with a constant feed conductivity and with varied electrode rinse solution conductivity also plotted as Michaelis-Menten type curves. The saturation coefficient for the electrode rinse solution also fitted well to {E sat } ∼ 22.1 mS/cm (2.21 S/m) and the values for i MAX were also linear with respect to applied voltage. Overall, the results from 12 data sets follow the formula i = αβi MAX where α and β are the ratios of the conductivity of the fluid divided by the sum of the conductivity plus the saturation, e.g., for electrode rinse solution α = {E}/({E} + {E sat }) and for the electrode rinse β = {D}/({D} + {D sat }), indicating the co-dependence of the feedstock and electrode rinse in the overall process dynamics. This interpretation is in opposition to the standard modeling of ED where the process is viewed as a series of resistances and the electrode rinse resistance is usually determined to be negligible. These data suggest that the effects of saturated feedstock and saturated rinse solution, measured as current or ion transport, are inextricably co-dependent. Three full-batch test runs of between 7.2 and 9-hr duration using different initial feedstock and electrode rinse water conditions were analyzed to estimate the relative effects of diluate and concentrate concentration on the test results. A third saturation coefficient was introduced, γ = {C}/({C} + {C sat }) where the terms {C} and {C sat } represent the conductivity and the saturation coefficient of the concentrate. The model for long duration batch tests, including the different effects of the concentrate and diluate, is well represented by i = (αβ 0.8 γ 0.2 i MAX ). The empirical power coefficients, β 0.8 and γ 0.2 , indicate that the diluate conductivity carries more influence on the rate of the process than does the conductivity of the concentrate. The model has the potential to allow for the scale-up of the ED process using at most one adjustable parameter (the saturation coefficient of the membrane measured as mS/cm or S/cm). Michaelis-Menten kinetics and the modifications by others represent a rich literature on the modeling of different types of inhibition, topics of concern for membrane transport phenomenon.