Transport modeling of membrane extraction of chlorinated hydrocarbon from water for ion mobility spectrometry

Abstract

Membrane-extraction Ion Mobility Spectrometry (ME-IMS) is a feasible technique for the continuous monitoring of chlorinated hydrocarbons in water. This work studies theoretically the time-dependent characteristics of sampling and detection of trichloroethylene (TCE). The sampling is configured so that aqueous contaminants permeate through a hollow polydimethylsiloxane (PDMS) membrane and are carried away by a transport gas flowing through the membrane tube into IMS analyzer. The theoretical study is based on a two-dimensional transient fluid flow and mass transport model. The model describes the TCE mixing in the water, permeation through the membrane layer, and convective diffusion in the air flow inside membrane tube. The effect of various transport gas flow rates on temporal profiles of IMS signal intensity is investigated. The results show that fast time response and high transport yield can be achieved for ME-IMS by controlling the flow rate in the extraction membrane tube. These modeled time-response profiles are important for determining duty cycles of field-deployable sensors for monitoring chlorinated hydrocarbons in water.

Appendix A - TCE partition coefficients

As discussed in the main text, the apparent permeability P is a lumped parameter used to conveniently describe the TCE partitioning, diffusion and desorption in membrane. In particular, its absolute value is dependent on the water-PDMS and air-PDMS partition coefficients for TCE, which are chosen to be 1 in the current model. Different K value will change the absolute TCE concentration in air but will not change the relative shape, as described in the following. Rewriting Eq. 3 results in:

$$ {J_{ss}} = {K_w}P\left( {{c_w} - \frac{{{K_a}{c_a}}}{{{K_w}}}} \right) = P\prime \left( {{c_w} - c_a^\prime } \right) $$

(A1)

where P′ = K_w P and $c_a^\prime = \frac{{{K_a}{c_a}}}{{{K_w}}}$. Dividing both sides of Eq. 8 by K _w and multiplying by K _a , it can be rewritten as:

$$ \frac{{\partial c_a^\prime }}{{\partial t}} + \frac{{\partial (uc_a^\prime )}}{{\partial x}} + \frac{{\partial (vc_a^\prime )}}{{\partial r}} = {D_a}\left[ {\frac{{{\partial^2}c_a^\prime }}{{\partial {x^2}}} + \frac{1}{r}\frac{\partial }{{\partial r}}\left( {r\frac{{\partial c_a^\prime }}{{\partial r}}} \right)} \right] $$

(A2)

Equations A1 and A2 are identical to Eqs. 3 and 8 except that $c_a^\prime$ and P′ replace c _a and P, respectively. The TCE concentration when K ≠ 1 can be calculated by multiplying the base concentration by $\frac{{{K_w}}}{{{K_a}}}$. Therefore, the relative shape of TCE distribution in air is not affected by the choice of K value.

It is noted that the above derivation is valid under linear condition. If the partition coefficients are concentration dependent, the actual values are required to solve the governing transport equations.