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It can be seen from Eq. (1.15) that the mass extracted is proportional to the sampling time, Dg for each analyte, and the bulk sample concentration and inversely proportional to d. This is consistent with the fact that an analyte with a greater Dg will cross the interface and reach the surface of the coating more quickly. Values of Dg for each analyte can be found in the literature or estimated from physicochemical properties. This relationship enables quantitative analysis. As mentioned above, non-reversible adsorption is assumed. Equation (1.15) can be modified to enable the estimation of the concentration of analyte in the sample for rapid sampling with solid sorbents:

where the amount of extracted analyte (n) can be estimated from the detector response.

The thickness of the boundary layer (d) is a function of the sampling conditions. The most important factors affecting d are the geometric configuration of the extraction phase, the sample velocity, temperature, and Dg for each analyte. The effective thickness of the boundary layer can be estimated for the coated fibre geometry by the use of Eq. (1.17), an empirical equation adapted from the heat transfer theory [1]:

where Re is the Reynolds number — 2usb/v, us is the linear sample velocity, v is the kinematic viscosity of the matrix, b is the outside radius of the fibre coating, and Sc is the Schmidt number — v/Ds. The effective thickness of the boundary layer in Eq. (1.17) is a surrogate (or average) estimate and does not take into account changes of the thickness that can occur when the flow separates, when a wake is formed, or when both occur. Equation (1.17) indicates that the thickness of the boundary layer will decrease with increasing linear sample velocity. Similarly, when the sampling temperature (Ts) increases, the kinematic viscosity decreases. Because the kinematic viscosity term is present in the numerator of Re and in the denominator of Sc, the overall effect on d is small. Reduction of the boundary layer and an increased rate of mass transfer for the analyte can be achieved in two ways—by increasing the sample velocity and by increasing the sample temperature. Increasing the temperature will, however, reduce the efficiency of the solid sorbent (reduce K). As a result, the sorbent coating might not be able to adsorb all of the molecules reaching its surface and it might, therefore, stop behaving as a 'perfect sink' for all of the analytes.

Further developments were made to provide accurate estimates of analyte concentrations using diffusion-based rapid SPME, and to this end a new mass transfer model was proposed [26] and this is illustrated in Fig. 1.10. When an SPME fibre is exposed to a fluid sample whose motion is normal to the axis of the fibre, the fluid is brought to rest at the forward stagnation point from which the boundary layer develops with increasing x under the influence of a favourable pressure gradient. At separation point, downstream movement is checked because fluid near the fibre surface lacks sufficient momentum to overcome the pressure gradient. In the meantime, the oncoming fluid also precludes flow back upstream. Boundary layer separation thus occurs, and a wake is formed in the downstream, where flow is highly irregular and can be characterized by vortex formation. Correspondingly, the thickness of the boundary layer (d) is minimum at the forward stagnation point. It increases with the increase of x and reaches its maximum value right after separation point. In the rear of the fibre where a wake is formed, d again decreases.

Instead of calculating d, the average mass transfer coefficient was used to correlate the mass transfer process. According to Hilpert

Boundary layer

Fig. 1.10. Schematic of rapid extraction with an SPME fibre in cross flow.

Boundary layer

Fig. 1.10. Schematic of rapid extraction with an SPME fibre in cross flow.

[27,28], the average Nusselt number Nu can be calculated by the following equation:

where h is average mass transfer coefficient, d is the outside diameter of the fibre, D is diffusion coefficient, Re is the Reynolds number, and Sc is the Schmidt number. Constants E and m are dependent on Reynoldsnumber and are listed in Table 1.2 [28].

Once hi is known, the amount of extracted analytes dn during sampling period dt can be calculated by the following equation:

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