A transient method for determining indoor chemical concentrations. based on SPME: model development and calibration

Transcription

1 Supporting Information A transient method for determining indoor chemical concentrations based on SPME: model development and calibration Jianping Cao 1,, Jianyin Xiong 3, *, Lixin Wang 4, Ying Xu 5, Yinping Zhang 1, 1 Department of Building Science, Tsinghua University, Beijing , China Beijing Key Laboratory of Indoor Air Quality Evaluation and Control, Beijing , China 3 School of Mechanical Engineering, Beijing Institute of Technology, Beijing , China 4 School of Environment and Energy Engineering, Beijing University of Civil Engineering and Architecture, Beijing , China 5 Department of Civil, Architectural and Environmental Engineering, The University of Texas at Austin, Texas , United States * Corresponding Author Address: School of Mechanical Engineering, Beijing Institute of Technology, Beijing , China; Phone: ; Fax: ; xiongjy@bit.edu.cn. This supporting information (14 pages) includes 4 sections, one table and 5 figures. S1

2 Section S1. Discussion about the error introduced by simplifying the cylindrical coating into a flat plate According to Figure 1 (a), the governing equation describing the mass transfer process of chemicals in the cylindrical coating of SPME is expressed as: C C 1 C t r r r m m m = Dm + (S1) where C m is the concentration of chemicals in the coating, µg/m 3 ; D m is the diffusion coefficient of the chemicals in the coating, m /s; t is the time, s; and r is the distance to the axis of SPME fiber, m. Similar to equations () and (3), the boundary conditions can be expressed as: C m = 0, r = R r Cm Cm Dm = hm ( Cin ), r= R+ L r K (S) (S3) where R is the radius of the fused silica (i.e., the inner radius of the coating), m; L is the thickness of the coating, m; C in is the concentration of chemicals in the sample matrix (liquid or gas phase), µg/m 3. The initial condition is: C ( r, t ) = 0, t= 0, R r R+ L (S4) m By solving equations (S1)-(S4), C m can be obtained. Then the amount of chemical adsorbed onto the coating can be obtained by: M t = rhc r t dr (S5) ( ) R + L π (, ) R where H is the length of SPME coating. Comparing M calculated by equation (S5) and that by equation (9) (or equation (10)), the error introduced by unfolding the cylindrical coating into a flat plate (ε) can be obtained. Dimensionless analysis of these two models (equations (S1)-(S5) and equations (1)-(9)) shows that M is a function of L/R, Bi m /K and Fo m, where Bi m is the Biot number of mass transfer (Bi m = h m L/D m ); and Fo m is the Fourier number of mass S m

3 transfer (Fo m = D m t/l ). Given that R is equal to 55 µm for commonly used SPME, Figure S1 shows the maximum deviation of M calculated by equation (S5) and that by equation (9) when increasing Fo m from 0 to infinity for different values of L (in the range of 1 µm and 100 µm) and Bi m /K (in the range of ). We can find from this figure that, when L is less than 7 µm, the maximum ε is less than 5% for all values of Bi m /K (10-3 ~10 5 ). It indicates that the treatment of unfolding the cylindrical coating into a flat plate is reasonable when L is small enough (L is much smaller than R, i.e., L/R 7/55). Figure S shows the comparison of M between calculated results of equation (S5) and that of equation (9) for different Fo m, when L = 7 µm and Bi m /K = 7. The good agreement also gives confidence in the treatment. Section S. Determination of the applied condition for equation (1) The followings are detailed expression of equations (10) and (1): sin q M ( t) = M M e n Dm L qnt equ equ (10) n= 1 qn sin qn+ qn cos qn q n are the positive roots of the following equations: h L q q = n= (11) m n tan n ( 1,,...) Dm K sin q 1 Dm L q1 t M ( t) = M equ 1 e q1 sin q1+ q1 cos q1 (1) Based on the principle of simplifying equation (10) into equation (1): if the time is sufficiently long, only the first term of equation (10) is significant, it is obvious that the deviation of M between equations (10) and (1) will decrease as time increases. If these equations are transformed to the dimensionless form, we get: M = 1 sin q n qn Fom e (S6) n= 1 qn sin qn+ qn cos qn q n are the positive roots of the following equations: Bim qn tan qn = ( n= 1,,...) (S7) K S3

4 Equation (S6) can be simplified into: sin q M = 1 e q q q q 1 1 sin 1+ 1 cos 1 qn Fom (S8) where, M * = M/M equ ; Bi m = h m L/D m, the Biot number of mass transfer; and Fo m = D m t/l, the Fourier number of mass transfer. From equations (S6) and (S8) we can see that the dimensionless mass of chemicals in the coating of SPME, M *, is a function of Bi m /K and Fo m. Therefore, the detailed procedure to determine the condition for simplifying equation (10) into equation (1) is as follows: (1) Calculate the relative deviation (= ǀM * (S6)-M * (S8)ǀ/M * (S6) 100%) of M * between equations (S6) and (S8) for a given Bi m /K by increasing the Fo m from zero with an appropriate interval (10-4 is used in our calculation). The relative deviation will decrease as Fo m increases; () Stop the calculation of step (1) when the relative deviation is no more than 5%, and the biggest value of Fo m is the applied condition, Fo m,c, for equation (1) for the given Bi m /K; (3) Repeat steps (1) and () for another value of Bi m /K. In our calculation, the value of Bi m /K is in the range of 10-3 to 10 5, and the selection principle of Bi m /K is increasing Bi m /K from 10-3 with an interval of ; (4) The biggest value of Fo m,c is treated as the applied condition for Bi m /K in the range of 10-3 to Figure S3 shows the determined Fo m,c for Bi m /K in the range of 10-3 to As shown, when Fo m > 0.1 (i.e., t 0.1L /D m ), the relative deviation is less than 5% when simplifying equation (10) into equation (1). In addition, this applied condition would be suitable for any value of Bi m /K, because Figure S3 shows that Fo m,c is close to zero as Bi m /K decreases from while close to 0.08 as Bi m /K increases from S4

5 Section S3. Analytical solution of the model of an unfolded cylindrical coating Applying the method of variable of separation, the analytical solution of equations (S1)-(S4) can be obtained: where J ( ) 1 βna Dmβn t ( βn, ) (S9) n= B J ( β a) V C = KC + πqkc R r e m in in n 0 ( β, ) ( β ) ( β ) R r = S J r V Y r (S10) 0 n 0 0 n 0 0 n ( ) ( β ) S = β Y β b + QY b (S11) 0 n 1 n 0 n ( ) ( β ) V = β J β b + QJ b (S1) 0 n 1 n 0 n h m Q= (S13) D K m B = Q + β (S14) n J and Y represent the Bessel functions of the first kind and the second kind, respectively. The subscripts, 0 and 1 represent the order of the Bessel functions. a and b represent the inner radius and the external radius of the SPME coating, respectively (i.e., a = R and b = R+L). β n are the roots of the following equations: ( β ) ( β ) ( ) S0J1 na V0Y1 na = 0 n= 1,,3, K (S15) Combining equations (S5) and (S9), we get: 1 ( βna) ( β ) 4Q J Dmβn t M = KCinVm 1 e b a n= 1 BJ1 na V (S16) 0 where V m is the volume of the SPME coating, m ( ) V = π H b a (S17) Similarly to equation (1), equation (S17) can also be simplified when the time is sufficiently long: 1 ( β1a) ( β ) M = KC V 4Q J e = M e in m 1 Dmβ1 t t equ 1 b a BJ1 1a V 0 S5 γ ( ϕ ) (S18)

6 where, M equ = KC in V m, φ = 4Q J1 ( β1a) ( b a )( BJ1 ( β1a) V0 ) Dmβ1, β 1 is the first positive root of equation (S15)., and γ = Comparing equations (S16) and (10), it is easy to find that equation (S16) is much more complicated than equation (10). Section S4. Using the calibration method to analyze data from the literature The main text analyzes the effectiveness of the calibration method for DBP and DEHP (SVOCs) tests. There are many experimental data reported in the literature for other SPME-chemical combinations. To further validate the calibration method, we have used data from two references as examples. 1, Bartelt and Zilkowski measured the time-dependent extraction of methyl tridecanoate with PDMS fiber coating (thickness: 100 µm) of the SPME. Tuduri et al. 1 tested the time-dependent extraction of many different chemicals using a polydimethylsiloxane-carboxen (PDMS-CAR) fiber coating (thickness: 75 µm) of the SPME. We will analyze only the results for p-xylene and toluene measured in static mode (i.e., the liquid in the main flow region is static). Since the thickness of the fiber coating in these extraction scenarios is comparable to (actually, larger than) the radius of the fused silica (55 µm), we would like to find out whether the proposed analytical model and calibration method is still valid in these circumstances. By applying the procedures described in the section Analytical model and calibration method to the experimental data, we can obtain the characteristic parameters (D m and K) of the PDMS/methyl tridecanoate, PDMS-CAR/p-xylene and PDMS-CAR/toluene combinations. The results are summarized in Table S1. This table shows that the values of R are all greater than 0.99, indicating high regression accuracy. Figure S4 gives the fitted curves of equation (1), and the comparison between the model predictions using equation (10) based on the determined characteristic parameters and the experimental data. For the three sets of SPME fiber-chemical combinations, the fitted curves of equation (1) S6

7 together with the model predictions all agree well with the experimental data, which further verifies the reliability of the calibration method. In addition, K for PDMS/methyl tridecanoate combination was determined to be by Bartelt and Zilkowski, which is consistent with the determined K with the proposed method as listed in Table S1 (relative deviation is about %). This also provides a confirmation on the reliability of the present method. Table S1. Determined characteristic parameters for different SPME-chemical combinations in the literature Fiber coating Chemicals D m (m /s) K R PDMS methyl tridecanoate a PDMS-CAR p-xylene b PDMS-CAR toluene c a h m is calculated as m/s by virtue of equation (4); b h m is calculated as m/s by virtue of equations (6) and (7); c h m is calculated as m/s by virtue of equations (6) and (7). S7

8 Figure S1. Maximum deviation between M calculated by equation (S5) and that by equation (9) when increasing Fo m from 0 to infinity for different values of L (in the range of 1 µm and 100 µm) and Bi m /K (in the range of ). S8

9 Figure S. Comparison of M between the results of equation (S5) and that of equation (9) for different Fo m, when L = 7 µm and Bi m /K = 7. S9

10 Figure S3. The determined Fo m,c (condition of using equation (1)) for Bi m /K in the range of 10-3 to S10

11 Figure S4. Comparison of the model predictions with the experimental data from the literature: (a) PDMS/methyl tridecanoate; (b) PDMS-CAR/p-xylene; (c) PDMS-CAR/toluene. (a) (b) S11

12 (c) S1

13 Figure S5. Sensitivity analysis of model parameters on sorption of chemicals onto the SPME coating (the baseline parameters are set as: K = , D m = m /h, h m = 0.01 m/s, and C in = 1 µg/m 3 ). (a) h m ; (b) K (M equ represents the equilibrium amount of chemicals extracted onto the coating); (c) D m. (a) (b) S13

14 (c) References 1. Tuduri, L.; Desauziers, V.; Fanlo, J. L. Dynamic versus static sampling for the quantitative analysis of volatile organic compounds in air with polydimethylsiloxane Carboxen solid-phase microextraction fibers. J. Chroma. A 00, 963 (1), Bartelt, R. J.; Zilkowski, B. W. Nonequilbrium quantitation of volatiles in air streams by solid-phase microextraction. Anal. Chem. 1999, 71 (1), S14