- Email: [email protected]
Theoretical approach to non-constant uptake rates for tube-type diffusive samplers
Talanta 54 (2001) 703– 713 www.elsevier.com/locate/talanta
Theoretical approach to non-constant uptake rates for tube-type diffusive samplers Bala´zs Tolnai a, Andra´s Gelencse´r b, Jo´zsef Hlavay a,* a
Department of Earth and En6ironmental Sciences, Uni6ersity of Veszpre´m, P.O. Box 158, H-8201 Veszpre´m, Hungary b Air Chemistry Group of the Hungarian Academy of Sciences, P.O. Box 158, H-8201 Veszpre´m, Hungary Received 31 August 2000; received in revised form 4 January 2001; accepted 5 January 2001
Abstract A simple theoretical model was developed for evaluating the validity of the simplified uptake model of diffusive sampling. In the model based on the plate theory diffusion to the adsorbent surface, phase equilibrium of the adsorbate and mass transport in the adsorbent bed were considered. It was found that in the early stage of sampling, the rate of sampling is close to its theoretical value. As sampling progresses, the concentration increases and the mass transfer front gradually moves into the adsorbent layer. Above a certain threshold limit, the mass uptake becomes a steady state process in which the diffusion in the air gap and the mass transport in the adsorbent bed are balanced. As uptake is a cumulative process, sampling should continue long enough to render the effects of these initial changes negligible. That is why constant uptake rates can still be obtained above a critical exposure dose. This critical exposure dose should be exceeded both in the determination of uptake rates and outdoor measurements, to obtain consistent and reliable analytical data. Evaluation of the time and concentration dependence of uptake rate in laboratory experiments and the time dependence of uptake rate in filed test was performed to justify the model results. Since the determination of uptake rates always takes places in the laboratory, where the exposure time is much shorter and the concentration is much higher than in the environment, the uptake rates are thus overestimated by 10–30%. Therefore, the uptake rates should be determined in the field under ambient conditions by means of an independent reference method. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Diffusive sampling; Tube-type sampler; Uptake rate; Volatile organic compounds
1. Introduction In the past few decades, the concentration of volatile organic compounds (VOC) in air, espe* Corresponding author. Tel.: +36-88-422022, ext. 4368; fax: +36-88-423203. E-mail address: [email protected] (J. Hlavay).
cially in urban locations, increased significantly. Several VOCs at the level existing in urban air have mutagenic, carcinogenic and teratogenic effects on humans and can cause acute and chronic diseases. Several GC-compatible air-sampling techniques are available for the determination of these compounds in air. Diffusive sampling, where air pollutants are collected on a sorbent by
0039-9140/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 9 - 9 1 4 0 ( 0 1 ) 0 0 3 2 2 - 8
B. Tolnai et al. / Talanta 54 (2001) 703–713
704
diffusion, is becoming more popular in environmental monitoring. The increasing acceptance of this technique is due to the increased sampling time, its low price, simplicity and unattended operation. The method is capable of delivering timeweighted average concentrations over a relatively long period of time. Diffusive sampling is a method in which volatile organic compounds are taken up by an adsorbent from the air at a rate controlled by diffusion. The differential equation describing the uptake by the adsorbent is the following [1]: dm D · A = (C0 −Ca) dt L
(1)
where m is the mass of the compound adsorbed (g); t is the exposure time (s); A, the cross sectional area of diffusion path (cm2); D, the diffusion coefficient of the compound in air (cm2 s − 1); L, the length of diffusion path (cm); C0 is the ambient concentration of the compound (g cm − 3); and Ca is the concentration of the compound above the adsorbent (g cm − 3). Assuming that the adsorbent acts as a perfect sink (Ca =0) and steady-state conditions hold, Eq. (1) is simplified to: C0 =
m ·L t ·D ·A
(2)
By introducing the term of uptake rate (u): u=
D ·A L
(3)
u: uptake rate (cm3 s − 1) and substituting Eq. (3) into Eq. (2) we get: C0 =
m t ·u
(4)
With this simple equation, the concentration of an analyte can be calculated from the adsorbed mass as determined by gas chromatography, the sampling time and the uptake rate. Ideally, uptake rates are constants that can be calculated from the geometry of the sampler and available diffusion coefficients. It was shown, however, that experimental uptake rates were significantly different from the theoretical values [2– 5]. The literature on the uptake rates of compounds for a given
adsorbent and sampler geometry is similarly contradictory. For benzene, for example, deviations up to 300% can be found in the literature [3–11]. This fundamental controversy in the literature may result from the differences in the conditions and methods in the laboratory. Furthermore, environmental conditions are substantially different from the controlled conditions in the laboratory, especially in terms of variability and concentration levels. Although some parameters can be varied in a laboratory trial, it is impossible to simulate environmental conditions. In a previous work [12], the reliability of diffusive sampling and the validity of its simplified uptake model were studied on site with benzene, toluene, ethylbenzene and m,p-xylene (BTEX compounds) on three different adsorbents (Tenax GR, Tenax TA, Carbopack). It was found that a critical exposure dose, which is the product of the time-weighted average concentration and sampling time, should be exceeded to obtain consistent and reliable analytical data and to ensure that the simplified uptake theory holds. Above this dose, the simplified uptake theory can be used with a fair degree of accuracy. The simplified model of diffusive sampling above this exposure dose seems to satisfy the requirements of environmental monitoring. Others [13] found that the uptake rate decreased exponentially with exposure dose at the initial stage of sampling. Non-constant uptake rates were reported for tube-type samplers [14–17]. It was found that the observed sampling rate was inferior to the theoretical value [18] and decreased as sampling proceeded. In our previous study [12], we implied that the reduced sampling efficiency is a function of both concentration and time. It is anticipated that the phenomenon of decreasing uptake rates is associated with the use of non-ideal adsorbents. In view of the effective desorption process required for on-line GC analysis, the use of strong adsorbent is not possible. Quite soon after the initial period of the sampling, the concentration of the analyte in the air layer adjacent to the adsorbent surface will not be zero. Theoretical treatment of this non-ideal situation is rather complex since it should account for the phase equilibrium of the adsorbate as well as for the mass transport in the adsorbent [19,20].
B. Tolnai et al. / Talanta 54 (2001) 703–713
A possible way to eliminate the theoretical problems might be to find a numerical solution to the differential Eq. (1), which can include all variables and parameters at any desired level of complexity. The results can give insight into the applicability of the simplified model. Since, in any application, only the use of the simplified diffusive uptake model is feasible, to have a prior knowledge on its limitations is a must. Several researchers have used computer programs for modelling mass uptake on diffusive samplers [14,20,21]. Coutant et al. [14] used sinusoidal function to model the effect of concentration variations on uptake rate on badge-type diffusive samplers. Posner et al. [15] also developed a mathematical description for the sampling on badge-type diffusive samplers and performed laboratory experiments at very high concentration range (mg/m3) to examine the sampling rate and reverse diffusion. van den Hoed et al. [21] used a computer model for the estimation of the uptake rates of tube-type diffusive samplers and Nordstrand et al. [20] made a computer program for simulating the performance of tube-type diffusive samplers, however they did not perform laboratory experiments. The previous studies used concentration levels of : 100 ppm for relatively short sampling times
705
and did not observe nearly constant mass uptake over a critical threshold. Moreover, none of these theoretical studies determined how the results translate into the simplified uptake process of diffusive sampling that is the measure of its applicability. The objective of this work is to model the uptake process of a tube-type diffusive sampler and to test the model experimentally, both in the laboratory and on the field. Understanding the process of uptake may help to establish the limits of the simplified model. Although the model is theoretical, its parameters are experimental data taken from the literature and its focus is kept on the environmental applications. For this reason, the effect of variation in ambient conditions is also considered.
2. Theoretical considerations A computer program was made to evaluate the uptake process of tube-type diffusive samplers. The principle of the rather simplified model is shown in Fig. 1. Three basic processes were taken into account: (a) diffusion to the surface of the adsorbent; (b) phase equilibrium of the adsorbate; and (c) mass transport within the adsorbent bed
Fig. 1. The model of the tube type diffusive sampler.
706
B. Tolnai et al. / Talanta 54 (2001) 703–713
Fig. 2. The mass adsorbed on the tube-type diffusive sampler and the changes in the calculated uptake rate of benzene as a function of exposure dose at constant (0.5 mg m − 3) ambient benzene concentration.
[20]. To simplify the calculations, the plate theory was used and the cells were considered as if they consisted of two continuous phases; the adsorbent phase and the interstitial air. Nordstrand et al. [20] also divided the adsorbent into slices that were calculated from the appropriate adsorption isotherm. For the numerical calculation in our study, the adsorbent was divided into 80 cells according to the theoretical plate model. This simplification is widely accepted in the models of chromatographic processes. In the air phase, only diffusion was taken into account. In the literature, several types of isotherms are used for describing the phase equilibrium between the adsorbent and the interstitial air in diffusive sampling. The Freundlich isotherm [20], the Dubinin– Radushkevich isotherm [20] and the linear part of the Langmuir isotherm [14,15] are equally used. In our study, to approximate the phase equilibrium, the linear part of the Langmuir isotherm was used. The linear part of Langmuir isotherm can be a reasonable approximation for a system, which is far from being saturated during sampling. A further sim-
plification was that only a single adsorption layer was assumed and pore diffusion was not taken into account. On the other hand, mass transport in the adsorbent bed was allowed [19,20] and the process was considered as quasi-diffusion. This is the case of mobile adsorption, when the adsorbent is not strong enough and the adsorption heat is not excessively high [22]. The simulations were performed for benzene. The diffusion coefficient of benzene was taken from the literature [23]. The parameters of standard Perkin– Elmer diffusive samplers, as cross-sectional area of diffusion path, A= 0.2 cm2, length of diffusion path, L=1.5 cm, mass of the adsorbent, madsorbens = 250 mg were used in the simulation. Tenax TA adsorbent was chosen for the evaluation. The partition coefficient K (slope of the linear part of the Langmuir isotherm) for benzene on Tenax TA adsorbent was calculated from breakthrough measurements of active sampling [22]. One week sampling time was simulated in steps of 0.02 s to bypass the numerical instability of the model [24,25].
B. Tolnai et al. / Talanta 54 (2001) 703–713
3. Experimental section The laboratory experiments were conducted with a vapour generator, which is described in details elsewhere [26]. A controlled stream of nitrogen, controlled by a Mass Track 810C (Sierra Instruments), is drawn at a low flow-rate through a thermostated quartz container filled with the individual standard substance, thermostated at −1690.1°C with a SCINICS CH-201 cool circulator (SCINICS, Japan) thermostat. In the quartz vessel, continuous stream of saturated vapour of the substance was produced. This stream was diluted in a mixing chamber by a large stream of pure nitrogen gas that was controlled using a Mass Track 810C (Sierra Instruments) mass flow controller (0– 70 l h − 1). The mixing chamber was also thermostated at −16 90.1°C with a SCINICS CH-201 cool circulator (SCINICS) thermostat. Depending on the carrier flow-rate of the dilution gas, different concentrations of the organic vapour can be prepared. Special PTFE tubes (E-06376-03, Kurt J. Lesker Co.) and PTFE fittings were used to eliminate the wall adsorption. The stability of the concentration generated was continuously checked using a piezoelectric chemical sensor system connected to the vapour generator device [27]. Samples were taken by diffusive sampling with standard Perkin– Elmer sample tubes (A=0.2 cm2; L=1.5 cm) filled with 250 mg of Carbopack B 60/80 and Tenax TA 60/80 adsorbents. For each measurement, three sampling tubes were applied simultaneously. Samples were analysed with a gas chromatograph, GC 6000 Vega Series 2, column SPB-1, 30 m, 0.53 mm i.d., film 1.5 mm, coupled to a Perkin– Elmer ATD 400 automatic thermal desorber. The data acquisition and processing was done on IBM compatible PC computer. The temperature program applied was 35°C for 10 min, 2 – 70°C per minute, 4– 130°C per minute and then 10– 250°C per minute, hold for 5 min. FID was used as a detector at a temperature of 250°C. The carrier gas was helium (T55, Messer Griesheim), with a flow rate of 2.45 ml min − 1, outlet split ratio was 1:5. Desorption parameters were as follows: primary desorption temperature: 280°C; primary
707
desorption time: 5 min; trap low temperature: − 30°C; trap high temperature: 300°C; trap hold time: 5 min. Sample tubes were conditioned at 300°C for 30 min prior to sampling. The blank levels were checked and found to be below the detection limit for the compounds studied. The limit of quantification were 0.5 ng for n-hexane, 0.5 ng for n-heptane, 0.4 ng for n-octane, 0.4 ng for n-nonane, 0.2 ng for benzene, 0.2 ng for toluene, 0.3 ng for ethylbenzene, 0.3 ng for m-xylene, 0.3 ng for o-xylene and 0.3 ng for styrene. For field sampling, the samplers were filled with 250 mg of Tenax TA adsorbent. Samples were collected at K-Puszta, Hungary (130 m asl.), between 29 June and 27 July, 1999. K-Puszta is situated on a forest clearing in the middle of the Great Hungarian Plain, relatively far from anthropogenic sources (it is 60 km SE from Budapest and the largest nearby city is 15 km S from K-Puszta). The geographical position is: 46°58% N and 19°33% E. The site is an established European Monitoring and Evaluation Programme (EMEP) station.
4. Results and discussion
4.1. E6aluation of the results of theoretical section With the model described in Section 2, the mass evolution of the adsorbed compound (benzene), its concentration in the air above the adsorbent (Tenax TA) and the uptake rates calculated for the whole period of sampling were established during sampling. First, the ambient concentration of benzene was held constant. In Fig. 3, the mass adsorbed on the tube-type diffusive sampler and the changes in the calculated uptake rate of benzene as a function of exposure dose at constant (0.5 mg m − 3) ambient benzene concentration can be seen. The uptake of benzene on the adsorbent at the beginning of the measurement was faster, as reflected by the higher initial slope of the curve showing the mass adsorbed. As sampling progresses, the uptake (slope) becomes constant, though the intercept differs from zero. Because of the non-linear dependence of the adsorbed mass
708
B. Tolnai et al. / Talanta 54 (2001) 703–713
on time, the calculated uptake rate of benzene was also varied. At the end of the first day of sampling, the uptake rate dropped by 70%, then it was gradually stabilized at a value of 1.20 ng/ (ppm·min). It can be seen in Fig. 2 that the uptake rate function can be divided into two parts. Initially, the changes in the uptake rate are fast, which would lead to unreliable measurements. However, above a certain threshold the slope becomes so small that the overall uptake rate can be nearly constant throughout a long sampling range. The explanation for these phenomena is the following: the sampling rate is governed by the concentration gradient through the diffusion gap and into the adsorption bed. In the early stage of sampling, the equilibrium concentration of a compound in the air layer adjacent to the adsorbent is low, thus the concentration gradient in the air gap is higher which results in a faster mass transfer (uptake), as indicated in Fig. 3. In Fig. 3, the changes in the equilibrium concentration of a compound with time in the sampler tube at con-
stant (0.5 mg m − 3) ambient benzene concentration (exposure time: 1: 1.16; 2: 2.33; 3: 3.5; 4: 4.66; 5: 5.83 and 6: 7.0 days) can be seen. The analyte starts to accumulate in the front stage of the adsorbent, becomes equilibrated with the gas phase and begins to diffuse further into the bed. As sampling advances, the concentration increases, as determined by the adsorption equilibrium and the mass transfer front gradually moves into the adsorbent bed. The sampling rate also decreases and it is :50% of the theoretical value after 1 week sampling period. This reduces the driving force and lowers the mass transfer rate. Above a certain threshold limit, the process reaches a steady state in which the rate of mass transfer in the diffusion gap equals that of the interstitial air and the adsorbent. As uptake is a cumulative process and uptake rate is calculated for the whole period of sampling, sampling should continue long enough to render the effects of these quick initial changes negligible. That is why constant uptake rates can be obtained only above a critical exposure dose. This limit has to be
Fig. 3. The changes in the equilibrium concentration of a compound with time in the sampler tube at constant (0.5 mg m − 3) ambient benzene concentration (exposure time: 1, 1.16; 2, 2.33; 3, 3.5; 4, 4.66; 5, 5.83; and 6, 7.0 days).
B. Tolnai et al. / Talanta 54 (2001) 703–713
709
Fig. 4. Changes in the calculated uptake rate of benzene on the tube-type diffusive sampler as a function of exposure time at different concentrations.
determined experimentally before the measurements. Based on these results and having some prior knowledge of the expected concentrations at a given sampling location, one can easily calculate the minimum sampling time required to obtain reliable concentration values. In Fig. 4 changes in the calculated uptake rate of benzene on the tube-type diffusive sampler as a function of exposure time at different concentrations can be seen. With varying concentrations, the uptake rate profile changed. The calculated uptake rates show an increase in uptake rate with decreasing concentration. In the environmental applications of diffusive sampling, the back diffusion of the adsorbed analyte can be a matter of concern, since ambient concentrations are usually variable, especially close to the emission source. Back diffusion takes place when the ambient concentration of the analyte drops below that in equilibrium with the surface of the adsorbent. This effect is negligible when adsorption forces are strong and the boundary layer concentration approaches zero, which seems not to be the case in the modelled process. In order to evaluate the effect of varying concentration of benzene, the concentration of ben-
zene was varied periodically to examine the effect of back diffusion on the uptake rate. The concentration of benzene was varied according to a sinus function, with amplitudes one and two orders of magnitude higher than the average concentration. In Fig. 5, changes in the calculated uptake rate of benzene on the tube-type diffusive sampler as a function of exposure time for periodically changing ambient concentration profiles can be seen. The calculated uptake rates were also stabilized above a critical exposure dose, as in the case of constant ambient concentration, but they are varied in phase with changing concentration. The amplitude of the changes in uptake rate decreased with increasing sampling time.
4.2. E6aluation of the results of the laboratory experiments For the determination of uptake rates on Carbopack B 60/80 adsorbent, three tubes were exposed for different periods of time at constant concentration and for constant time at different concentrations in the gas generator device. In Fig. 6, the measured uptake rates of various compounds as a function of exposure time at constant concentrations can be seen. The concentrations of
710
B. Tolnai et al. / Talanta 54 (2001) 703–713
the components in the air generated for the experiments were 45.94, 56.11, 15.71, 3.87, 45.98, 34.07, 48.45, 14.73, 8.10 and 4.63 mg m − 3 for n-hexane,
n-heptane, n-octane, n-nonane, benzene, toluene, ethylbenzene, m-xylene, o-xylene and styrene, respectively. In Fig. 7, the measured uptake rates of
Fig. 5. Changes in the calculated uptake rate of benzene on the tube-type diffusive sampler as a function of exposure time for periodically changing ambient concentration profiles.
Fig. 6. Measured uptake rates of various compounds as a function of exposure time at constant concentrations.
B. Tolnai et al. / Talanta 54 (2001) 703–713
711
Fig. 7. Measured uptake rates of various compounds as a function of exposure dose at constant exposure time.
various compounds as a function of exposure dose at constant exposure time can be seen. The concentrations of the components in the air generated for the experiments varied between 3 and 105 mg m − 3. The relative S.D. for three parallel measurements was between 10 and 20% for the uptake rates determined. Decreasing uptake rates were observed as a function of both exposure time and concentration. In this range, the uptake rates fall rapidly with time. This, together with the inherent uncertainties of the laboratory measurements may account for the deviations from reported uptake rates measured [3,4] of the ppb concentration range. During the laboratory determination of uptake rates, the exposure time is shorter and the concentrations are higher than those in the environment. According to the model discussed in Section 2, these experiments took place in the range, where the uptake rates fell rapidly and were far from being constant. It was previously observed that diffusive sampling usually underestimated the ambient concentrations as compared to active sampling [5,28,29]. In the laboratory, constant, low concentrations are difficult to maintain for a pe-
riod of several days and uptake rates are usually determined for a few hours of exposure. The resulting systematic error in the uptake rate was estimated to be about + 10 to + 30%. Therefore, it is not possible to compare the uptake rate obtained in the laboratory experiments to other values published elsewhere, which usually refer to lower concentrations and shorter sampling times [3,4]. To overcome the problem, the uptake rates should be determined in the field under environmental conditions in comparison with an independent reference method.
4.3. E6aluation of the results of the field experiments Field tests were performed to justify the results of the computer simulation. The samples were collected at K-Puszta, Hungary. A 28-day sampling period was sampled as follows: 7·4, 4·7, 2·17 and 1·28 days. In Fig. 8, the total adsorbed mass of benzene, toluene, ethylbenzene and m,p-xylene for the whole sampling period are shown, as obtained by the different sampling schemes. The concentrations were very low since the sampling
712
B. Tolnai et al. / Talanta 54 (2001) 703–713
References
Fig. 8. Adsorbed mass of benzene, toluene, ethylbenzene and m,p-xylene during field test.
site is a background station. With these experiments, although absolute concentrations are not known, the quasi constant part of the uptake rate curve in Fig. 2 can be tested. As predicted by the model, the shorter the sampling time, the higher the collected mass of compound. There are differences between the total collected masses collected during 28 days, but the differences are not significant (9 15%). Unfortunately, the results of the model and experiments are not readily comparable, since the latter yield only four data points for each compound. Furthermore, the sampling periods of 7, 14 and 28 days exceeded the time span covered in the model. The qualitative agreement observed concerns the general tendency of exponentially decaying uptake efficiencies as sampling progresses in harmony with the prediction of the model.
Acknowledgements The authors are indebted to the OTKA T 029250, OTKA T 030186, FKFP 0084/1999 and AKP 97-112 2,5 for their financial support.
[1] E.D. Palmes, A.F. Gunnison, Am. Ind. Hyg. Assoc. J. 34 (1973) 78 – 81. [2] R.H. Brown, J. Charlton, K.J. Saunders, Am. Ind. Hyg. Assoc. J. 42 (1981) 856 – 869. [3] ATD 400 User’s Manual, Thermal Desorption Data Sheet No. 2, Appendix A, Perkin – Elmer, 1991, Beaconsfield. [4] MDHS 80: VOCs in Air: Laboratory Method Using Diffusive Solid Sorbent Tubes and TD-GC, 1995, Health and Safety Executive, Sheffield, 1 – 12. [5] Gy Kiss, A. Gelencse´ r, J. Hlavay, Mag. Ke´ m. Folyo´ ir. 100 (2) (1994) 81 – 86. [6] X.-L. Cao, C.N. Hewitt, Atmos. Env. 27A (12) (1993) 1865 – 1872. [7] A. Gelencse´ r, Gy Kiss, J. Hlavay, Th.L. Hafkenscheid, R.J.B. Peters, E.W.B. de Leer, Talanta 41 (1994) 1095 – 1100. [8] B. Tolnai, A. Gelencse´ r, Gy Barko´ , J. Hlavay, Analyst 124 (1999) 1859 – 1863. [9] R.H. Brown, J. Env. Monit. 1 (1999) 115 – 116. [10] S.F. Patil, S.T. Lonkar, J. Chromatogr. A 688 (1994) 189 – 199. [11] P. Rosmanith, in: A. Berlin, R.H. Brown, K.J. Saunders (Eds.), Diffusive Sampling, An Alternative Approach to Workplace Air Monitoring, Royal Society of Chemistry, London, UK, 1986, pp. 161 – 162 Luxembourg CEC Publication No. 10555EN. [12] B. Tolnai, A. Gelencse´ r, C. Ga´ l, J. Hlavay, Anal. Chim. Acta. 408 (1 – 2) (2000) 117 – 122. [13] A. Roche, R. Thevenet, V. Jacob, P. Kaluzny, C. Ferrari, P. Baussand, P. Foster, Atmos. Env. 33 (12) (1999) 1905 – 1912. [14] R.W. Coutant, R.G. Lewis, J. Mulic, Anal. Chem. 57 (1985) 219 – 223. [15] J.C. Posner, G. Moore, Am. Ind. Hyg. Assoc. J. 46 (5) (1985) 277 – 285. [16] N.M. Bradshaw, J.A. Ballantine, Env. Technol. 16 (1995) 433 – 444. [17] N. Kilic, J.A. Ballantine, Analyst 123 (1998) 1795 – 1797. [18] H.R. Brown, K.T. Walkin, Anal. Proc. 18 (1981) 205. [19] N. van den Hoed, M.T.H. Halmans, in: A. Berlin, R.H. Brown, K.J. Saunders (Eds.), Diffusive Sampling, An Alternative Approach to Workplace Air Monitoring, Royal Society of Chemistry, London, UK, 1987, p. 131 CEC Publication No. 10555EN, Luxembourg. [20] E. Nordstrand, J. Kristensson, Am. Ind. Hyg. Assoc. J. 55 (10) (1994) 935 – 941. [21] N. Van den Hoed, Van Asselen, Ann. Occupat. Hyg. 35 (1991) 273 – 285. [22] A.R. Mastrogiacomo, E. Pierini, L. Sampaolo, Chromatographia 41 (1995) 599 – 603. [23] G.A. Lugg, Anal. Chem. 40 (1968) 1072 – 1077. [24] K.W. Morton, D.F. Mayers, Numerical Solution of Partial Differential Equations, Cambridge University Press, UK, 1994.
B. Tolnai et al. / Talanta 54 (2001) 703–713 [25] G.D. Smith, Numerical Solution of Partial Differential Equations, Oxford University Press, UK, 1985. [26] B. Tolnai, G. Barko´ , R. Ne´ meth, J. Hlavay, Microchem. J. 67 (2000) 157 –161.
.
713
[27] G. Barko´ , Hlavay, J. Anal. Chim. Acta. 367 (1998) 135 –143. [28] Y.S. Fung, Z. Wu, Analyst 121 (1996) 1955 – 1961. [29] J. Mowrer, P.A. Svanberg, A. Potter, A. Lindskog, Analyst 121 (1996) 1295 – 1300.
Theoretical approach to non-constant uptake rates for tube-type diffusive samplers Bala´zs Tolnai a, Andra´s Gelencse´r b, Jo´zsef Hlavay a,* a
Department of Earth and En6ironmental Sciences, Uni6ersity of Veszpre´m, P.O. Box 158, H-8201 Veszpre´m, Hungary b Air Chemistry Group of the Hungarian Academy of Sciences, P.O. Box 158, H-8201 Veszpre´m, Hungary Received 31 August 2000; received in revised form 4 January 2001; accepted 5 January 2001
Abstract A simple theoretical model was developed for evaluating the validity of the simplified uptake model of diffusive sampling. In the model based on the plate theory diffusion to the adsorbent surface, phase equilibrium of the adsorbate and mass transport in the adsorbent bed were considered. It was found that in the early stage of sampling, the rate of sampling is close to its theoretical value. As sampling progresses, the concentration increases and the mass transfer front gradually moves into the adsorbent layer. Above a certain threshold limit, the mass uptake becomes a steady state process in which the diffusion in the air gap and the mass transport in the adsorbent bed are balanced. As uptake is a cumulative process, sampling should continue long enough to render the effects of these initial changes negligible. That is why constant uptake rates can still be obtained above a critical exposure dose. This critical exposure dose should be exceeded both in the determination of uptake rates and outdoor measurements, to obtain consistent and reliable analytical data. Evaluation of the time and concentration dependence of uptake rate in laboratory experiments and the time dependence of uptake rate in filed test was performed to justify the model results. Since the determination of uptake rates always takes places in the laboratory, where the exposure time is much shorter and the concentration is much higher than in the environment, the uptake rates are thus overestimated by 10–30%. Therefore, the uptake rates should be determined in the field under ambient conditions by means of an independent reference method. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Diffusive sampling; Tube-type sampler; Uptake rate; Volatile organic compounds
1. Introduction In the past few decades, the concentration of volatile organic compounds (VOC) in air, espe* Corresponding author. Tel.: +36-88-422022, ext. 4368; fax: +36-88-423203. E-mail address: [email protected] (J. Hlavay).
cially in urban locations, increased significantly. Several VOCs at the level existing in urban air have mutagenic, carcinogenic and teratogenic effects on humans and can cause acute and chronic diseases. Several GC-compatible air-sampling techniques are available for the determination of these compounds in air. Diffusive sampling, where air pollutants are collected on a sorbent by
0039-9140/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 9 - 9 1 4 0 ( 0 1 ) 0 0 3 2 2 - 8
B. Tolnai et al. / Talanta 54 (2001) 703–713
704
diffusion, is becoming more popular in environmental monitoring. The increasing acceptance of this technique is due to the increased sampling time, its low price, simplicity and unattended operation. The method is capable of delivering timeweighted average concentrations over a relatively long period of time. Diffusive sampling is a method in which volatile organic compounds are taken up by an adsorbent from the air at a rate controlled by diffusion. The differential equation describing the uptake by the adsorbent is the following [1]: dm D · A = (C0 −Ca) dt L
(1)
where m is the mass of the compound adsorbed (g); t is the exposure time (s); A, the cross sectional area of diffusion path (cm2); D, the diffusion coefficient of the compound in air (cm2 s − 1); L, the length of diffusion path (cm); C0 is the ambient concentration of the compound (g cm − 3); and Ca is the concentration of the compound above the adsorbent (g cm − 3). Assuming that the adsorbent acts as a perfect sink (Ca =0) and steady-state conditions hold, Eq. (1) is simplified to: C0 =
m ·L t ·D ·A
(2)
By introducing the term of uptake rate (u): u=
D ·A L
(3)
u: uptake rate (cm3 s − 1) and substituting Eq. (3) into Eq. (2) we get: C0 =
m t ·u
(4)
With this simple equation, the concentration of an analyte can be calculated from the adsorbed mass as determined by gas chromatography, the sampling time and the uptake rate. Ideally, uptake rates are constants that can be calculated from the geometry of the sampler and available diffusion coefficients. It was shown, however, that experimental uptake rates were significantly different from the theoretical values [2– 5]. The literature on the uptake rates of compounds for a given
adsorbent and sampler geometry is similarly contradictory. For benzene, for example, deviations up to 300% can be found in the literature [3–11]. This fundamental controversy in the literature may result from the differences in the conditions and methods in the laboratory. Furthermore, environmental conditions are substantially different from the controlled conditions in the laboratory, especially in terms of variability and concentration levels. Although some parameters can be varied in a laboratory trial, it is impossible to simulate environmental conditions. In a previous work [12], the reliability of diffusive sampling and the validity of its simplified uptake model were studied on site with benzene, toluene, ethylbenzene and m,p-xylene (BTEX compounds) on three different adsorbents (Tenax GR, Tenax TA, Carbopack). It was found that a critical exposure dose, which is the product of the time-weighted average concentration and sampling time, should be exceeded to obtain consistent and reliable analytical data and to ensure that the simplified uptake theory holds. Above this dose, the simplified uptake theory can be used with a fair degree of accuracy. The simplified model of diffusive sampling above this exposure dose seems to satisfy the requirements of environmental monitoring. Others [13] found that the uptake rate decreased exponentially with exposure dose at the initial stage of sampling. Non-constant uptake rates were reported for tube-type samplers [14–17]. It was found that the observed sampling rate was inferior to the theoretical value [18] and decreased as sampling proceeded. In our previous study [12], we implied that the reduced sampling efficiency is a function of both concentration and time. It is anticipated that the phenomenon of decreasing uptake rates is associated with the use of non-ideal adsorbents. In view of the effective desorption process required for on-line GC analysis, the use of strong adsorbent is not possible. Quite soon after the initial period of the sampling, the concentration of the analyte in the air layer adjacent to the adsorbent surface will not be zero. Theoretical treatment of this non-ideal situation is rather complex since it should account for the phase equilibrium of the adsorbate as well as for the mass transport in the adsorbent [19,20].
B. Tolnai et al. / Talanta 54 (2001) 703–713
A possible way to eliminate the theoretical problems might be to find a numerical solution to the differential Eq. (1), which can include all variables and parameters at any desired level of complexity. The results can give insight into the applicability of the simplified model. Since, in any application, only the use of the simplified diffusive uptake model is feasible, to have a prior knowledge on its limitations is a must. Several researchers have used computer programs for modelling mass uptake on diffusive samplers [14,20,21]. Coutant et al. [14] used sinusoidal function to model the effect of concentration variations on uptake rate on badge-type diffusive samplers. Posner et al. [15] also developed a mathematical description for the sampling on badge-type diffusive samplers and performed laboratory experiments at very high concentration range (mg/m3) to examine the sampling rate and reverse diffusion. van den Hoed et al. [21] used a computer model for the estimation of the uptake rates of tube-type diffusive samplers and Nordstrand et al. [20] made a computer program for simulating the performance of tube-type diffusive samplers, however they did not perform laboratory experiments. The previous studies used concentration levels of : 100 ppm for relatively short sampling times
705
and did not observe nearly constant mass uptake over a critical threshold. Moreover, none of these theoretical studies determined how the results translate into the simplified uptake process of diffusive sampling that is the measure of its applicability. The objective of this work is to model the uptake process of a tube-type diffusive sampler and to test the model experimentally, both in the laboratory and on the field. Understanding the process of uptake may help to establish the limits of the simplified model. Although the model is theoretical, its parameters are experimental data taken from the literature and its focus is kept on the environmental applications. For this reason, the effect of variation in ambient conditions is also considered.
2. Theoretical considerations A computer program was made to evaluate the uptake process of tube-type diffusive samplers. The principle of the rather simplified model is shown in Fig. 1. Three basic processes were taken into account: (a) diffusion to the surface of the adsorbent; (b) phase equilibrium of the adsorbate; and (c) mass transport within the adsorbent bed
Fig. 1. The model of the tube type diffusive sampler.
706
B. Tolnai et al. / Talanta 54 (2001) 703–713
Fig. 2. The mass adsorbed on the tube-type diffusive sampler and the changes in the calculated uptake rate of benzene as a function of exposure dose at constant (0.5 mg m − 3) ambient benzene concentration.
[20]. To simplify the calculations, the plate theory was used and the cells were considered as if they consisted of two continuous phases; the adsorbent phase and the interstitial air. Nordstrand et al. [20] also divided the adsorbent into slices that were calculated from the appropriate adsorption isotherm. For the numerical calculation in our study, the adsorbent was divided into 80 cells according to the theoretical plate model. This simplification is widely accepted in the models of chromatographic processes. In the air phase, only diffusion was taken into account. In the literature, several types of isotherms are used for describing the phase equilibrium between the adsorbent and the interstitial air in diffusive sampling. The Freundlich isotherm [20], the Dubinin– Radushkevich isotherm [20] and the linear part of the Langmuir isotherm [14,15] are equally used. In our study, to approximate the phase equilibrium, the linear part of the Langmuir isotherm was used. The linear part of Langmuir isotherm can be a reasonable approximation for a system, which is far from being saturated during sampling. A further sim-
plification was that only a single adsorption layer was assumed and pore diffusion was not taken into account. On the other hand, mass transport in the adsorbent bed was allowed [19,20] and the process was considered as quasi-diffusion. This is the case of mobile adsorption, when the adsorbent is not strong enough and the adsorption heat is not excessively high [22]. The simulations were performed for benzene. The diffusion coefficient of benzene was taken from the literature [23]. The parameters of standard Perkin– Elmer diffusive samplers, as cross-sectional area of diffusion path, A= 0.2 cm2, length of diffusion path, L=1.5 cm, mass of the adsorbent, madsorbens = 250 mg were used in the simulation. Tenax TA adsorbent was chosen for the evaluation. The partition coefficient K (slope of the linear part of the Langmuir isotherm) for benzene on Tenax TA adsorbent was calculated from breakthrough measurements of active sampling [22]. One week sampling time was simulated in steps of 0.02 s to bypass the numerical instability of the model [24,25].
B. Tolnai et al. / Talanta 54 (2001) 703–713
3. Experimental section The laboratory experiments were conducted with a vapour generator, which is described in details elsewhere [26]. A controlled stream of nitrogen, controlled by a Mass Track 810C (Sierra Instruments), is drawn at a low flow-rate through a thermostated quartz container filled with the individual standard substance, thermostated at −1690.1°C with a SCINICS CH-201 cool circulator (SCINICS, Japan) thermostat. In the quartz vessel, continuous stream of saturated vapour of the substance was produced. This stream was diluted in a mixing chamber by a large stream of pure nitrogen gas that was controlled using a Mass Track 810C (Sierra Instruments) mass flow controller (0– 70 l h − 1). The mixing chamber was also thermostated at −16 90.1°C with a SCINICS CH-201 cool circulator (SCINICS) thermostat. Depending on the carrier flow-rate of the dilution gas, different concentrations of the organic vapour can be prepared. Special PTFE tubes (E-06376-03, Kurt J. Lesker Co.) and PTFE fittings were used to eliminate the wall adsorption. The stability of the concentration generated was continuously checked using a piezoelectric chemical sensor system connected to the vapour generator device [27]. Samples were taken by diffusive sampling with standard Perkin– Elmer sample tubes (A=0.2 cm2; L=1.5 cm) filled with 250 mg of Carbopack B 60/80 and Tenax TA 60/80 adsorbents. For each measurement, three sampling tubes were applied simultaneously. Samples were analysed with a gas chromatograph, GC 6000 Vega Series 2, column SPB-1, 30 m, 0.53 mm i.d., film 1.5 mm, coupled to a Perkin– Elmer ATD 400 automatic thermal desorber. The data acquisition and processing was done on IBM compatible PC computer. The temperature program applied was 35°C for 10 min, 2 – 70°C per minute, 4– 130°C per minute and then 10– 250°C per minute, hold for 5 min. FID was used as a detector at a temperature of 250°C. The carrier gas was helium (T55, Messer Griesheim), with a flow rate of 2.45 ml min − 1, outlet split ratio was 1:5. Desorption parameters were as follows: primary desorption temperature: 280°C; primary
707
desorption time: 5 min; trap low temperature: − 30°C; trap high temperature: 300°C; trap hold time: 5 min. Sample tubes were conditioned at 300°C for 30 min prior to sampling. The blank levels were checked and found to be below the detection limit for the compounds studied. The limit of quantification were 0.5 ng for n-hexane, 0.5 ng for n-heptane, 0.4 ng for n-octane, 0.4 ng for n-nonane, 0.2 ng for benzene, 0.2 ng for toluene, 0.3 ng for ethylbenzene, 0.3 ng for m-xylene, 0.3 ng for o-xylene and 0.3 ng for styrene. For field sampling, the samplers were filled with 250 mg of Tenax TA adsorbent. Samples were collected at K-Puszta, Hungary (130 m asl.), between 29 June and 27 July, 1999. K-Puszta is situated on a forest clearing in the middle of the Great Hungarian Plain, relatively far from anthropogenic sources (it is 60 km SE from Budapest and the largest nearby city is 15 km S from K-Puszta). The geographical position is: 46°58% N and 19°33% E. The site is an established European Monitoring and Evaluation Programme (EMEP) station.
4. Results and discussion
4.1. E6aluation of the results of theoretical section With the model described in Section 2, the mass evolution of the adsorbed compound (benzene), its concentration in the air above the adsorbent (Tenax TA) and the uptake rates calculated for the whole period of sampling were established during sampling. First, the ambient concentration of benzene was held constant. In Fig. 3, the mass adsorbed on the tube-type diffusive sampler and the changes in the calculated uptake rate of benzene as a function of exposure dose at constant (0.5 mg m − 3) ambient benzene concentration can be seen. The uptake of benzene on the adsorbent at the beginning of the measurement was faster, as reflected by the higher initial slope of the curve showing the mass adsorbed. As sampling progresses, the uptake (slope) becomes constant, though the intercept differs from zero. Because of the non-linear dependence of the adsorbed mass
708
B. Tolnai et al. / Talanta 54 (2001) 703–713
on time, the calculated uptake rate of benzene was also varied. At the end of the first day of sampling, the uptake rate dropped by 70%, then it was gradually stabilized at a value of 1.20 ng/ (ppm·min). It can be seen in Fig. 2 that the uptake rate function can be divided into two parts. Initially, the changes in the uptake rate are fast, which would lead to unreliable measurements. However, above a certain threshold the slope becomes so small that the overall uptake rate can be nearly constant throughout a long sampling range. The explanation for these phenomena is the following: the sampling rate is governed by the concentration gradient through the diffusion gap and into the adsorption bed. In the early stage of sampling, the equilibrium concentration of a compound in the air layer adjacent to the adsorbent is low, thus the concentration gradient in the air gap is higher which results in a faster mass transfer (uptake), as indicated in Fig. 3. In Fig. 3, the changes in the equilibrium concentration of a compound with time in the sampler tube at con-
stant (0.5 mg m − 3) ambient benzene concentration (exposure time: 1: 1.16; 2: 2.33; 3: 3.5; 4: 4.66; 5: 5.83 and 6: 7.0 days) can be seen. The analyte starts to accumulate in the front stage of the adsorbent, becomes equilibrated with the gas phase and begins to diffuse further into the bed. As sampling advances, the concentration increases, as determined by the adsorption equilibrium and the mass transfer front gradually moves into the adsorbent bed. The sampling rate also decreases and it is :50% of the theoretical value after 1 week sampling period. This reduces the driving force and lowers the mass transfer rate. Above a certain threshold limit, the process reaches a steady state in which the rate of mass transfer in the diffusion gap equals that of the interstitial air and the adsorbent. As uptake is a cumulative process and uptake rate is calculated for the whole period of sampling, sampling should continue long enough to render the effects of these quick initial changes negligible. That is why constant uptake rates can be obtained only above a critical exposure dose. This limit has to be
Fig. 3. The changes in the equilibrium concentration of a compound with time in the sampler tube at constant (0.5 mg m − 3) ambient benzene concentration (exposure time: 1, 1.16; 2, 2.33; 3, 3.5; 4, 4.66; 5, 5.83; and 6, 7.0 days).
B. Tolnai et al. / Talanta 54 (2001) 703–713
709
Fig. 4. Changes in the calculated uptake rate of benzene on the tube-type diffusive sampler as a function of exposure time at different concentrations.
determined experimentally before the measurements. Based on these results and having some prior knowledge of the expected concentrations at a given sampling location, one can easily calculate the minimum sampling time required to obtain reliable concentration values. In Fig. 4 changes in the calculated uptake rate of benzene on the tube-type diffusive sampler as a function of exposure time at different concentrations can be seen. With varying concentrations, the uptake rate profile changed. The calculated uptake rates show an increase in uptake rate with decreasing concentration. In the environmental applications of diffusive sampling, the back diffusion of the adsorbed analyte can be a matter of concern, since ambient concentrations are usually variable, especially close to the emission source. Back diffusion takes place when the ambient concentration of the analyte drops below that in equilibrium with the surface of the adsorbent. This effect is negligible when adsorption forces are strong and the boundary layer concentration approaches zero, which seems not to be the case in the modelled process. In order to evaluate the effect of varying concentration of benzene, the concentration of ben-
zene was varied periodically to examine the effect of back diffusion on the uptake rate. The concentration of benzene was varied according to a sinus function, with amplitudes one and two orders of magnitude higher than the average concentration. In Fig. 5, changes in the calculated uptake rate of benzene on the tube-type diffusive sampler as a function of exposure time for periodically changing ambient concentration profiles can be seen. The calculated uptake rates were also stabilized above a critical exposure dose, as in the case of constant ambient concentration, but they are varied in phase with changing concentration. The amplitude of the changes in uptake rate decreased with increasing sampling time.
4.2. E6aluation of the results of the laboratory experiments For the determination of uptake rates on Carbopack B 60/80 adsorbent, three tubes were exposed for different periods of time at constant concentration and for constant time at different concentrations in the gas generator device. In Fig. 6, the measured uptake rates of various compounds as a function of exposure time at constant concentrations can be seen. The concentrations of
710
B. Tolnai et al. / Talanta 54 (2001) 703–713
the components in the air generated for the experiments were 45.94, 56.11, 15.71, 3.87, 45.98, 34.07, 48.45, 14.73, 8.10 and 4.63 mg m − 3 for n-hexane,
n-heptane, n-octane, n-nonane, benzene, toluene, ethylbenzene, m-xylene, o-xylene and styrene, respectively. In Fig. 7, the measured uptake rates of
Fig. 5. Changes in the calculated uptake rate of benzene on the tube-type diffusive sampler as a function of exposure time for periodically changing ambient concentration profiles.
Fig. 6. Measured uptake rates of various compounds as a function of exposure time at constant concentrations.
B. Tolnai et al. / Talanta 54 (2001) 703–713
711
Fig. 7. Measured uptake rates of various compounds as a function of exposure dose at constant exposure time.
various compounds as a function of exposure dose at constant exposure time can be seen. The concentrations of the components in the air generated for the experiments varied between 3 and 105 mg m − 3. The relative S.D. for three parallel measurements was between 10 and 20% for the uptake rates determined. Decreasing uptake rates were observed as a function of both exposure time and concentration. In this range, the uptake rates fall rapidly with time. This, together with the inherent uncertainties of the laboratory measurements may account for the deviations from reported uptake rates measured [3,4] of the ppb concentration range. During the laboratory determination of uptake rates, the exposure time is shorter and the concentrations are higher than those in the environment. According to the model discussed in Section 2, these experiments took place in the range, where the uptake rates fell rapidly and were far from being constant. It was previously observed that diffusive sampling usually underestimated the ambient concentrations as compared to active sampling [5,28,29]. In the laboratory, constant, low concentrations are difficult to maintain for a pe-
riod of several days and uptake rates are usually determined for a few hours of exposure. The resulting systematic error in the uptake rate was estimated to be about + 10 to + 30%. Therefore, it is not possible to compare the uptake rate obtained in the laboratory experiments to other values published elsewhere, which usually refer to lower concentrations and shorter sampling times [3,4]. To overcome the problem, the uptake rates should be determined in the field under environmental conditions in comparison with an independent reference method.
4.3. E6aluation of the results of the field experiments Field tests were performed to justify the results of the computer simulation. The samples were collected at K-Puszta, Hungary. A 28-day sampling period was sampled as follows: 7·4, 4·7, 2·17 and 1·28 days. In Fig. 8, the total adsorbed mass of benzene, toluene, ethylbenzene and m,p-xylene for the whole sampling period are shown, as obtained by the different sampling schemes. The concentrations were very low since the sampling
712
B. Tolnai et al. / Talanta 54 (2001) 703–713
References
Fig. 8. Adsorbed mass of benzene, toluene, ethylbenzene and m,p-xylene during field test.
site is a background station. With these experiments, although absolute concentrations are not known, the quasi constant part of the uptake rate curve in Fig. 2 can be tested. As predicted by the model, the shorter the sampling time, the higher the collected mass of compound. There are differences between the total collected masses collected during 28 days, but the differences are not significant (9 15%). Unfortunately, the results of the model and experiments are not readily comparable, since the latter yield only four data points for each compound. Furthermore, the sampling periods of 7, 14 and 28 days exceeded the time span covered in the model. The qualitative agreement observed concerns the general tendency of exponentially decaying uptake efficiencies as sampling progresses in harmony with the prediction of the model.
Acknowledgements The authors are indebted to the OTKA T 029250, OTKA T 030186, FKFP 0084/1999 and AKP 97-112 2,5 for their financial support.
[1] E.D. Palmes, A.F. Gunnison, Am. Ind. Hyg. Assoc. J. 34 (1973) 78 – 81. [2] R.H. Brown, J. Charlton, K.J. Saunders, Am. Ind. Hyg. Assoc. J. 42 (1981) 856 – 869. [3] ATD 400 User’s Manual, Thermal Desorption Data Sheet No. 2, Appendix A, Perkin – Elmer, 1991, Beaconsfield. [4] MDHS 80: VOCs in Air: Laboratory Method Using Diffusive Solid Sorbent Tubes and TD-GC, 1995, Health and Safety Executive, Sheffield, 1 – 12. [5] Gy Kiss, A. Gelencse´ r, J. Hlavay, Mag. Ke´ m. Folyo´ ir. 100 (2) (1994) 81 – 86. [6] X.-L. Cao, C.N. Hewitt, Atmos. Env. 27A (12) (1993) 1865 – 1872. [7] A. Gelencse´ r, Gy Kiss, J. Hlavay, Th.L. Hafkenscheid, R.J.B. Peters, E.W.B. de Leer, Talanta 41 (1994) 1095 – 1100. [8] B. Tolnai, A. Gelencse´ r, Gy Barko´ , J. Hlavay, Analyst 124 (1999) 1859 – 1863. [9] R.H. Brown, J. Env. Monit. 1 (1999) 115 – 116. [10] S.F. Patil, S.T. Lonkar, J. Chromatogr. A 688 (1994) 189 – 199. [11] P. Rosmanith, in: A. Berlin, R.H. Brown, K.J. Saunders (Eds.), Diffusive Sampling, An Alternative Approach to Workplace Air Monitoring, Royal Society of Chemistry, London, UK, 1986, pp. 161 – 162 Luxembourg CEC Publication No. 10555EN. [12] B. Tolnai, A. Gelencse´ r, C. Ga´ l, J. Hlavay, Anal. Chim. Acta. 408 (1 – 2) (2000) 117 – 122. [13] A. Roche, R. Thevenet, V. Jacob, P. Kaluzny, C. Ferrari, P. Baussand, P. Foster, Atmos. Env. 33 (12) (1999) 1905 – 1912. [14] R.W. Coutant, R.G. Lewis, J. Mulic, Anal. Chem. 57 (1985) 219 – 223. [15] J.C. Posner, G. Moore, Am. Ind. Hyg. Assoc. J. 46 (5) (1985) 277 – 285. [16] N.M. Bradshaw, J.A. Ballantine, Env. Technol. 16 (1995) 433 – 444. [17] N. Kilic, J.A. Ballantine, Analyst 123 (1998) 1795 – 1797. [18] H.R. Brown, K.T. Walkin, Anal. Proc. 18 (1981) 205. [19] N. van den Hoed, M.T.H. Halmans, in: A. Berlin, R.H. Brown, K.J. Saunders (Eds.), Diffusive Sampling, An Alternative Approach to Workplace Air Monitoring, Royal Society of Chemistry, London, UK, 1987, p. 131 CEC Publication No. 10555EN, Luxembourg. [20] E. Nordstrand, J. Kristensson, Am. Ind. Hyg. Assoc. J. 55 (10) (1994) 935 – 941. [21] N. Van den Hoed, Van Asselen, Ann. Occupat. Hyg. 35 (1991) 273 – 285. [22] A.R. Mastrogiacomo, E. Pierini, L. Sampaolo, Chromatographia 41 (1995) 599 – 603. [23] G.A. Lugg, Anal. Chem. 40 (1968) 1072 – 1077. [24] K.W. Morton, D.F. Mayers, Numerical Solution of Partial Differential Equations, Cambridge University Press, UK, 1994.
B. Tolnai et al. / Talanta 54 (2001) 703–713 [25] G.D. Smith, Numerical Solution of Partial Differential Equations, Oxford University Press, UK, 1985. [26] B. Tolnai, G. Barko´ , R. Ne´ meth, J. Hlavay, Microchem. J. 67 (2000) 157 –161.
.
713
[27] G. Barko´ , Hlavay, J. Anal. Chim. Acta. 367 (1998) 135 –143. [28] Y.S. Fung, Z. Wu, Analyst 121 (1996) 1955 – 1961. [29] J. Mowrer, P.A. Svanberg, A. Potter, A. Lindskog, Analyst 121 (1996) 1295 – 1300.