Robust Nonfitting Way To Determine Mass Diffusivity and Initial

Jul 19, 2013 - Robust Nonfitting Way To Determine Mass Diffusivity and Initial Concentration for VOCs in Building Materials with Accuracy Estimation. ...
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Robust Nonfitting Way To Determine Mass Diffusivity and Initial Concentration for VOCs in Building Materials with Accuracy Estimation Min Li* School of Energy Science and Engineering, Central South University, Changsha 410083, China ABSTRACT: This paper presents a novel procedure for estimating mass diffusivity and initial concentration for VOCs in building materials. In contrast to methods that fit data to an organic emission model, this new method determines these two parameters by two observations from a chamber test in a nonfitting and sequential way, which defines a well-posed problem and requires no iterative procedure as well as is robust to initial guess and random uncertainties. The most outstanding feature of this method is that multiple estimates of the parameters can be obtained when more than two experimental data are available; thus, these estimates constitute a sample of population of parameter estimates involving experimental errors. The averages of the sample can be regarded the best estimates of the parameters, and, more importantly, variances (standard deviations) and confidence intervals can be readily and naturally estimated making no assumption about the distribution of experimental errors. Two easy and direct ways are suggested for determining confidence limits: one is percentile method, and the other is the normal approximation. This feature highlights the major difference between the new method and common curve-fitting procedures that generally assume random errors being Gaussian distribution.

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sensitive to small perturbations. 21 Finally, it is a great challenge for the curve-fitting methods to provide accuracy estimation for parameters if the probability distribution of experimental errors is unknown in advance. Error or accuracy estimation is an important issue that goes beyond the mere finding of best-fit parameters. Linear fitting procedures can usually provide variances and standard errors of estimated parameters but presume the distribution of experimental errors is normal. 22 These error estimates would fail in the cases where the normality assumption is invalid or nonlinear curve-fitting is required. To be genuinely useful, an estimating procedure should provide both parameter and error estimations for the parameters (or an approach to sampling from errors probability distribution), but, to the best knowledge of the author, little attention has been paid to the accuracy estimation of characteristic parameters of VOCs emission in building materials, particularly in the situations where the probability distribution of experimental errors is unknown. This paper presents a robust nonfitting method for both parameter and accuracy estimations for VOCs in diffusioncontrolled materials. It requires no fitting data to a model. In this context, the new procedure should be more convincing than the fitting procedures for validating mass-diffusion models.

INTRODUCTION The continuing concern about indoor emission of volatile organic compounds (VOCs) has heightened the need for characterizing organic emission from building materials.1−4 Of particular interest and difficulty is the determination of the three characteristic parameters of emitting materials: diffusivity D,5,6 initial concentration C0,7,8 and air-material partition coefficient K.9−11 These parameters are key inputs of mass diffusion models for predicting organic emission in indoor environments.12−15 Various methods for estimating these parameters have been proposed, among which a small but important point is that models are generally fit to measured data to obtain parameter estimates.16−20 The methods using steady-state diffusion models usually require a relatively long testing time to approach a quasi-steady-state;6,16 thus methods built on a transient concentration profile have been paid considerable attention in recent years.17−20 In general, two or all of the three characteristic parameters are determined simultaneously by fitting measured data to a transient emission model.8,9,17−20 There are several problems associated with the fitting fashion. The first is, as Haghighat et al. noted, 5 that the parameters determined by fitting experimental data are insufficient for validating theoretical models; a set of independently measured parameters is necessary for a complete validation. The second is that determining multiple parameters from a measured profile is an inverse problem, which is highly probably an ill-posed problem, i.e., the solution being not unique or extremely © 2013 American Chemical Society

Received: Revised: Accepted: Published: 9086

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This method determines in a sequential way parameters (D and C0) from two observations of air-phase concentration, so it is essentially well-posed and thus robust to small perturbations. The most distinguishing feature of the new method is that accuracy estimation of parameters is readily available from experimental data of more than two observations, making no assumption about the nature of experimental errors; if measurements more than two are available, this nonfitting method can yield a set of estimates of D and C0, equivalently, providing a sample of the probability distribution of experimental errors.

or equivalently tan(αL) m − Dα 2 = 2 αL KRDα

Here, R is defined as AL/V. The series solutions converge very fast for large Fourier number Fo = Dt/L2, thus in this situation only the first term is significant (if Cin = 0): Ca(t ) =

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METHODS We develop the new procedure from a mass-diffusion model for a ventilated chamber because independent emission data of a prototype reference material in such a chamber is available and can serve to illustrate and validate our method (Section 3); but it should be noted that the underlying idea of this method is rather general, suitable for tests in not only conditioned chambers but also closed chambers and even other pollutant emission processes. VOCs Emission Model. Figure 1 shows a schematic layout of a conditioned environmental chamber equipped with a

n=1

(4)

Ca(t ) = NC0 exp( −nt )

(5)

where N is only a function of α1 N=

2R tan(α1L) α1L{f1 + f2 α1L[cot(α1L) + tan(α1L)]}

(6)

Here, the definitions f1 = (m+n)/n and f 2 = (m−n)/n are used. Nonfitting Parameter Estimation Method. As shown in the next section, eq 5 is extremely insensitive to partition ratio K for diffusion-controlled materials; it is virtually impossible to accurately estimate K by measured air-phase concentrations on the one hand, the uncertainty of K has no impact on the estimates of other characteristic parameters on the other. Therefore, we can reasonably assume an approximate value for K and only estimate D and C0 on the basis of the new form of the model equation, eq 5. To determine D and C0, we need two measurements of airphase concentration. First, the value of n can be estimated from eq 5 by two observations Ca(t1) and Ca(t2):

C (t ) 1 ln a 1 t 2 − t1 Ca(t 2)

(7)

Then, remembering that 0 < α1L < π/2, α1 can be easily determined from eq 2b because the left-hand side of eq 2b is known with the estimated K and the determined n (n = Dα21). Next, the known α1 and n gives D from eq 4, and N is also found from eq 6. Finally, C0 is obtained from eq 5 using one of the two observations

C0 =

2exp( − Dαn 2t )[χC0αn tan(αnL) − mCin] 2

αn{αn(2D + χKL) + tan(αnL)[(m − Dαn )L + χK ]}

Ca(ti) N exp( −nti)

(8)

where ti denotes the used sampling time (i = 1 or 2). In summary, only two observations are used to determine two parameters (C0, D) in a sequential order. No initial guesses are required. Since many more than two observations are generally available, a great number of estimates of D and C0 are probably obtained by repeating this procedure on each possible combination of two observations. These estimated D and C0 form a sample of parameter estimates involving experimental errors. In other words, this method provides a natural and convenient approach to sampling from the distribution of experimental errors. As shown below, this function should be clearer by applying this method to a set of actually data.

(1)

where Ca is concentration of VOCs in the chamber air; t denotes time; Cin is inlet concentration of pollutants if any; χ is defined as DA/V; A denotes area of material contacting with the air; V is chamber volume; L is material thickness; m is air change rate defined as Q/V; and Q is volumetric flow rate of the air flows. The initial air- and material-phase concentrations are assumed to be zero and a constant C0, respectively. αn are the nth positive roots of the following transcendental equation m − Dα 2 = χKα tan(αL)

(3)

and substituting eq 2 into eq 3, we can rewrite eq 3 as follows

material emitting some gaseous organic pollutants. Transportation of these pollutants in this chamber consists of molecular diffusion within the material, which can be described by the Fick’s second law, partition on the material-air interface, which can be defined by a partition ratio, K, and convective mass transfer in the chamber, which is controlled by the resistance of convective mass transfer on the surface.12−15 Several authors have solved this mass transfer problem, and one of analytical solutions is12 ∞

α1L(2D + DKR ) + tan(α1L)[(m − Dα12)L2 + DKR ]

n = Dα12

n=

∑

2C0DR tan(α1L) exp(− Dα12t )

Here, α1 is the first positive root of eq 2. Using the definition

Figure 1. Simplified representation of VOC emission in a conditioned chamber or room.

Ca(t ) = Cin +

(2b)

(2a) 9087

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RESULTS Sensitivity of Partition Coefficient. Equation 5 is very insensitive to parameter K. This conclusion is drawn from Figure 2, which shows the relative sensitive coefficients of

∂Ca(t ) Δt = −nΔtCa(t ) ∂t

(9)

This sensitivity is proportional to the sampling interval Δt and decays exponentially as concentration Ca(t). It is also equal to the variation (decrease) of concentration during interval Δt. Provided that the measurements have a (absolute) error ε, having the dimension of concentration, to obtain reliable parameter estimates, Δt should be great enough that the variation of VOCs concentration during Δt (i.e., relative sensitivity coefficient) is larger than random error ε. Otherwise, the variation in Ca is dominated by errors not emission processes. This requirement can be expressed as ε ε Δt > = exp(nt ) nCa(t ) nNC0 (10) Equation 10 indicates that the sampling interval should increase exponentially because of the exponential decrease in concentration. Although not all parameters are available and Δt cannot be accurately determined, it is still possible to estimate the order of magnitude of Δt empirically. Application to VT/NIST Reference Material. To illustrate and validate the new method, this work uses the reference material developed recently by a group of the Virginia Tech (VT) and the National Institute of Standards and Technology (NIST).23,24 This material is loaded with a specific amount of toluene, and toluene air-phase concentrations are sampled in a small environmental-chamber test. It has homogeneous and stable characteristic parameters D, K, and C0 (these original parameters are listed in Table 1). More

Figure 2. Sensitivity coefficients of characteristic parameters of organic emission materials D, K, and C0.

parameters D, K, and C0. The definition of relative sensitive coefficients is β ∂Ca/∂β, having the dimension of concentration, where β denotes a parameter such as D or K. Sensitivity of a parameter is proportional to its sensitivity coefficient. Figure 2 illustrates that despite K varying from 100 to 10000 its sensitivity coefficient is all nearly equal to zero, implying that concentration Ca keeps essentially invariant with the variation of K no matter what the value of K is. Previous numerical and analytical model studies have also proved, from different aspects, that the extremely insensitivity of K indeed exists, particularly at the late emission stage.12,25 Our estimating method uses this feature, whereby K is not determined, but an approximation is specified. The influence of the guessed K on the performance of the estimation procedure is examined below. Sensitivity Analysis of C0 and D. C0 is a relative sensitive parameter among the three characteristic parameters, and it is easy to verify from eq 5 that the sensitivity of C0 is independent of its values. So, the relative sensitivity curves of C0 of various values have the same shape as that shown in Figure 2. In contrast, the sensitivity of D depends heavily on its value, increasing with its value (Figure 2). In consequence, determination of D becomes difficult for diffusion-controlled materials, which have very small D. The major difficulty is that determining D could be an ill-posed problem. The sensitivity coefficients of D and C0 decay with time exponentially, suggesting that the measurements of concentration should be performed as early as possible to improve the accuracy of estimation, while the requirement of a large Fo number (eq 5), say 0.1, must be fulfilled. Sensitivity Analysis of Time. Another important sensitivity is that of time (not shown in Figure 2 for clarity). The sensitivity of time determines the effective intervals of measurements used in the parameter estimation methods, including both fitting and nonfitting methods. If the interval of sampling interval is Δt, the time sensitivity can be expressed as eq 9:

Table 1. Some Important Estimates and Statistics Derived from the Nonfitting Methoda original parameters averages variances standard deviations accuracy estimation (percentile method)

accuracy estimation (normal approxima tion) a

data data data data data data data

1 2 1 2 1 2 1

K

D

C0

500 × × × × × × ×

3.60 × 10−14 2.58 × 10−14 2.42 × 10−14 1.15 × 10−30 5.04 × 10−30 1.07 × 10−15 2.25 × 10−15 (2.40 × 10−14, 2.83 × 10−14) (2.02 × 10−14, 3.02 × 10−14) (2.37 × 10−14, 2.79 × 10−14) (1.98 × 10−14, 2.86 × 10−14)

7.86 × 108 8.73 × 108 9.24 × 108 1.24 × 1014 1.90 × 1015 1.11 × 107 4.36 × 107 (8.50 × 108, 8.92 × 108) (8.40 × 108, 9.99 × 108) (8.52 × 108, 8.95 × 108) (8.38 × 108, 10.09 × 108)

data 2

×

data 1

×

data 2

×

The accuracy estimations given here are the 95% confidence intervals.

importantly, these parameters are determined by independent microbalance procedures, for instance, D and K are determined by independent microbalance sorption and desorption procedures. 10 So, this reference material is very suitable for validating our estimation method. Two sets of concentration data of the small chamber tests are used. The configuration of the small chamber is the same as that shown in Figure 1 with the exception of a mixing fan equipped in the chamber. 23 One of the data sets is sampled with the fan being operated; the other one is obtained without the fan (see Figure 10 of ref 24). Only the data of testing times larger than 20 h are used as input of this determining 9088

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procedure. Data set 1 consists of 12 effective data, whereas set 2 is of 31 data. Although any combination of two observations of each data set can produce an estimated parameter pair (D, C0), the choice of observations should be somewhat cautious to improve the precision of estimates. An examination of the data shows that the uncertainties of measurements in data 1 and 2 are slightly larger than 5%. The thickness of the reference material is 0.0254 cm. From eq 10, Δt should be on the scale of about 18 h by assuming D of order of magnitude 1.0 × 10−14. Data sets 1 and 2 comprise data sampled on different days. So only observations sampled from different days (Δt > 24 h) are used as the inputs of the procedure to estimate D and C0. Any combination of observations from the same day is abandoned. In this way, data set 1 yields 44, and data set 2 yields 412 estimates of (D, C0). The 44 and 412 estimates of (D, C0) constitute two samples of the population of estimates, having an unknown probability distribution. The statistics of the samples such as averages, variances, standard deviations, and so on can be obtained as usual, and some of them are reported in Table 1. Since a real estimate sample is available, the statistics are determined without any assumption about error distribution. This is the most important characteristic of this method. Among all statistics, particularly important are the sample averages (not any individual of the sample) that can be reasonably regarded as the best estimates of parameters D and C0. As can be seen from Table 1, the differences between the estimates of D and C0 derived from this new method and the microbalance procedure are not significant. This point is also demonstrated by Figures 3 and 4.

Figure 4. Comparison between model predictions with the whole emission profiles. The inputs of the models are the same as used in Figure 3 except that K = 500.

Figure 4 compares the modeling prediction (eq 1) with all data sampled during the whole tests, including data measured before 20 h (not used in the estimating procedure). All models somewhat overrate the early stage emission profile, which is probably due to the mass-diffusion model (eq 1) ignoring the convective mass-transfer resistance on the air-material surface; but, the overall matches between the data and predictions are very favorable, validating the mass-diffusion model and the estimated parameters. Moreover, the predictions using the new values of D and C0 match experimental data better than using original values, though the differences between them are minor. Robustness. The term “robust method”, according to Box,22 refers to a method that is insensitive to specific assumptions or uncertainties. By this definition, the new nonfitting procedure for estimating D and C0 is robust to uncertainty of guessed K. The estimates summarized in Table 1 are obtained by assuming a rough estimate for K, say 1000. The original value of K estimated by the microbalance procedure is 500. 24 Our computation shows that variation in the guessed K ranging from 10 to 10000 yields the same estimates for D and C0; the influence of uncertainty of K can be reasonably totally ignored. As has been pointed out, this robustness is contributed to K having the almost zero sensitivity coefficient. On the other hand, the insensitivity of K also implies that any attempt at estimating K from such an opening environmental chamber test is likely to prove a failure. The sensitivity of a parameter defines the accuracy of the parameter estimation, and it depends only on the form of the model (or the nature of emission processes) and is independent of used estimation methods. So it is impossible for both fitting and nonfitting methods to accurately estimate K in such an ventilated chamber test. Unlike the fitting methods, this nonfitting procedure rules out the possibility of existence of multiple best-fit solutions and dependence of initial estimates and inputs. Estimation of multiple parameters, such as D, K, and C0, by a least-squares regression analysis is usually an ill-posed problem because K and sometimes D are not sensitive parameters, as shown above. The well-known difficulties associated with ill-posed problems are that solution is not unique or very sensitive to small perturbations of inputs. 26 Our procedure, determining two parameters D and C0 by two observations of organic emission, defines a well-posed problem; its solution is unique and

Figure 3. Comparison of model predictions with experimental data. Three sets of parameters D and C0 are used: the black solid line is the prediction with the original values of D and C0 determined by using independent microbalance sorption and desorption procedures,10,24 and the other two lines are obtained by using the parameters estimated by the present method from the data sets 1 and 2.

Figure 3 compares the experimental data with the model predictions (eq 5) inputted with the estimated D and C0, including the original values determined by the microbalance procedure. 24 It shows that there are some discrepancies between the data and all the predictions. These discrepancies can be attributed to two sources of errors, errors associated with the small chamber tests and the estimation procedures. 9089

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Figure 5. Frequency diagrams (histograms) for the two samples of the estimates of D and C0.

1.96 × 2.25 × 10−15, that is, (1.98 × 10−14, 2.86 × 10−14). All the 95% normal confidence limits are also provided in Table 1 for comparison with those derived by the percentile method. The accuracy of the normal approximation method depends critically on the similarity of the distributions of normal and samples of estimates (not distribution of errors). Figure 5 shows the histograms of estimate samples of D and C0 derived from the data sets. These frequency diagrams differ more or less from normal distribution in different ways, though the same data sets are used. Moreover, and perhaps most importantly, ones should understand that common curve-fitting methods provide estimates of variances (and standard deviations) of parameters by assuming experimental errors being Gaussian distribution, whereas the nonfitting procedure provides these statistics making no assumption about the error probability distribution. This point emphasizes the difference between this method and other curve-fitting methods. Consequently, the confidence limits given by the normal approximation also differs from those given by line-fitting methods. In practice, the distribution of experimental errors is not known; thus the new procedure is general and suitable for real situations. The percentile method gives confidence intervals more directly, making no assumption not only about distribution of errors but that of estimates. Another advantage of the percentile method is no invalid parameter values can be included in the interval. Thus this method should be a more general method than the normal approximation, but the coverage error of this method is often substantial if the sample size of estimates is small. Large samples of estimates are necessary to improve the accuracy. It is advisable that confidence limits of parameter estimates should be determined by the percentile method if a great sample of estimates is available. The differences between confidence limits given by the percentile and normal approximation methods are a consequence of the distribution of estimates (D, C0) not being strictly normal; fortunately, these differences are minor in this case, implying that the two methods are nearly equivalent. The two methods, however, may not be always equivalent to each

requires no initial guesses about D and C0. In contrast, these initial guesses are required by methods fitting data to a nonlinear equation, and thus the converged results depend on the initial guesses, especially on those of D and K. This seriously decreases the robustness of least-squares methods. In addition, averages of estimates, not individual estimates, are used as the parameter estimates, enabling the estimation to be insensitive to measurement errors. This insensitivity to random errors can be further enhanced as size of the sample increases. To conclude, the new uncoupled method is robust to uncertainties associated with initial values and random errors. Accuracy Estimation. A central issue in parameter estimation is the quantification of the uncertainties associated with the estimation; determination of parameters is absolutely not the end-all of model parameter estimation. The nonfitting method may provide many estimates of D and C0 from one experimental run, which constitute a sample of parameter estimates involving measurement errors; therefore, this method provides a natural approach to estimating the uncertainty of experimental errors. This uncertainty can be summarized by a confidence limit on each parameter. Two methods, percentile method and normal approximation, for estimating confidence intervals are suggested because of their simplicity. The steps of the percentile method for determining confidence intervals are: 1) sort the samples of estimates into an ascending order; 2) find the upper end point of the onesided 97.5% percentile interval, βu, which is estimated by the 0.975Bth estimate of the sorted sample, where B denotes the size of the samples, say 412 for the data set 2. If 0.975B is not an integer, the nearest integer larger than 0.975B is used. A similar approach can find the lower end point βl; 3) the 95% interval is therefore (βl, βu). Other intervals of different levels can also been obtained by the same procedure except using different percentiles. Table 1 gives 95% confidence intervals for D and C0. Since we have standard errors of parameter estimates (Table 1), standard 95% normal approximation confidence intervals for D and C0 can also be estimated, for example, 2.42 × 10−14 ± 9090

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chamber tests provide rapid and feasible approaches to this purpose. Therefore, in the future a similar procedure deserves to be explored for closed chamber environments and other environmental processes involving organic emissions.

other: in particular, they may differ considerably when distributions of estimates differ from Gaussian distribution greatly.

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DISCUSSION Our new developed method cannot provide the estimation for partition ratio K. From the mathematical perspective, our method utilizes two observations of air-phase concentrations, i.e., two known conditions or equations, thus only two parameters (D, C0) can be determined as a well-posed problem. Fortunately, on one hand, K is extremely insensitive to the used model, eq 5, thus the guessed value of K has no influence on the performance of the method; unfortunately, on the other, whatever methods are used, such opening chamber tests cannot be used to estimate K. Although K depends on the chemical and physical properties of VOCs, many studies have shown that K should increase as the VOC vapor pressure decreases, and some relationships have been created for approximately estimating K based on vapor pressure.9,25,27 According to the sensitivity analysis, these rough approximations should be sufficient for predicting indoor organic emissions under practical circumstances. Functionally, by far the most interesting feature of the new estimation procedure is the ability to provide estimation of accuracy without assumption about experimental errors: multiple estimates of the parameters form a sample of estimate population involving experimental errors, obtained by repeating the procedure on each observation pair if multiple observations are available. This sample is precious information and generally not available from the normal estimation methods. To obtain similar information, considerable effort has been devised, among which Monte Carlo (MC) simulations have been widely used, whereby large numbers of estimates can be generated by computational algorithms and thus error estimations can be estimated from the artificial sample. But, the nature (i.e., probability distribution) of measurement errors must be known to perform a credible MC simulation. Another increasing popular method is the bootstrap method, 28 which is similar to MC simulations but the underlying nature of errors is not required. Ones, however, must watch out for cases where some theoretical and technical assumptions of bootstrap methods are violated. Compared to these computational methods, our method provides a sample of estimates from the actual measurements in a natural and direct way, without recourse to any assumption or computational algorithm. The only potential problem is the size of the sample, which depends on the number of effective measurements. A sample of greater size can provide more reliable estimates for accuracy. Since one estimate of D and C0 is determined from two data, the sample size can increase fast with the available data. For N effective data, N(N − 1)/2 estimates can be found theoretically; we have the sample size on the scale of N2, a very fast increasing rate. In conclusion, by a skillful arrangement of an emission model (eq 5), we get a well-defined procedure for determining D and C0 from two air-phase organic concentrations. The idea of the new approach should be theoretically feasible for both ventilated and closed chambers, except some mathematical details and expressions. This paper develops the method for VOCs emission in a conditioned chamber because reliable experimental data are available for such a situation. Determination of key material parameters can facilitate the applications of mechanistic models for characterizing emitting behaviors of building materials.12−15 Small environmental

AUTHOR INFORMATION

Corresponding Author

*Phone: 86 15575800109. E-mail: [email protected], [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS I am deeply grateful to Professor John C. Little of the Virginia Tech for providing their environmental-chamber testing data. REFERENCES

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