Predicting the emissions of VOCs/SVOCs in source and sink materials: development of analytical model and determination of the key parameters

Abstract

The emissions of volatile organic compounds (VOCs) and semi-volatile organic compounds (SVOCs) from indoor materials pose an adverse effect on people’s health. In this study, a new analytical model was developed to simulate the emission behaviors for both VOCs and SVOCs under ventilated conditions. Based on this model, we further presented a hybrid optimization method to accurately determine the key parameters in the model: the initial emittable concentration, the diffusion coefficient, and the material/air partition coefficient (for SVOCs the surface/air partition coefficient is also included). Experiments for VOC emissions from solid wood furniture were performed to determine the key parameters. We also evaluated the hybrid optimization method with the data of flame retardant emissions from polyisocyanurate rigid foam and VOC emissions from a panel furniture in the literature. The correlation coefficients are high during the fitting process (R²=0.92–0.99), demonstrating effectiveness of this method. In addition, we observed that chemical properties could transfer from SVOC-type to VOC-type with the increase of temperature. The transition temperatures from SVOC-type to VOC-type for the emissions of tris(2-chloroethyl) phosphate (TCEP) and tris(1-chloro-2-propyl) phosphate (TCIPP) were determined to be about 45 °C and 35 °C, respectively. The present study provides a unified modelling and methodology analysis for both VOCs and SVOCs, which should be very useful for source/sink characterization and control.

1. Introduction

Human exposure to volatile organic compounds (VOCs) and semi-volatile organic compounds (SVOCs) in polluted indoor environments can result in severely adverse health effects (Tang et al., 2009; Liu et al., 2013; Landrigan et al., 2018; Tian et al., 2020; Wang et al., 2020a; Zhang et al., 2021; Liu and IV, 2021). According to ASTM standard D8141 (2017), VOCs are classified as a class of organic compounds with a boiling point ranging from 50 °C to 240 °C, while SVOCs with a boiling point ranging from 240 °C to 400 °C. Compared with VOCs, SVOCs have higher boiling points and lower saturated vapor pressures. Due to high surface/air partition coefficient, SVOCs can be easily sorbed to indoor surfaces, such as walls, floors, furniture surfaces, and even human skin. Thus, sorption effect is significant and should be considered for SVOC transport, which differs greatly from that of VOCs. Understanding the emission/sorption mechanisms of VOCs and SVOCs is essential for characterizing their fate and transport in the indoor environment, and further for proposing strategies to reduce human exposure to these compounds.

Modelling investigation on the emissions of VOCs from indoor building materials and products has gained much success, and the fully-analytical solutions have been obtained in many scenarios by excluding the sorption effect on chamber or indoor surfaces (Little et al., 1994; Huang and Haghighat, 2002; Xu and Zhang, 2003; Deng and Kim, 2004; Kumar and Little, 2003; Xiong et al., 2012; He et al., 2019). For SVOC transport in indoor environments, the sorption effect must be taken into account. However, up to now no fully-analytical model is derived due to the complexity of incorporation of surface partitioning. Xu and Little (2006) established the governing equation of SVOC transport by including the sorption of chemicals on chamber surface and particles in the chamber air and obtained a semi-analytical solution. Liu et al. (2013) developed a fully-analytical model by incorporating interface partitioning and external convection on the surfaces, but ignored the internal diffusion process to simplify when getting the analytical solution. Gong (2015) proposed a mass transfer model for SVOCs on human skin and obtained an analytical solution. This model assumed the multi-layer diffusion inside the skin but neglected the external sorption of SVOCs on indoor surfaces. To date, the modelling studies on VOCs and SVOCs are performed separately. There is no single analytical model to predict the emission/sorption characteristics in the chamber or indoor environment for both VOCs and SVOCs.

The fate and transport of VOCs from materials can be characterized by three key parameters in analytical models, i.e., the initial emittable concentration C₀, the diffusion coefficient D_m, and the material/air partition coefficient K_ma (Little et al., 1994; Yang et al., 2001; Liu et al., 2013; Zhang et al., 2016; Wang et al., 2021). A significant challenge is to determine the three parameters simultaneously and accurately. To achieve this goal, some researchers proposed the linear regression methods. Xiong et al. (2011) developed a C-history method to determine the C₀, D_m and K_ma simultaneously for VOC emissions from building materials in an airtight chamber, by performing linear curve fitting between the dimensionless concentration and emission time. This method requires multiple samplings inside the chamber, which will cause measurement error if the chamber volume is very small. To overcome this limitation, some improved methods have been proposed (Huang et al., 2013; Zhou et al., 2018). There are also some studies working on nonlinear regression methods. Zhang et al. (2018) applied the Levenberg-Marquardt algorithm for nonlinearly fitting the VOC emission curve under ventilated conditions to get two parameters (D_m K_ma), and then used the VOC equilibrium concentration under airtight condition to further obtain the C₀. The multi-parameter fitting process using traditional algorithms probably produces multiple solutions, and the uniqueness of the key parameters is difficult to be guaranteed. All methods mentioned above are confined to VOCs.

Different from VOCs, the emission behaviors of SVOCs from materials can be generally characterized by a new parameter y₀ (the gas-phase SVOC concentration adjacent to the material surface), together with D_m and K_ma (Xu and Little, 2006; Zhang et al., 2016; Wei et al., 2018). The introduction of key parameter y₀ is based on the assumptions that the SVOC concentration inside the material doesn’t change over time and the internal diffusion can be neglected. A few methods have been developed to measure y₀. Xu et al. (2012) performed experiments to measure di-(2-ethylhexyl)-phthalate (DEHP, a typical SVOC) emitted from vinyl flooring, which could shorten the experimental time from 150 days to 20 days. Clausen et al. (2012) assumed that y₀ of DEHP from vinyl flooring is close to the vapor pressure of the pure DEHP and measured the y₀ of DEHP under different temperatures. Some researchers found that the above assumptions are not suitable for other SVOCs emitted from different materials, thus proposed different methods to determine y₀ (Little et al., 2012; Cao et al., 2017; Liang et al., 2018; Yang et al., 2020). Recently, Pei et al. (2017) observed that y₀ could decrease if the experimental time was relatively long, e.g., the y₀ of DEHP from polyvinyl chloride (PVC) flooring decreased 38% in 1.5 years. This means it might be more appropriate to use C₀ rather than y₀ as a key parameter, since y₀ is changeable and the internal diffusion should be considered for SVOC emissions. In terms of determining D_m and K_ma of SVOCs, Liang et al. (2019) used a numerical model and proposed a nonlinear regression method to obtain the two parameters on the basis of known C₀ At present, there is no method that could determine the three key parameters for SVOCs (C₀, D_m, K_ma) simultaneously and accurately.

The main objectives of this study are to: (1) develop a fully-analytical model that could characterize the emissions of both VOCs and SVOCs from indoor materials and products; (2) present a method to accurately determine the key parameters of VOCs and SVOCs by applying a hybrid optimization method.

2. Methods

2.1. Development of the fully-analytical model

Figure 1 shows the schematic of VOC/SVOC emissions in a ventilated chamber. In this figure, the building material is placed at the bottom of the chamber. For VOCs, the chamber wall sorption could be ignored; while for SVOCs, a certain proportion of SVOCs will be sorbed by the chamber wall. In addition, boundary layers exist immediately adjacent to the chamber and material surfaces.

Figure 1.

Schematic of VOC/SVOC emissions in a ventilated chamber.

To develop a comprehensive model to characterize both VOC and SVOC emissions, the following assumptions are applied: (1) the initial concentration of the VOCs/SVOCs in the material, C₀, is uniformly distributed; (2) the inlet and initial VOC/SVOC concentrations of the chamber are zero; (3) mass transfer is one-dimensional, vertical to the chamber or material surfaces; (4) the diffusion coefficient and partition coefficient are constant; (5) the chamber air is well mixed; (6) instant equilibrium always exists at the material/air and chamber/air interfaces.

The governing equations, together with the boundary and initial conditions, as well as the mass conservation in the chamber can be described as follows:

$$\frac{\partial C(x,t)}{\partial t}=D_{\text{m}}\frac{{\partial }^{2}C(x,t)}{\partial x^2}$$ (1)

$$C(x,0)=C_0\text{ for }0\le x\le L$$ (2)

$$\frac{\partial C(x,t)}{\partial x}=0\text{ for }t>0,x=0$$ (3)

$$-D_{\text{m}}\frac{\partial C(x,t)}{\partial x}=h_{\text{m}}\left(y_0-C_{\text{a}}\right)\text{ for }t>0,x=L$$ (4)

$$C(x,t)=K_{\text{ma }}{y}_0\text{ for }t>0,x=L$$ (5)

$$K_{\text{sa}}\frac{\text{d}{y}_{\text{s}}}{\text{d}t}=h_{\text{s}}\left(C_{\text{a}}-y_{\text{s}}\right)$$ (6)

$$V\frac{\text{d}{C}_{\text{a}}}{\text{d}t}=Q\left(y_{\text{in }}-C_{\text{a}}\right)+h_{\text{m}}{A}_0\left(y_0-C_{\text{a}}\right)-h_{\text{s}}{A}_{\text{s}}\left(C_{\text{a}}-y_{\text{s}}\right)$$ (7)

where, C_a is the gas-phase VOC/SVOC concentration in the chamber, μg/m³; yin is the inlet gas-phase concentration, μg/m³, which is assumed to be zero; y₀ is the gas-phase SVOC concentration adjacent to the material surface, μg/m³; y_s is the gas-phase concentration in the boundary layer adjacent to the chamber surface, μg/m³; C(x,t) is concentration inside the material at thickness of x, μg/m³; C₀ is the initial emittable concentration inside the material, μg/m³; h_m and h_s are the mass transfer coefficients across the material surface and chamber surface, respectively, m/s; K_ma is the material/air partition coefficient at equilibrium, dimensionless; K_sa is the chamber surface/air partition coefficient at equilibrium, m; Q is the ventilation rate, m³/s; A₀ is the total emission area of the material in the chamber, m²; As is the area of the chamber wall, m²; L is the thickness of the material, m; V is the volume of the chamber, m³; D_m is the material-phase diffusion coefficient, m²/s.

By virtue of Laplace transformation, the following fully-analytical solution can be derived:

$$C_{\text{a}}=2C_0\beta \sum _{n=1}^{\infty }\frac{\left(q_n-B_s{q}_n^3\right)\sin q_n}{A_n}\times e^{-D_{\text{m}}{L}^{-2}{q}_n^{2}t}$$ (8)

$$C=2C_0\sum _{n=1}^{\infty }\frac{A_{\mathfrak{u}}\cos \frac{q_{n}x}{L}}{A_n}\times {\text{e}}^{-D_{\text{m}}{L}^{-2}{q}_n^{2}t}$$ (9)

with

$$\begin{gathered} A_n=q_n\sin q_n\left\{A_{\text{u}}+K_{\text{ma}}\beta \left(1-3B_{\text{s}}{q}_n^2\right)+K_{\text{ma}}B{i}_{\text{m}}^{-1}\left[\alpha -3\left(1+\alpha B_{\text{s}}+\xi \right)q_n^2+5B_{\text{s}}{q}_n^4\right]\right\} \\ +q_n^2\cos q_n\left[K_{\text{ma}}\beta \left(1-B_{\text{s}}{q}_n^2\right)+K_{\text{ma}}B{i}_{\text{m}}^{-1}{A}_{\text{u}}+2\left(1+\alpha B_{\text{s}}+\xi \right)-4B_{\text{s}}{q}_n^2\right] \end{gathered}$$ (10)

$$A_{\text{u}}=\alpha -\left(1+\alpha B_{\text{s}}+\xi \right)q_n^2+B_{\text{s}}{q}_n^4$$ (11)

where, Bi_m is the Biot number for mass transfer in the material, $B{i}_{\text{m}}=h_{\text{m}}L/D_{\text{m}}$; α is the dimensionless air exchange rate, $\alpha =N{L}^2/D_{\text{m}}$; β is the ratio of the material volume to chamber volume, $\beta =L{A}_0/V$; ξ is the ratio of the equivalent volume of the sorption surface to the chamber volume, $\xi =K_{\text{sa}}{A}_{\text{s}}/V$; $B_{\text{s}}=\frac{K_{\text{sa}}{D}_{\text{m}}}{L^2{h}_{\text{s}}}$; q_n are the positive roots of the following equation:

$$q_n\tan q_n=\frac{A_{\text{u}}}{K_{\text{ma}}\beta \left(1-B_{\text{s}}{q}_n^2\right)+K_{\text{ma}}B{i}_{\text{m}}^{-1}{A}_{\text{u}}}$$ (12)

In the solutions (8)-(12), ξ and B_s are parameters related to the sorption properties of chamber surface. B_s can be represented as $K_{\text{sa}}{L}^{-1}B{i}_{\text{s}}^{-1}$ (Bi_s is the Biot number for mass transfer onto the chamber wall, $1/B{i}_{\text{s}}=D_{\text{m}}/L{h}_{\text{s}}$). The physical meaning of 1/B_s is the ratio of internal diffusion mass transfer resistance to external convective mass transfer resistance. Based on equations (5) and (9), y₀ is related to time, and is not a constant for a given temperature for many cases, so it is not appropriate to use y₀ as a key parameter, while C₀ is more suitable. According to the equation (8), D_m also influences gas-phase SVOC concentrations. Therefore, we can use C₀, D_m, and K_ma as the key parameters for SVOCs, the same as VOCs.

Compared with Deng and Kim (2004)’s model for VOC emissions, the present model includes both the material emission and chamber wall sorption processes, making it applicable for both VOC and SVOC emissions, thus can be regarded as a significant improvement in model development. For the developed model (equation (8)), if we neglect the chamber surface sorption (K_sa=0), this fully-analytical solution can be simplified into the solution in Deng and Kim’s study. This not only implies that the Deng and Kim’s model is a special case of the present model, but also preliminarily demonstrates the correctness of the derivation process.

It should be noted that the model proposed includes source emission, sink sorption and ventilation, but excludes sorption related to airborne particles. The impact of particle and dust is ignored in the model because clean air is generally used in laboratory chamber testing. When assessing the exposure of SVOCs in a realistic environment, particle and dust play an important role (Liu et al., 2012; Shi and Zhao, 2015) and should be considered. Under these conditions, no analytical model could be derived, and numerical model must be used.

2.2. Determination of the key parameters via a hybrid optimization method

In this study, the gas-phase VOC/SVOC concentrations (C_a) in the chamber over time were measured, which are then used to determine the key parameters (C₀, D_m, and K_ma) in the model via a hybrid optimization method. For SVOCs, the chamber surface/air partition coefficient (K_sa) is also determined. Particle swarm optimization (PSO) is a widely used intelligent algorithm to solve highly complex optimization problems, especially for continuous function extremum problems and nonlinear problems with strong global search ability. The basic strategy for PSO is to adjust search direction and speed based on the best location in search history and population’s history (Kennedy and Eberhart, 1995). However, for multi-peak functions and ill-posed equations, this algorithm probably cannot obtain the best solution. The accuracy will not enhance greatly with the increase of number of iterations and populations, in other words, it is easy to get premature convergence (Shi and Eberhart, 2002).

To address this problem, Ghamisi and Benediktsson (2015) suggested that a small part of particles should be selected for hybridization in the iterative process. The ant colony optimization (ACO) can guide the search direction according to the intensity of the pheromone (Colorni et al., 1991), which is a good choice for hybridization. ACO has good optimization performance, but it is easy to occur stagnation and the updated speed is slow, while PSO can make up this shortcoming. Considering that, we use particle swarm optimization, coupled with ant colony algorithm, and cross-mutate the results. The equations of the hybrid algorithm for updating key parameters are as follows:

Define a state transition probability function:

$$S_i=\frac{{\text{FIT}}_i-{\text{FIT}}_{\text{best }}}{{\text{FIT}}_{\text{best }}}$$ (13)

For ACO:

$$X_i(n+1)=X_i(n)+(\text{rand }-0.5)\frac{1}{n},S_i<S_0$$ (14)

$$X_i(n+1)=X_i(n)+(\text{rand }-0.5)(\text{Upper }-\text{Lower }),S_i>S_0$$ (15)

For PSO (no iteration for S_i <S₀ ):

$$v_i(n+1)=w{v}_i(n)+c_1{r}_1\left(p_{\text{best }}(n)-X_i(n)\right)+c_2{r}_2\left(g_{\text{best }}(n)-X_i(n)\right),S_i>S_0$$ (16)

$$X_i(n+1)=X_i(n)+v_i(n+1)$$ (17)

The fitness function is

$${\text{FIT}}_i=\frac{1}{W}\sum _{n=1}^W\frac{{\left[C_{\text{pre}}\left(t_n\right)-C_{\exp }\left(t_n\right)\right]}^2}{P_n}\{\begin{array}{l}{\text{ for VOCs: FIT}}_i=f\left(X_i(n)\right)=f\left(C_0,D_{\text{m}},K_{\text{ma}}\right)\hfill \\ {\text{ for SVOCs: FIT}}_i=f\left(X_i(n)\right)=f\left(C_0,D_{\text{m}},K_{\text{ma}},K_{\text{sa}}\right)\hfill \end{array}$$ (18)

where, S_i is the state transition probability of X_i, and S₀ is the transition probability constant; X_i is the key parameters to be fitted; v_i is the velocity of the particles or ants, and the total number is (40–50); w is the inertia weight; c₁ and c₂ are the individual and social learning factors, respectively, in the range of (0, 4); r₁ and r₂ are random numbers, in the range of (0, 1); p_best(n) is the optimal position of the particles in the n iteration; g_best(n) is the global optimal location in the n iteration; Upper and Lower are the upper and lower limits of the key parameters, respectively; FIT_i is the fitness function, representing the residual between the predicted results obtained using the key parameters and experimental data, and is used to judge whether the values of the key parameters need to be updated during the iterative process; FIT_best is the optimal value of FIT_i calculated for all particles during the nth iteration; C_pre(t_n) is the calculated gas-phase VOC/SVOC concentration at t_n; C_exp(t_n) is the measured gas-phase VOC/SVOC concentration at t_n; W is the number of experimental data; P_n is a weighting factor, $P_n=\log \left[C_{\exp }\left(t_n\right)\right]$.

This method combines two kinds of intelligence algorithms, thus is called as the hybrid optimization method, which can be applied to various conditions and complex scenarios. The flow chart of the hybrid optimization method is shown in Figure S1 of the supplementary material. In this study, we use the developed fully-analytical model to predict the gas-phase VOC/SVOC concentration with the hybrid optimization method.

3. Experimental section

Experiments were performed to evaluate the effectiveness of the fully-analytical model and the hybrid optimization method. Figure S2 in the supplementary material shows a schematic of the experimental chamber system to determine key parameters of VOCs from materials. The environmental chamber was made of stainless steel with the volume of 1 m³. A fan was placed at the top for mixing the air inside the chamber, with the air velocity of 0.2 m/s. Prior to tests, the inner surface of the chamber was scrubbed with acetone. After scrubbing, clean air with high air exchange rate (10 h⁻¹) was flushed into the chamber until the acetone was completely evaporated. The background concentration of VOCs in the chamber was measured at the beginning of the experiment. The concentration levels of VOCs (< 2 μg/m³) met the requirement described in the ASTM D6670 standard (2018). During experiment, the ambient air passed through an activated carbon filter, followed by a high efficiency particulate air (HEPA) filter, to reduce the contaminations of VOCs and particles. Then the cleaned air was introduced into the test chamber by a flow controller at an air exchange rate of 0.5 h^-1. The temperature was controlled at 23 ± 0.5 °C via an air-conditioning system, and the relative humidity was controlled at 45 ± 5% via a humidifier.

Two kinds of solid wood furniture (designated as solid wood furniture 1 and solid wood furniture 2) were selected for the tests. Five typical pollutants commonly emerged in Chinese home decoration were chosen as the target pollutants for analysis, including toluene, butyl acetate, m-xylene, p-xylene, and formaldehyde. The chamber air was sampled at regular time intervals with a sampling flow rate of 0.3 L/min. Tenax-TA tubes were used for collecting VOC samples (excluding ketones and aldehydes), which were then analyzed by gas chromatography/mass spectrometry (GC/MS) for identification and quantification. For ketones and aldehydes, 2,4-dinitrophenylhydrazine (DNPH) tubes were used, which were then analyzed by high performance liquid chromatography (HPLC). The total experimental time lasted for about seven days. For the first three days, the sampling time for each interval was 10 min, and was increased to 30 min for the following four days due to relatively low VOC concentrations. Duplicate samples were carried out for reducing the measurement error. In addition, quality assurance (QA) and quality control (QC) activities were also performed for the experimental system. Detailed information of the experiments is included in Table S1 of the supplementary material.

4. Results and discussion

4.1. Determination of the key parameters of VOCs with the hybrid optimization method

We measured the gas-phase concentrations of five VOCs (toluene, butyl acetate, m-xylene, p-xylene and formaldehyde) from the two kinds of solid wood furniture, and these data were used for determining the three key parameters (C₀, D_m, K_ma) via the hybrid optimization method. For the adsorption of VOCs onto chamber surface, the surface/air partition coefficient K_sa was previously measured, with most of K_sa values less than 0.05 m (Liu, 2013; Wang et al., 2020a), which are relatively small compared to SVOCs’ sorption. This implies that the wall adsorption could be ignored during the VOC emission process. Table 1 lists the measured results for solid wood furniture 1, while the results for solid wood furniture 2 are given in Table S2 of the supplementary material. Table 1 and S2 show that all the square of correlation coefficient (R²) for data fitting to the hybrid optimization method are over 0.92, meaning relatively high accuracy according to the ASTM D5157 standard (2019). Figure 2 shows the model fitting results of furniture 1. During the nonlinear fitting process with the hybrid optimization method, we randomly ran the program five times. Table 1 and Table S2 indicate that the determined key parameters are relatively stable for all the different running times, which is a significant improvement upon previous nonlinear fitting methods for key parameter determination.

Table 1.

Determined key parameters for solid wood furniture 1 by hybrid optimization method.

VOCs	Run No.	Hybrid optimization method
		C0 (μg/m3)	Dm (m2/s)	K ma	R2
Toluene	1	3.79 × 105	1.42 × 10−10	3.68 × 102	0.94
	2	3.79 × 105	1.42 × 10−10	3.68 × 102	0.94
	3	3.79 × 105	1.42 × 10−10	3.68 × 102	0.94
	4	3.79 × 105	1.42 × 10−10	3.68 × 102	0.94
	5	3.79 × 105	1.42 × 10−10	3.68 × 102	0.94
	Avg	3.79 × 105	1.42 × 10−10	3.68 × 102
Butyl acetate	1	4.00 × 105	2.63 × 10−10	2.40 × 103	0.99
	2	4.00 × 105	2.63 × 10−10	2.40 × 103	0.99
	3	4.00 × 105	2.63 × 10−10	2.40 × 103	0.99
	4	4.00 × 105	2.63 × 10−10	2.40 × 103	0.99
	5	4.00 × 105	2.63 × 10−10	2.40 × 103	0.99
	Avg	4.00 × 105	2.63 × 10−10	2.40 × 103
M-xylene	1	8.88 × 105	8.32 × 10−10	5.41 × 103	0.99
	2	8.89 × 105	8.19 × 10−10	5.40 × 103	0.99
	3	8.89 × 105	8.18 × 10−10	5.40 × 103	0.99
	4	8.89 × 105	8.23 × 10−10	5.40 × 103	0.99
	5	8.89 × 105	8.21 × 10−10	5.40 × 103	0.99
	Avg	8.89 × 105	8.23 × 10−10	5.40 × 103
P-xylene	1	1.31 × 105	2.02 × 10−10	1.62 × 103	0.99
	2	1.31 × 105	2.02 × 10−10	1.62 × 103	0.99
	3	1.31 × 105	2.02 × 10−10	1.62 × 103	0.99
	4	1.31 × 105	2.02 × 10−10	1.62 × 103	0.99
	5	1.31 × 105	2.02 × 10−10	1.62 × 103	0.99
	Avg	1.31 × 105	2.02 × 10−10	1.62 × 103
Formaldehyde	1	7.33 × 105	1.72 × 10−9	6.46 × 103	0.92
	2	7.33 × 105	1.94 × 10−9	6.49 × 103	0.92
	3	7.33 × 105	1.62 × 10−9	6.44 × 103	0.92
	4	7.33 × 105	2.12 × 10−9	6.52 × 103	0.92
	5	7.33 × 105	1.65 × 10−9	6.44 × 103	0.92
	Avg	7.33 × 105	1.81 × 10−9	6.47 × 103

Figure 2.

Fitting results for VOC emissions from solid wood furniture 1 with the hybrid optimization method.

To further verify the robustness (stability) of the algorithm, we perform a sensitivity analysis. A measurement error in statistics is introduced to represent the concentration fluctuation in realistic experiments (Li and Niu, 2005). Then, the updated concentration of VOCs/SVOCs (C_a) can be expressed as:

$$C_a=C_{\exp }+\omega \sigma$$ (19)

where, σ is the standard deviation of the concentration measurements (measurement error); ω is the random variable with normal distribution (zero mean and unitary standard deviation); C_exp is the measured concentration data in the experiments.

We choose p-xylene in solid wood furniture 1 as an example for the analysis. Measurement error of 5% and 10% are chosen. The updated concentrations in equation (19) are then used to determine the key parameters via the hybrid optimization method. Results show that when the σ is up to 10%, the relative deviations (RDs) of the three key parameters are less than 10% (RDs of C₀, D_m, K_ma are 0.76%, 7.92%, 6.17%, respectively). The RDs decrease further when σ is 5%. This means, the smaller the measurement error, the higher the accuracy of the determined key parameters. This sensitivity analysis convincingly demonstrates the robustness of the algorithm in this study.

In addition, we compare the hybrid optimization method and a typical optimization method, the genetic algorithm (GA) method. The results by the GA method to fit the experimental data are also summarized in Tables S2 and S3. From these two tables, we can see that: (1) the R² of the GA method for the same VOCs from the same solid wood furniture is less than that of the hybrid optimization method; and (2) the determined key parameters for different running times of the GA method are different especially for the diffusion coefficient D_m. This means that the GA method will probably produce multiple solutions, which cannot escape the problem of non-uniqueness of solution when the three key parameters (C₀, D_m, K_ma) are intercorrelated during the optimization process. The present hybrid optimization method can solve this challenging problem well. It couples multiple algorithms during the optimization process to ensure that a stable solution can be obtained. This is the most important feature of the hybrid optimization method. If we take the average values of the five running times as the final results, we can see that the determined key parameters between the hybrid optimization method and GA method are different. The C₀ differs slightly, while the D_m and K_ma differ greatly. The differences in D_m and K_ma will cause the deviations in the predicted mid- and long-term (mass transfer Fourier number, Fo_m>0.2) gas-phase VOC concentration and emission rate, which will accordingly influence the evaluation of exposure level of people in the environment.

Moreover, we conducted an analysis on the experimental data in literature for VOC emissions under ventilated condition via the present hybrid optimization method, and compared the obtained key parameters with those measurements by other methods in literature (Wang et al., 2020b). For the emissions of some common VOCs (ethylbenzene, butyl acetate, styrene, m-/p-xylene, benzaldehyde, formaldehyde) from a panel furniture, the RDs of the determined key parameters for the same dataset with the results in literature are all less than 20%. For example, the RDs of C₀, D_m, K_ma for formaldehyde are 0.38%, 18.4%, 5.19%, respectively. The consistency in the obtained key parameters between different approaches further proves the reliability of the present method.

It should be noted that it is always a challenging problem on how to shorten the experimental time in obtaining the key parameters of VOCs. For this consideration, we took VOC emissions from solid wood furniture 1 as an example and performed an analysis to examine the influence of experimental time on the key parameter determination. When the experimental time was reduced from 150 h to 96 h, the determined key parameters via the hybrid optimization method (using data in 96 h) didn’t change greatly, with RDs less than 14% compared with the values in 150 h. When the experimental time was further reduced to 78 h, most of RDs were less than 26% (except for the D_m of m-xylene), as shown in Table S4 of the supplementary material. We also attempted to shorten the experimental time to 72h, but found that the RDs of the determined key parameters became unreasonably large, which was probably due to too few sampling points. The above analysis indicates that the experimental time could be reduced from 150 h to 78 h while we can still get acceptable results.

4.2. Determination of the key parameters of SVOCs with the hybrid optimization method

Compared with VOCs, the SVOCs have lower vapor pressure and stronger sorption properties, making it a big challenge for laboratory testing and key parameter determination. We analyzed the SVOC emission characteristics by using the experimental data in prior studies conducted by U.S. EPA with the hybrid optimization method. Liang et al. (2019) measured the emissions of several SVOCs, focusing on TCEP, TCIPP, and tris(1,3-dichloro-2-propyl) phosphate (TDCIPP) from customized polyisocyanurate rigid (PIR) foam materials under ventilated conditions. The chamber size, material dimensions, and test conditions were directly extracted from this reference. As an example, we used the measured gas-phase SVOC concentrations in microchamber 1 for analysis. The SVOC emission behaviors can be described by equation (8) developed in Section 2.

By applying the hybrid optimization method, the key parameters (C₀, D_m, K_ma, K_sa) for TCEP, TCIPP, and TDCIPP emitted from the PIR foam were determined, and the results under different temperature are listed in Table S5 of the supplementary material. For comparison, the measured parameters for the same SVOCs from the same material by other methods are also summarized in this table. The fitting results with the hybrid optimization method are shown in Figure S3 of the supplementary material. We calculated the normalized mean square error (NMSE) described in ASTM D5157 standard (2019). The values of the NMSE with the hybrid optimization method are 0.02, 0.015, 0.011, 0.043, 0.017, 0.026, 0.053 for the cases listed in Table S5, indicating an adequate accuracy (more details are included in Section S1 of the supplementary material). Liang et al. (2018, 2019) used a different fitting approach to obtain the key parameters of SVOCs from the same PIR materials, and the results are also summarized in Table S5. In the table, for the same SVOC at the same temperature (e.g., 25 °C), the measured key parameters between the hybrid optimization method and the Liang et al. (2018, 2019)’s methods are different.

In Liang et al.’s methods, a material extraction experiment was conducted to measure the total SVOC concentration, which was used as the parameter of C₀. Nonlinear regression approach was applied to fit the experimental data (gas-phase SVOC concentrations) with a numerical model to obtain the other two key parameters D_m and K_ma. Huang et al. (2015) examined the emissions of VOCs from building materials and found that the initial emittable concentration (C₀) of formaldehyde was much less than the total concentration in the material. A theoretical correlation between the ratio of emittable concentration to total concentration and temperature was then derived from kinetic molecular theory. This phenomenon and emission mechanism are probably applicable for SVOCs. By analyzing the data in Table S5, we find that ratio of emittable concentration to total concentrations of TCEP and TCIPP at room temperature is in the range of 9.4%−14.2%, which is consistent with the range for formaldehyde emissions (Huang et al., 2015). Liang et al. (2018, 2019) used the total concentration to substitute emittable concentration for data fitting, which would introduce some deviations. While for the present hybrid optimization method, the obtained C₀ is the initial emittable concentration, thus can ensure the accuracy of D_m and K_ma. The C₀ in the material will increase with the temperature increasing. Liang et al. (2019) also pointed out that the overestimation of C₀ may result in the underestimation of D_m and the overestimation of K_ma, especially at low temperatures. These comments are consistent with the tendency of results determined by the hybrid optimization method. It should be noted that, the determined chamber surface/air partition coefficient (K_sa) in Table S5 is very close to the value measured via independent experiment in literature.

4.3. Transition temperature from SVOCs to VOCs

Generally, the time-dependent emission curves for VOCs and SVOCs are different (Liu et al., 2013; Zhang et al., 2016). For VOCs, the gas-phase concentrations will increase rapidly, reach a peak, and then decrease gradually. While for SVOCs, the gas-phase concentration will increase at a much slower rate, and then reach an equilibrium (steady state). The main reason for such difference is the partition coefficient K_ma. We can see from Table S5, as temperature increases, the values of C₀ and D_m determined by the hybrid optimization method increase while that of K_ma decreases. The reduction in K_ma helps the transition from SVOCs (lgK_ma = 6∼11) to VOCs (lgK_ma = 2∼6) (Zhang et al., 2016). We examined this interesting phenomenon via the developed model and determined key parameters. Previously, Zhang et al. (2007) proposed a correlation between K_ma and temperature (T) by experiments and theoretical analysis. Based on this, Deng et al. (2009) developed a correlation to represent the relationship between D_m and T. By fitting D_m and K_ma with T reported in Table S5, we obtained the quantitative correlations for TCEP and TCIPP, as shown in Table 2. We also determined D_m and K_ma at the other temperatures (e.g., 30 °C, 40 °C, 45 °C, and 50 °C) by using the above correlations. Then, we substituted D_m and K_ma at different temperatures into the developed model in Section 2 to predict gas-phase TCEP/TCIPP concentrations, which are shown in Figure 3. During the calculation process, given that C₀ does not change much and has little influence on the trend of the concentration curve in the range of 23 °C to 55 °C, we assume C₀ is in a linear relationship with temperature for the simplification of the analysis.

Table 2.

Correlations between the key parameters (D_m, K_ma) of SVOCs and temperature.

SVOCs	Correlations	R2
TCEP	ln(Dm / T1.25) = −6460/T −15.59	0.98
	ln(Kma / T0.5) = 7933 / T −13.53	0.90
TCIPP	ln(Dm /T1.25) = −5351/T −18.68	0.94
	ln(Kma /T0.5) = 4626 / T − 4.68	0.80

Figure 3.

Predicted gas-phase concentrations of SVOC emission from PIR foam under different temperatures (a) TCEP; (b) TCIPP.

Figure 3 reveals that temperature has a significant influence on TCEP and TCIPP emissions. When the temperature increases from 23 °C to 45 °C, the tendency of the gas-phase TCEP concentration remains almost unchanged over time (increase and reach equilibrium), which is the characteristic of SVOCs with low volatility and large K_ma (lgK_ma = 6∼12). When the temperature increases from 45 °C to 55 °C, the concentration after the peak time decays gradually over time, which is the characteristic of VOCs with small K_ma. Thus, the transition temperature of TCEP changing from SVOC-type to VOC-type is about 45 °C. Similar analysis indicates that the transition temperature of TCIPP is about 35 °C, which is lower than that of TCEP. It implies that TCIPP is more volatile with increasing temperature.

In Figure 3, the gas-phase TCEP and TCIPP concentrations at 600 h are almost unchanged compared to the concentrations at 500 h, so they can be assumed to be steady state concentrations. According to the concentrations at the temperature range of 23–55 °C, the steady state concentrations with temperatures for TCEP and TCIPP are calculated. The calculated steady state gas-phase concentrations and the fitting results with temperature are given in Figure 4. The relation of the steady state concentrations with temperature follows an exponential function (R²>0.96). The correlations in Figure 4 can be used to predict the steady state concentrations of SVOCs from PIR materials at different temperatures in the range of 23 °C-55 °C.

Figure 4.

Calculated steady state concentration of SVOC emission from PIR foam at different temperatures and model fitting.

4.4. Comparison with the numerical model of SVOCs in the literature

We further perform a comparison on the predictions of gas-phase SVOC concentrations between the developed analytical model and numerical model in the literature (Liang et al., 2019). The calculated results in Figure 5, using 55 °C data as an example, indicate that both R² are equal for the two kinds of models. The good agreement between the analytical model and numerical model demonstrates the effectiveness of the developed model. For other temperatures (23 °C, 35 °C), the comparison results are also good, but are not shown in Figure 5, for the purpose of making the figure clear. Since the analytical model explicitly represents the function between the gas-phase VOC or SVOC concentrations and emission time, it can be regarded a great improvement on model development.

Figure 5.

Comparison of gas-phase concentrations between the analytical model and numerical model at 55 °C for (a) TCEP and (b) TCIPP.

5. Conclusions

In this study we developed a novel analytical model to characterize both VOC and SVOC mass transfer behaviors in source and sink materials in an indoor environment. The predictions of the analytical model are consistent with that of previous numerical models and experimental data. Based on this model, a hybrid optimization method is presented to determine the three key parameters (C₀, D_m, K_ma) of VOCs and SVOCs from indoor materials accurately, by combining particle swarm optimization with an ant colony algorithm. The improvement of the hybrid optimization method is that the determined key parameters are demonstrated to be stable, based on the experimental data of VOC and SVOC emissions under ventilated conditions. This is the first attempt to combine the analysis of VOCs and SVOCs by using a unified analytical model and measurement method. Through detailed analysis on the correlation between D_m, K_ma, and temperature, as well as the developed model, we observe the existence of transition temperature from SVOC-type to VOC-type. The transition temperatures are determined to be about 45 °C and 35 °C for TCEP and TCIPP emissions from PIR foam, respectively. Future study will focus on determining the key parameters for other material-VOC/SVOC combinations, as well as exploring the transition mechanism.