1.2. State of Knowledge–Emission Modelling
In order to make basic assumptions about the shape of the VOC emission model and its components for heated floors (considering a higher temperature and humidity), it is necessary to present the existing state of the art on emission modelling. Based on a provided review in this section, the authors in the next section, Methods, propose a model that is used later to determine the VOC concentration regression profiles based on the experimental results (VOC emission test in chambers).
The prediction of VOC concentration in a chamber or room is attempted by modelling carried out on the basis of the analysis of the influence of variables, e.g., temperature, air humidity and air change rate on the characteristic parameters of the physical processes of emission: initial VOC concentration in solid material (µg m−3), diffusion factor of a given VOC in the emitting building material Dm (m2/s) and the dimensionless mass partition coefficients Kma of the diffusing agent into the part of the mass remaining in the porous material and the part of the mass released into the environment above the emitting surface. These parameters are the output of models of convergence of heat diffusion and the VOC migration processes in porous materials (single and multilayer, dry and wet environments).
Existing emission models typically fall into one of the following types: empirical, analytical or mass-transfer-equation-based. The literature analysis suggests the need to refine this division of models considering verified assessments of their reliability, the possibility of generalisation and the ease of use in practice, leading to the presentation of their classification proposals (see
Figure A1,
Appendix A).
The first group of publications contains five basic publications on emission models based on the theory of mass transfer of VOC compounds from building materials, in which models of the first researchers of emission phenomena from a single homogeneous layer and a double layer are presented. Little et al. [
10] first presented an emission analysis with a model to predict diffusion-controlled VOC emissions from homogeneous building materials. Although the model is useful and straightforward in many cases, an extension of the assumption neglecting convective mass transfer resistance through the air boundary layer was not justified under all conditions. This model’s parameters were the initial VOC concentrations in the material, the material–air partition coefficient and the material’s VOC diffusion coefficient. The presented model has been verified by comparing it with the carpet emission tests results and analysing the determined model parameters′ influence on the emission profile runs. Cox et al. [
11] used this analytical model to calculate the emission rate of vinyl flooring and found a relatively good agreement. Based on the model of Little et al. [
10], Zhao et al. [
12] developed an analytical model to study the transition state and reproduce the reversible effect of secondary adsorption as a response to the simultaneous emissions of pollutants. Huang and Haghighat [
13] developed a numerical model taking into account the mass transfer resistance by introducing mass transfer coefficients; however, they assumed that the VOC concentration in the air should be zero in the gas phase.
Xu and Zhang [
14] presented an improved model that overcame the two previous models′ limitations and obtained an analytical solution. However, as the authors′ reserve, Xu and Zhang’s solution is not fully analytical because it is coupled with the air concentration and needs to be solved simultaneously.
Later, Deng and Kim [
15] developed a fully analytical solution of the given system of Xu and Zhang’s equations [
14]. The Deng model was later the most frequently cited and left the largest trace in the existing models of physical VOC emission processes from building materials. The Deng Kim model [
15] takes into account both material diffusion and mass transfer through the air boundary layer. As the authors themselves claimed, a general characteristic equation is developed which would reduce to that of Little et al. [
10] when the gas phase mass transfer coefficient becomes infinite. The Deng Kim model [
15] shows a good agreement with the experimental data, while the model of Little overestimated the concentration of the emitting VOC in the air.
When building their model, used until today, Deng Kim [
15] needed basic parameters of the mass transfer process, i.e., they used the experimental data of mass transfer: the diffusion coefficients of VOCs
Dm in the materials, the initial concentrations in the materials
Co and the material–air partition coefficients
Kma (verified by the CFD method) from Yang et al. [
16]. Another study by Huang Haghighat [
17] looked at the effect of air velocity on the amount of emissions (measured as chamber
C(
t) in the air) due to an increase in the VOC diffusion coefficient in the material. For a material with a diffusion coefficient
Dm > 10
–10 m
2/s, the VOC emission rate increased with increasing velocity, with air velocity having a significant effect on VOC emissions. For a material with a diffusion coefficient of
Dm < 10
−10 m
2/s, the VOC emission rate increased with the increase in air velocity only in a short period of time <24 h. The source publications in the diagram (
Figure A1,
Appendix A) show reference to the research by Yang [
9] and the more recent publication by Xiong, Huang and Zhang [
18], which describes a new quick method for measuring two key parameters,
Kma and
Dm, as well as convective mass transfer coefficient (
hm). Compared to traditional methods, it has the advantage that the
Kma,
Dm and
hm factors can be obtained in one experiment simultaneously, making it convenient to use. The frequently cited research on the theoretical dependence of the VOC emission rate from building materials on temperature was carried out by the team of Xiong, Wei, Huang and Zhang [
19] focused on the example of formaldehyde. Their validated correlation “shows that the logarithm of the emission rate by a power of 0.25 of the temperature is linearly related to the reciprocal of the temperature”. The emission rate at temperatures other than the test condition can be obtained using the correlation, greatly facilitating engineering applications. However, theoretical work on the procedures for determining the parameters of physical emission models is still ongoing. For example, Liu, Nicolai et al. [
20] used the Least Square and Global search algorithm with multi-starting points to achieve a good agreement in the normalised VOC concentrations between the model prediction and experimental data. They estimated the effects of experimental uncertainty of chamber-measured concentration.
Returning to the work of the Deng Kim [
15] team, their new approach was based on the fact that they adopted some known empirical relations for the determination of the mass transfer coefficient in the gas phase. For laminar flow, it was assumed [
13] (1):
where
Sh = hl/Da is Sherwood’s number,
Sc = ν/Da is Schmidt’s number,
Re = ul/ν is Reynolds’ number,
ν is the kinematic viscosity of air,
u is the velocity of air above the material,
l is the characteristic length of the material,
Da is the VOC diffusion coefficient in the air and
h is the gas phase mass transfer coefficient (m·s
−1).
Apart from these uniform initial condition models, Kumar and Little [
21] also developed a model for the two-layer system with the assumption thatan initial condition is given by a general non-uniform concentration profile in each material layer. This model was useful for studying the effects of adopting different initial VOC concentrations in the emitting layers. Two years later, the team of Qian [
22], based on the solution (Deng and Kim [
15]) and the equations derived from dimensionless analyses by Xu and Zhang [
14], normalised these equations and obtained a group of dimensionless correlations for VOC emissions from dry building materials. The team of Qian [
22] showed that the time dependence of the dimensionless quantity characterising quantitatively the emission (total VOC mass emitted per unit area of building material at time
t (mg m
−2) and the time dependence of the dimensionless quantity characterising the total emission rate (depending on the VOC emission rate per unit area of building material at time
t (mg m
−2 s
−1) are functions of only four (not seven, as according to Deng, Kim [
15]) dimensionless parameters, i.e., the ratio of the mass transfer number to the partition coefficient (
Bim/Kma), the Fourier number (
Fom), dimensionless air change number coefficient (
α = Nδ2/Dm) and the ratio of building material volume to chamber volume or room volume (
β =
Aδ/V). With these correlations, the emission rate could be easily estimated. In addition, the team of Qian et al. [
22] determined experimentally expected ranges of values of dimensionless parameters useful in the fitting procedures, e.g., emission rates in time using the least-squares method.
In recent years, too often, ventilation systems provide nominal airflow rates regardless of the actual need for dilution. In this way, there will be more cold air in the heating season that needs to be heated to ensure thermal comfort, which increases heat loss through ventilation. An effective way to reduce these heat losses through ventilation is to install an intelligent ventilation system, which is characterised by the possibility of continuous adaptation of ventilation, which consequently leads to energy savings without lowering the air quality IAQ. One application of intelligent ventilation systems is demand-controlled ventilation (DCV). In the past, most DCV systems only considered CO
2 and H
2O as good indicators of comfort, ignoring the other VOC emitting contaminants that determine the health aspects of IAQ. New systems may include: (1) a thermal model; (2) a ventilation airflow model and models of selected pollutants levels. However, the condition for the use of intelligent ventilation systems is the availability in databases of the characteristics of materials emitting in various hygro-thermal conditions of the environment. Then, the emission parameters
Dmi and
Coi parameters for the
i-th emitting material enter the system as input data for the model. Such a model for the emission source, which is a layer of building material with a thickness of δ, with the diffusivity
Dm (variable depending on air temperature and humidity) and the initial concentration C
0 of volatile VOC gases in the material (variable depending on temperature and absolute air humidity), was developed by De Jonge and Laverge [
23]. The difficulties in applying this model are that the reference emission data was poorly available (the model excluded the partition coefficient
Km).
The problem of evaluating the impacts of VOC emissions from building materials on the indoor pollution load and indoor air quality beyond the standard chamber test conditions and test period, with the use of mechanistic emission source models, was recognised by two research teams, Liu et al. [
20] and Rode et al. [
24].They co-worked with Project IEA EBC Annex 68,”Indoor Air Quality Design and Control In Low Energy Residential Buildings” (completed in 2020). Scientific research may involve an explanation through the mechanistic description. Mechanisms comprise entities, the physical actorsof a system, and activities that the entities perform. These entities and activities are then organised temporally and spatially in such a way as to give rise to the overarching behaviour of the mechanism [
25].The project considered the problem to provide a comprehensive set of data and tools whereby buildings′ indoor environmental conditions can be optimised. Research teams of Liu et al. [
20] developed a procedure for estimating the mechanistic emissions model parameters using VOC emission data from standard small chamber tests.
In the second group of models presented in the literature (see
Figure A1,
Appendix A), models are also presented, partly based on the theory of mass transfer, but these are specialised models of VOC emissions from systems of multilayer building materials; these systems in our research report are very important because the subject of these studies were multilayer sets of products-emitting VOCs, representing as faithfully as possible the practical sets of floor layers. Although our discussion’s subject is currently emission models from multilayer systems, the researchers from Lawrence Berkeley National Laboratory, Hodgson, Wooley, Daisey [
26] should be considered a precursor. However, their research is limited to providing the captured elapse time emission curves of several compounds. Such tests were performed by observing the emissions from one layer of solid material (in this case, a carpet), but in fact, it was a two-layer system in which the double layer was represented by material/air interface.
As a result of this work, volatile VOC emission profiles from several new carpets were recorded. More comprehensive was the research carried out at Concordia University by Haghighat, de Bellis [
27]. Their scope can be included in the same group of observation of phenomena, although there is a reference to the physical model according to Fick’s second law and the related need to diversify the description of emission phenomena:
where
δCa/δt is a rate of change in the concentration of compound
a (mg/m
3 h
−1),
Dm = diffusion coefficient (m
2/h) and ∇
2 = the Laplacian operator of
Ca (concentration of compound
a in the overlying air (mg/m
3). Each compound under given environmental conditions has its own diffusion coefficient, depending on its molecular weight, molecular volume, temperature and the material’s characteristics within which the diffusion occurs.
Haghighat and Huang [
28], followed by Zhang Niu [
29] and Kim et al. [
30], continued their work, now on a physical model of a multilayer material based on the theory of mass transport, although, according to other authors [
31,
32], it is not possible to develop such a model. This would be a model from which the final concentration of VOC emitted from the surface of a multilayer sample could be determined by simply superimposing the concentrations of pollutants emitted from the individual layers of building materials. This is not possible because the model uses four parameters specific to each material of each layer: the diffusion coefficient of each material layer (
Dm,i), material/air partition coefficient for each material layer (
Kma,i), initial concentration in each material layer (
C0,i) and convective mass transport coefficient (
hma). Such a model would ignore the phenomenon of mutual suppression of VOC diffusion through layers.
The models are to be used to predict the rate of VOCs emission from a multilayer material: the rate of VOCs absorption by the material, the concentration of VOCs in the air, both emitted from VOC sources and materials post-adsorbing (sink) VOC gases from the surrounding air (
Cair–chamber concentration) and the spatial distribution of VOC concentration within the material. Typical layered building materials are in the form of composites such as wall-layering systems (paint/plasterboard/water vapour insulation) or floor-layering systems (wax/vinyl/adhesive/concrete). The model validation results [
28,
29,
30,
31,
32] showed that the multilayer material exhibits the same or similar emission properties as the top layer material. The top layer strongly retards the emission of VOCs from the bottom layer material. A multilayer material has a much longer VOC emission time than a single-layer material.
Deng et al. [
32] investigated the influence of the multilayer material system’s parameters on VOCs emission. The results showed that the inner layer could act as a VOC adsorbent or emission source to the top layer depending on the initial VOC concentration in thelayer’s material. In the case that the inner layer is an emission source for the outer layer, the outer layer becomes a barrier layer reducing the rate of VOC emission from the source.
The emission profile characteristic determined experimentally for the layered building material was presented by Weigl et al. [
33]. TVOC emissions from OSB wood-based boards and the same boards covered with gypsum fibreboards (GKF boards) were investigated. VOC emissions from uncoated OSBs were higher in the beginning. In the presented case, the maximum (TVOC concentration peak in emission profile) was reached after 3 days, followed by a sharp decrease in the emission profile curve in the time period until the seventh day of the measurement, in which the second decay curve breaking down and a slow but continuous decrease of emissions until the end of the experiment could be read. Considering both curves of the emission profile, it can be stated that the gypsum covering the surface of the OSB board caused a reduction and significant delay in the first peak of TVOC concentration.
Another approach to building a model of emissions from many coexisting emitting surfaces in one space, which also applies to layered building materials systems, is presented by Guo et al. [
34]. When different building materials release VOCs at the same time, the indoor VOC concentration increases and vice versa decreases due to the mutual inhibition of the emissions released from the material. Therefore, the whole process is dynamic, and the final indoor VOC concentration cannot be obtained by simply superimposing the concentrations of pollutants emitted from the individual layers of dry building materials [
31]). An equation describing the equilibrium of mass transfer through all thin layers of material [
34] would be Equation (3):
where
V is the chamber volume (m
3),
Ca is the concentration of compound a in the overlying air (mg/m
3),
Q is the amount of ventilating air, m
3 h
−1,
Ai is the emitting surface of the
i-th building material, m
2 and
Dm,1 is the diffusion coefficient of the compound transfer through the material
a of the lower layer with a thickness of δ
1 of the layer system,
Dm, and diffusion factor of the compound
a through the material of any layers
i. However, this equation does not take into account the mutual inhibition effect of emissions and assumes constant values of
Cm,i,; therefore, it would only be valid at very short intervals. For the numerical determination of the time course of the increase in the concentration of VOC in the chamber caused by the emission of VOC from building materials coexisting in the chamber (in the floor layers), Guo et al. [
34] used the finite difference method (Saul’ev finite difference method). They validated the model in the testing of chamber VOC emissions from wall and floor sandwich systems. Interestingly, for both models, in the Deng and Kim physical model completed in 2004 [
15] based on mass transfer equations and the Guo et al. model [
34], the same assumptions were made.
The third group of models provided in the literature (
Figure A1,
Appendix A) is related to the emission from wet materials. When considering the study of emissions from wet material, it should be emphasised that the generally accepted definition of wet material is a solid, homogeneous material covered with a layer of liquid coating. Furthermore, for such materials, the Yang et al., [
35] models were built, which take as a variable the thickness of the liquid’s emitting top layer. As demonstrated by testing VOC emissions from liquid coating materials in a small test chamber in accordance with standard ASTM D 5116−06 [
36], the emission profile consists of two stages: an early stage (stage I), during which the material is still quite wet, has a high emission level whichdecays quickly and another dry stage (stage 2) during which the VOCs are released much more slowly [
35]. Rapid emission occurs mainly by evaporation from the surface of the material, with internal diffusion relatively negligible at this early stage, while in the subsequent dry stage, internal diffusion becomes the controlling factor. Usually, there is a transitional phase between the two steps, which makes the prediction of VOC emissions from liquid materials more complicated. Studies by Yang et al. [
35] and Haghighat and Huang [
28] have shown that the emissions of “wet” materials will depend on environmental conditions (e.g., temperature, air velocity, turbulence, humidity and VOC concentration in the air), as well as the physical properties of the material and the substrate (e.g., diffusivity). Since many factors can influence the emission behaviour of “wet” materials, testing of emissions by laboratory experimentation is usually necessary and expensive (time-consuming).
Altkinkaya [
37] built an emission model from a single homogeneous layer of wet material that accommodates this layer’s changing thickness during emission and considers both internal and external mass transfer resistances through the moving wet shell/air interface. However, the simplifying assumptions introduced, related to the assumption of homogeneity of the wet material layer (constant
Cm and
Kma values of the wet layer), limit this model’s scope of application.
A frequently cited work by Deng, Zhang and Qiu, [
38] presents a model of emission from a moist material with thickness varying during the emission process. This model is not exponential but expressed by a fairly simple differential equation, the variable of which is the thickness of the material layer, but its solution is complicated by the adopted variables (reduction) of the
Cm values (VOC concentration in the material) as well as the partition coefficients
Kma. Therefore, in multilayer systems, the Deng equation for emissions from wet materials would be difficult to use by a difficult mathematical model because the Deng, Zhang and Qiu [
38] equation is solved using the generalised integral transform technique (GITT).
All the above-mentioned mathematical models of emission processes are troublesome to use, although they were created to predict the rate and mass balance of emissions from building materials of different structures and physical conditions. Moreover, it is also difficult to find a relationship between the models. The experimental VOC emission characteristics are inconvenient for engineering applications due to the relatively slow emission rates under chamber test conditions. Therefore, it remains to simulate emission processes, which is related to the need to develop simpler models.
The fourth group of publications (
Figure A1,
Appendix A) presents analytical models for the simulation of VOC emissions from building materials. For example, the Deng Kim [
15] model’s mass transfer model parameters have a practical application for understanding the mechanism of VOC emissions′ physical processes from building materials. However, these parameters usually involve additional physical parameter measurements needed to perform a simulation using the model. Consequently, it is crucial to establish a simple and practical model to predict building material emissions.
Research on the development of analytical models for engineering applications was carried out assuming constant values of the emission rate of pollutants or assuming that the concentration of TVOCs at the boundary of building material is exponentially decaying (adopted constantly, e.g., concerning formaldehyde HCHO emissions).
The ASTM D 5116-06 Standard Guide [
36] is important in this group of publications (Small-Scale Environmental Chamber Determinations of Organic Emissions From Indoor Materials/Products) based on a study by Tichenor, Sparks and Jackson [
39] (of which the emission profile is cited in
Figure A1), which proposes an analytical model useful in calculating the VOC chamber concentration during an emission test. The model includes the exponential equation to calculate the emission factor
EF and the chamber concentration
Ci of the VOC emitting from a material/product sample in a small, normalised chamber used in the standard chamber test. This model, called the first-order decay source model, is one of the most frequently used models of empirical emission processes carried out in a ventilated chamber (small-scale Environmental Test Chamber). The ASTM 5116-06 Standard Guide [
36] presents the first-order decay source model
C(
t) with the following solution under the condition of
t = 0 and
C = 0:
where
C is chamber VOC concentration (mg m
−3),
N is air change rate (h
−1),
L is chamber loading factor (m
2·m
−3) and
k is first-order decay rate constant (h
−1).
EF0 is the initial emission factor from the equation (in mg m
−2 h
−1) calculated in the formula:
where
EF (
ti)–emission factor at time
ti and Δ
Ci/Δ
ti—the slope/gradient of the time concentration curve at time
ti (h). This model was also adopted in a simplified form. As a result of the research carried out at the LBNL in Berkeley (Willem [
40] and Hodgson et al. [
41]) on the optimisation of the strategy of removing air pollutants in residential residences, it was found for most VOC compounds (except, e.g., formaldehyde) that if the emission factor
EF (
t) was constant, the emitted VOC concentration in rooms/chambers was inversely proportional to the time-averaged air-change rate. Therefore, the equation simplified by Willem et al. [
40] for constant source is adopted in the following form:
where
EF is emission factor mg m
−2 h
−1;
A is floor surface m
2;
V-is chamber volume m
3;
N is air change rate h
−1,
Cout is outdoor VOC concentration mg m
−3 and
C is chamber VOC concentration mg m
−3;
t-time h.
Two recently published analytical models of emissions were developed in 2020 by two Chinese scientists’ teams with a very similar composition Zhang, Niu, Liu et al. [
42] and Zhang, Liu, Wu et al. [
43]. These are exponential mass transfer models that eliminate some of the disadvantages of the empirical model, and their parameters have a justified reference to the parameters of physical models. Efforts were made to develop a practical and straightforward model to simulate the VOC emission characteristics in an environmental chamber, taking into account convective mass transfer and equalising mass balance in the indoor environment. The mass transfer model considers the release characteristics of formaldehyde and VOCs from the following three aspects: (1) VOCs diffuse in materials, (2) diffusion from the surface of building materials to the air boundary layer, and (3) convection and diffusion of the air layer. According to both models, the concentration of
Cair in the environmental chamber can be expressed:
where,
where
Kma is an interface partition coefficient of TVOC between the material and the interface,
h is the convective mass transfer coefficient m h
−1and
B is constant. The authors of the model Zhang, Niu, Liu et al. [
42] ignored the process of filling the environmental chamber with TVOC without chemical reactions, Equation (13), which allowed to simplify to the following exponential decay model:
The authors of the model Zhang, Liu, Wu et al. [
43] ignored only the chemical reactions during the emission process and wrote the equation as an empirical model of decay of emissions doubly exponential in the form:
This very analytical model of the double exponential decay empirical model became an inspiration for the new emission model described in this paper, with the additional assumption of trying to eliminate the partition parameter Kma from the equation of the physical value, which would be difficult to determine in a multilayer floor system composed of several materials.
It should also be added that the authors are aware of the potential and possible use of finite element program (CHAMPS) for a thermodynamic VOC equilibrium modelling also in a multilayer structure. In recent years, FDFD methods were used for research on heat and moisture transfer in construction and the selection of parameters of thermal insulation layers of partitions, also for heated wall units, e.g., [
44]. The authors are interested in applying the FDFD method to the selection of layer parameters in the close future.