The Sink-Effect in Indoor Materials:
Mathematical Modelling and Experimental Studies
Doctoral Thesis Peter Hansson
Gävle, Sweden October, 2003 KTH Research School
Centre for Built Environment
Department of Technology and Built Environment, University of Gävle
Department of Technology and Built Environment University of Gävle
SE-801 76 Gävle Sweden
Printed in Sweden by AB Öberghs Ljuskopia, Gävle 2003 ISBN 91-7283-590-7
The Sink-Effect in Indoor Materials:
Mathematical Modelling and Experimental Studies
Peter Hansson
Department of Technology and Built Environment University of Gävle
KTH Research School Centre for Built Environment Abstract
In this thesis the sink-effect in indoor materials was studied using mathematical modelling and experimental studies. The sink-effect is a concept which is commonly used to characterise the ability of different indoor materials to sorb contaminants present in the indoor air. The sorption process is more or less reversible, i.e. molecules sorbed in materials at high contaminant concentrations may again be desorbed at lower concentrations.
Knowledge of the sorption capacity of materials and the rate at which sorption and desorption takes place is of fundamental importance for mathematical simulation of indoor air quality.
The aim of this work is to contribute with knowledge about how the sink-effect can be described in mathematical terms and how the interaction parameters describing the sorption capacity and sorption/desorption kinetics can be determined. The work has been of a methodological nature. The procedure has been to set up physically sound mathematical models of varying complexity and to develop small-scale chamber experiments.
Two different dynamic chamber methods have been used. One is based on a modified
standard FLEC-chamber while the other uses a chamber with two compartments, one on each side of the material. The "twin-compartment" method was designed due to the observation that the contaminant readily permeated straight through the selected materials, which resulted in uncontrolled radial losses in the FLEC-chamber. In order to be useful for comparison between experiments and calculations and parameter fitting, the boundary conditions in the chambers must be precisely known and controlled. This matter has shown to be the most crucial and difficult problem in the research.
A variety of mathematical models for the sink-effect have been proposed. In some models advanced fluid simulations were used in order to test the influence of ill-defined flow boundary conditions. The aim of the modelling is to find a formulation with a minimum of interaction parameters, which is generally useful, i.e. both in small-scale laboratory
environments and in full-scale like an office room. Estimated model parameters are shown to be able to yield a reasonably good fit to experimental data for the sorption process but a less satisfactory fit for the desorption process.
Keywords: sink-effect, sorption, adsorption, diffusion, indoor air quality, volatile organic compounds, VOC, contaminants, building materials
Förord
Under min tid i industrin väcktes intresset för inomhusklimat och luftkvalitet. Intresset gjorde att jag valde att studera ämnet djupare. Efter studier vid Högskolan i Gävle och KTH fick jag möjlighet att forskarstudera. Denna avhandling baseras på forskarstudier utförda vid
avdelningen för Inomhusmiljö vid Institutionen för Teknik och Byggd Miljö i Gävle vid Högskolan i Gävle. Avhandlingen behandlar sorption och diffusion av gasformiga
föroreningar i inomhusmaterial. Arbetet bakom denna avhandling har delvis publicerats vid fem vetenskapliga konferenser.
Först och främst vill jag tacka min handledare Docent Hans Stymne. Hans vetenskapliga skicklighet och pedagogiska förmåga har varit till stor hjälp vid forskningsarbetet. Det har varit mycket lärorikt och roligt att arbeta med Hans.
Avhandlingen hade inte kunnat utföras utan hjälp av medarbetarna på avdelningen.
Jag vill tacka Larry Smids och Ragnvald Pelttari för hjälp med tillverkning av kammare och annan experimentell utrustning. Dessa herrar har en fantastisk förmåga att tillverka högklassig experimentell utrustning utifrån enkla skisser och samtal vid fikabordet.
Hans Lundström har hjälpt till att lösa invecklade elektriska problem med analysutrustningen.
Claes Blomquist har bistått med hjälp att lösa mättekniska problem.
Jag tackar Professor Mats Sandberg för givande diskussioner och intressanta kommentarer.
Tack John Airey för språkgranskning av avhandlingen.
Tack alla musiker i Gävle och Stockholm jag fått den stora äran att musicera med. Utan er och hade denna avhandling aldrig blivit färdig.
Ett stort tack riktar jag även till min familj som haft stor förståelse och givit mig sitt starka stöd under avhandlingsarbetet. Tack Cynthia, mor, far, bror och syster.
Slutligen vill jag tacka de organisationer som finansierat projekten avhandlingen baseras på.
Tack Formas/BFR och KKS.
Contents
1 Introduction 1
2 Involved processes and contaminants 7
2.1 Boundary layer diffusion 7
2.2 Surface adsorption 8
2.2.1 Interionic and intermolecular forces 9
2.2.2 Adsorption kinetics 10
2.2.3 Adsorption isotherms 11
2.2.4 Indoor air adsorption isotherm 18
2.3 Diffusion inside the material 18
2.3.1 Molecular diffusion 18
2.3.2 Diffusion in porous media 19
2.3.3 Knudsen diffusion 19
2.3.4 Surface diffusion 20
2.4 Absorption 20
2.5 Organic pollutants 21
2.6 VOC concentrations indoors 22
2.7 Health effects 23
3 Previous work 25
3.1 Adsorption and diffusion models 25
3.1.1 The adsorption model by Tichenor et al. 25
3.1.2 An alternative interpretation of ka and kd 27
3.1.3 The magnitude of ka 27
3.2 Equilibrium sorption models 30
3.3 Diffusion models 31
3.3.1 The diffusion model by Crank 32
3.3.2 The diffusion model by Lee, Haghighat and Ghaly 33
3.3.3 The diffusion model by Axley 34
3.3.4 The diffusion model by Little and Hodgson 35
3.3.5 The diffusion model by Bodalal, Zhang and Plett 36 3.3.6 The hybrid and K-diffusion models by Dunn & Chen 37 3.3.7 The compartment model by Christiansson et al 40 3.3.8 Similarities and differences between some diffusion-sorption models 42
3.4 Static methods 43
3.5 Gravimetric methods 45
3.6 Sensory methods 45
3.7 Full-scale simulations (CFD) 46
3.8 Factors influencing the sink effect and results from adsorption/desorption studies 47
3.8.1 Influence of compound volatility 47
3.8.2 Influence of temperature 47
3.8.3 Influence of humidity 48
3.8.4 Influence of compound polarity, surface affinity and hydrogen bonding 48
3.8.5 A molecular model for the sink effect 48
3.8.6 Mixtures of VOCs versus single VOCs 49
3.8.7 Diffusion through and sorption of VOC in indoor materials 50
3.8.8 Correlations for diffusion 51
3.8.9 Effect of boundary layer diffusion 52
3.9 Demands on mathematical models for the sink effect 52
3.10 Conclusions 53
4 Experimental conditions for FLEC and Twin Compartment experiments 55
4.1 Air generation 55
4.2 Compound 55
4.3 Compound generation 55
4.4 Compound compartment 56
4.5 Flow chart and flow control 57
4.6 Concentration measurement methods 58
4.6.1 Mass spectrometry 58
4.6.2 Gas chromatography 59
4.7 Air flow rate measurements 63
4.8 Test material 63
4.9 Temperature & humidity 63
5 Mathematical models for the experiments 65
5.1 The 1-D fully mixed model for the FLEC experiments, model A 65
5.1.1 Mathematical description of the processes 65
5.1.2 Setting up the 1-D fully mixed model 67
5.1.3 A numerical scheme for the system of equations 70
5.1.4 Solving the numerical scheme 73
5.2 The 1-D fully mixed model for the Twin Compartment experiments, model B 77 5.2.1 Setting up the model for the Twin Compartment experiments 77 5.2.2 A numerical scheme for the system of equations 78 5.3 The 3-D fully mixed model for FLEC experiments with gypsum board, model C 78 5.4 2-D and 3-D coupled mass transport models for twin compartment experiments 81 5.4.1 A 2-D coupled mass transport model for twin compartment experiments, model D 81 5.4.2 A 3-D coupled mass transport model for the single compartment which forms the twin
compartment chamber, model E 84
5.5 A 2-D coupled mass transport model for FLEC experiments with stainless steel plate and gypsum
board paper respectively, model F 85
5.5.1 Description of the model 85
6 Results from FLEC-studies 89
6.1 Experimental method 89
6.2 Climate chamber 89
6.3 Sealing method 90
6.4 Evaluation Methods 90
6.5 The empty chamber 91 6.5.1 Experimental conditions for the empty chamber 91
6.5.2 Results 91
6.6 Simulation of the empty chamber 93
6.6.1 Fully mixed simulation 93
6.6.2 An alternative approach 1– radial simulation 95 6.6.3 An alternative approach 2– radial simulation with laminar flow 97
6.6.4 Results 98
6.6.5 Discussion 100
6.6.6 Conclusions 101
6.7 The gypsum board paper 102
6.7.1 Experimental data 102
6.7.2 Experimental results 102
6.8 Simulation of the gypsum board paper 104
6.8.1 Simulation results (fully mixed model) 104
6.8.2 An alternative approach – radial simulation with laminar flow 107
6.8.3 Conclusions 109
6.9 The gypsum board 109
6.9.1 Experimental data 109
6.9.2 Results from single experiments 110
6.9.3 Conclusions from single experiments 112
6.9.4 Results – from repeated exposure 113
6.9.5 Conclusions from repeated exposures 114
6.10 1-D Simulation of the gypsum board 115
6.10.1 Adsorption phase 1-D 115
6.10.2 Desorption phase 1-D 117
6.10.3 Conclusions 1-D simulation 118
6.11 3-D Simulation of the gypsum board 118
6.11.1 Adsorption phase 3-D 119
6.11.2 Desorption phase 3-D 120
6.11.3 Simulation of repeated exposure of the gypsum board 121 6.11.4 Conclusions from gypsum board simulation 122
6.12 The painted gypsum board 123
6.12.1 Experimental data 123
6.12.2 Results 124
6.12.3 Conclusions for the painted gypsum board experiments 126
6.13 3-D simulation of the painted gypsum board 126
6.13.1 Results 126
6.13.2 Conclusions 3-D simulation painted gypsum board 129
7 Permeability, diffusion and sorption of toluene in three building materials
investigated with the Twin Compartment method 131
7.1 INTRODUCTION 131
7.2 EXPERIMENTS AND SIMULATIONS, PHASE 1 131
7.2.1 Data for Twin compartment experiments 131
7.2.2 Results for PHASE 1 experiments and 1-D fully mixed simulations 132 7.2.3 Comparison of permeation constant obtained using gravimetric method 136 7.2.4 Conclusions for phase 1 experiments and 1-D fully mixed simulations 137
7.3 Experiments and simulations, PHASE 2 137
7.3.1 Data for decay experiments, Twin Compartment chamber 138 7.3.2 Data for the Twin Compartment experiment with gypsum board and GC 138 7.3.3 Results for decay experiments and simulations 138 7.3.4 Conclusions for decay experiments and simulations 139 7.3.5 Comparison between 3-D simulation and 2-D simulation for decay experiments 139 7.3.6 2-D simulation of phase 1 Twin Compartment experiment with gypsum board 140 7.3.7 Conclusions for 2-D simulation of phase 1 Twin Compartment experiment with gypsum board142 7.3.8 2-D simulation of phase 1 Twin Compartment experiment with MDF-board 143 7.3.9 Conclusions for 2-D simulation of phase 1 Twin Compartment experiment with MDF-board 144 7.3.10 Experimental results and 2-D simulation of phase 2 Twin Compartment experiment with gypsum
board and the modified GC 144
7.3.11 Conclusions for experiment and simulation of phase 2 Twin Compartment experiment with
gypsum board and the modified GC 147
7.4 CONCLUSIONS FROM TWIN COMPARTMENT METHOD 148
8 Permeation and sorption capacity of three building materials investigated
with a simple gravimetric technique 151
8.1 Gravimetric investigation of permeation rate 151
8.1.1 Results 151
8.2 Gravimetric investigation of sorption capacity 152
8.2.1 Method 152
8.2.2 RESULTS 152
8.3 Conclusions 156
9 VOC sorption of activity related contaminants – influence of boundary
layer diffusion 159
9.1 Introduction 159
9.2 Methods 159
9.2.1 Mathematical model 159
9.2.2 Boundary layer diffusion 159
9.2.3 System time constant 159
9.3 RESULTS 160
9.3.1 Simulation results 161
9.4 DISCUSSION 162
9.5 CONCLUSION AND IMPLICATIONS 163
10 Conclusions 165
10.1 Overall results from the work 168
10.1.1 Method applicability 168
10.1.2 Model behaviour 168
10.1.3 Gravimetric methods 168
10.1.4 Influence of boundary layer diffusion 168
10.1.5 Implications for indoor air quality 169
10.1.6 Topics, which need further research, include: 169
11 References 171
1 Introduction
Indoor materials are normally thought of as sources for contaminants of different kinds polluting the indoor air. The contaminants released are often so called VOCs (Volatile Organic Compounds). The concentration of (and exposure to) the emitted contaminants is dependent of the emission rate, the air velocity above the material and the dilution rate of the ventilation. The material emission varies over time and depends on several factors, which are often accounted for in advanced emission models. Emission models are essential tools for prediction of the contaminant concentration in ventilated spaces. When air quality is to be analysed during different ventilation strategies, the emission models need to be coupled with models for ventilation and airflow.
The physical mechanisms describing building material emissions are similar to the
mechanisms present when a compound sorbs on/in indoor materials (the sink-effect). This continuous interaction between contaminants in room air and indoor materials will influence the contaminant concentration over time considerably compared to the concentration
predicted using traditional dilution based models for air quality, which normally do not include the sink-effect. The sink-effect smoothes out the concentration variations. It reduces the increase in room air concentration when conditions in a space are changed (when a new material is introduced, a paint is applied on a surface or ventilation is reduced etc). But the sink-effect also contributes to a prolonged low level exposure for humans, since the
contaminants are more or less re-emitted from the materials to the room air. The secondary emission from indoor sinks can proceed for a very long time (Saarinen and Saarela 2000, Tirkkonen and Saarela 1997, Zellweger et al 1995, Tichenor et al 1991).
The most pronounced consequences of the sink-effect occur when room air concentrations change quickly. This happens for example when chemicals are released during activities in the indoor environment (painting, cooking, smoking, use of cleaning agents and use of consumer chemicals). As chemicals are often released repeatedly in the indoor environment, there will probably be a build up of compounds in the materials (compound depots). The subsequent re-emission from depots contributes to an increased background concentration of contaminants in the indoor environment as well as an increased exposure risk to contaminated dust particles.
Modelling of the-sink effect is essential to obtain knowledge of how important emissions from activities are in relation to emissions from materials for the contamination of indoor air.
The ongoing work with emission testing and classification of materials with respect to their health-risk will, in the long run, result in a decreased risk for emissions from materials to be responsible for severe indoor contamination. The corresponding work on emissions from activities (with the exception of the influence from smoking) has not been so successful.
Therefore it is important to clarify the basic principles behind this type of contamination and exposure. In the long run this will also lead to an increased awareness about the significance of emissions generated by activities for indoor air quality. Materials will be classified according to their ability to interact with air-borne compounds as well as their emissions.
Another aspect, in which the knowledge of indoor sinks may be a question of vital importance for guaranteeing good air quality, is the ventilation strategy in buildings. Ventilation is often strongly reduced during weekends and nights in buildings where people are normally not present under these periods. Primary emissions from materials continue to contaminate the air even during periods of time when ventilation is reduced. As a result, air concentrations of contaminant increase and a considerable amount of these contaminants are sorbed to the surfaces of all indoor materials due to the sink-effect. When ventilation is normalised again sorbed contaminants will be released more or less reversibly together with contaminants from primary sources. The total emission rate from materials after a period of reduced ventilation will thus be higher than the primary emission rate. Consequently, by reducing the ventilation during weekends and nights air contamination will increase during daytime. There is also a risk that frequent use of reduced ventilation may result in a build up of contaminants in the materials. This subsequent re-emission from depots contributes to an increased background concentration of contaminants in the indoor environment for a long time.
The aim of the underlying work of this thesis has been to contribute to knowledge about how the interaction between contaminants and materials can be described in mathematical terms (i.e. the sink-effect). Such models are essential tools for analysing indoor air quality when the indoor environment is subject to repeated contaminant exposure or decreased ventilation rate for example.
Models should be of such a nature that they can describe how the air concentration of a contaminant is influenced by the presence of materials indoors, both with respect to time and concentration level.
The most important property of such models is the ability of scaling. In other words, models should be of general validity. It should be possible to use the models in small-scale laboratory environments as well as in an office room, dwelling or industry.
Thus, it is not sufficient to adapt small-scale experimental data to a suitable mathematical function when the sink effect is evaluated using a dynamic technique. The validity of the results from such evaluations is probably isolated to the chamber in which the experiment was performed and under identical boundary conditions.
There is a distinction between variables and parameters in mathematical modelling. Variables represent quantities that change with the environment, which is intended to be described.
Parameters, on the other hand, are specific for the contaminant or material included in the description of the problem. Parameter values are thus equal in small scale and full scale while variables are not.
A mathematical model for the sink-effect should include variables describing room volume, air flow rate and air velocities (a detailed description of the air flow pattern), air temperature (and air humidity). The strength and duration of the emission source is also described by variables. The model should also include variables describing the amount of material, which interacts with contaminants, the exposed area of the material and the thickness of the material.
Parameters relevant for the sink effect are for example those describing the equilibrium between the compound concentration in the gas phase and the amount sorbed on the surface and in the interior of the material respectively, the porosity of the material and the diffusion of the compound in the material.
The main approach for the research within this project is to to set up physically reasonable models for the interaction and fix the variables in a model experiment. Thereafter the unknown material parameters are determined by fitting the mathematical model to the
experimental concentration data obtained from dynamic experiments. The ambition is to find a combination of parameter values that yield a best fit. In some cases static methods have been used as well, as a complement to the dynamic experiments.
The following processes are considered to be the basis of the sink effect:
• Boundary layer diffusion (contaminants in room air diffuses to air close to the surface)
• Surface adsorption
• Diffusion of contaminants in vapour phase and solid phase
• Adsorption and absorption inside the material
The processes are, among other things, dependent on: temperature, ventilation (air velocity, turbulence), surface structure, air humidity, air concentration of reactive species (ozone, nitrogen oxides), UV-light, material properties (porosity, water content, age of the material, content of compounds that may react with the interacting contaminant) and contaminant properties (boiling point, polarity, hydrogen bonding ability etc).
Re-emission of sorbed contaminants is assumed to follow the same scheme in reverse order.
In addition to the above mentioned processes for the sink effect, chemical reactions on the surface and in the interior of the material may occur. Chemical reactions, however, often show limited reversibility and the reverse process is, as a rule, not observed at all. If the reverse process occurs, contaminants other than the previously adsorbed contaminants may be emitted from the material.
Only one chemical compound (toluene) has been used in the dynamic chamber experiments within this work. This stable molecule is expected not to undergo chemical reactions, at least not during the experimental conditions in this work. A limited number of materials have also been evaluated namely gypsum board with and without paint, wood particleboard and MDF- board.
The reason why so few compound/material combinations have been evaluated is that the work has been of a methodological nature. The intention of this work was not to experimentally determine interaction parameters. The aim was to set up physically sound mathematical models of varying complexity together with the development of a small-scale experimental technique for determination of the interaction parameters.
Useful models should not be too complicated. Another feature of a useful model is that it should not include too many unknown parameters. It is often possible to decide if a model is functional by examining the ability to fit the model to experimental data. A great deal of the work described in this thesis deals with comparison between simulations and experiments.
The work in this thesis was performed in five steps:
Step 1
Selecting experimental method.
Design of the experimental set-up.
Performing experiments with the FLEC-chamber.
Proposing a mathematical model for the sink-effect.
Evaluation of the FLEC-experiments with the mathematical model.
The work behind this thesis started by selecting an experimental method for the sink-effect. A dynamic method was chosen. This means that the chamber and material are subject to a step change in inlet concentration of a VOC. By studying the concentration in the outlet of the chamber as a function of time it is possible to get information about the sink-effect. If a mathematical model is fitted to the concentration data, values of the material parameters are obtained. The first series of experiments were performed with the FLEC-chamber. The FLEC- chamber was chosen because it has a high loading factor, only one side of the material is exposed to the compound as in a real room and because it was believed that the FLEC was equipped with well defined boundary conditions for the flow. The results showed that the VOC toluene diffused into porous materials. Therefore, a mathematical model including boundary layer diffusion, surface sorption, interior diffusion and interior sorption had to be developed to interpret the experimental results.
Step 2
Design of a new Twin Compartment chamber to meet the demands discovered in the FLEC- study.
Performing experiments with the Twin Compartment chamber.
Evaluation of the experiments with the mathematical model.
Step 2 of the experimental work focused on the findings from step 1. Thus, diffusion and sorption of VOC inside building materials was investigated systematically. For this purpose, a new small-scale chamber, equipped with two compartments on each side of the material, was developed. The demands on the new chamber, called the Twin Compartment chamber, were high. Firstly, the chamber needs to be both inert and leak free. Secondly, the projected surface area of the test material has to be well defined. Additionally, the edges of the material have to be sealed so that VOC cannot enter or escape from the material or chamber that way. An aluminium sealing ring was chosen to hold the specimen and to seal the chamber from the surroundings. The material was fixed to the centre of the seal with a solvent free epoxy resin.
The previously proposed mathematical model was fitted to the experimental results obtained from the new chamber.
Step 3
Development of a concentration measurement device for measurements of low levels of VOC to improve the experimental findings.
The results from the Twin Compartment study showed that the desorption phase of the experiments was not recorded in a satisfying way by the mass spectrometer (MS). Therefore, a gas chromatograph (GC) equipped with a flame ionisation detector was modified to be able to measure low levels of VOC, reasonably fast, directly in the outlet of the chamber. The GC was modified by equipping it with a sorbent tube and a cryogenic trap designed within the project.
Step 4
Development of detailed models to be able to interpret the experiments more precisely.
Evaluation of the dynamic experiments with the detailed models.
In order to attain an increased certainty on the magnitude of the material parameters in the model information about the detailed flow conditions inside the chambers were needed.
Therefore, several advanced mathematical models were developed. In these models the mass transport equations were coupled together with equations for the airflow.
Step 5
Direct measurements of sorption and diffusion using gravimetric techniques.
An alternative approach for determination of the material parameters was tested in the project as well. The aim was to determine material parameters independently from each other.
Sorption of VOC in materials and diffusion of VOC through materials were determined by two simple gravimetric techniques.
The outline of the thesis is as follows:
In chapter 2 of the thesis a short description of the fundamental processes for the interaction is made.
A review of previous research on the sink-effect is given in chapter 3. This part also includes a comparison between existing diffusion models for the sink effect.
A description of how the contaminant is generated and distributed to the small-scale chambers is described in chapter 4. The chemical analyses are also described here. Some of the results were reported in a conference paper, Hansson & Stymne (1998a) and in a licentiate thesis, Hansson (1999).
Chapter 5 describes how the mathematical models used for the experiments are formulated and solved. Some of the results in chapter 5 were reported in a conference paper, Hansson &
Stymne (1998b) and in a licentiate thesis, Hansson (1999).
A description of the experimental layout and results from the FLEC-study is detailed in
chapter 6. The model parameters described in chapter 5 are adjusted to achieve a best fit to the experimental results. Some of the results in chapter 6 were reported in two conference papers and in a licentiate thesis (Hansson & Stymne 1998b, Hansson & Stymne 1999 and Hansson 1999).
Chapter 7 illustrates the experiments with the Twin Compartment method. Similarly, chapter 6 model parameters described in chapter 5 are adjusted to achieve a best fit to the
experimental results. Some of the results in chapter 6 were reported in a conference paper, Hansson & Stymne (2000).
The results from two simple gravimetric techniques for determination of permeation of a contaminant through and sorption in a material are displayed in chapter 8.
In chapter 9 the influence of boundary layer diffusion on sorption of activity related contaminants is discussed. The results in chapter 9 were reported in a conference paper, Hansson & Stymne (2002)
The conclusions from this work are summarised in chapter 10.
2 Involved processes and contaminants
The most common processes involved when gaseous compounds interact with building materials are described in this section. Additionally, a brief description of some relevant indoor contaminants and their health effects is made to complete the picture.
2.1 Boundary layer diffusion
Boundary layers are normally present between two phases that are not in equilibrium with each other for example with respect to temperature or chemical composition. Thus, if a solid material is in contact with room air, transport of a chemical compound will occur through this boundary layer as long as there is a lack of chemical equilibrium between the compound in the air and in the material. The thickness of and concentration variation in the boundary layer is strongly influenced by the air convection characteristics along the material surface.
Material
Cc Cs d
Figure 2.1. The boundary layer above a material. d is the thickness of the boundary layer. Cc
is the bulk concentration and Cs is the vapour concentration close to the surface.
Mass transport from bulk air to the surface is governed by boundary layer diffusion or film diffusion (assuming that there is no pressure difference induced flow of air through the material surface). Indoors this process is mainly dependent on the air velocity and turbulence above the material surface, the diffusion coefficient and the compound concentration. If the concentration does not vary too rapidly it is appropriate to approximate this mass transfer process using a steady state mass transfer correlation from boundary layer theory (Axley 1991). The mass flux from the bulk air to an air layer very close to the surface then becomes:
) (
)
( c s g c s
s C C K C C
d
J =D − = − (2.1)
where D is the diffusion coefficient in air, d is the thickness of the boundary layer, Cc is the bulk concentration and Cs is the vapour concentration close to the surface, see figure 2.1. The mass transfer coefficient Kg can be obtained from mass transfer literature see for example Cussler (1997). For isothermal conditions it is common to express the average mass transfer coefficient in dimensionless form by the following numbers: The average Sherwood number, (Sh), the Reynolds number (Re) and the average Schmidt number (Sc).
Sherwood number (Sh) =
velocity diffusion
velocity transfer
=mass
⋅ D
L Kg
Reynolds number (ReL) =
forces viscous
forces inertial
=
⋅ ∞
ν U L
Schmidt number (Sc) =
mass of y diffusivit
momentum of
y diffusivit
=
D ν
where ν is the kinematic viscosity for air, L is a characteristic length and U∞ =bulk velocity.
For room sorption dynamics Axley (1991) uses the following well established correlations:
) 10 5 Re ( plate flat a over flow laminar for
Sc Re 646 . 0
Sh= ⋅ L1/2 1/3 L < ⋅ 5 (2.2)
) 10 5 Re ( plate flat a over flow ent for turbul
Sc Re 037 . 0
Sh= ⋅ L4/5 1/3 L > ⋅ 5 (2.3)
2.2 Surface adsorption
When gas molecules are in direct contact with a surface of a solid they are partly adsorbed there. The gas binds to the surface of the solid. The molecules that are in condensed phase on the surface are called adsorbate and the substance that holds the molecules is called adsorbent, (Masel 1996). There is a difference between adsorption and absorption. Absorption is a process where the gas molecules dissolve or become chemically bonded into the bulk of a liquid or solid. It is also observed that the mass absorbed is proportional to the volume of the absorbent while the mass adsorbed is proportional to the surface area of the adsorbent.
The processes associated with adsorption are generally surface bindings and pore
condensation. Gases can adsorb on surfaces forming a monolayer or multilayer of molecules, the latter being similar to condensation. Multilayer adsorption is controlled by adsorbate – adsorbate interactions and it occurs when the temperature is close to the boiling point of the adsorbate at a given vapour pressure, (Masel 1996). Monolayer adsorption, on the other hand, can occur hundreds of degrees above the boiling point of the adsorbate, Masel (1996). The density of the monolayer is close to that in a liquid. The coverage for monolayer adsorption never exceeds 1-2·1019 molecules/m2, (Masel 1996). If one mole of gas is to be adsorbed to a monolayer thickness on a flat surface one needs 30.110 - 60.220 m2. Suppose that the
monolayer-hypothesis can be adapted to indoor air quality. Consider an office room (V=36 m3) which is equipped with smooth flat surfaces (A=66 m2) and that a VOC at a vapour concentration of 1 ppm is present in the room. Furthermore, suppose that the VOC is adsorbed to a concentration of 1·1019 molecules/m2 on the surfaces. Then, 1.1·10-3 moles of VOC are adsorbed by the surfaces. The surfaces accumulate the amount of VOC held in 26 m3 of air or 73 % of the total amount of VOC in the room. The partitioning coefficient between sorbed VOC and VOC in gas phase [moles/m2 / moles/m3] becomes 0.4 m, for this particular case.
However, indoor surfaces are seldom perfectly smooth and flat. The effective surface area for the majority of indoor materials is probably several orders of magnitude larger than the projected surface area. Thus, the simple calculation of maximal surface accumulation based on monolayer coverage of molecules is only a rough estimate for real situations indoors.
There is a difference between non-dissociative adsorption and dissociative adsorption. The former is sometimes called molecular adsorption and means that the adsorbing molecule is unaffected by the adsorption process. Dissociative adsorption is a process where the adsorbing molecule undergoes a change during the adsorption process (bonds could break for example:
B2→B+B).
Physical adsorption or physisorption mainly involves van der Waals interactions between sorbate and sorbent. These forces are long range but weak. The electronic structure of the sorbed molecules is largely unaffected when they are physisorbed and physical adsorption is a reversible process. Chemical adsorption on the other hand means that the sorbate sticks to the sorbent by chemical bonding, usually covalent. Electrons are more or less shared between the sorbate and sorbent yielding perturbed sorbate molecules.
Most adsorption processes are exothermic. Enthalpies (∆H) of physical adsorption are negative and of the same magnitude as for condensation (8 – 40 kJ/mole) while enthalpies of chemisorption are considerably greater (60 – 400 kJ/mole). The negative enthalpy is a natural consequence of the thermodynamics for adsorption. Since adsorption is a spontaneous process
∆G (change in free energy) for the sorbing molecule has to be negative, (Atkins 1995). The translational freedom of the molecule is reduced when the molecule is adsorbed. Therefore ∆S (change in entropy) is negative. Thus, ∆H must be negative to fulfil the equation:
S T H G=∆ − ∆
∆ . Exceptions to this rule occur. H2 chemisorbs endothermically on glass, for example. The hydrogen molecules dissociate into hydrogen atoms that move freely over the surface, Atkins (1995).
2.2.1 Interionic and intermolecular forces
Van der Waals forces (intermolecular forces) consist of attractive and repulsive forces . An attractive interaction which is present between all types of species (ions, polar molecules and non-polar molecules) is the London interaction (dispersion interaction), (Atkins & Beran 1992). This attraction originates from fluctuations in the electron distributions in the species.
Instantaneous dipoles which are continuously changing direction are created due to the
fluctuations. These instantaneous dipoles occur even for nonpolar molecules, (Atkins & Beran 1992). The London interaction is always attractive. The strength of the force increases with increasing molar mass and molecular radius. The strength depends on the polarizabilities of the interacting molecules. The London interaction between polar molecules is usually stronger than their dipole-dipole interaction according to Atkins & Beran.
Dipole-dipole interaction is another type of dispersion interaction. It is the attraction between the electric dipoles of polar molecules, (Atkins & Beran 1992). Permanent dipoles cause the molecules to attract each other. HCl for example has a permanent dipole because chlorine is more electronegative than hydrogen. The attraction between dipoles is somewhat stronger than it is between non-polar molecules but it is weaker than the attraction between oppositely charged ions.
The strength of attractive forces between atoms, ions and molecules varies with type of substance, see table 2.1, (Atkins & Beran 1992). Table 2.1 shows the potential energy for a distance between species of 0.5 nm. Ion-ion interactions show the strongest attractive forces and the strength of ion-ion interactions decreases comparatively slowly with increasing distance, (Atkins & Beran 1992).
Table 2.1 Multipole interaction potential energies for interacting species at a distance of 0.5 nm.
Interaction Typical energy
[kJ/mole]
Distance dependence
Ion-ion 250 1/r
Ion-dipole 15 1/r2
Dipole-dipole (stationary molecules) 2 1/r3 Dipole-dipole (rotating molecules) 0.3 1/r6
London (dispersion) 2 1/r6
Hydrogen bonding 20 -
Hydrogen bonding occurs when a hydrogen atom is situated between two strongly
electronegative atoms, (Atkins & Beran 1992). In the case of neutral molecules, hydrogen bonding only occurs together with the sufficiently electronegative N, O and F. The hydrogen bond is a very strong intermolecular force, it has about a tenth of the strength of an average covalent bond. The bonding is a contact like interaction which is turned on when the molecules touch each other and turned off when the bonding is broken. Liquid water and molecules containing OH-groups are examples of substances forming hydrogen bonds.
2.2.2 Adsorption kinetics
The fate of a molecule colliding with a surface is illustrated in figure 2.2. Picture A in the figure represents a case with an undeformable (hard) surface. Picture B in figure 2.2
represents a case with a deformable soft surface. Figure C depicts a case with a highly porous surface, (Eriksson 1998). When the surface is undeformable the molecule bounces without losing translational energy, figure 2.2a. This is called elastic scattering, (Masel 1996). When the molecule collides with a soft surface it loses its translational energy and is trapped there for a while. The time spent on the surface during trapping is small (a few hundred
microseconds) yet it is much longer than the time between collisions, (Masel 1996). When the molecule is trapped the molecule stays on the surface but it is in a weakly bonded state. A small amount of thermal energy can cause the molecule to desorb from the surface. Sticking is a process where molecules collide with a surface, lose energy and remain on the surface for a reasonable time, figure 2.2c. In the case of sticking the molecule bounces around until it finds a site where it can adsorb. The rate of trapping is determined by energy transfer rates, while the rate of sticking is determined both by energy transfer rates and the ability of the surface to form bonds. Trapping rates go down with increasing gas temperature, while sticking rates are usually unaffected by gas temperature, (Masel 1996).
Figure 2.2 A molecule colliding with a surface. A) undeformable hard surface B) soft surface and C) porous surface. The processes are named scattering (A), trapping (B) and sticking (C).
2.2.2.1 Calculation of adsorption rate for sticking molecules
According to Nix (1999) it is possible to express the adsorption rate, Ra, as a product of the rate of arrival of molecules at the surface (incident molecular flux), F, and the proportion of incident molecules that stick to the surface (sticking probability), S:
F S
Ra = ⋅ (2.4)
where Ra has the units of molecules / m2, s. According to Masel (1996) the sticking probability, S, is defined as:
surface a
on impinge that
molecules of
Number
stick that molecules of
Number
The equation for the molecular flux is as follows:
T k m F P
⋅
⋅
⋅
= ⋅ π
2 (2.5)
where P is the partial pressure of the gas [Pa], m is the mass of one molecule [kg], k is Boltzmanns constant and T is the absolute temperature. F has the units of molecules/ m2, s.
The sticking probability varies between 0 and 1. It depends upon many factors (type of gas, surface composition, and temperature). The most obvious factor is the number of adsorbed molecules or occupied sites for adsorption on the surface. The sticking probability varies with the coverage of the surface when a surface is exposed. The initial sticking probability, S0, (when the surface is free from sorbate) is high. Thereafter S drops with increasing coverage.
However, when the coverage of molecules is low the sticking probability can be assumed to be basically constant, (Nix 1999). Masel (1996) reports that the initial sticking probability usually lies between 0.1 and 1 and a gas with S0 < 0.01-0.0001 is not expected to stick. Masel also gives some examples: H2 on Ni(100), S0 = 0.06 and CO on Pt(111), S0 = 0.67 etc.
2.2.3 Adsorption isotherms
Adsorption isotherms are used to quantify the adsorption process. The isotherms are plots of the amount of gas that adsorbs on a surface as a function of concentration (or pressure) of the gas at constant temperature, (Masel 1996). Adsorption isotherms can be determined by several different methods. One technique involves putting a piece of sorbent (initially free of sorbate) into a airtight container which is filled with a known amount of sorbing gas. A point on the adsorption isotherm is obtained at steady state through the decrease in air phase concentration of sorbate in the container or weight gain of the sorbent test piece.
The shape of the isotherms can give information about the adsorption process. One example of this is the classification system for isotherms proposed by Brunauer in 1945 described in Masel (1996). Brunauer classified five types of adsorption isotherms displayed in figure 2.3.
Type 1 and 2 are normally observed with flat non-porous adsorbent surfaces. Type 1 is characteristic for monolayer adsorption on flat surfaces or a porous sorbent where the pores fill up quickly. Type 2 is characteristic for multilayer adsorption. Type 3 represents a case with strong adsorbate-adsorbate interactions. Adsorption is very slow in the beginning but after a while when a part of the surface is occupied with adsorbate molecules (islands or droplets), the adsorption rate increases. Types 4 and 5 are essentially types 2 and 3 but on a porous adsorbent (pore filling saturates the adsorbent).
Figure 2.3 Five types of adsorption isotherms according to Brunauer. Graphs have molecules adsorbed on the y-axis and vapour phase concentration on the x-axis. Type 1 describes a case with a monolayer coverage on the adsorbent. Type 2 shows multilayer coverage. Type 3 represents a case with strong adsorbate-adsorbate interactions. Types 4 and 5 are essentially types 2 and 3 but on a porous adsorbent (pore filling saturates the adsorbent).
The most common adsorption isotherms are described below.
2.2.3.1 The linear adsorption isotherm
A linear adsorption isotherm is applicable when the surface concentration of sorbate is low.
The linear relation between vapour phase concentration and surface concentration is:
s s
s C K
Y = ⋅ (2.6)
where Ys is surface concentration (coverage) of adsorbate, Cs is vapour phase concentration close to the surface and Ks is slope of the linear isotherm.
2.2.3.2 The Langmuir isotherm for noncompetitive, nondissociative adsorption
The most important model for reversible monolayer adsorption is the Langmuir isotherm. It is based on an ideal gas adsorbing onto an idealised surface. This one-parameter model was developed between 1913 to 1918 by Langmuir with the following assumptions: One layer adsorption at distinct sites of the sorbent. Occupied neighbouring sites do not influence adsorption of one molecule at a given site of the sorbent. The number of sites available for adsorption is limited (Atkins 1995 and Masel 1996).
If we consider the free gas, the surface of an adsorbing material and the system of sorbed gas molecules on the surface, the reaction formula for the adsorption process when adsorption is treated as a reversible chemical reaction is:
Type 1
Type 4
Type 3 Type 2
Type 5
AM M
A
d a
k k
←
→
+ (2.7)
where A is the symbol for the free gas, M is the surface and AM is the system of sorbed gas on the surface. ka is the rate constant for adsorption (proportional to free gas concentration) and kd is the rate constant for desorption (proportional to sorbed concentration). The equations for mass transport rates are as follows:
[ ]
M J k[ ]
AMC k
JAds = a⋅ s⋅ and Des = d⋅ (2.8)
where JAds is the mass transport rate of adsorption, Cs is the gas concentration, [M] is the concentration of bare sites in number per square metre (available sites for adsorption), JDes is the mass transport rate of desorption, [AM] is the surface concentration of A in molecules per square metre.
At equilibrium there is no net-exchange of mass between surface and gas. The corresponding equations at equilibrium are:
[ ] [ ] [ ]
[ ]
d e a sd s
a K
k k M C AM AM
k M C
k = =
⇒ ⋅
=
⋅
⋅ (2.9)
where Ke is the equilibrium constant between sorbed compound and compound in the gas phase.
We need a balance for the adsorption sites:
] [ ] [ ]
[M = M0 − AM (2.10)
where [Mo] is the concentration of available sites for adsorption in number per square metre.
Equation 2.9 and 2.10 yields ] [ ] ] [
[
M0
K AM C
AM
e s
=
⋅ + (2.11)
Introducing the surface coverage θ as
] [
] [
M0
= AM
Θ (2.12)
We finally get the Langmuir adsorption isotherm from equation 2.11 and 2.12
s e
s e
C K
C K
⋅ +
= ⋅
Θ 1 (2.13)
In this equation Ke is the Langmuir adsorption coefficient. When the adsorbate concentration is very low in the gas phase the coverage varies linearly with concentration since Ke·Cs << 1,
(Masel 1996). Equation 2.13 can be rewritten. It becomes analogous to equation 2.6 at low concentrations since Ys (the adsorbed mass per unit area on the surface) is directly
proportional to θ.
s s s s
e C Y K C
K ⋅ ⇔ = ⋅
=
Θ (2.14)
At high concentrations the coverage saturates since Ke·Cs >> 1
1 =1
⋅
≈ ⋅
⋅ +
= ⋅ Θ
s e
s e s e
s e
C K
C K C K
C
K (2.15)
The heat of adsorption can be obtained from thermodynamics (The Clausius Clapeyron equation). At equilibrium (no net exchange between vapour phase and adsorbed phase) the chemical potential, (µ), (or molar Gibbs free energy) is equal in vapour and adsorbed phases:
dp ad V dT ad S dp g V dT g S ad
g) ( ) ( ) ( ) ( ) ( )
( =µ ⇒ − + =− +
µ (2.16)
where g and ad are gaseous and adsorbed phases respectively, S is entropy V is volume, p is pressure and T is absolute temperature. Equation 2.16 can be re-arranged to:
dp V dT
S =∆
∆ (2.17)
Since V(ad) << V(g) one can set ∆V≈V(g). Furthermore, suppose that the ideal gas law can be employed for the gaseous phase:
dp p RT dT
S =( / )
∆ (2.18)
where R is the universal gas constant. The left hand side of equation 2.18 can be re-arranged using ∆G= ∆H-T∆S as:
) (ln )
/
( 2 R d p
T dT dp H
p T RT
dT
H = ⇒ −∆ =
−∆ (2.19)
The temperature term on the left-hand side can be re-written by manipulating the differentials (Atkins 1995):
) / 1 1 (
) / 1 (
2
2 d T
T dT T
dT T
d =− ⇒ =− (2.20)
We get:
) / 1 (
) ) (ln
) (ln / 1 (
T d
p d R p H
R d T d
H = ⇒ ∆ =
∆ (2.21)
Thus, the heat of adsorption is obtained from the slope of d(ln p) / d(1/T). The equation is only valid for a system in chemical equilibrium. Plots of ln p over 1/T have to be for identical coverage (Atkins 1995).
The relation between the equilibrium constant for adsorption and enthalpy can be derived from a modified equation 2.13 (Masel 1996). The Langmuir relation is re-arranged to:
Θ
−
= Θ
⋅ s 1
e C
K (2.22)
At a fixed coverage equation 2.22 becomes:
constant ln
ln
constant ⇒ + =
=
⋅ s e s
e C K C
K (2.23)
The pressure p in equation 2.21 is identical to the vapour concentration Cs in equation 2.23.
Therefore equation 2.23 can be used in conjunction with equation 2.21 to derive an equation for the equilibrium constant for adsorption as a function of enthalpy at a fixed coverage:
) / 1 (
) (ln )
/ 1 (
) const (ln
ln
ln d T
K d R
H T
d C d R C H
Ke+ s = + ∆ = s ⇒ ∆ =− e (2.24)
For a linear isotherm the equilibrium constant Ks is constant and independent of the coverage.
The equation for heat of adsorption (eq 2.21) is therefore valid for every possible coverage in the case of a linear isotherm.
2.2.3.3 The Langmuir isotherm for competitive, nondissociative adsorption
In some cases more than one species adsorbs on a surface. Consider a case where two gases (A and B) compete for the same adsorption sites (competitive adsorption). Langmuir proposed a model for competitive nondissociative adsorption with the following assumptions: all sites are equivalent, each site can hold only one molecule of either A or B and adsorption of one molecule at a given site of the sorbent is not influenced by occupied neighbouring sites (Masel 1996). At equilibrium we get the following relation for the two species:
[ ] [ ] [ ]
[ ]
eBsB A
e sA
M K C K BM M
C
AM =
= ⋅
⋅ (2.25)
where CsA and CsB are the vapour concentrations of A and B respectively close to the surface.
The site balance becomes:
] [ ] [ ] [ ]
[M = M0 − AM − BM (2.26)
The surface coverage factors, θA and θB are
] [
] [ ]
[ ] [
0
0 M
BM M
AM
B
A = Θ =
Θ (2.27)
Equations 2.25-2.27 give the isotherm for species A as:
B e sB A
e sA
A e sA B
e sB A e sA A e sA
A e sA
A e sA A
e sA
A e sA
A e sA A
A e sA A
e sA
K C AM K
C AM AM
K C AM M
K C K C K C AM AM
K C AM
K C BM K
C AM AM
K C AM BM
K AM C
AM
AM M
AM
BM K AM
C M AM BM
AM K M
C M AM
⋅
⋅ +
⋅
⋅ +
⋅
= ⋅
⋅
⋅
⋅
⋅ +
⋅
⋅ +
⋅
= ⋅
⋅ =
⋅ +
⋅
⋅ +
⋅
= ⋅ +
⋅ +
=
= Θ
+
⋅ +
=
⇒
−
−
⋅ =
=
] [ ]
[ ] [
] [ ]
[ ]
[ ] [
] [
] [ ]
[ ] [
] [ ]
[ ] ] [
[
] [ ]
[ ] [
] [ ] ] [
] [ [ ]
[ ] [ ] ] [
] [ [
0
0 0
B e sB A e sA
A e sA
A C K C K
K C
⋅ +
⋅ +
= ⋅
Θ 1 (2.28)
The isotherm for species B is similar to equation 2.28
B e sB A e sA
B e sB
B C K C K
K C
⋅ +
⋅ +
= ⋅
Θ 1 (2.29)
2.2.3.4 The Langmuir isotherm for dissociative adsorption
Dissociative adsorption occurs when a molecule dissociates into two atoms during the adsorption process. The model was derived with the following assumptions: the molecule P2
dissociates to two P-atoms during adsorption, all sites are equivalent, each site can hold only one P-atom and adsorption of one atom at a given site of the sorbent is not influenced by occupied neighbouring sites (Masel 1996). At equilibrium the following equation is valid:
[ ] [ ]
ePsP
M K C
PM =
⋅
2 / 1
2
(2.30) where the exponent acting on the vapour concentration term, C, is due to the fact that one gas molecule yields two adsorbed species. Surface coverage and site balance equations for dissociative adsorption are identical to the original Langmuir equations (eq. 2.10 and 2.12).
The isotherm for dissociative adsorption becomes:
2 / 1 2 / 1
2 2
1 eP sP
sP P e
P K C
C K
⋅ +
= ⋅
Θ (2.31)
2.2.3.5 The Freundlich isotherm
Though empirically derived this two-parameter model for monolayer adsorption has been widely used to interpret experimental data for rough surfaces (Masel 1996). The surface coverage θ is expressed as:
/ 2
1 1
c
Cs
=c
Θ (2.32)
where c1 and c2 are fitting parameters associated with the Freundlich model.
