Experimental and numerical investigations of the eects of air leakage on temperature and moisture elds in porous insulation Clément Belleudy
, Ahmad Kayello , Monika Woloszyn , Hua Ge
a,b,∗
c
a
c
a LOCIE, Université Savoie Mont Blanc, Campus scienti que Savoie Technolac, 73376 Le Bourget du Lac, France b Centre Scienti que et Technique du Bâtiment, 24 rue Joseph Fourier, 38400 Saint Martin d'Hères, France c Department of Building, Civil and Environmental Engineering, Concordia University, Montreal, Canada
Abstract
Air leakage through the building envelope can lead to an increase in energy consumption and to potential moisture damages. This paper investigates the impact of air leakage on the hygrothermal eld in a ceiling section insulated with blown-in cellulose, and separating an attic space from a heated indoor space. This ceiling section, part of a full scale test-hut built in the Environmental Chamber of Concordia University, is tested experimentally with and without air leakage. The temperature in the cellulose is measured at dierent locations along the air leakage path and compared with temperature outputs from recently developed HA (Heat Air) and HAM (Heat Air Moisture) numerical models. The HAM model, which shows good agreement with experimental data, is nally used to perform an analysis of the hygrothermal eld in the cellulose. The ability of this HAM model to calculate 2D-hygrothermal elds in the presence of air ow is helpful to predict consequences of poor workmanship or bad design and to move toward moisture safe buildings. Keywords: HAM model, moisture, transient, building envelope, air leakage, porous insulation
1. Introduction
Airtightness has become a major challenge over the past few decades in creating low-energy and durable buildings. Bad design and poor workmanship can lead to unintentional air leakage through the building envelope. Besides signi cantly increasing the energy loss [1], leaking air may reach its dew point inside building assemblies and thus cause additional moisture damage [2, 3]. Therefore, it is of importance to better understand the impact of air transfer on the hygrothermal eld to ensure moisture safe buildings. Although good airtightness can be achieved regardless of the construction technology [4], lightweight constructions are particularly sensitive to airtightness defects, because of their structure and numerous joints, that may lead to unintentional air ows through the porous insulation. Related physical phenomena can be analysed using numerical models for heat and mass transfers as well as experimental measurements. ∗ Corresponding
author
Email addresses:
[email protected] (Clément Belleudy),
[email protected] (Ahmad
Kayello),
[email protected] (Monika Woloszyn),
[email protected] (Hua Ge)
Preprint submitted to Building and Environment
October 28, 2015
Numerical models to assess heat and moisture (HM) transfer through porous material have been released since the nineties [5, 6, 7]. By taking into account vapour and liquid transport, transient moisture buering and moisture dependent material properties, they predict potential moisture risks more reliably than the well-known Glaser method [8]. The coupling with air transfer was deemed to be necessary, as experimental work showed the signi cant impact of air ow on the hygrothermal eld in building assemblies [9, 10, 11]. In this regard, several approaches were proposed to model heat air and moisture (HAM) transfer in building components. Apart from the chosen potential to model HM transfer, these approaches mainly dier by the number of dimensions in the case study (1D, 2D or 3D), by the ability to deal with air permeable materials, and by the strategy to implement air transfer in the global system of equations. One of the major numerical di culties is the dierence of time scale between air transfer (rapid process), and HM transfer (slow process) [12]. A rapid dynamic imposes the use of small time steps, which is not compatible with ensuring decent processing time for long term simulations. A great amount of experimental and modelling work has been done to assess the interactions between hygrothermal transfer in building materials and indoor air moisture balance [13, 14, 15, 16]. Successful modelling attempts couple CFD air ow simulations in the indoor volume with heat and moisture transfer in building materials [17, 18, 19, 20]. When it comes to account for air transfer in building materials, the most simpli ed approach is presented by [21, 22]: air transfer is indirectly modelled by adding a source in the moisture conservation equation, reproducing the additional amount of moisture brought by air ow. The position of this source in the assembly is determined by practical experience (e.g. contact of air with a cold layer). When air is considered as an active mass component in the assembly, conservation of air mass and momentum must be added to the HM system of equations. This is the case with HAMFit developed by Tariku et al. [23], where air transfer is seen as forced convection through 1D multilayered porous material. If natural convection is disregarded, air conservation equations are decoupled from the energy equation. Alternatively, the 2D model HAM-BE [24] includes solely natural convection in the porous material. The Boussinesq approximation is used to capture natural convection while limiting the complexity of the model: the temperature dependence of air density is limited to the gravity term of the momentum equation (Darcy's law here). Some authors combine forced and natural convection, such as [12, 25]. This last work aimed to implement air transfer in DELPHIN software, more speci cally to deal with air channels in contact with air permeable porous material. Successful attempts to deal with HAM transfer with 3D geometries have been published by van Schijndel [26]. The present paper reports experimental and numerical investigations of the heat, air and moisture transport in blown-in cellulose insulation, and the impact of the moist air ow through air leakage on the hygrothermal eld is assessed. The experimental setup involves a large scale climatic chamber to impose realistic boundary conditions. These measurements are compared with outputs from recently developed bidimensional (2D) HA and HAM models in order to extend the analysis. 2
The rst part of the paper describes the experimental setup. Then, the conservation equations of both HAM and HA models are presented as well as the corresponding boundary conditions. The third part compares the experimental measurements and simulation outputs on the temperature eld, in order to gain con dence in the numerical model's capabilities. This comparison also helps in determining whether a HAM model instead of a HA model provides more accurate simulations. Additionally it provides some elements of validation of the model. In the last part, HAM-Lea is used to conduct a comprehensive analysis of the hygrothermal eld in the cellulose insulation. Nomenclature
c
heat capacity [J/(kg.K)]
Greek symbols
cpair
heat capacity at constant pressure of dry
β
vapour transfer coe cient [s/m]
air [J/(kg.K)]
δ
vapour permeability [s]
λ
thermal conductivity [W/(m.K)]
µair
air dynamic viscosity [Pa.s]
Dw
moisture diusivity of material [m /s]
g
moisture ux [kg/(s.m2 )]
H
enthalpy by volume of material [J/m3 ]
h
heat transfer coe cient [W/(m2 .K)]
ρ
density [kg/m3 ]
k
permeability [m2 ]
ϕ
relative humidity [-]
Lv
latent heat of evaporation of water [J/kg]
Subscripts
M
molar mass [kg/mol]
adv
advection
n
outward pointing normal vector
att
attic space
P
total air pressure [Pa]
cond
conduction
pv
partial pressure of water vapour [Pa]
conv
convection
Psat
saturation vapour pressure [Pa]
dif f
q
heat ux [W/m ]
diusion
universal gas constant [J/(mol.K)]
int
interior space
R
surface of the ceiling section [m2 ]
lat
S
latent
temperature [K]
liq
T
liquid
u
Darcy velocity [m/s]
mat
dry material
w
moisture content by volume of material
surf
surface
[kg/m3 ]
w
liquid water
2
2
3
2. Experimental setup
2.1. Setup
The experimental setup of our study is a full-scale test hut built inside the environmental chamber in Concordia University ( g. 1, left), part of a research program aiming to demonstrate the feasibility of an unvented cold attic in extreme cold climates, and more speci cally for the Canadian North [27]. Two main construction typologies exist while insulating roof sections: cathedral ceilings and cold attics. In the rst, insulation is placed in the cavities between the rafters. In the latter, an horizontal insulation layer above the ceiling separates heated indoor space from the attic space. Higher insulation levels can be attained with cold attics compared to cathedral ceilings, and the volume of the heated space is eectively reduced. Cold attics are commonly ventilated but in the context of arctic climate, ne snow particles may be introduced by incoming air and cause snow accumulation on the upper part of the insulation layer, hence the necessity to demonstrate the feasibility of unvented attics. For this purpose, the cold attic of the experimental test hut has been separated into two zones: a ventilated one and an unvented one. Throughout the paper, we are exclusively focusing on the unvented attic. The environmental chamber measures 4.66 m by 8.77 m by 7.17 m in width, length, and height, respectively. The temperature in the environmental chamber can be controlled between -40◦ C and 50◦ C. The test hut measures 4.27 m by 3.05 m (14' by 10') in length and width ( g. 1, right). The test hut contains an unvented attic space insulated at the ceiling level with 380 mm of blown-in cellulose ( g. 2) above 38 mm of rigid polyisocyanurate (PIR). The PIR also acts as the air and vapour barrier of the ceiling. 2.2. Instrumentation
To realistically reproduce air leakage, sampler pumps installed in the indoor space are used to deliver indoor air into the attic space at a controlled rate. The air is supplied at the bottom of the cellulose insulation through an ori ce in the PIR using a tube with a 6.4 mm inner diameter. In the case where there is no air ow into the attic, the ori ce is sealed with tape at bottom of the PIR. ( g. 3) shows the location of the temperature and relative humidity sensors in the indoor, ceiling, and attic space. RH/Tatt and RH/Tint are the relative humidity with embedded temperature sensors for the attic and indoor spaces, respectively. A way to indirectly map air leakage through building components is to measure temperature in the vicinity of air leakage path [9], so the remaining sensors are located within the cellulose insulation at three dierent heights and at various horizontal distances away from the air leakage ori ce in the PIR. RH/T3A and RH/T3∞ are relative humidity sensors with embedded resistive temperature detectors installed in the top layer of the cellulose. The relative humidity sensors have a an accuracy of 4% while the temperature sensors have an accuracy of ±0.3◦ C . The other temperatures are measured by thermocouples (Type T, 30 gauge) with an accuracy of ±0.5◦ C . The temperature sensors in the cellulose insulation are supported by a 4
Figure 1: Full scale test hut in the Environmental Chamber of Concordia University (left), isometric view of the test hut with dimensions (right)
Figure 2: Attic of the test hut insulated with loose ll cellulose
thin wooden structure to ensure proper placement of the sensors without risk of displacement ( g. 3, right). Horizontal wood members less than 10 mm in diameter support the sensors at each of the three heights in the cellulose insulation; the horizontal members are themselves supported by vertical members at least 100 mm away from the sensors.
5
RH/Tatt
RH/T3A T3B T3C T3D
RH/T3∞
T2∞
T1
T1∞
e/2
T2A T2B
Attic space
e =38 cm
3e / 4 e/2 e/4
52 cm
RH/Tint
Airflow
3.8 cm
5 cm
Cellulose insulation Polyisocyanurate insulation
Interior heated space Figure 3: Location of temperature sensors across the ceiling (left), held in place with a low-pro le wooden support system (right)
2.3. Conducted experiments
The indoor space is maintained at 22◦ C and 60% relative humidity, while the outdoor space is maintained at 5◦ C and 70% relative humidity. Air leakage rates used are 0, 2 and 5 L/min. With the available sampler pumps, it was di cult to ensure constant ow rate below 2 L/min. Additional simulations showed that 2 L/min and 5 L/min leakage rates correspond to a pressure dierence across the ceiling of 10 Pa and 26 Pa, respectively. According to [28], typical values of resulting mean pressure dierences across the building envelope are within [0-10 Pa]. As a consequence, chosen ow rates remain reasonably close to typically measured values in practice. A summary of the boundary conditions is provided by ( g. 4). The overall experiment duration was over 600 hours and measurements were taken every minute. 3. Numerical model
A 2D numerical model for simulating coupled HA transfer through porous medium has been developed in COMSOL Multiphysics software and presented in previous work [29], and is abbreviated as HA-Lea throughout the paper. Since then, moisture has been implemented, leading to a newly-developed HAM model called HAM-Lea. HAM-Lea uses a similar approach as HAMFit from [23], but applied in a 2Daxisymmetric environment instead of a 1D environment. Both models consider forced convection for air transfer, and use relative humidity as moisture potential. They dier in some assumptions (e.g. HAM-Lea neglects sensible heat carried by vapour and liquid water, while accounting for the latent heat ux carried by water vapour) and the way equations are implemented into COMSOL. The nal aim of HAM-Lea is to model complex 2D air leakage geometries, combining thin air channels in contact with air-permeable 6
1.0
24
0.9
22
0.8
20
0.7 T_int
0.6
16
0.5
14
0.4
12 10
no airflow
0.3
airflow 2L/min
airflow 5L/min
no airflow
φ (-)
T (°C)
18
T_att phi_int phi_att
0.2
8
0.1
6
0.0 0
100
200 145
Time (h)
300 313
400
500
600
388
Figure 4: Boundary conditions: indoor and attic temperatures (black lines) and relative humidities (gray lines), imposed ow rate at the ori ce in the PIR (indicated in red)
materials. This implies a speci c modelling strategy, described in a subsequent paper. In the present paper, the experimental setup is used to obtain a rst experimental validation of HAM-Lea's system of equations implemented in COMSOL, in case of 2D HAM transfer through a porous air permeable material without air channels. HAM-Lea has already been validated in 1D with the numerical HAMSTAD benchmarks [30]. The question whether natural convection should be implemented or not may be discussed. On a similar con guration without any imposed air ow, Wahlgren [31] shows that natural convection may occur in loose ll insulation when subjected to high temperature gradients, which alters its apparent thermal resistance. This natural convection may also in uence moisture redistribution in the material according to [32]. However, in our case, as we deal with temperature dierences far below 40◦ C, natural convection is expected to be of minor importance in porous insulating material [33] and will therefore be disregarded. In contrast, forced air convection is likely to have a signi cant impact on the hygrothermal eld. This section rstly presents HAM-Lea governing equations, and those from HA-Lea, after making the suitable simpli cations. The boundary conditions are discussed based on the defect geometry of ( g. 3). Finally, the models implementation into COMSOL's modelling interface is brie y mentioned. 3.1. Governing equations
HAM-Lea is formulated using the continuous medium approximation : material properties and local elds are averaged over Representative Elementary Volumes (REV), which enable conservation laws to be written in local form using Partial Derivative Equations (PDE). HAM-Lea's conservation equations for energy (1),
7
moisture (2), mass (3) and momentum (4) are given below, with (T, ϕ, u, P ) as variables: ∂H(T, ϕ) = −∇ · [qcond (T, ϕ) + qconv (T ) + qlat (T, ϕ)] ∂t ∂w(ϕ) = −∇ · [gdif f (T, ϕ) + gadv (T, ϕ) + gliq (ϕ)] ∂t ∇·u = 0 kmat ∇P u = − µair
(1) (2) (3) (4)
The simple form of continuity equation (3) is due to low air velocities encountered in building physics, implying the assumption of incompressible ow. Darcy's law (4) is a simpli ed form of Navier-Stokes equation in the porous material to describe momentum conservation. This law is valid for low velocities, i.e. pore Reynolds number of order of unity [34]. Natural convection is said to be of limited importance when indoor-outdoor temperature dierences do not exceed 40◦ C [33]. As HAM-Lea is rstly dedicated for temperate climates, natural convection is not implemented. This choice also enhances simulation performance because Darcy's law can be solved independently, prior to moisture and energy equations. For the energy equation, a one-temperature approach is adopted, assuming therefore thermal equilibrium between air and solid material. This approximation is valid for building insulation materials with low air velocities, as proved by [35]. In the energy equation (1), the enthalpy variation rate of a REV may be expressed as: ∂T ∂H = [ρmat cmat + w(ϕ)cw ] ∂t ∂t
(5)
This energy variation is driven by three net heat in uxes as shown in (1). qcond is the heat conduction ux density, described by the well-known Fourier's law (6) with a moisture dependent thermal conductivity. Air is regarded as an ideal mixture between dry air and water vapour following the ideal gas law. From there, the heat convection ux density qconv corresponds to the heat ux carried by dry air ow and is expressed by (7). Finally, the latent heat ux density qlat describes heat transfer occurring during moisture sorption and desorption in the porous medium (8). The sensible heat carried by liquid and vapour uxes is commonly neglected in calculations [5, 36]. qcond (T, ϕ)
= −λmat (ϕ) ∇T
(6)
qconv (T )
= ρair cpair T u
(7)
= Lv (gdif f (T, ϕ) + gadv (T, ϕ))
(8)
qlat (T, ϕ)
Moisture conservation states that the increase rate of moisture content in an REV equals the sum of three net in ux of moisture which are vapour diusion, vapour advection and liquid transport. Vapour diusion is described by Fick's law (9). Vapour advection refers to the vapour ow carried by air ow (10). The 8
humidity by volume of air, ρvap , can be linked with temperature and vapour pressure via the ideal gas law (12). Liquid transport is driven by capillary suction, in water lled pores. This phenomena occurs in smaller pores rst, and becomes dominant for high relative humidities (ϕ > 0.98). It can be written with relative humidity as potential (11). The de nition of relative humidity ϕ is needed (13) in order to add the relationship between ϕ and vapour pressure pv . gdif f (T, ϕ)
=
−δmat (ϕ) ∇pv (T, ϕ)
gadv (T, ϕ)
=
gliq (ϕ)
=
ρvap (T, ϕ)
=
pv (T, ϕ)
=
ρvap (T, ϕ) u ∂w(ϕ) −Dw (ϕ) ∇ϕ ∂ϕ Mw pv (T, ϕ) RT ϕ Psat (T )
(9) (10) (11) (12) (13)
To solve the coupled system of PDE constituting HAM-Lea, the commercial nite-element software COMSOL Multiphysics is used [37]. For this purpose, energy and moisture equations have been formulated according to a given general form PDE (25), whose description is given in the appendix. Simplifying HAM-Lea into a HA model is straightforward: the moisture conservation equation and the latent ux in the energy equation (1) are removed. The resulting PDE system of equations for the simpli ed HA-model is recalled below: ∂T = ∂t ∇·u =
0
u =
−
ρmat cmat
−∇ · [qcond (T ) + qconv (T )]
(14)
kmat ∇P µair
3.2. Boundary conditions
This section presents the boundary conditions implemented in HAM-Lea and HA-Lea to simulate the geometry presented in ( g. 3, left). To reduce computational time, the transfer phenomena are considered to be rotationally symmetric about the longitudinal axis that starts at the air ori ce and points upward, neglecting therefore the thermal bridge induced by the truss and wooden structures supporting the sensors ( g. 3, right). The simulation is therefore performed on a 2D axisymmetric plane, creating the 3D rotational polar coordinate system. The modelled geometry implemented in COMSOL, as well as equations and boundary conditions are summarized in ( g 5). Boundary conditions for air and heat must be chosen on each boundary. The measured temperatures Tint (t) and Tatt (t) are used as boundary conditions for the model. To simplify the geometry, polyisocyanurate insulation has been integrated as air tight and vapour tight boundary conditions, and as thermal boundary condition with an equivalent global heat transfer 9
coe cient on the interior surface: 1
(15)
heq = 1 Rpolyiso +
hint
where Rpolyiso [m .K/W] is the polyisocyanurate insulation layer thermal resistance. 2
Air velocity was applied at air inlet, and a reference pressure at air oulet. At airtight boundaries, a slip boundary condition is applied: (16)
u·n=0
For heat boundary conditions, temperature is imposed at air inlet (17) and a heat ux is imposed at air oulet (18). The heat ux applied at the interface between polyisocyanurate and ambient interior air is described by (19). At adiabatic boundaries, the total heat ux is set to zero (20). The Neumann boundary conditions are written in accordance with the PDE formulation described in appendix. Tsurf = Tint
(17)
−n · (−α20 ∇T − α30 T + α40 ) = hatt (Tatt − Tsurf ) +Lv βatt [pv (Tatt , ϕatt ) − pv (Tsurf , ϕsurf )]
(18)
−n · (−α20 ∇T − α30 T + α40 ) = heq (Tint − Tsurf )
(19)
−n · (−α20 ∇T − α30 T + α40 ) = 0
(20)
Similarly, moisture boundary conditions consist in a xed relative humidity at air inlet (21) and a moisture ux at air outlet (22). At moisture impermeable boundaries, the total moisture ux is set to zero (23). ϕsurf = ϕint
(21)
−n · (−α2 ∇ϕ − α3 ϕ + α4 ) = βatt [pv (Tatt , ϕatt ) − pv (Tsurf , ϕsurf ))]
(22)
−n · (−α2 ∇T − α3 T + α4 ) = 0
(23)
3.3. Numerical simulation
The previously described equations and boundary conditions are implemented via the COMSOL "Mathematics" module, in which PDE equations are described by their general formulation with coe cients. The hygrothermal properties were either obtained from the manufacturer or found in [38] and in the Fraunhofer IBP database accessible via the WUFI software. Moisture dependent properties and parameters are extensively presented in the appendix. COMSOL's built-in meshing interface is used and the meshing is re ned in narrow regions and in regions where high temperature/velocity gradients are expected. The mesh size is re ned until the ux and temperature calculated become stable. The geometry has a total of 2720 meshes and the 638 h-HAM simulation 10
L=1 m
Attic space temperature Tatt Heat flux (eq. 18) Pressure Vapour flux (eq. 22) Adiabatic (eq. 20) Airtight (eq. 16) Vapour tight (eq. 23)
Axisymmetry
e=38 cm
Conservation laws:
Air inlet
Energy (eq. 27) Continuity (eq. 3) Momentum (eq. 4) Moisture (eq. 25)
Temperature Tint (eq. 17) Velocity Relative humidity fint (eq. 21)
Heat Air
Heat flux (eq. 19) Airtight (eq. 16) Vapour tight (eq. 23)
Moisture Temperature sensor Air velocity field
Interior space temperature Tint Figure 5: Domain equations, computational domain, and boundary conditions of HAM-Lea, applied to the ceiling section insulated with loose ll cellulose
requires approximately 3 minutes using an Intel Xeon E5-1650 CPU v2 at 3.5 GHz. However, only 1.50 GB RAM are needed to perform the HAM simulations of the present case study. 4. Results and discussion
In this section experimental results are presented, together with values obtained with numerical simulation. Both HA-Lea and HAM-Lea models are used, in order to determine whether considering moisture transfer increases the model agreement with experimental data. Then, HAM-Lea is used as a numerical tool to assess the impact of moist air ow on the hygrothermal eld. 4.1. Temperatures on the mid-plane of the cellulose: comparison between simulations and measurements
A rst comparison between simulations and measurement is done by examining temperature T2A , located in the middle plane in the cellulose. This plane is a relevant location to assess the accuracy of the models, as it is located at the farthest position from both horizontal interfaces (top and bottom of the cellulose insulation) where boundary conditions are imposed. The temperature value at point T2A calculated with HAM-Lea and HA-Lea simulations, and experimentally measured are plotted on ( g. 6). Before dealing with the comparison between simulated data and experimental measurements, the general plots tendencies are explained in correspondence of the boundary conditions of the experiment. This will facilitate understanding the impact of boundary conditions on T2A . At initial stage, the whole experimental setup is at an initial temperature of roughly 25◦ C. HVAC systems of the environmental chamber and those located inside the 11
24 22
T ( °C)
20
T_int
18
T2A HAM-Lea
16
T2A HA-Lea T2A Exp.
14
T_att
12 10 airflow 5L/min
no airflow
8
airflow 2L/min
no airflow
6 0
100
200 145
300 Time (h) 313
400 388
500
600
Figure 6: Comparison between the measured T2A and the simulation results, obtained with both newly-developed HA and HAM models
experimental test hut are turned on at t = 0, with set point temperatures of 5◦ C and 22◦ C, respectively. The measured temperature in the unvented attic (6.8◦ C) is slightly above the one imposed in the environmental chamber because of the roof thermal resistance. A temperature gradient is thus created in the cellulose insulation. T2A is located at the middle height in the cellulose, which explains that measured and simulated values at this point are within the interval [5, 22◦ C]. At t=145 h, air ow from heated indoor space is injected from a hole in the PIR to the cellulose at a 5 L/min ow rate, which leads to a logical increase of T2A . The ow rate provided by the sample pump is changed to 2 L/min at t=313 h: less warm air from indoor space is thus supplied to the cellulose, hence a temperature decrease. 75 hours later, at the 388th hour, the pump is completely switched o, which makes T2A decrease. Finally, shortly before the 600th hour, all HVAC systems are switched o, and all temperatures retrieve their initial values. In addition some punctual discrepancies between set-point and measured values were observed. This is due to experimental di culties to ensure stable level of relative humidity. As all boundary conditions were precisely measured and used in the model, these discrepancies do not in uence model precision. When comparing more nely the dierent plots, ( g. 6) clearly shows that HAM-Lea outputs are at all times closer to experimental data than HA-Lea outputs. The same tendency is also observed for other temperatures such as T2B and T2∞ on the mid-plane in the cellulose, which are not presented here. These results con rm that accounting for moisture transfer in addition to heat and air transfers enhances the model accuracy. Latent heat uxes, which are accounted for in the HAM model only, may play a signi cant role in the overall heat transfer. This latent heat eect was also experimentally observed by [39] for hygroscopic materials subjected to HM transfer. Until the 388th hour, the dierence between HAM-Lea outputs and experimental measurements are below 12
the sensor accuracy. However, after the 388th hour, HAM-Lea overestimates the experimental measurements by approximately 1◦ C. This might be due to the inability of the model to detect moisture accumulation for high relative humidity levels. Indeed, moisture accumulation has been reported for a similar assembly, subjected to winter conditions [10]. In the authors' view, possible moisture accumulation could have been likely in our case if high relative properties were attained, suggesting a dierent behaviour due to liquid water and capillary uxes. However, the maximum relative humidity is 67% in our case, and this value is already attained in the vicinity of the air inlet before the 388th hour. A more likely explanation would be a local alteration of the density of the loose- ll cellulose layer, due to maintained air ow over more than 200 hours, and consequently an alteration of its physical properties. Experimental investigations support this assumption by indicating that outdoor conditions or mechanical vibrations may in uence settlement behaviour of loose- ll cellulose insulation [40]. To check further the plausibility of this assumption, experimentally measured temperatures T2A , T2B and T2∞ are plotted in ( g. 7). T2B and T2∞ are located on the same mid-plane in the cellulose at a horizontal
distance of 9 cm and 52 cm from T2A , respectively. The plots shows that measurements of T2∞ are identical just before the 145th hour and after the 388th hour, coinciding with time periods without air ow. This implies that the structure of the cellulose has not been altered in this region less impacted by the air ow, contrary to the vicinity of the hole where T2A and T2B are located. 24 22 20 T_int
T ( °C)
18
T2A Exp.
16
T2B exp.
14
T2inf exp.
12
T_att
10 airflow 2L/min
airflow 5L/min
no airflow
8
no airflow
6 0
100
200 145
Time (h)
300 313
400 388
500
600
Figure 7: Comparison between measured temperatures values T2A , T2B , and T2∞ which is designated as T2inf
4.2. Analysis of hygrothermal eld through cellulose insulation with moist air ow
In section 4.1, a comparison between measured and simulated temperatures on the mid-plane in the cellulose has been made, and close agreement has been obtained. Now, the model is used to better understand 13
transport processes in the cellulose subjected to moist air ow. This section presents the hygrothermal eld of cellulose insulation i.e. vapour pressure and moisture content simulated by HAM-Lea. Vapour pressures calculated by the model at dierent heights in the cellulose are shown on ( g. 8). They have been calculated according to (13), either from temperature and relative humidities measurements (Tint , Tatt , ϕint , ϕatt ), or by simulation (T1 , T2A , T3A , ϕ1 , ϕ2A , ϕ3A ). Vapour pressure values in the cellulose, range from around 500 Pa in the attics to 1400 Pa in the test hut. Consequently, vapour pressure gradient appears in cellulose insulation. As a result, the relative position of the curves pvint > pv1 > pv2A > pv3A > pvatt is consistent. The portions of the chart where no air ow is injected ([0 - 145 h] and after the 388th hour)
show that this vapour pressure gradient is mainly concentrated across the polyisocyanurate insulation, which is consistent with the fact that it is used as vapour barrier. Because of air leakage, the moisture content in 1900 1700
pv (Pa)
1500
pv_int
1300
pv_1
1100
pv_2A pv_3A
900
pv_att 700 500
airflow 5L/min
no airflow
airflow 2L/min
no airflow
300 0
100
200 145
300 Time (h) 313
400 388
500
600
Figure 8: Vapour pressures obtained from simulation at dierent heights above the air inlet in the cellulose. pvint and pvatt are obtained from experimental measurements (Tint , ϕint ) and (Tatt , ϕatt ), respectively
the cellulose changes over time. With HAM-Lea, moisture content can be evaluated on every point in the cellulose insulation. Its averaged moisture content can be calculated as follows: RR wavg (t) =
S
w(x, y, t) dx dy S
(24)
with S the surface of the computed ceiling section, depicted in ( g. 5). wavg (t) is plotted together with w2A (t) and w2∞ (t) versus time in ( g. 9). The chart shows that the air ow causes a signi cant increase in
the averaged moisture content in the cellulose. This increase is more pronounced at regions close to the air inlet, that is why w2A > w2∞ . In order to have a overall view of the transfer phenomena, it is relevant to focus on snapshots (at t = 100 h, 200 h, 350 h, 550 h) of each of the temperature, vapour pressure and moisture content elds
( g. 10). Compared to the case with no air leakage where the isotherms are horizontal and equidistant 14
2.9 2.8 2.7
w (kg/m3)
2.6 w_avg
2.5 2.4
w_2A
2.3
w_2inf
2.2 2.1
airflow 5L/min
no airflow
airflow 2L/min
no airflow
2.0 0
100
200 145
Time (h)
300 313
400 388
500
600
Figure 9: Evolution of total averaged moisture content in the cellulose wavg , and ponctual values w2A and w2∞ . w2∞ is designated as w2inf in the plot. Values obtained by simulation with HAM-Lea
in the cellulose insulation (a1. and a4.), imposing air leakage shifts the isotherms upwards at the axis of the ori ce, creating bell-shaped curves (a2. and a3.). The increase in temperature in the cellulose is more pronounced with higher ow rates, as expected. These snapshots also illustrate the dierence of time constant between heat and moisture transfers. After the pump is being switched o at the 388th hour, the temperature eld at the 550th hour (a4.) is fairly identical to the one at the 100th hour before the pump is switched on (a1.). On the contrary, the eld of moisture content are signi cantly dierent on the 100th hour and the 550th hour. This clearly brings to light that moisture transfer is a much slower process than heat transfer.
15
Axisymmetry
Temperature
Vapour pressure
Water content
t=100h
a1.
b1.
c1.
a2.
b2.
c2.
a3.
b3.
c3.
a4.
b4.
c4.
t=200h
t=350h
t=550h
T (°C)
pv (Pa)
w (kg/m3)
Figure 10: 2D snapshots and contour lines of temperature (a1-a4), vapour pressure (b1-b4) and moisture content (c1-c4) at dierent time points : t=100 h (no air ow), t=200 h (5 L/min air ow), t=350 h (2 L/min air ow), and t=550 h (no air ow)
5. Conclusion
In this article, we assess the impact of an airtightness defect on the hygrothermal eld in a ceiling section insulated with loose ll cellulose, and separating an heated indoor space from an unheated attic. Temperature sensors located above the air inlet and at a further distance enable the air path to be mapped indirectly. Experimental results compare well with newly developed HA and HAM models, as far as global tendencies are concerned. However, the HAM model outputs better reproduces experimental temperature variations with a maximum discrepancy of order of 1◦ C, which suggests that latent heat uxes have a large contribution in the overall heat transfer. Moreover the results show that air ow aects moisture eld with a longer time constant than temperature eld. After switching o sample pumps, a permanent oset appears between experimental and simulated temperature from HAM model, which suggests that the structure of the cellulose may have changed because of maintained air ow. The results show unambiguously that even a relatively limited air ow through construction elements has a strong impact on hygrothermal elds within building envelope. This outcome is helpful in increasing designers' and builders' awareness about the importance of airtightness, as stressed by [41]. 16
This work con rms the relevancy considering HAM models to better describe coupled transfers in the building envelope. The present HAM model is able to predict the modi cation of the hygrothermal eld in an insulation material caused by air leakage, and could be used to detect interstitial condensation. However, it should be noted that the model is based on some assumptions and that actual eld air leakage geometries may dier from the simulated ones [42]. Further work could focus on moisture and heat ux outputs, in order to quantify their modi cation consequently to air leakage. Accurate measurement of the material properties combined with a parameter analyses can also be of interest and may partly explain the slight disagreement between measured and simulated values. Acknowledgements
This work is nancially supported by ADEME (the French Environment and Energy Management Agency), CSTB (Building Scienti c and Technical Centre), and the Région Rhône-Alpes. The experimental data is taken from a project under the NSERC Smart Net-Zero Energy Buildings Strategic Research Network (SNEBRN), sponsored by the Natural Sciences and Engineering Research Council of Canada (NSERC) and 14 industrial partners including KOTT Group. The project is also supported through a NSERC discovery grant. The materials and installation of the test hut are supplied by KOTT Group. The authors would like to hereby pay tribute to the late Dr. Paul Fazio, who passed away in September 28, 2014. Concordia's Centre for Building Studies and its unique facility - the Solar Simulator and the Environmental Chamber - owe their existence to him. Appendix
HAM-Lea formulation in COMSOL Multiphysics
Moisture conservation equation (2) is rewritten as: α1
∂ϕ + ∇ · (−α2 ∇ϕ − α3 ϕ + α4 ) + α5 ∇ϕ + α6 ϕ = α7 ∂t
The dierent coe cients αi (with i ∈ J1, 7K) are given below:
17
(25)
∂w(ϕ) ∂ϕ
α1
=
α2
= δmat (ϕ)Psat (T ) + Dw (ϕ)
α3
= δmat (ϕ)
α4
=
α5 α6 α7
∂w(ϕ) ∂ϕ
dPsat (T ) ∇T dT
0 Mw Psat (T ) = u RT Mw ∇T · u 1 dPsat (T ) Psat (T ) = − R T dT T2 = 0
(26)
Energy conservation equation (1) is rewritten as: α10
∂T + ∇ · (−α20 ∇T − α30 T + α40 ) + α50 ∇T + α60 T = α70 ∂t
(27)
The dierent coe cients αi0 (with i ∈ J1, 7K) are given below: α10 α20 α30 α40 α50 α60 α70
w(ϕ) = ρmat cmat + cw ρmat dPsat (T ) = λmat (ϕ) + Lv δmat (ϕ)ϕ dT = 0 = −Lv δmat (ϕ)Psat (T )∇ϕ Lv Mw ϕ dPsat (T ) Lv Mw ϕPsat (T ) = ρair cpair + − u RT dT RT 2 = 0 Lv Mw Psat (T ) = − u · ∇ϕ RT
(28)
HA-model formulation in COMSOL Multiphysics
Coe cients of (27) are modi ed as follows: α10
∂T + ∇ · (−α20 ∇T − α30 T + α40 ) + α50 ∇T + α60 T = α70 ∂t
18
(29)
α10
=
α20
= λmat
α30
=
0
α40
=
0
α50
= ρair cpair u
α60
=
0
α70
=
0
ρmat cmat
19
(30)
Material properties and parameters Table 1: Constant input parameters Symbol
Unit
Value
Source/Note
λcellulose
W/(m.K)
0.038
[38]
λpolyiso
W/(m.K)
0.022
[38]
Thermal conductivity of air
λair
W/(m.K)
0.026
at 20◦ C
Thermal capacity of cellulose
ccellulose
J/(kg.K)
1400
WUFI database
Thermal capacity of air
cpair
J/(kg.K)
1006
Assumed constant
Thermal capacity of liquid water
cw
J/(kg.K)
4.18
-
Density of cellulose
ρcellulose
kg/m3
30
[38]
Air density
ρair
kg/m3
1.2
at 20◦ C
Density of liquid water
ρw
kg/m3
1000
-
Permeability of cellulose
kcellulose
m2
4.35 × 10−9
[38]
Vapour permeability of air
δ0
kg/(s.m.Pa)
1.96 × 10−10
[43]
Dynamic viscosity of air
µair
Pa.s
1.8 × 10−5
at 20◦ C
Sorption heat of vapour water
Lv
J/kg
2491 × 103
[44]
Molar mass of water
Mw
kg/mol
18 × 10−3
-
Ideal gas constant
R
J/(K.mol)
8.314
-
βatt
kg/(s.m.Pa)
18.5 × 10−9
[45]
hint
W/(m2 .K)
4.32
[38]
hatt
W/(m2 .K)
9.26
[38]
Parameter
Thermal conductivity of cellulose (for HA-Lea) Thermal conductivity of polyisocyanurate
Surface lm coe cient for vapour transfer attic Surface lm coe cient for heat transfer (inside) Surface lm coe cient for heat transfer (attic)
20
Table 2: Variable properties of Klima ockr cellulose, Fraunhofer IBP material database of WUFI Parameter
Water content of cellulose
Symbol
Relative humidity [-]
Unit 0
0.5
0.8
0.9
0.98
1
wcellulose
kg/m3
0
2.7
5.5
6.6
10.1
426
λcellulose
W/(m.K)
0.038
0.039
0.040
0.041
0.043
0.200
0
9.82
20.0
21.3
25.3
500
3.53
3.53
3.53
3.53
3.53
0
Thermal conductivity of cellulose (for HAM-Lea) Liquid water coe cient
m2 /s
Dw
(×10−10 )
Vapour permeability of
kg/(s.m.Pa)
δcellulose
(×10−10 )
cellulose
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