Experimental and numerical determination of convective
vapour transfer coefficients.
Citation for published version (APA):
Neale, A., Derome, D., Blocken, B. J. E., & Carmeliet, J. E. (2008). Experimental and numerical determination of convective vapour transfer coefficients. In C. Rode (Ed.), Proceedings of the 8th Symposium of Building Physics in the Nordic Countries, 16-18 June 2008, Copenhagen, Denmark (Vol. 2, pp. 707-714). (DTU Byg Report; Vol. R-189). Technical University of Denmark.
Document status and date: Published: 01/01/2008
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Experimental and numerical determination of convective vapour
transfer coefficients
Adam Neale, M.A.Sc.,
Department of Building, Civil and Environmental Engineering, Concordia University, Canada; aj_neale@encs.concordia.ca
Dominique Derome, Ph.D.,
Empa, Swiss Federal Laboratories for Materials Testing and Research, Wood Laboratory, Dübendorf
Bert Blocken, Ph.D.,
Building Physics and Systems, Technische Universiteit Eindhoven, The Netherlands; b.j.e.blocken@tue.nl
Jan Carmeliet, Ph.D.,
Chair of Building Physics, Swiss Federal Institute of Technology ETHZ, Zürich
Empa, Swiss Federal Laboratories for Materials Testing and Research, Laboratory for Building Technologies, Dübendorf
KEYWORDS: CFD, vapour transfer, laminar flow, forced convection. SUMMARY:
An experimental setup was designed for the purpose of validating a coupled diffusion model. In the CFD-diffusion model, heat and mass transport in the air domain is solved using CFD, while, in the material domain, vapour transport is modelled using a control-volume vapour diffusion model. This CFD-diffusion model allows the prediction of the convective vapour transfer coefficient for developing momentum and moisture boundary layers. In the traditional vapour diffusion model, a convective vapour transfer coefficient is used. This model allows the indirect determination of the vapour transfer coefficient from experiments. The experiment consisted of a wind tunnel placed in an environmental chamber with climate control capability. The air flow in the wind tunnel was driven by a variable-control fan that allows for a range of speeds in the laminar regime. Convective vapour transfer coefficients were indirectly determined based on the experimenta measured moisture content changes in the material for a number of air speeds. The results were compared with CFD-diffusion model for the purpose of validation and sensitivity analysis.
1. Introduction
Convective heat and vapour transfer coefficients, also called surface transfer coefficients, are generally applied as boundary conditions for hygrothermal calculations in building applications. Surface coefficients are theoretically dependent on velocity and type of the air flow, surface temperature, reference temperature of the air, surface relative humidity, reference relative humidity of the air and porosity at the surface of the material. Since the boundary layers for heat and moisture content are similar for specific conditions, it is often assumed that there exists an equivalence relationship between the heat and vapour transfer coefficients, such as the one proposed by Chilton and Colburn (1934). However, studies have shown discrepancies between equivalence equations and measurements, which suggest the need for further study on the determination of convective vapour transfer coefficients. Authors such as Masmoudi and Prat (1990), Wadsö (1993), Derome (1999), Hukka and Oksanen (1999), and Salin (2003) have reported errors as high as 300% between experimentally determined vapour transfer coefficients and values obtained using analogy equations.
In this paper, an investigation on numerical and indirect experimental determination of convective vapour transfer coefficients is presented. Convective vapour transfer for laminar air flow over gypsum samples is studied for a number of air speeds. The corresponding convective vapour transfer coefficients are compared for experimental and numerical cases. The models are presented in Section 2. The material property data used in the numerical simulations are then presented in Section 3. Next, Section 4 presents the experimental setup, the methodology used to determine indirectly the convective vapour transfer coefficients, and the comparison
between the indirect results and CFD-diffusion simulations. Finally, some general conclusions are presented in Section 5.
2. Numerical models and preliminary model validation
The convective vapour transfer between air and a porous material can be defined by the following relationship:
(
vs vf)
m y v p p h y p g = − ∂ ∂ − = =0δ
(1)where g is the mass flux per unit area (kg/m2s), δ is the vapour permeability of the material (s), (∂pv/∂y) is the
vapour pressure gradient in the direction normal to the surface (Pa/m), pvs and pvf are, respectively, the partial
vapour pressure at the surface and the fluid reference vapour pressure (Pa), and hm is the convective vapour
transfer coefficient, which in this case is derived with vapour pressure as the driving potential (s/m). Note that the convective vapour transfer coefficient, sometimes referred to as β, will henceforth be referred to as hm.
In order to model the vapour transfer between a moving air layer and a porous material, a number of numerical methods exist. Two models are proposed to simulate the moisture uptake of gypsum samples: a vapour diffusion model with an imposed convective vapour transfer coefficient as a boundary condition, and a coupled CFD-vapour diffusion model that resolves the moisture transfer in the boundary layer.
2.1 Vapour diffusion model
The diffusive vapour flux within a solid material was expressed in part earlier in equation 1. When considering diffusion in a porous material, the storage of moisture must be considered. If one considers vapour transport in one direction (x), the resulting equation can be expressed as:
∂ ∂ − ∂ ∂ = ∂ ∂ x p A x t w A δ v (2)
where w is the moisture content (kgmoisture/m3), δ is the vapour permeability of the material (s), A is the area
perpendicular to the vapour flow, pv is the partial vapour pressure (Pa), and x is the direction of the vapour flow
(m). The gradient ∂w ∂t represents the vapour storage within the material. Material properties are often expressed in terms of relative humidity, and therefore equation 2 can be transformed as:
∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ x T T p A x p A x t w A vsat vsat
δ
φ
φ
δ
φ
φ
(3)where
φ
is the relative humidity, pvsat is the saturation vapour pressure (Pa), and the gradient ∂w∂φ representsthe slope of the sorption isotherm for the material. Equation 3 together with the heat balance equation was subsequently implemented in Matlab using a control volume discretization scheme.
The primary purpose of the vapour diffusion model is to determine the change in mass of the material due to moisture adsorption or desorption in order to indirectly determine the vapour surface coefficient from
experimental findings.
2.2 Coupled model
Since the convective vapour transfer at the surface of a material is dependent on the boundary layer development, it is interesting to obtain a highly accurate resolution of the air flow field in order to study the effects of varying air speeds on the convective vapour transfer. In commercial CFD codes such as Fluent 6.3.26, a highly accurate solution of the momentum boundary layer can be obtained. In addition, an equally accurate vapour boundary layer can be calculated. However, in Fluent, it is difficult to solve for vapour transport in hygroscopic porous materials without significantly altering the program functionality through the use of user-defined functions. Therefore, in order to accurately solve the air flow domain and the material domain, CFD is coupled with the vapour diffusion model mentioned previously. Details on the coupling methodology can be found in Neale et al (2007). In brief, the commercial software Matlab is used as a controller to iterate between the vapour diffusion model and the CFD solution. It is important to note that the coupled model does not make
use of any surface coefficients to obtain a solution, though the surface coefficients may be calculated from the solution data. In addition, while the cases presented in this paper are all isothermal, the coupled model can be used to solve non-isothermal problems.
3. Material property data
The average moisture content for the gypsum panels used in this study was measured at 30% and 80% relative humidity for 8 samples. The resulting sorption curve was interpolated using the function that resulted from a round-robin sorption curve determination for gypsum board (IEA 2008). It was assumed that the behaviour of the gypsum board used in the tests would be the same as reported in IEA (2008). However, additional testing showed that the gypsum board attains higher moisture. The difference is likely due to the different fabrication process and materials found in North American gypsum board panels. The different sorption curves are presented in Figure 1 (left), which show how the moisture content varies in the range of relative humidity of interest. Based on the two measured points the actual hygroscopic curve was corrected using a same functional description as in IEA (2008). Note that the gypsum board was assumed to be a homogeneous material for the purpose of this study, but in reality it is a composite material of paper and gypsum, with some coatings on the paper for fabrication purposes.
0 1 2 3 4 5 6 7 8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Relative humidity (-) M o is tu re c o n te n t (k gm o is tu re /m 3)
Average measured moisture content Interpolated sorption curve IEA Annex 41 gypsum data
0 5E-12 1E-11 1.5E-11 2E-11 2.5E-11 3E-11 3.5E-11 4E-11 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Relative humidity (-) P e rm e a b il it y ( s )
IEA Annex 41 permeability data
Figure 1. a) Sorption isotherm and b) permeability data for gypsum board.
The permeability of the gypsum board as a function of relative humidity was obtained from the Annex 41 measurements, which are shown in Figure 1 (right).
The properties of air were calculated for a mean ambient temperature of 20°C and implemented accordingly.
4. Experimental measurements
In order to describe the methodology used to determine the convective vapour transfer coefficients, the experimental setup is presented first. The diffusion model results are subsequently presented, which illustrate how the experimental measurements are used to indirectly determine hmvalues. Finally, the experimental data is
compared with the coupled model data and the results are discussed.
4.1 Experimental setup
The experimental setup was designed to measure the convective vapour transfer coefficients for laminar air flow over a given porous material. The material selected for the present study was gypsum boards cut in to 20 cm x 20 cm specimens. The experimental setup consisted of five main components shown in Figure 2: 1) a variable control humidifier, 2) a variable speed fan with a working range of 0.05 to 1 m/s, 3) the gypsum sample bed, 4) a PMMA open circuit wind tunnel with a relative humidity probe upstream and manual anemometer downstream, and 5) an adiabatic/impermeable sealed environmental chamber. The air tunnel was 0.025 m high
by 0.4 m by 2.8 m long. Due to sufficient length of the windtunnel before the test section, the small height of the windtunnel, the low air speeds varying between and the 0.1 and 0.35 m/s, the air flow is fully developed and laminar. Note that the set-up components and the specimens were all at the same temperature as the air at the start of the test.
Figure 2. Schematic of the experimental setup (not to scale).
The gypsum board samples were pre-conditioned at a relative humidity of 30%. The edges of the samples were sealed with wax to ensure one-dimensional moisture transport. While there was 0.8 m of gypsum panel in the experimental setup, only the first sample (0.2 m) was used for measurement purposes. The air inside the environmental chamber was maintained at 77.5% RH, with an accuracy of ±2.5%. Once the air inside the chamber reached the desired setpoint, the humidifier was shut off and the samples were placed in the test bed. The velocity of the air passing over the samples was maintained at the desired setpoint (0.1 to 0.35 m/s) with an accuracy of ±0.01 m/s. The samples remained in the tunnel for a period of 10 minutes, after which the change in mass of the gypsum board was recorded. Note that over the course of 10 minutes, the relative humidity in the chamber would only decrease between 3% and 5% from the starting value, which allows the assumption that the boundary conditions were constant during the test..
The convective vapour transfer process described in equation 1 can be rearranged to isolate the hm value:
(
vs vf)
m p p g h − = (4)The mass flux g is expressed in terms of the change in mass per unit time (kg/s per unit area), but the relationship between the uptake in moisture versus time is not linear. Consequently, the flux cannot be calculated by simply dividing the measured change in mass by the time period. In addition, the surface vapour pressure pvscannot be
directly measured, particularly for a material such as gypsum board. Therefore, the value of hm was determined
indirectly based on the experimental results using the vapour diffusion model.
4.2 Diffusion model numerical simulations
In this section, we explain the procedure to indirectly determine the vapour transfer coefficients from measurements. The diffusion model allows the determination of the accumulation of moisture in a porous material for different boundary conditions. At the surface of the gypsum board, a convective vapour transfer boundary condition was implemented in the diffusion model for a number of different convective surface coefficients (hm). The reference relative humidity was set to be equal to 77.5% RH, which is the average of the
relative humidity in the chamber for the different experiments. The gypsum samples were initialized to 30% relative humidity, which is equal to the laboratory RH. The material properties of the gypsum panels were implemented as described in Section 2. The values of hm were varied from 0.5x10-8 s/m to 10x10-8 s/m and the
resulting accumulation of moisture in the gypsum was determined for each case by numerical simulation. The results are shown in Figure 3, which are ten curves showing the change in mass of the gypsum board samples over time.
Fan
Gypsum board specimen
Impermeable wind tunnel
0.8 m
2.0 m
Humidifier
Adiabatic/impermeable chamber
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Time (min) M o is tu re a c c u m u la ti o n ( g ) hm = 0.5 E-8 s/m hm = 10 E-8 s/m
Figure 3. Diffusion model results – moisture accumulation for varying values of hm over time.
The data points (●) in Figure 3 indicate the change in mass of the gypsum panel for various values of hm after 10
minutes. When the change in mass is graphed versus hm, the result is a curve that allows the prediction of the hm
value for a given change in mass of a sample loaded under specific conditions, which is shown in Figure 4.
0 0.05 0.1 0.15 0.2 0.25 0.3
0.E+00 1.E-08 2.E-08 3.E-08 4.E-08 5.E-08 6.E-08 7.E-08 8.E-08 9.E-08 1.E-07
Vapour transfer coefficient (s/m)
M o is tu re a c c u m u la ti o n ( g )
Figure 4. Diffusion model results - moisture accumulation vs vapour transfer coefficient after 10 minutes.
Using the relationship shown in Figure 4, the experimentally measured change in mass of the samples can then be associated with a convective vapour transfer coefficient.
4.3 Results
Eight different gypsum samples were used in the experimental setup described previously. The samples were all obtained from the same gypsum panel and cut to be approximately the same dimension. The surface
area for each sample was measured and taken into consideration. Experiments were performed for three different air speeds: 0.1 m/s, 0.2 m/s and 0.35 m/s. The change in mass of the gypsum specimens was measured after 10 minutes, which was converted into an hm value using the relationship shown in Figure 4. The estimated
error for the velocity and mass measurements was ±0.01 m/s and ±0.01 g, respectively.
The coupled model was used to simulate the experimental setup for a range of air speeds from 0.05 m/s to 0.5 m/s. The computational domain of the coupled simulation matched the experimental conditions in most aspects, except that the length of the gypsum board simulated was longer: 0.5 m instead of 0.2 m. The extra length did not affect the simulation results in any way, but provided extra information as will be shown below. The material properties for gypsum and air were as described in Section 2. A fully developed laminar air velocity profile was imposed at the inlet of the computational domain. The water vapour concentration profile was expected to evolve to a fully developed boundary layer along the length of the gypsum board. For each case of velocity, the average values of hm were calculated above each grid cell in the computational domain.
The values of hm as obtained indirectly from the diffusion model (dots) are compared to the coupled
model simulations in Figure 5. The error bars for the x- and y-directions were calculated based on the estimated error for the air speed and mass measurements, respectively. Note that for an error of ± 5-7% in the mass measurement there was a corresponding error of ±12-15% in the value of hm, which illustrates the sensitivity of
the results on the mass measurements. The results from the coupled model simulation were analyzed at two locations along the length of the panel: hm value was averaged for the first 0.2 m of the simulation domain
(denoted as developing ) and also averaged from 0.3 m to 0.5 m ( denoted as developed region). Note that the notations developing and developed are adopted as a naming convention, and further analysis has to confirm these notations. 0.0E+00 1.0E-08 2.0E-08 3.0E-08 4.0E-08 5.0E-08 6.0E-08 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Air speed (m/s) C o n vect ive vap o u r tr an sf er co e ff ici en t (s/ m
) Coupled model - developing region
Coupled model - developed region Experimental + diffusion model
Figure 5. Convective vapour transfer coefficient results for various air speeds resulting from the two calculation approaches.
The coupled model results for the region where the mass concentration profile was expected to be developing tend to overestimate the convective vapour transfer coefficient when compared with the
experimentally-based results. The ‘developed’ results show a closer agreement with the experimentally-based results, which indicates that the mass concentration profile is becoming developed faster than expected. One
explanation of the faster response could be due to the fact that the layered composition of the gypsum board consists at the surface of a very hygroscopic paper layer with limited thickness, while on the other hand gypsum is less hygroscopic but very vapour permeable. This means that in reality, the paper will quickly stabilize the boundary layer, which would have the effect of creating a more uniform surface condition. For this reason, after a certain amount of time elapses, the case of moisture uptake for gypsum panels behaves closer to a developed moisture boundary layer, which is illustrated by the ‘developed region’ results shown in Figure 5. Further work is on-going to analyse the influence of the composite structure of gypsum board and to further validate the model for other materials.
5. Conclusions
An experimental setup was designed and the determination of convective vapour transfer coefficients (hm)
was performed for laminar air flow at around 80%RH over gypsum panel samples that had been in equilibrium with 30%RH. The moisture content of the gypsum samples was determined experimentally for two relative humidities, which were used to adapt the Annex 41 round-robin gypsum panel sorption curve data for the samples used in this study. The gypsum properties were assumed to be homogeneous across the panel thickness.
A vapour diffusion model was used to determine the predicted change in mass for gypsum panels exposed to convective vapour transfer for a period of 10 minutes. The values of hm were varied from 0.5 x10-8 to 10 x10-8
s/m and imposed as a surface boundary condition. The change in mass of the samples was calculated for each case and a function describing the relationship between the change in mass vs hm was then established for then
indirectly determining the vapour transfer coefficient from experimental data.
Experimental measurements were performed for eight gypsum panel samples exposed to laminar air flow for forced convection vapour transport. The change in mass of the gypsum samples was recorded after 10 minutes for three different air speeds. The corresponding hm values were indirectly determined based on the
change in mass of the samples using the vapour diffusion model. The estimated error for the hm values was
between 12 and 15% for the different samples. The experimentally determined hm results were compared with
values calculated using a coupled CFD. The coupled model tends to over-predict the convective vapour transfer coefficient if the moisture boundary layer is assumed to be developing. If the moisture boundary layer is assumed to be developed after 10 minutes, the coupled model predicts the hm values within the experimental
uncertainty. It is suspected that the influence of the paper layer in the experiment causes the surface relative humidity to be more uniform, which explains the good agreement between the developed case and the measurements. In future work it is planned to test the effect of modeling the gypsum panels as a composite material with paper and gypsum layers, instead of modeling gypsum panels as a homogeneous material.
6. References
Chilton T.H., Colburn A.P. (1934). Mass transfer (absorption) coefficients. Industrial and engineering chemistry 26, 1183-1187.
Derome D. (1999). Moisture occurrence in roof assemblies containing moisture storing insulation and its impact on the durability of the building envelope. Ph.D. Thesis, Montreal: Concordia University.
Hukka A., Oksanen O. (1999). Convective mass transfer coefficient at wooden surface in jet drying of veneer. Holzforschung 53, 204-208.
Masmoudi W., Prat M. (1991). Heat and mass transfer between a porous medium and a parallel external flow. Application to drying of capillary porous materials. Int. J. Heat Mass Transfer 34, 1975-1989.
Neale A., Derome D., Blocken B., Carmeliet J. (2007). Coupled simulation of vapour flow between air and a porous material. Proceedings of Performance of Exterior Envelopes of Whole Buildings X Conference, ASHRAE, Atlanta, 11 pages.
Roels S. (2008) Subtask 2 Report, IEA Annex 41: Whole building heat, air and moisture response, to be
published.
Salin J-G. (2003). External heat and mass transfer – some remarks. 8th Int. IUFRO Wood Drying Conference, 343-348.
