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Abstract and Figures

This study aims at providing an accurate description of the thermal behaviour of solar shading devices in order to develop a simple modeling which can be integrated in a thermal simulation platform. A 1D model is developed by considering precisely the radiative exchanges in short and long wavelengths, and by integrating ascending laminar flow which takes place between the shading device and the wall (structural skin). The accuracy of this model is confronted with the results of measurements for a textile screen and a simple model based on a parameterization is proposed and discussed.
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Antoine Dugué is a research fellow in Nobatek-INEF4. Alain Sommier is a research engineer at the I2M laboratory, Denis Bruneau is
Senior Researcher at ParisTech -Institut de Mécanique et d'Ingénierie (I2M). Philippe Lagière is the scientific director of Nobatek-INEF4.
Characterization and valorization of
shading devices: proposition of a simple
and flexible model
DUGUÉ A1
SOMMIER A2
BRUNEAU D2
LAGIÈRE P12
1Nobatek-INEF4, 67, rue de Mirambeau 64600 ANGLET
2 Arts et Métiers ParisTech, I2M, UMR 5295. Univ. Bordeaux, I2M, UMR 5295. CNRS, I2M, UMR 5295.
F-33400 Talence, France.
Email of corresponding author: adugue@nobatek.com
ABSTRACT
This study aims at providing an accurate description of the thermal behaviour of solar shading devices
in order to develop a simple modeling which can be integrated in a thermal simulation platform. A 1D
model is developed by considering precisely the radiative exchanges in short and long wavelengths, and
by integrating ascending laminar flow which takes place between the shading device and the wall
(structural skin). The accuracy of this model is confronted with the results of measurements for a textile
screen and a simple model based on a parameterization is proposed and discussed.
INTRODUCTION
The increasing development of highly glazed buildings exposed to high solar gain induces a greater
use of solar protection solutions such as textile screens or outside climbing plants which allow passive
solar gain in winter. However current thermal simulation platforms do not accurately integrate those
architectural solutions, especially the induced air flow between the wall and the shading device. This low
consideration of those systems implies a weak valorization of their impacts on the building. The study
presents a model of heat transfers which are observable at the scale of a wall. Then a simple model based
on the use of the total solar energy transmittance of the shading device is proposed and discussed. Such a
model allows an easy and quick way to integrate those systems in a thermal study of a building.
Furthermore the use of a total solar energy transmittance to characterize those shading devices would
help the architect in the design phase, the engineer in order to choose the best solution to fit its criteria
and the industries which develop those systems.
EXPERIMENTAL PROTOCOL
In order to quantify the heat flows which take place in the studied configuration, an experimental
platform has been built and instrumented (see figure 1) in Bordeaux (Atlantic climate). As represented
on figure 2, 5 cm thick glass wool (B) was fixed on an existing wall (A) with two 13 mm thick plaster
boards (C); the studied shading device (E) is placed in front of it, delimiting a small ventilated cavity
(D). The width of this cavity can be set to 33 mm, 66 mm or 100 mm. The whole system is called ESAW
(Exterior Shading Device Air gap Wall).
The instrumentation (figure 3) is made of 18 thermocouples placed at three different heights and at
different depths, a pyranometer operating on the spectral range 400 to 1100 nm measuring the incident
solar flux on the wall, a cup anemometer for the external wind speed and three hot-wire anemometers
located in the middle of the air gap at different height. Two shading devices can be monitorized at the
same time with an air gap between them that act as a thermal barrier.
Figure 1: Photograph of
the instrumented wall for
the two wooden claddings
Figure 2:
Representation of the
layers of materials of
the ESAW system
Figure 3: Representation of sensors location.
Thermocouple. Hot wire anemometer.
Pyranometer and anemometer
Measurements were carried out in June 2012 for two opaque wooden claddings (a light one made
of hardwood and a red one made of chipboard), one grey textile screen and one bright expanded metal. A
24 hour span is presented in the figures 4 and 5.
We can notice that maximum air speed in the cavity is 1 m/s and that this velocity is always above
0,1 m/s which traduces the permanent movement in the cavity. Also, the measured velocity at the bottom
varies more than the ones measured at medium and high height, this can be explained by turbulences at
the bottom of the cavity while the air flow is laminar in the upper part.
Figure 4: Incident normal radiative flux (W/m²)
Figure 5: Vertical air velocity inside the cavity in
the case of a wooden cladding.
DESCRIPTION OF THE MODEL
The wall is discretized into a set of volumes for which the flux balance is written in a dynamic
regime (see figure 6). The temperature in the cavity is considered uniform and corresponds to an average
temperature. Modelling is one-dimensional (horizontal direction) and vertical gradients outside the
ventilated air gap are neglected. The general model describes radiative, conductive, and convective heat
fluxes.
Figure 6: Thermal network for the external part (left) and for the air cavity (right) using the
rheoelectrical analogy.
General model
The short and long wave radiative fluxes are differentiated, and we consider the multi-reflexion
phenomenon as presented by Rodriguez et al (2007).
For the external convective heat transfer coefficient, we use McAdams (1994) expression:
      (1)
The sky temperature is given by Duffie and Beckman (1974). And the heat loss in the cavity due to
the advection can be written such as:
  - (2)
Air flow model, determination of the air velocity and the heat transfer coefficient
Types of models. Different models of natural convection between two plates exist - mostly
established for a steady state- which fall into two types. First are numerical approaches that consider the
flow (possibly asymmetric) in the walls adjacent to the cavity to establish correlations between Nusselt
and Rayleigh numbers. These numerical models were first established experimentally and next using
CFD (Aung et al, 1972). In 2011, Gan (2011) sets those correlations more generally for a bigger set of
configurations. Although those are limited to situations for which heat fluxes given to the air are
between 100 and 1000W/m² and an aspect ratio between 5 and 60. Others approaches are analytical.
Considering the chimney effect, the supposed linear pressure profiles outside and inside the cavity can
be written and used to determine the air flow. Bansal (1993) and Ong (2003) established such
correlations for solar chimneys. The air flow can also be determined by calculating the pressure losses
along the channel, especially frictions one. This model by its flexibility and the quality of the predictions
it gave, has been selected and will be used in the following.
Determination of the air velocity. In the air cavity in a static regime the driving pressure
difference between the air in the cavity and the exterior air balances the resistance to the air flow along
the cavity. Considering the air as a perfect gas we can write the driving pressure with air temperatures as
made in the norm EN 13363. The resistance to the air pressure is the sum of the pressure losses at the
openings (bend and reduction at the bottom and enlargement at the top of the cavity) and along the
channel (linear pressure loss by friction), that are here determined using Idel’chik (1994).
Influence of the exterior wind. Measured data shows an influence of the wind velocity on the
vertical air speed in the cavity as already showed by Mayer and Künzel (1983). Indeed the wind can
affect the air pressure at the outside of the system, and this variation can be taken into account in the
model by adding a term in the driving pressure difference as done by Falk (2013). One limit of our study
is that the direction of the wind was not measured, so this coefficient is a constant while it should depend
on the wind direction.
Convective heat transfer. Once the air flow is calculated, the convective heat transfer coefficient
is determined using a correlation between Nusselt and Grashof numbers established for natural
convection within the cavity. Here we use Fishenden and Saunders (1950):
   
(3)
Evacuated heat flux. In order to determine the heat flux evacuated through the air cavity by
convection we use an experimental law determined by Hirunlabh (1999) linking the average temperature
to the air temperatures at the bottom and at the top of the air cavity.
Resolution
The heat balance on each node gives a linear differential equation (4):

    (4)
Previous set of equations can be written in a matrix form to give equation (5) distinguishing internal and
external loads, respectively the matrices F and G, and the vector of temperatures in the wall T and U
which corresponds to the incident solar flux and the outside temperature.

   (5)
The numerical solution of this differential equation is obtained with the implicit Euler method with
a time step of 1 minute using Matlab, where matrix F must be recalculated at every step.
VALIDATION OF THE MODEL
In order to validate the proposed modelling we compare the results of the simulation against
measurements. Data to be compared are chosen to characterize the behaviour of the system: the heat
flow evacuated by convection and the heat flow transmitted through the wall. The first one can be
described by the air velocity in the ventilated cavity and the difference of temperature between the top
and the bottom of the cavity. The latter by the difference of temperature between the two sides of the
second plaster board.
We compare the measured data with results from the model in the case of a textile screen for an air
gap width of 100 mm during two sunny days. Air velocity in the air gap is plotted in figure 7, while the
gradient of air temperature in the air gap is in figure 8 and the difference of temperature between the two
sides of the second plaster board in figure 9.
Figure 7: Comparison between estimated and measured air velocity in the cavity
We can see on figure 7 the increase during the daytime of the air velocity inside the air cavity. It is
also notable that during the night the calculated air velocity can be negative which means that the sign of
the driving pressure can change and would correspond to an air flow from the top to the bottom. But the
measurements don’t indicate the direction of the air flow, so it could not have been confirmed and it
does not have a big impact on the overall behaviour of the system. During daytime the air velocity is
quite well estimated by the model.
Figure 8: Comparison between estimated and
measured gradient of air temperature in the air gap
Figure 9: Comparison between estimated and
measured difference of temperature between the
two sides of the second plaster board.
On figure 8 and 9, we can also see the great variation of temperatures during the daytime due to the
solar radiation. The estimations are quite good during the daytime. Still, we can see that the gradient of
air temperature is overestimated. It is due to the permeability of the textile screen as there can also be a
horizontal air flow that is not taken into account in our model
Results were even better for the two wooden claddings but are quite limited for the expanded
metals. The latter has a high reflectivity and a complex 3d shape that are at the limit of the hypothesis of
this model. For the three first shading devices both the transmitted heat flux within the wall and the
evacuated heat flux by convection are well estimated.
APPLICATION TO A SIMPLE MODEL
Simplified model with the use of the solar transmittance
Definition. The total solar energy transmittance (fs) of a system is defined as the proportion of
incoming solar energy transmitted to the thermal zone behind the system as described in the EN ISO
13363. It depends on the properties of the system but also on the outdoor conditions, indoor conditions,
and thermal properties of the elements. The EN ISO 13363 proposes a static calculation which is limited
in the case of complex shading devices.
We here consider the characterization proposed by Hellstrom et al (2007). For the total solar
transmittance (fs) of a shading device, the average values are obtained from:

 (6)
where Co and He are the cooling and heating demands for a period of the associated thermal zone
maintained at a uniform temperature, indices 1 and 0 indicate with and without solar irradiation and S*Is
is the total solar irradiation incident on the shading device during the same period.
In our case the whole system can be decomposed as a shading device with a ventilated air gap and a
wall. The total solar energy transmittance of the system can be written as:
  
  
(7)
Method of calculation. The method of calculation of the total thermal energy transmittance is as
follows. We consider a shading device for a given sunny period, the inside air temperature is fixed at
20°C and we carry four simulations, where heating and cooling loads are recorded:
1. Simulation of the ESAW system taking into account the solar radiation
2. Simulation of ESAW system without taking into account the solar radiation
3. Simulation of the Wall taking into account the solar radiation
4. Simulation of the Wall without taking into account the solar radiation.
The first two simulations are used to calculate the solar factor of the whole ESAW system, and the next
two for the solar transmittance of the wall. We then obtain the total solar energy transmittance of the
shading device using the equation (7).
Results. Total solar energy transmittance of the 4 considered shading devices that are made of
materials with low thermal mass have been calculated using this method for 3 different air cavity widths.
Results appear on figure 10. The two wooden claddings differ a lot. It is due to differences in their
properties. A lower coefficient of absorption and a lower conductivity for the light hardwood cladding
limits the solar transmittance compared to the red dense chipboard. We can also see the influence of the
air width. The wider the gap is, the lower the solar energy transmittance is as the heat flux evacuated by
convection is greater.
Figure 10: Results of the total solar energy transmittance for the 4 considered shading devices for
three different air gap widths.
Discussion. The intrisic nature of this defined total energy transmittance of a shading device has
been discussed by analyzing the sensibility of the obtained values to some parameters of the method of
calculation such as the insulation width, whether it is placed on the inner part of the concrete or the outer
part, the weather file and the fixed temperature of the thermal zone. Results show that the sensitivity is
very low. The total solar energy transmittance of a shading device can be useful for the characterization
of its impact on the thermal gains of a wall or glazing.
Application to sensibility analysis. In order to evaluate the influence of some design parameters
of a shading device on its performance it is possible to analyse the sensibility of the model to those
parameters.
Figure 11: Sensibility analysis of the solar factor of the wooden cladding varying thermal resistance and
air gap width (left) and thermal resistance and absorption coefficient (right)
Two examples are shown in figure 11 for a wooden cladding with in the first case the thermal
resistance and the cavity width as variables and in the second thermal resistance and air gap width. It is
then possible for example to evaluate the way a wooden cladding performance will age with the
deterioration of the color.
CONCLUSION
The model presented was aiming at easing the taking into account of a shading device in the
modelling of a building. Experiences have been carried out and the set of equations used to determine
the velocity in the cavity and the associated convection has been selected among different models. The
estimated results from the model are in good agreements with the measurements for the two wooden
claddings and the textile screen. But the expanded metal seems to be at the limit of the domain of
validity of the model.
Then a simple model based on the use of the total solar energy transmittance has been presented.
The shading device can then be replaced by this coefficient in a building simulation platform in order to
consider its impact on the building. Furthermore, this total solar energy transmittance allows the
characterisation of its impact and as so help architects and engineers in quickly assessing its impact
without using complicated models which can’t be used at the early phase of a the design of a building.
Finally the complete model can be used to make some optimization in order to produce a shading device
that will correspond to a set problem.
NOMENCLATURE
= absoption coefficient (-)
e = width cavity (m)
fs = Total solar energy transmittance (-)
Is = Total solar incident radiation (W/m²)
Gr = Grashof number (-)
Nu = Nusselt number (-)
R = Thermal resistance (m².K/W)
T = Temperature (K)
v = Air velocity (m/s)
Subscripts
b = bottom
m = medium
h = top
ext = exterior
cv = associated to convection
rad = associated to radiation
lame, L = air gap
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