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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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