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Consideration of permeable and semi-transparent

shading devices into the modeling of the exchanges

between the building and its environment: proposition of

a simple and flexible model

Antoine Duguéa, Denis Bruneaub, Alain Sommierb

aNobatek, bINSTITUT de MECANIQUE et d’INGENIERIE (I2M),

Esplanade des Arts et Métiers33405 TALENCE CEDEX

adugue@nobatek.com

Abstract

This investigation aims at providing an efficient description of solar shading devices

behaviour which can be integrated into modelling platforms currently on the market.

A 1D model is developed by considering precisely the radiative exchanges in short

and long wavelengths, and integrating ascending laminar flow which takes place

between the shading device and the wall (structural skin). The model describes the

heat and mass flows through the solar textile screen by considering it permeable and

semi-transparent.

Two different types of correlation which explicit the characteristics of the mass flow in

the cavity are presented. The first one is based on the driving force as a difference of

pressure between the air in the cavity and the air outside. The second one relies on

experimental and numerical studies that propose correlations between the Nusselt,

Rayleigh and Reynolds numbers. Their pertinences are discussed based on the first

measurements realized on the experimental protocol which has been built and

instrumented.

Study is still currently underway to refine the chosen correlations and to validate the

model for natural convection with a permeable shading device.

Keywords: shading devices, natural convection, heat transfer coefficient, air flow

Nomenclature

Transmission coefficient

h

Convective heat exchange coefficient

Reflexion coefficient

v

Air speed

Absorption coefficient

Mass air flow

Heat capacity

Wind speed

Sh

Relative to the shading device

Reynolds number

w

Relative to the wall

Rayleigh number

Solar incident radiation flux

Nusselt number

1. Introduction

The increasing development of highly glazed buildings subject 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.

The study presents a model of heat transfers which are observable at the scale of a

wall. A system composed of a shading device placed in front of a wall is instrumented

by flux and temperature sensors. Part of the measurements is used for model settings

definition and others for model validation. We propose here an analysis of different

flow patterns and an interpretation of preliminary results which justifies the choice of

models and their settings.

2. Experimental protocol

In order to quantify the heat flows which take place in the studied configuration, an

experimental platform has been built and instrumented. 5cm thick glass wool was

fixed on an existing wall with two 13mm thick plaster boards; the studied shading

device is placed in front of it, delimiting a ventilated cavity.

The instrumentation is made of 18 thermocouples placed at three different heights and

at different depths, a pyranometer operating on the spectral range 400 to 1100nm

measuring the incident solar flux on the wall, a cup anemometer for the external wind

speed and a hot-wire anemometer placed in the middle of the air gap at a median

height (see scheme).

Measurements were carried out in June for two opaque wooden claddings (see photo)

with a 6cm air gap, and others in the month of September 2011 with solar screens.

Figure 1: Photograph of the

instrumented wall

Figure 2: Representation of sensors location

3. Description of the model

The wall is discretized into a set of volumes for which the flux balance is written in a

dynamic regime. The temperature in the cavity is considered uniform temperature and

corresponds to an average temperature.

3.1 General model

3.1.1 Radiative fluxes

The short wave and long wave radiative fluxes are differentiated, and we consider the

multi-reflexion phenomenon as presented by Rodriguez [4].

Figure 3: Scheme illustrating the infinite reflexion phenomenon

The total short wave flux absorbed by the wall can be written in a fully developed form:

(E1)

Recognizing a geometric series, we can write the total:

(E 2)

3.1.2 Convective fluxes

For the external convection heat transfer coefficient, we consider a “medium smooth”

surface according to Ashrae 1989 manual to write:

(E3)

And the heat loss in the cavity due to the advection can be written as:

(E4)

3.1.3 Resolution

Figure 4: Thermal network for the external part of the studied experimental platform

The heat balance on every node gives a linear differential equation (E5):

(E5)

(E5) can be written in a matrix form distinguishing internal and external loads,

respectively the matrices F and G, and the vector of temperatures in the wall T and U

which consists of the incident solar flux and the outside temperature.

(E6)

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.

3.2 Air flow model, determination of the air speed and the heat transfer

coefficient

Different models of natural convection between two plates exist - mostly established

for a steady state- which fall into two types. The first comes from the writing of the

driving pressure gradient in the air gap as the difference between the temperature in

the cavity and the outside temperature. It is found in the ASHRAE Handbook and

European standards EN13363. Ong [2] provides the following form for a solar

chimney:

(E7)

The latter type considers 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 [1]

wrote in 1972 the correlations between the numbers of Nusselt, Rayleigh and flow. In

2011 Gan [3] sets those more generally for a set of configurations in the form of two

relations to explicit the heat transfer coefficient and the mass flow in the cavity.

(E8)

4. First results

To study the respective relevance of these two models, we draw the driving terms

depending on air speed in the cavity: the one integrating the temperature difference

and the term including the solar radiative flux.

Figure 5 shows that the distribution of pairs corresponding to the Aung’s model [1] is

irregular while the identity function would be expected. On the opposite, the terms

which correspond to relationships based on heat flows from the walls to the air gap

reflect a strong correlation whose shape can be explained by the inertia of the layer of

wooden cladding.

We can conclude that the predictions of the second type correlation are more in

adequation with the measurements, and thus such type of correlation should be used

in the model.

Figure 5: Analysis of the correlations of two types of models calculating the air flow

speed.

5. Perspective

The model has been validated for a non-permeable shading device and considering a

forced air flow in the cavity. Current work aims at refining the existing correlations for a

better integration which takes into account the natural convection in the cavity.

Meanwhile, measurements are underway for solar screens, for which the mass flow

prediction must be adjusted taking into account the charge losses associated to the

permeability.

References

[1] Aung, W., LS Fletcher, and V. Sernas. “Developing laminar free convection

between vertical flat plates with asymmetric heating.” International Journal of

Heat and Mass Transfer 15, no. 11 (1972): 2293–2304.

[2] Ong, K. “A mathematical model of a solar chimney.” Renewable Energy 28, no.

7 (June 2003): 1047-1060.

[3] Gan, Guohui. “General expressions for the calculation of air flow and heat

transfer rates in tall ventilation cavities.” Building and Environment 46, no. 10

(April 2011): 2069-2080.

[4] Rodriguez, Julio (2006) “Déshydratation par effet de serre d'un produit emballé

dans un film polymère perméable aux molécules d'eau: approche

expérimentale et de modélisation ». Doctorat Génie energétique, Paristech >

ENSAM 2006ENAM0002.

[5] Cole, R. J., and N. S. Sturrock. 1977. The Convective Heat Exchange at the

External Surface of Buildings. Building and Environment, Vol. 12, p. 207.