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Thermal analysis of architectural glazing in uneven conditions based on Biot’s variational principle: Part II—validation and case-studies

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Abstract

The thermal state in architectural glazed panes strongly depends on the properties of the components, on inclination, on solar radiation and, most of all, on projected shadows on the surface. The unevenness of temperature distribution in the pane, often referred to as “thermal shock”, can produce stresses leading to breakage. Determining the temperature field is therefore of primary interest in the design. In Part I of this work, a semianalytical FEM formulation, based on Biot’s variational principle for heat transfer (hence, called in short BVM-FEM) has been presented. Here, the BVM-FEM formulation is first validated by means of comparison with the direct solution of the differential form of the thermal problem, analyzed under either fixed or variable outdoor conditions. Successively, it is used to calculate the temperature field in the paradigmatic examples of rectangular monolithic glass panes, with cast shadows and contour frames. The worked case-studies emphasize the importance of heat exchange by conduction between regions receiving a different amount of solar radiation, which is found to be localized in narrow strips across the interface. The code recognizes the temperature and heat-flux peaks occurring at the corners of differently irradiated regions, which pose the greatest risk to glass integrity.

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Notes

  1. The Matlab tool does not allow to define the nodal locations and the elements shape, but only to define a maximum and minimum value for the element size. This represents a limitation, because the body dimensions and the boundary conditions are strongly different in the in-plane and through-the-thickness directions.

  2. As previously stated, the definition of the heat displacement carries an indeterminacy, since only the heat flux, related to its time derivative, defines the thermal state.

  3. For a precise evaluation of the temperature in the thickness, the number of elements in this direction should be high. However, since the Matlab tool prescribes the same maximum/minimum element size in all directions, also the number of elements in the y direction should be increased. By halving the element size, the computational time for the PDEs approach increases of more than \(20\%\) in the 2D case.

  4. The very first values of \(\dot{{\overline{H}}}_z\) obtained with the Matlab-PDE tool are outside the plot range.

  5. This confirms another hypothesis at the base of the engineered model described in Galuppi et al. (2021).

Abbreviations

a, b :

length and width of the glass pane [mm]

\(c_p\) :

specific heat [J/(Kg K)]

\(h_{eC}, h_{iC}\) :

external and internal convective heat transfer coefficients [W/(\(\hbox {m}^2\) K)]

\(h_{eR}, h_{iR}\) :

external and internal radiative heat transfer coefficients [W/(\(\hbox {m}^2\) K)]

\(h_{e}, h_{i}\) :

external and internal total heat transfer coefficients [W/(\(\hbox {m}^2\) K)]

s :

thickness of the glass pane [mm]

t :

time [s]

x, y, z :

coordinates [mm]

G :

solar radiation [\(\hbox {W/m}^2\)]

\(\overline{H}_{x},\overline{H}_{y},\overline{H}_{z}\) :

components of the heat displacement field [\(\hbox {J/m}^2\)]

T :

absolute temperature [K]

\(T_{int}\) :

temperature of the internal environment [K]

\(T_{ext}\) :

temperature of the external environment [K]

\(T_{sky}\) :

sky temperature [K]

\(\tilde{T}_{ext}\) :

fictitious temperature [K]

\(\alpha \) :

material absorptivity \([-]\)

\(\varepsilon \) :

material emissivity \([-]\)

\(\theta \) :

shading coefficient \([-]\)

\(\lambda \) :

thermal conductivity [W/(m K)]

\(\rho \) :

mass per unit volume [\(\hbox {Kg/m}^3\)]

\(\sigma \) :

Stefan-Boltzmann constant [W/(\(\hbox {m}^2\) \(\hbox {K}^4\))]

\(\textbf{b}\) :

global driving force vector [\(\hbox {m}^2\) K]

\(\textbf{C}_{tot}\) :

global “damping” matrix [\(\hbox {m}^4\) K/W]

\(\textbf{F}\) :

vector of the generalized coordinates [\(\hbox {J/m}^2\)]

\(\overline{\textbf{H}}^{(i)}\) :

heat displacement field in the i-th element [\(\hbox {J/m}^2\)]

\(\textbf{K}_{tot}\) :

global “stiffness” matrix [\(\hbox {m}^4\) K/J]

\(\varvec{\Phi }^{(i)}(\varvec{\xi })\) :

global interpolation matrix for the heat displacement field \([-]\)

\(\varvec{\Psi }^{(i)}(\varvec{\xi })\) :

global interpolation vector for the temperature field \([-]\)

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Acknowledgements

This research was internally supported by Maffeis Engineering SpA, Solagna (Vi), Italy, in collaboration with the University of Parma, Italy.

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Authors and Affiliations

  1. Department of Engineering and Architecture, University of Parma, Parco Area delle Scienze 181/A, 43124, Parma, Italy

    Laura Galuppi & Gianni Royer-Carfagni

  2. Construction Technologies Institute - Italian National Research Council (ITC-CNR), Via Lombardia 49, 20098, San Giuliano Milanese, Milan, Italy

    Gianni Royer-Carfagni

Authors
  1. Laura Galuppi
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Correspondence to Laura Galuppi.

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Appendix: A Detailed comparison of heat flux and temperature field for the benchmark example

Appendix: A Detailed comparison of heat flux and temperature field for the benchmark example

With reference to the benchmark example of Sect. 3.1, Fig. 33 shows the temperature profile in the y direction at different times for the same constant environmental conditions of Sect. 3.1, evaluated with either the Matlab-PDE or BVM-FEM approaches. The graphs on the r.h.s. show a magnification in the neighbourhood of the interface between the fully irradiated region A and the shaded region B. Figure 33a and 33b refer to the external (\(z=0\)) and internal (\(z=s\)) surfaces of the glass pane, respectively.

Fig. 33
figure 33

Spatial temperature distribution (in the y direction) at different times, for a the external and b the internal surfaces, with a magnification in neighbourhood of the interface. Comparison between the Matlab-PDE and the BVM-FEM solutions

Full size image

s Figure 34a and 34b represent the trend along y of the through-the-thickness heat flux \(\dot{{\overline{H}}}_z\), respectively evaluated at the external and internal surfaces, with a magnification across the interface between the fully irradiated region A and the shaded region B.

Fig. 34
figure 34

Spatial distribution (in the y direction) of the heat flux \(\dot{{\overline{H}}}_z\) at different times, for a the external and b the internal surfaces, with a magnification in neighbourhood of the interface. Comparison between the Matlab-PDE and the BVM-FEM solutions

Full size image

The comparisons confirm the excellent agreement between the proposed BVM-FEM method and the numerical solution of the thermal problem via PDEs.

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Galuppi, L., Royer-Carfagni, G. Thermal analysis of architectural glazing in uneven conditions based on Biot’s variational principle: Part II—validation and case-studies. Glass Struct Eng 8, 57–80 (2023). https://doi.org/10.1007/s40940-023-00217-0

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