Engineered calculation of the uneven in-plane temperatures in Insulating Glass Units for structural design

Abstract

Insulated Glazed Units (IGUs) are composite elements formed by two or more glass panes held together by structural edge seals, entrapping a gas for thermal and acoustic insulation. There is a wealth of evidence that they are prone to cracking due to an uneven temperature distribution within each glass pane, caused in particular by solar radiation, enhanced by the shielding of contour-frame and cast shadows. The accurate assessment of the temperature field in the glass is the input datum to calculate the thermal strain and the consequent state of stress. An engineered method for calculating the temperature field in the glass panes of multiple IGUs, of arbitrary composition, is developed here. This method considers the various sources of thermal exchange: conduction between the exposed, shadowed and border regions of each pane, convection and radiation with the surrounding environment and between the panes, heat storage. Proper coefficients are defined for multiple reflective phenomena of radiant energy between the various glass layers. The proposed model is applied to explanatory case studies, accounting for the daily variations of the external temperature and solar radiation.

Appendix. Evaluation of the radiant coefficients

As indicated in Sect. 2.2.3, when a multiple glazing unit is irradiated by the solar radiation as per (2.8), the energy absorbed by each panel depends upon the coefficients $\alpha ^*_k$, defined as the ratio between the energy absorbed by the $k-$th glass pane and the total energy. Furthermore, to evaluate the amount of energy emitted by the various panels that is absorbed by each panel, as per (2.11), what is needed is the value of the coefficients ${\overline{\varepsilon }}_{lk}$, defined as the part of energy emitted by the $l-$th glass element that is absorbed by the $k-$th element.

The amount of radiation reflected and transmitted by a multiple glazing, as well as the energy absorbed by each glass pane, can be found by considering the fluxes of radiant energy flowing between the $i-$th region of the $k-$th glass panel and the adjacent ones, as schematically shown in Fig. 22 where, for clarity, the indexes i are omitted. Let us denote with $\Phi _{k,k+1}$ the radiant heat flux transmitted by the $k-$th to the $(k+1)$-th glass element (i.e., the transmitted heat flux, from left to right, towards the internal environment), and by $\Phi _{k,k-1}$ that transmitted by the $k-$th to the $(k-1)$-th glass element (reflected flux, towards the external environment). The heat flux absorbed by the $k-$th element is denoted to as $\Phi _{k,k}$. Each element emits an energy proportional to the fourth power of its temperature, given by the Stefan-Boltzmann law (2.11) and compactly denoted to as ${\mathscr {E}}_k$ in the same Figure.

Fig. 22

Energy flux transmitted by the glass plies

When the solar radiation is considered, the heat fluxes are strongly related one another by the absorptivity, transmissivity and reflectivity coefficients $\alpha _k$, $\tau _k$ and $r_k$ of the glass panes¹⁰. For each glass panel, one can write the system of equations (Wright 1998; Hollands et al. 2001; Edwards 1977)

$$\begin{aligned} \left\{ \begin{array}{l} \Phi _{k,k+1}=\tau _k\Phi _{k-1,k}+r_{k}\Phi _{k+1,k}+{\mathscr {E}}_k\,,\\ \Phi _{k,k-1}=\tau _k\Phi _{k+1,k}+r_{k}\Phi _{k-1,k}+{\mathscr {E}}_k\,,\\ \Phi _{k,k}=\alpha _k(\Phi _{k+1,k}+\Phi _{k-1,k})\,.\\ \end{array}\right. \end{aligned}$$

(A.1)

On the other hand, to evaluate the flows due to the energy emitted by the glass panels (2.12), one has consider that the aborptivity of the $k-$th glass pael coincides with its emissivity $\varepsilon _k$, while the reflectivity is $1-\varepsilon _k$, being null the transmissivity. Hence, in this case system (A.1) reduces to

$$\begin{aligned} \left\{ \begin{array}{l} \Phi _{k,k+1}=(1-\varepsilon _k)\Phi _{k+1,k}+{\mathscr {E}}_k\,,\\ \Phi _{k,k-1}=(1-\varepsilon _k)\Phi _{k-1,k}+{\mathscr {E}}_k\,,\\ \Phi _{k,k}=\varepsilon _k(\Phi _{k+1,k}+\Phi _{k-1,k})\,.\\ \end{array}\right. \end{aligned}$$

(A.2)

This kind of problem is usually solved by means of various numerical methods, which can be based on the “net radiation” (Siegel 1973), recursive (Edwards 1977), matrix (Wright and Kotey 2006) and “ray tracings” (Wright 1998) approaches. Here, in order to evaluate $\alpha ^*_k$ and ${\overline{\varepsilon }}_{kl}$, the problem is split in two different parts, by considering separately the effects of the energies from the solar radiation and from the glass panel themselves.

1.1 Solar radiation

Consider the case where the front surface of the fenestration (panel $k=1$) is irradiated by the energy ${\mathscr {E}}$, representative of the contribution by the sun. The relevant heat fluxes can be evaluated by posing $\Phi _{0,1}={\mathscr {E}}$ and ${\mathscr {E}}_k=0$ in (A.1), and by solving this system of equations by assuming that the heat coming from the internal environment is null ($\Phi _{N+1,N}=0$).

Let us define the overall reflectivity coefficients of the multiple glazing $r^*$ as the ratio between the total reflected energy and the total energy, and analogously define the overall transmissivity coefficient $\tau ^*$. Furthermore, with reference to Fig. 22, denote as $\alpha ^*_k$ the part of the total energy absorbed by the $k-$th glass ply, $k=1 \dots N$, i.e.,

$$\begin{aligned} r^*= & {} \Phi _{1,0}/\Phi _{0,1}\,,\ \ \tau ^*=\Phi _{N,N+1}/\Phi _{0,1}\,,\ \ \nonumber \\ \alpha _k^*= & {} \Phi _{k,k}/\Phi _{0,1}\,. \end{aligned}$$

(A.3)

These coefficients can be analytically calculated by solving (A.1). Consider, first, the simplest case of a double glazing, with glass panels characterized by the same thermal properties ($\alpha _k=\alpha$, $\tau _k=\tau$ and $r_k=r$, for $k=1,2$). The relevant coefficients read

$$\begin{aligned} r^*= & {} r\left[ 1+\tau \sum _{n=0}^\infty r^{2n}\right] =r\left[ 1+\frac{\tau }{1-r^2}\right] \,, \ \ \ \nonumber \\ \tau ^*= & {} \tau ^2\sum _{n=0}^\infty r^{2n}=\frac{\tau ^2}{1-r^2}\,,\nonumber \\ \alpha ^*_1= & {} \alpha \left[ 1+\tau r\sum _{n=0}^\infty r^{2n}\right] =\alpha \left[ 1+\frac{\tau r}{1-r^2}\right] \,, \ \ \ \nonumber \\ \alpha ^*_2= & {} \tau \alpha \left[ \sum _{n=0}^\infty r^{2n}\right] =\frac{\tau \alpha }{1-r^2}\,. \end{aligned}$$

(A.4)

A graphical interpretation of these results is given in Fig. 23.

Fig. 23

Energy fluxes in a double glazing, with glass panels with the same thermal properties $\alpha$, $\tau$ and r

Figure 24 shows the coefficients $\alpha _1^*$ and $\alpha _2^*$, evaluated as per (A.4), plotted as a function of $\alpha$ and r. As expected, for given values of $\alpha$ and r, one finds that $\alpha _1^*>\alpha _2^*$: the energy absorbed by the front panel is higher than that absorbed by the back panel. Furthermore, while $\alpha _1^*$ monotonically increases as $\alpha$ increases, $\alpha _2^*$ shows a strong decrease for high values of $\alpha$ and r, due to the decrease of $\tau =1-\alpha -r$. Notice that, when $\tau =0$, no thermal energy is transmitted to the second panel, and $\alpha _2^*=0$.

Fig. 24

Coefficients $\alpha _1^*$ and $\alpha _2^*$, for double glazing with glass panel with the same thermal properties, as a function of $\alpha =\alpha _1=\alpha _2$ and $r=r_1=r_2$

For two glass elements with different thermal properties, it can be calculated that the overall coefficients take the form

$$\begin{aligned} r^*= & {} r_1+\frac{r_2\tau _1^2}{1-r_1r_2}\,,\ \ \ \tau ^*=\frac{\tau _1\tau _2}{1-r_1r_2}\,,\ \ \ \ \nonumber \\ \alpha ^*_1= & {} \alpha _1\left[ 1+\frac{\tau _1 r_2}{1-r_1r_2}\right] \,, \ \ \alpha ^*_2=\frac{\tau _1\alpha _2}{1-r_1r_2}\,. \end{aligned}$$

(A.5)

In case of three glass elements, all with the same thermal properties, one finds

$$\begin{aligned} \nonumber r^*= & {} r\left[ 1+\frac{\tau ^2(1+\tau ^2-r^2)}{(1+r \tau -r^2)(1-r \tau -r^2)}\right] \,,\ \ \ \\ \tau ^*= & {} \frac{\tau ^3}{(1+r \tau -r^2)(1-r \tau -r^2)}\,,\nonumber \\ \nonumber \alpha ^*_1= & {} \alpha \left[ 1+\frac{\tau r(1+\tau ^2-r^2)}{(1+r \tau -r^2)(1-r \tau -r^2)}\right] \,, \ \ \\ \alpha ^*_2= & {} \frac{\tau \alpha }{1-\tau r-r^2}\,,\nonumber \\ \alpha ^*_3= & {} \frac{\tau ^3\alpha }{(1+r \tau -r^2)(1-r \tau -r^2)}\,. \end{aligned}$$

(A.6)

Figure 25 shows the tridimensional plot of $\alpha _1^*$, $\alpha _2^*$ and $\alpha _3^*$, as given by (A.6).

Fig. 25

Coefficients $\alpha _1^*$, $\alpha _2^*$ and $\alpha _3^*$, for triple glazing with glass panel with the same thermal properties, as a function of $\alpha$ and r

It is clear that $\alpha _1^*>\alpha _2^*>\alpha _3^*$. Notice as well that $\alpha _2^*$ and $\alpha _3^*$ are strongly affected by the reflectivity and the transmissivity. In particular, when $\tau =1-\alpha -r=0$, no thermal energy is transmitted to the second and the third panels, and $\alpha _2^*=\alpha _3^*=0$.

Finally, for a unit composed of four glass panels, one has

$$\begin{aligned}&r^{*}=r\left[ 1+\frac{\tau ^2(1+\tau +\tau ^2-r^2)(1-\tau +\tau ^2-r^2)}{f(\tau ,r)}\right] \,\nonumber \\&\tau ^*=\frac{\tau ^4}{f(\tau ,r)}\,,\nonumber \\&\alpha ^{*}_{1} =\alpha \left[ 1+\frac{\tau r(1+\tau +\tau ^2-r^2)(1-\tau +\tau ^2-r^2)}{f(\tau ,r)}\right] \,\nonumber \\&\alpha ^*_2=\tau \alpha \ \frac{1-r^2+r(\tau -r)(1+\tau ^2-r^2)}{f(\tau ,r)}\,,\nonumber \\&\alpha ^{*}_{3} =\tau ^2\alpha \ \frac{1+r\tau -r^2}{f(\tau ,r)}\,, \ \ \alpha ^*_4= \frac{\tau ^3\alpha }{f(\tau ,r)}\,, \nonumber \\ \end{aligned}$$

(A.7)

where $f(\tau ,r)=(1+r-r^2+r\tau ^2-r^3)(1-r-r^2-r\tau ^2+r^3)$.

1.2 Energy emitted by the glass plies

Consider now the energy emitted by a single glass panel, that is proportional to the fourth power of its temperature according to (2.11). The energy absorbed by the different glass panes can be evaluated by assuming that the heat coming from both the external and the internal environment are null, i.e., by setting $\Phi _{0,1}=\Phi _{N+1,N}=0$ in (A.2). Solving (A.1), the energy absorbed by the $k-$th glass pane¹¹ can be found in the form

$$\begin{aligned} \Phi _{k,k}=\sum _{l=1}^N{\overline{\varepsilon }}_{lk} {\mathscr {E}}_l\,. \end{aligned}$$

(A.8)

Fig. 26

Coefficients ${\overline{\varepsilon }}_{lk}$ for IGU composed by glass panels with the same thermal properties as a function of $\varepsilon =\varepsilon _k$, for a double glazing and b triple glazing

As discussed in Sect. 2.2.3, due to reflection phenomena (Fig. 4), the $k-$th glass absorbs a part of the energy emitted by itself, denoted as ${\overline{\varepsilon }}_{kk}\ne 0$. Obviously, for a single glazing ($N=1$), one has that ${\overline{\varepsilon }}_{11}=0$, since in this case there are no reflection phenomena. For a double glazing ($N=2$), reflections play a decisive role, as schematically shown in Fig. 4. The relevant coefficients read

$$\begin{aligned} \nonumber&{\overline{\varepsilon }}_{11}=\frac{\varepsilon _1 (1-\varepsilon _2)}{\varepsilon _1+\varepsilon _2-\varepsilon _1\varepsilon _2}\,, \ \ \ \ {\overline{\varepsilon }}_{21}=\frac{\varepsilon _1 }{\varepsilon _1+\varepsilon _2-\varepsilon _1\varepsilon _2}\,, \\&{\overline{\varepsilon }}_{22}=\frac{\varepsilon _2 (1-\varepsilon _1)}{\varepsilon _1+\varepsilon _2-\varepsilon _1\varepsilon _2}\,, \ \ \ \ {\overline{\varepsilon }}_{12}=\frac{\varepsilon _2 }{\varepsilon _1+\varepsilon _2-\varepsilon _1\varepsilon _2}\,. \nonumber \\ \end{aligned}$$

(A.9)

For the case of a double glazing where glass panels have the same thermal properties ($\varepsilon _k=\varepsilon$ for $k=1,2$), Eq. (A.9) can be simplified in the form

$$\begin{aligned} {\overline{\varepsilon }}_{11}={\overline{\varepsilon }}_{22}=\frac{1-\varepsilon }{2-\varepsilon }\,, \ \ \ \ {\overline{\varepsilon }}_{12}={\overline{\varepsilon }}_{12}=\frac{1}{2-\varepsilon }\,. \end{aligned}$$

(A.10)

Figure 5a shows the coefficients ${\overline{\varepsilon }}_{11}={\overline{\varepsilon }}_{22}$ and ${\overline{\varepsilon }}_{12}={\overline{\varepsilon }}_{12}$, evaluated as per (A.10), plotted as a function of $\varepsilon$. In general, coefficients ${\overline{\varepsilon }}_{11}$ and ${\overline{\varepsilon }}_{22}$, accounting for the “self-reflection”, i.e., the phenomenon for which a glass panel absorb a part of the energy emitted by itself as shown in Fig. 4, are lower than those accounting for the energy exchange between adjacent panels. It may be also noticed that, when $\varepsilon \rightarrow 1$, the reflected part is negligible, and hence ${\overline{\varepsilon }}_{12}={\overline{\varepsilon }}_{12}\rightarrow 1$ and ${\overline{\varepsilon }}_{11}={\overline{\varepsilon }}_{22}\rightarrow 0$, i.e., all the energy emitted by panel 1 is absorbed by panel 2, and viceversa.

In case of three glass elements, all with the same thermal properties, one obtains

$$\begin{aligned} \nonumber&{\overline{\varepsilon }}_{11}={\overline{\varepsilon }}_{33}=\frac{1-\varepsilon }{2-\varepsilon }, {\overline{\varepsilon }}_{21}={\overline{\varepsilon }}_{12}={\overline{\varepsilon }}_{23}={\overline{\varepsilon }}_{32} =\frac{1}{2-\varepsilon }\,,\nonumber \\&{\overline{\varepsilon }}_{22}=2 \ \frac{1-\varepsilon }{2-\varepsilon }\,,\ \ {\overline{\varepsilon }}_{13}= {\overline{\varepsilon }}_{31}=0\,. \end{aligned}$$

(A.11)

Figure 5b is the counterpart of Fig. 5a for a triple glazing, showing ${\overline{\varepsilon }}_{lk}$, evaluated as per (A.11), as a function of $\varepsilon$. Observe that, due to the symmetry of the problem, the role of the external elements ($k=1$ and $k=3$) is the same: this provides ${\overline{\varepsilon }}_{k1}={\overline{\varepsilon }}_{k3}$ and ${\overline{\varepsilon }}_{1k}={\overline{\varepsilon }}_{3k}$. Furthermore, it may be verified that ${\overline{\varepsilon }}_{kl}={\overline{\varepsilon }}_{lk}$, $k,l=1 \dots N$: the heat interchange between two surfaces is symmetrical. These relations hold also for the case of a glazing fenestration composed by four glass panels, for which the relevant coefficients are

$$\begin{aligned} \nonumber&{\overline{\varepsilon }}_{11}={\overline{\varepsilon }}_{44}=\frac{1-\varepsilon }{2-\varepsilon }\,,\ \ \ {\overline{\varepsilon }}_{22}={\overline{\varepsilon }}_{33}=2 \ \frac{1-\varepsilon }{2-\varepsilon }\,,\\ \nonumber&{\overline{\varepsilon }}_{21}= {\overline{\varepsilon }}_{12}= {\overline{\varepsilon }}_{34}= {\overline{\varepsilon }}_{43}={\overline{\varepsilon }}_{32}={\overline{\varepsilon }}_{23}=\frac{1}{2-\varepsilon }\,,\\&{\overline{\varepsilon }}_{31}= {\overline{\varepsilon }}_{13}= {\overline{\varepsilon }}_{24}= {\overline{\varepsilon }}_{42}={\overline{\varepsilon }}_{41}={\overline{\varepsilon }}_{14}=0\,. \end{aligned}$$

(A.12)

Both in the case $N=3$ [Eq. (A.11)] and $N=4$ [Eq. (A.12)], notice that the coefficients ${\overline{\varepsilon }}_{11}={\overline{\varepsilon }}_{NN}$, accounting for the self-reflection of the external panes, are in general lower than those accounting for the thermal energy exchange between adjacent panels. On the other hand, the heat exchange between not adjacent panes is null, due to the null transmissivity of the interposed panel. The “self-reflection” coefficient of inner panels, i.e., ${\overline{\varepsilon }}_{kk}\,,\ k=2\dots N-1$, takes quite high values, since it accounts for the energy reflection on both sides.