Comparative Thermal Losses of One, Two and Three -Layer Windows in Winter

Reducing thermal losses (heat gains/losses) through transparent building enclosures (windows, glazed facades) is one of the important tasks in building energy saving [1,2]. To reduce these losses, a number of solutions have been proposed - insulating glass units (including gas fills), low-emissivity coatings on inner glass surfaces (reducing heat loss from the room by increasing long-wave reflectance), semi-transparent screens outside or inside insulating glass units, ventilated facades, external blinds, etc. [3-5]. Calculations and experimental studies are ongoing to evaluate these solutions [6-10]. Because the problem is multifactorial, many studies use the steady-state approach and mainly refer to a generalized parameter- the equivalent thermal resistance R₀. This parameter is standardized for different climatic zones and seasons [11,12]. The convective heat transfer coefficients entering R₀ are also standardized; they indirectly account for radiative exchange coefficients.

Such a generalized approach complicates the determination of the influence of individual window parameters (number of glass layers, thickness, radiative characteristics) and of external conditions (outdoor temperature, solar radiation, sky and surroundings radiation) on heat losses. References [13-15] note that the equivalent thermal resistance is generally variable and that analysis of heat fluxes through windows and facades should be carried out using un-steady approaches that account for both variabilities of external parameters and thermal inertia of transparent enclosures, and that separate convective and radiative heat fluxes should be considered. The aim of this work is to develop a one-dimensional unsteady model of heat exchange in windows and facades and to conduct a comparative study of the influence of the number of glass layers on thermal losses in winter, assuming constant indoor air and wall temperatures.

The calculation scheme for a triple-glazed window, problem parameters and convective and radiative heat fluxes are shown in Figures 1a,b: 1-3 are the transparent elements (glasses), 4 is a fictitious plane with wall temperature and effective absorption and emission properties α_∋ (ε_∋ ), characterizing multiple reflections between window and room, 5-room, ES- total radiative flux incident on the window (hereafter specific fluxes or flux densities in W/m²), including solar radiation (EС), and combined radiation of the sky and surrounding structures (their effective density q_r and temperature t_r).

Let tn denote the outdoor air temperature, t₂. t_r2 the temperatures of the room air and walls (temperatures with small letter t are in ^o С, with capital T in K),α_p (εp ) are the wall absorption and emission coefficients. Figure. 1b shows the scheme for determining the effective radiation fluxes in the system. The mathematical model is based on the assumption that temperature gradients within the glazing panes are small, i.e., their mean (mass-averaged) temperatures are used, so the problem re-duces to equations for changes of the panes’ enthalpies over an elementary time step Δτ . As in the finite difference method [16], it is assumed that heat exchange during a small-time interval Δτ occurs at constant element temperature and changes abruptly at the end of the interval Δτ . The model also includes equations that describe notable (order-of-magnitude) changes in the glass absorption coefficient β in the wavelength region up to 2.7 μm and beyond. For ordinary window glass, in the shorter-wavelength range (index “v”), β_v≈0.6 , (1/сm), while in the longer-wavelength thermal range (index “n”), β_n ≈6 , (1/cm) [17].

For example, for glazing pane 1 the enthalpy at time step j+1, , equals

where is the enthalpy of glass 1 at time j, and dQ¹ is the enthalpy change over Δt , equal to

Where α_v,α_n are the glass absorption capabilities in the first and second spectral ranges (and correspondingly their emission capabilities ε_v =α_v,ε_v=α_p; q_k1, q_k12 – are convective losses of the pane;

“black” emission at the temperature of pane 1; q_1n ( q_1n = b_n∗q₁ ) − portions of radiation fluxes incident on pane 1: incident, effective and self-emitted in the first (coefficients c_v,b_v) and second (c_n,b_n) ranges. Coefficients c and b characterize the fractions of the radiation flux in those ranges and equal, for solar radiation, cv=0.97, c_n=0.03, while for thermal radiation b_v=0, b_n=1. Convective losses of pane 1 to the outside and inward toward the interspace are

Convective heat transfer coefficients, except for interspaces, are determined for natural convection using formulas for a vertical plate, and for forced convection using formulas for flow past a flat surface with air speed w [18-20]. Convective heat transfer within the interlayer between panes is determined by a conduction-type mechanism [18,21]. To determine effective radiative fluxes in a three-layer window including the fictitious plane, we generally have the system of equations [22]

where are self-emission fluxes of the elements; K₁ ÷ K₃ , r₁ ÷ r₃ are the total normal (angle of incidence not accounted for) transmission and reflection coefficients of panes 1-3-equal to the ratios of transmitted or reflected radiation by the pane to the incident radiation (for glass they are set by refractive index or normal reflection coefficient of a surface ρ₀ , absorption coefficients β and thickness h). Coefficients K, r and α (ε ) for comparative evaluation of heat losses (gains) are determined approximately-taking into account three internal reflections at the pane surfaces for the case of normal incidence. For glass panes, expressions (5)–(10) are written separately for the first and second wavelength ranges. The system (5)–(10) is coupled (at time j the temperatures of all system elements and incident fluxes are known) and was solved by the Gauss method. It should be noted that for low-emissivity glasses analogous equations should be written for at least three ranges: visible (up to 0.78 μm), thermal (0.78-2.7 μm) and thermal (2.7–25 μm). Effective radiative characteristics of the fictitious plane (indices α_∋ and ε_∋ ) are approximated as [23]

Where Θ = F/F_c is the ratio of window area F to the area of room walls Fc. The temporal variability of the solar flux Ес incident on the window was modeled as

where E₀ is the normal solar irradiance at noon (it remains fairly constant through most of the day [24]), and i is the angle of incidence of solar rays on the window, which depends on latitude, season and the window orientation. Daily variation of outdoor air temperature t_n for comparative assessment of heat losses (gains), accounting for its asymmetry, was approximately modeled by two periodic functions with temperature minimum t_min at 4.00 and maximum t_max at 16.00 [25]. The model separately accounts for convective and radiative flows. Based on this model a program written in Delphi was developed to compute temperatures and resultant heat fluxes through single-, double- and triple-glazed windows. The total resultant heat fluxes were determined in two sections-at the window outer face (q_res1) and at the room side (q_res2), i.e.,

where q_fall1, q_fall2 are the fluxes incident on the window and from the window onto the fictitious plane, q_los1, q_los2 are heat losses of the window to the outside and from room to window, including respective differential fluxes by convection q_k, radiation q_iz and transmission qtr. Fluxes at the second section allow accounting not only for the thermal inertia of the system but serve as a consistency check of the program-in steady state these resultant fluxes should be equal. Checks for the steady case showed discrepancies not exceeding 0.1%.

Studies of thermal losses (gains) for south-facing single-, double- and triple-glazed windows were carried out for winter conditions (solar declination δ =-23.5, t_max=-10, t_min=-20, sky temperature t_r1=0, ground t_z=-10, resulting effective temperature -4.9) for clear and overcast days (E₀=700,0 W/m²) with and without wind (w = 0, 5 m/s). Glass parameters: thickness 4 mm, β¹ = 0.6 and β² = 6, single-face reflectance across the spectrum ρ₀= 0.04 so that the full fractions of incident radiation-reflection, absorption and transmission- were: r_v=0.068 and r_n=0.043, a_v=0.2115 and a_n=0.8761, transmissions K_v=0.7257 and K_n=0.0836. Indoor air and wall temperatures t₂ and t_r2 were taken constant at 20⁰С,α =ε = 0.5 , Θ =F / F_π = 1 (i.e., window area equals a reference fraction of wall area). Inter-pane spacing was 1.5 cm. Figure 2 shows the change of resultant fluxes q_res (q_res>0 means heat gains into the window, q_res<0 means heat losses through the window) for single-, double- and triple-glazed windows during two days for E₀=700 in calm conditions (w=0). As seen in Figure 2, resultant fluxes and their differential components-convection, radiation and transmission-depend not only on the number of panes but also strongly on external parameters. Moreover, even in winter on sunny day resultant fluxes are positive for a single-glazed window. In general, steady state in windows establishes fairly quickly even for a triple-glazed window (qres2, Figure. 2c). With increasing number of panes, the resultant daytime flux slightly increases.

At night the largest heat losses, as expected, occur for the single- glazed window. Figure. 3a shows resultant fluxes of single- and triple-glazed windows relative to the double-glazed window, and Figure. 3b shows temperatures of the outer panes of these windows.

It can be seen that in this case (w=0) the presence of a second pane reduces losses by about 60%, and adding a third pane further reduces losses by about 24%. From Figure 3b the outer pane temperature of the single-glazed window at night is always higher than that of the double- and triple-glazed windows. Moreover, for the triple-glazed window the outer pane temperature may become even lower than the effective radiation temperature of the sky and ground, so qiz (see Figure 2c) may become positive. Figures 4a and 4b show thermal losses of the double-glazed window on an overcast day with w=5 m/s. Here resultant fluxes (losses) are larger (more negative) than in Fig. 2b, but because the outer pane temperature drops significantly, the differential radiative flux qiz becomes noticeably positive. That is, simple summation of convective and radiative coefficients is not always possible and one must account for the signs of the individual fluxes. Also note that here the influence of the number of panes is greater than in the windless day. Thus, the second pane reduces losses of a single pane by almost 90%, and the addition of a third pane reduces losses by a further ≈30%. Investigation of other cases (sun and wind; no sun and no wind) showed that generally the effectiveness of the second and third panes is similar, allowing use of these results under other external parameter sets.

Knowledge of the resultant fluxes also permits evaluation of actual values of equivalent thermal resistance R₀ for single-, doubleand triple-glazed windows. From the general definition of R₀ [18] it follows that

Values of R₀ for two characteristic cases are shown in Figure 5.

The obtained R₀ values (Figure 5a) for a winter overcast day with wind, when resultant fluxes are negative (window loses heat), generally agree with their normative values [11]. It should also be noted (Figure 5b) that on a sunny, windless winter day resultant fluxes can be positive (see Figure 3b) and R₀ becomes very small and practically independent of the number of panes. That is, standardized R₀ values reflect specific climatic conditions and requirements for indoor air temperatures. Their variation with external parameters is secondary; in practice the more informative quantities are the resultant heat fluxes, which in the first approximation determine the heating power required to compensate heat losses through transparent enclosures. It is also worth noting that R₀ weakly depends on outdoor air temperature changes in winter.

• One-dimensional transient thermal models and computational programs have been developed to calculate thermal losses of single-, double- and triple-glazed windows, taking into account variations of outdoor air temperature and solar radiation. The model and programs allow determination of glass temperatures and the influence on thermal losses of glass thickness and absorption coefficients, indoor air and wall temperatures, and-in a first approximation-the effect of the window-to-room area ratio.
• A comparative investigation of winter thermal losses for single-, double-and triple-glazed windows was carried out. It was found that a double-glazed window reduces losses of a single-glazed window by almost 80-90%, and adding a third pane to a double-glazed window reduces losses by 25-30%. Evaluation of the windows’ equivalent resistances R₀ showed that they are comparable to normative values. The model and program permit analysis of the structure of heat losses-convection, radiation and transmission.

The work was carried out within the framework of state fundamental research of the Scientific and Technical Center with a design bureau and pilot production of the Academy of Sciences of the Republic of Uzbekistan.

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