1. Research question and system boundary

How does additional multilayer dynamic thermal protection change the calculated thermal transmittance of an existing window? The earlier version addressed this question through a parametric study of assumed additional thermal resistances ΔR. This revision retains that basic structure but first derives the central range of thermal resistance from conduction, longwave radiative heat exchange and natural convection.

The closed state of an interior system is investigated. The calculation describes the average, one-dimensional heat flow through the area. Frames, glass edges, mounting components, lateral openings and air exchange are treated separately. The steady-state results establish neither annual energy savings nor a product U-value determined according to a standard.

The starting configuration comprises five thermally functional films of the current type, metallised on both sides. Between them are four directly formed air cavities, each approximately 10 mm deep. Additional decorative or protective films are not automatically counted as equivalent thermal protection layers.

2. From an air gap to effective cavity resistance

2.1 Why R = d / λ is only the first step

For pure conduction in still air, R = d / λ. With d = 0.010 m and λ = 0.026 W/(m·K), this gives R = 0.385 m²K/W. However, this value is only the conduction-only limit. In a real air cavity, longwave radiative exchange and, depending on geometry and temperature difference, natural convection act in parallel with conduction.

The model calculation uses air properties at approximately 20 °C: λ = 0.026 W/(m·K), ρ = 1.204 kg/m³, μ = 1.81·10⁻⁵ Pa·s, cp = 1006 J/(kg·K), β ≈ 1/T. The cavity height is set to 1.40 m. For the internal 10-mm cavity, a temperature difference of 5 K is used as the calculation case; Fraunhofer ISE investigated the same range of dimensions and showed a conduction-dominated regime for 1-cm cavities [1].

2.2 Natural convection according to ISO 15099

For a vertical, enclosed gas layer, the Rayleigh number is first determined:

Ra = g · β · ΔT · d³ / (ν · α)

where ν = μ/ρ and α = λ/(ρ·cp). ISO 15099 uses a Nusselt correlation for vertical cavities. For the range relevant here, the following expressions apply, among others:

Nu₁ = 1 + 1.7596678·10⁻¹⁰ · Ra²·²⁹⁸⁴⁷⁵⁵ for Ra ≤ 10⁴

Nu₁ = 0.028154 · Ra⁰·⁴¹³⁴ for 10⁴ < Ra ≤ 5·10⁴

Nu₁ = 0.0673838 · Ra¹ᐟ³ for Ra > 5·10⁴

Nu₂ = 0.242 · (Ra/A)⁰·²⁷²; Nu = max(Nu₁, Nu₂); A = H/d

The combined convective-conductive heat-transfer coefficient is h_c = Nu · λ/d. For a depth of 10 mm, a height of 1.40 m and a temperature difference of 5 K, the results are Ra ≈ 5.18·10² and Nu ≈ 1.0003. In this model, the cavity is therefore effectively in the conduction-dominated regime; h_c ≈ 2.601 W/(m²K).

2.3 Longwave radiative heat exchange

Linearised longwave radiative exchange between two large parallel surfaces is described by

h_rad = σ · (T₁ + T₂) · (T₁² + T₂²) / (1/ε₁ + 1/ε₂ − 1)

where ε is the longwave emissivity of the respective surface. For the sensitivity calculation, ε = 0.03 is assumed as a low emissivity for a metallised surface. This value comes from the historical Fraunhofer measurement of the aluminium side used at that time; it is not a new measurement of the current functional film metallised on both sides [1]. For the historical PET side, Fraunhofer used ε ≈ 0.48; a value of ε = 0.84 is assumed here for a typical high-emissivity reference surface.

Table 1. Modelled effective thermal resistances of vertical air layers

Units: h_c and h_rad in W/(m²K), R_eff in m²K/W. The results are model values for enclosed cavities and the stated boundary conditions.

3. What metallising the second side changes in the calculation

In the 10-mm calculation case, cavity resistance increases from approximately 0.361 m²K/W for the historical metal/PET pairing to approximately 0.372 m²K/W for metal/metal. In this specific model, that is only around 2.9 % more cavity resistance. The physical reason is plausible: even one surface with very low thermal emission strongly limits longwave exchange; the second low-emissivity surface further reduces the remaining radiative component, but not in proportion to the number of coatings.

This statement is consistent with the qualitative conclusion of the Fraunhofer report: for the film metallised on one side that was investigated at the time, a second low-e boundary surface within a single cavity provided only a comparatively small additional improvement [1]. This explicitly does not mean that the current film metallised on both sides has already been measured to have ε = 0.03. A separate material measurement is required for that.

4. Four internal 10-mm cavities: the central 1D model case

Four identical internal metal/metal cavities give the following result in the idealised central area:

ΔR_intern = 4 × 0.372 ≈ 1.488 m²K/W.

This value is physically derived but remains idealised: it assumes enclosed, uniform 10-mm cavities without lateral bypass airflow. It is therefore not a measured product property of the present-day movable FVS.

Table 2. Existing windows with the idealised resistance of the four internal 10-mm cavities

Calculation: U_neu = 1 / (1/U_alt + ΔR). The heat-transfer rates apply to A = 1.40 m² and ΔT = 20 K. Here, ΔR is the additional thermal resistance per unit area of the film stack in the central region. With an additional interior system, the room-side surface boundary condition moves to the new interior surface; it must not be counted a second time as additional resistance.

5. The 60–70-mm spacing from the window: not a fifth 10-mm cavity

An air space approximately 60–70 mm deep may lie between the existing glazing and the film stack. This spacing must not be represented as a still air layer using R = d/λ. At this depth, natural circulation within an enclosed vertical cavity is already well developed.

For a representative spacing of 65 mm, H = 1.40 m and a glass/metal surface pairing, the ISO 15099 correlation gives, at 5 K, approximately Nu = 3.52 and R_eff ≈ 0.634 m²K/W; at 15 K, convection increases to Nu ≈ 5.07 and the resistance falls to approximately 0.454 m²K/W. An ideal, fully enclosed complete assembly comprising four internal cavities plus this window cavity would therefore have a calculated resistance of approximately ΔR = 2.12 m²K/W in the 5-K case or 1.94 m²K/W in the 15-K case.

These two values are deliberately identified as enclosed-cavity cases. If the space between the window and film stack is connected to the room through lateral, upper or lower openings, this additional effect is not assured. Air exchange must then be modelled or measured separately. For the actual movable system, this publication therefore derives no fixed product R-value from the 60–70-mm cavity.

Table 3a. Sensitivity: 15-K case for the enclosed 65-mm cavity

Table 3b. Sensitivity: 5-K case for the enclosed 65-mm cavity

6. Lateral air gaps: consider geometry and airflow separately

6.1 Direct geometric contribution of the edge strips

For two unprotected lateral edge strips of width a and a window width B, the simplified parallel-area model gives:

f = 2a/B

U_eff = (1 − f) · U_geschützt + f · U_alt

Additional loss = 100 · (U_eff/U_geschützt − 1)

The following matrix uses U_alt = 1.30 W/(m²K) and the central model value ΔR_intern = 1.488 m²K/W, giving U_geschützt ≈ 0.443 W/(m²K).

Table 4. Calculated additional loss due solely to the geometric edge strips

Example: for a window width of 1.00 m and a 3 mm gap on each side, the geometric area fraction is 0.6 %. In the parallel-area model, heat transfer increases by around 1.16 % relative to the fully protected case. This must not be used to conclude that ‘a 3 mm gap causes only a 1 % loss’.

6.2 Additional natural air circulation

A perimeter gap can also form part of an airflow path. Warm room air can enter behind the covering, cool against the cold glazing and leave at another location. The magnitude and direction of this air exchange depend on lateral, upper and lower openings, cavity depth, height, temperature difference, airtightness and flow resistance.

Hart, Goudey and Curcija investigated precisely such naturally ventilated cavities with perimeter openings through measurements, CFD and simplified ISO 15099 correlations. The reported 2.5 ± 7 % is the mean deviation between modelled and measured heat flow; it is explicitly not a general heat loss caused by a particular perimeter gap [2]. No unsupported convection surcharge is therefore added to the geometric table here.

7. Context from measurements and the literature

Fraunhofer ISE 2016

The Fraunhofer ISE report describes an early, fixed specimen with ten vertical films, air spacing of approximately 9 mm, two Plexiglas covers and eleven air spaces in total. The 12-µm PET film used at the time was metallised with aluminium on one side. Under five measurement boundary conditions, the measured U-values of the specimen ranged from 0.35 ± 0.03 to 0.45 ± 0.04 W/(m²K). The report also notes that the temperature dependence was probably influenced by cavities that were not fully airtight [1].

These measured values apply to the fixed specimen used at that time, not to the present-day movable system. They are nevertheless directly relevant to the physical modelling: Fraunhofer separated convection and radiation, used ISO 15099 for the overall analysis and selected 1 cm as a suitable cavity depth in the conduction-dominated regime [1].

SINTEF / NTNU 2023

Misiopecki, Grynning and Gustavsen investigated ten box-window configurations in hot-box tests. The largest measured improvement in the assembly investigated was achieved with a low-emissivity roller-blind attachment and limited perimeter air exchange: the U-value fell from 1.9 to 1.0 W/(m²K). A reflective covering positioned between the window frames reduced the U-value from 2.0 to 1.3 W/(m²K). The authors also investigated condensation risks [3]. These values demonstrate the effect of the systems tested there, not the performance of the present-day FVS.

ORNL / LBNL 2022

Kunwar, Bhandari and Curcija experimentally investigated fabric cellular shades over two heating seasons and supplemented the measurements with building simulations. In two comparable test rooms, daily heating-energy savings of 17–36 % were measured relative to the condition without a covering. Annual and national potential was subsequently simulated [4]. The study shows that window attachments can have technically significant effects, but it provides no factor for transferring these results to metallised film stacks.

Adaptive building envelopes

Zhang et al. place switchable building envelopes within a broad field of research. For dynamic systems, not only thermal transmittance in the closed state but also time-dependent switching between thermal and optical states is crucial [5].

8. Comparison with 50 mm EPS

For 50 mm EPS with λ = 0.035 W/(m·K), R = 0.050/0.035 ≈ 1.43 m²K/W. This is of the same order of magnitude as the idealised calculated resistance of the four internal 10-mm metal/metal cavities considered here, approximately 1.49 m²K/W.

The comparison concerns thermal resistance alone. EPS is a static, opaque insulation material. A movable film assembly is periodically removed from the heat-transfer path; airtightness, perimeter junctions, moisture behaviour, mechanical construction and optical properties differ fundamentally.

9. From the steady-state U-value to the window energy balance

A low U-value in the closed state is only one aspect of a dynamic window system. Its annual effect also depends on when the system is closed, when solar energy is admitted, how long heat losses dominate and which control strategy is used. No whole-building or annual energy saving may therefore be derived from the steady-state tables calculated here.

The time-dependent balance is addressed in the separate articles ‘Windows: Energy Balance Rather Than U-Value’ and ‘Window Heat Losses and Solar Gains – When Does a Window Become Energy-Positive?’. Research findings on interior window attachments are discussed in ‘Interior Thermal Protection for Windows – What Scientific Studies Actually Show’.

10. Conclusion

For four 10-mm air cavities between films with low-emissivity surfaces on both sides, the 1D model documented here, at a mean temperature of 20 °C and a temperature difference of 5 K per cavity, gives an effective cavity resistance of around 0.372 m²K/W and thus ΔR_intern ≈ 1.49 m²K/W. For an existing window with U_alt = 1.30 W/(m²K), this gives a calculated U_neu ≈ 0.443 W/(m²K); steady-state heat transfer decreases by around 65.9 % in the central-area model. At an area of 1.40 m² and a temperature difference of 20 K, the instantaneous heat-flow rate falls from 36.4 to approximately 12.4 W.

An additional, fully enclosed 65-mm cavity between the glazing and film stack can contribute further resistance in the model. Because of natural convection, however, this contribution depends strongly on temperature difference and geometry and is not assured where actual perimeter openings are present. Lateral edge strips cause a directly calculable geometric bypass contribution; additional air exchange must be modelled or measured separately.

The previously used calculation range around ΔR ≈ 1.5 m²K/W can therefore be physically substantiated for an idealised central four-cavity assembly, provided the assumed low-emissivity surface properties are achieved. Values approaching 2.0 m²K/W are plausible only where additional enclosed cavities are genuinely thermally effective. The actual system performance of the present-day movable FVS must be validated through measurements of the complete installed system.