Passively Maintained Closed Cavity Façade—Experimental Validation of the Mathematical Thermal Model

Abstract

Although glass façades have been on the market for over a century, new improvements, following sustainable standards, are still being invented. An improvement of the actively maintained CCF has occurred in passive maintenance with natural ventilation of the cavity and insulation glass unit placed on the external side, which has served as a true motivation for further research. To develop the idea, a new type of CCF was invented, followed by the creation of the software, whose purpose is to determine optimal CCF façade components. During this research, an experimental and mathematical model was made regarding the thermal behavior, later validated by the measurements on-site in Rugvica, Croatia. Using simplified but unconventional methods, numerous formulae and variables, a simulation of climatic loads onto the CCF was conducted. Validations of the thermal model were made during winter and summer periods for southern and western façade orientation, explaining how heat transfers from the environment to close cavity façade elements. It was found from the analysis that air temperatures of the façade elements follow the outer air temperature, by constant air exchange with the outer space. The results showed great potential with up to 3 °C (5–10%) of difference in experimental and calculated results, thus creating a basis for further improvement of the software with the addition of structural and hygric behavior of the façade element, regarding climate conditions.

1. Introduction

A close cavity façade is a type of double-skin façade (DSF), which was developed by improving the benefits of double-skin façades (such as protected sun blinds, lowered cooling, and heating needs with a U value < 0.7 W/m²K) with the advantages of single-skin façades (such as clean air cavity providing lower maintenance costs, having only two glass surfaces to maintain, and less reduction of floor area having larger depth to the outside). The greatest weaknesses of the close cavity façades are unavoidable excess temperatures and pressures in the façade and the risk of condensation between two skins requiring the ventilation of the cavity [1]. Standard closed cavity façades are composed of an insulated glass unit (IGU) on the inside, a closed cavity (maintained by mechanical air supply), an external single glass pane, and an aluminium frame with a thermal bridge break (Figure 1a) [2]. During the research of the “Development of a double facade with a hermetically sealed cavity H-CCF” project, made by the University of Zagreb Faculty of Architecture, and KFK company, enhancements of the close cavity façade were made to nullify energy needs for active maintenance of the air gap. The newly developed—passively maintained closed cavity façade, commercially named ACS, consists of an external IGU unit, a naturally ventilated cavity, an internal single glass pane, and an aluminium frame with a thermal bridge break (Figure 1b). A CTE mismatch is solved by placing a PVB foil between the laminated glass layers. The major steps forward, outcoming from this research are the passive moisture control by thin capillary and IGU on the outside. Capillary is placed between outer air and air cavity, which provides the natural airflow to and from the cavity, eliminating excessive pressures and minimizing the risk of condensation and dust inflow. Condensation above 6% is passively controlled by the desiccant placed inside the frame. Insulated glass unit placed on the outside, provides lowered temperatures during summer, and consequently pressures, while also contributes to higher temperatures during winter reducing the risk of condensation. Argon and air gap together create double barrier conditions for better thermal comfort of the interior.

Energy efficiency and sustainability in the building sector are crucial for achieving the 2030 Agenda for Sustainable Development Goals (SDGs), especially regarding Goal 7: Affordable and Clean Energy and Goal 12: Responsible Consumption and Production. Buildings play a significant role in contributing to climate change and environmental degradation due to their energy consumption, resource usage, and waste generation [4]. Passive solutions in the building industry have a primary role in achieving greater sustainability without excessive energy consumption. These solutions often involve strategic building orientation, and optimal thermal performance to reduce the need for heating, cooling, and ventilation. Physical processes happening inside the close cavity façades are mostly affected by the climate conditions and the façade’s response to them (Figure 2). Solar irradiance absorption affects heat transfer and consequently, pressure equalization, moisture control, air supply, and glass deflection. All these processes mutually affect the façade, which can be observed in its thermal, structural, and hygric behavior. Thermal behavior serves as a starting point for observing the interspace of the façade, as the rise in temperature increases the pressure, while a drop in temperature decreases the pressure. Equalizing the cavity and pressure with the surrounding atmospheric air causes over—or under-pressure inside the cavity. Due to air suction and deflation regarding pressure equalization, structural behavior of the glass panes results in deflection under the influence of cyclical changes in climatic conditions. As the glass deforms, there is a change in air volume inside the cavity, the pressure decreases and most of the load disappears.

Hygric behavior is directly related to air supply from the outside, as the air supply also introduces humidity. The greatest factor influencing thermal behavior, tending towards the thermal equilibrium, is solar irradiation on the facade, consisting of direct and diffuse irradiance. Accumulated solar energy and heat from the building transfer through the façade elements by radiation, convection, and conduction.

Since their first installment in 2009 by Gartner at the Roche Headquarters in Rotkreuz (CH), sustainable façades have been the subject of numerous experimental, mathematical, and on-site measurements. In 2003, Huveners, Herwijnen, and Soetens conducted research on glass and closed air cavities within the insulated double glass units, investigating structural behavior, heat transfer, and climatic loads. Their study also examined termo-mechanical loads of cavity pressure (effective cavity pressure, humidity, and temperature) in relation to glass stiffness [5]. Laverge, Schouwenaers, and Janssens conducted research in 2009 on Gartner’s invention of the CCF and its application in real buildings in Zürich. Their findings demonstrated improved thermal insulation properties for winter conditions and high overload risks, with the management of condensation risk achieved by introducing a mechanical air supply. Gartner validated the dynamic behavior of the model by measuring all surface temperatures, air temperature, relative humidity, pressure differences, windspeed, and other parameters [1]. Continuing the investigation of condensation risks in CCF, the same authors conducted research in 2010, examining the thermal and moisture behavior of the façade in response to environmental conditions [6]. Regarding Laverge’s research, a series of three research papers, led by the group authors, G. Lori, K. Van Den Brande, N. Van der Bossche, H. De Bleecker, and Jan Belis, were conducted on pressure equalization-thermal-mechanical model of climatic and wind loads in Closed Cavity Façades. In 2022 they represented an experimental model outcome with a reference software for the analysis of IGU. According to them, the main difference between IGU and CCF is the dry air flow continuously pumped into the cavity, in order to control the relative humidity and avoid the risk of condensation. The façade skin is permeable to the interior (mechanical supply) and exterior (controlling the excessive air—temperature and structural performance). Their numerical design tool was based on validated analytical findings and Finite Element commercial code, generating the shape functions of the single skin surfaces under the net pressures [7,8,9,10]. In 2016, Wüest and Luible explored thermally induced climatic loads concerning radiative, convective, and conductive thermal transmittance to interspace air. Their research also involved a comparison between naturally ventilated (DSF) and non-ventilated (CCF) façades. In a close cavity façade without ventilation, the air cavity is provided by a low flow of conditioned air to avoid condensation, although a cooling effect can be achieved, and higher temperatures occur. They also discussed double and triple glass units, as well as the sun blinds, and color affecting the heat transmittance [11]. Respondek et al. [12] and Rashid et al. [13,14,15] made reviews and research on heat transmission regarding Reynolds’, Nusselt’s, and Prandtl’s numbers in nano- and phase-change materials. P. Gallo and R. Romano investigated in 2017 the ‘SELFIE system’, developing a multifunctional, adaptive, and dynamic façade aimed at enhancing the energy performance of buildings in the Mediterranean region. Their work included the integration of smart glazing, movable solar shading, phase change materials, and multifunctionality of the façade. The study also considered aspects of structural safety, mechanical strength, shock resistance, fire resistance, and deformation resistance, along with indoor comfort factors such as air permeability, water tightness, thermal transmittance, hygrothermal insulation, thermal inertia, daylighting, solar protection and acoustic insulation [16]. Pastori et al. investigated passive solutions in 2021, such as highly energy-efficient building envelopes reaching greater sustainability, and improving thermal comfort without energy consumption, mainly focusing on ventilated façades providing a mathematical, radiation, energy conservation and physical model [17].

The existing calculation methods currently used for verifying double-skin façade glazing under wind and climatic loads are not accurate enough to capture the specific behavior of the passively maintained CCF, since DSF cavity’s permeability is high enough to prevent pressure and temperature build-up. Furthermore, current standards lack guidelines for the optimized design of impermeable double skins. They are designed with a safety-first approach leading to potential result in unnecessary material usage and increased costs. To address this issue, researchers have developed a comprehensive dynamic model of thermal behavior during the whole year, which would later be the basis for the structural and hygric behavior, all coupled together. The aim of this research is to create a mathematical model of passively maintained CCF and validate it through physical measurements on-site.

2. Materials and Methods

The project’s scope encompassed three main stages: conducting the experimental research, creating analytical and mathematical models, and conducting a thorough comparison of the results obtained from these methods.

2.1. Experimental Model

The experimental test box was situated in Rugvica, Croatia (N 45.75°, E 16.24°), near Zagreb, where average temperatures in July do not exceed 37 °C (Figure 3a), and solar global radiation reaches 180 kWh/m² (Figure 3b).

Experimental model was built, as shown in Figure 4, with an outer width of 149.7 cm, and a height of 258.7 cm. The glass layers, argon, air cavity, sun blinds and frame parameters are listed in

Table 1. Material properties used in the experimental test box and the mathematical model.

Material	Dimensions Width (cm)/ Height (cm)/ Thickness (mm)	Specific Heat Capacity (J/kgK)	Solar Factor (-)	Emissivity (-)	Reflectivity (-)	Absorption (-)
Inner glass—inner pane	133.3/235.9/5	840	0.71	0.89	0.45	0.19
Inner glass—outer pane	133.3/235.9/5	840	0.71	0.12	0.27	0.19
Middle glass—inner side	133.3/235.9/8	840	0.84	0.89	0.08	0.08
Middle glass—outer side	133.3/2 35.9/0	840	0.84	0.89	0.08	0.08
Outer glass—inner pane	149.7/258.7/6	840	0.36	0.01	0.11	0.18
Outer glass—outer pane	149.7/258.7/6	840	036	0.89	0.12	0.17
Sun blinds	133.3/235.9/1.5	880	-	0.94	0.59	0
Argon	149.7/258.7/14	520	-	-	-	-
Air cavity	133.3/235.9/155	1000	-	-	-	-
Peripheral element—down	133.3/12.8	880	-	0.94	0.59	0.41
Peripheral element—left	235.9/12.8	880	-	0.94	0.59	0.41
Peripheral element—right	235.9/12.8	880	-	0.94	0.59	0.41
Peripheral element—up	133.3/12.8	880	-	0.94	0.59	0.41
Façade frame		880	-	0.94	0.59	0.41
Frame cladding —overhang depth—down	30.8	880	-	0.94	0.59	0.41
Frame cladding —overhang depth—left	18	880	-	0.94	0.59	0.41
Frame cladding —overhang depth—right	30.8	880	-	0.94	0.59	0.41
Frame cladding —overhang depth—up	30.8	880	-	0.94	0.59	0.41

, including dimensions, solar factors, and heat capacity along with emissivity-, reflectivity-, and absorption coefficients.

For the numerical analysis, various measurements were taken from the site during 2020 and 2021, which can be divided into three categories: climate conditions, model characteristics, and façade behavior. Climate conditions were measured by weather station, captured in Figure 4a, which measured air temperature, relative humidity of the outer air, atmospheric pressure, global radiation, wind speed and direction. The necessary climate conditions encompassed global solar radiation, which was later divided into direct, diffuse, reflected, and extraterrestrial radiation for the calculations. Model characteristics included the location coordinates (time zone), dimensions of the facade, the orientation of the sample model, as well as optical properties of the glass panes. Façade behavior was determined through measuring probes placed on the façade, as shown in Figure 5, where temperature sensors (T_x), relative humidity sensors (H_x), strain gauge sensors (S_x), sensors for measuring differential pressure (P₁-ΔP_GI P2-ΔP_GA) are shown. P_GA is the differential pressure between surrounding atmospheric pressure and gap pressure, while P_GI is the differential pressure between internal and gap pressure, and D_x is the glass deflection measurement point.

Data regarding physical properties (temperatures of the internal, middle, and external glass, air cavity, and interior temperature) and structural model characteristics (glass deflection, stresses, and air cavity pressure) were directly measured on-site, as shown in Figure 6.

2.2. Mathematical Model

The mathematical model of the CCF was developed using Visual Basic/Visual Studio 2017 programming software, consisting of over 150 variables regarding properties of the façade elements and more than 100 formulae. This comprehensive model facilitates the calculations of energy flows within the CCF and between the CCF and its surroundings. It takes into account numerous physical properties of the façade elements and meteorological data. The model allows for temperature calculation considering various compositions of glazing, sunshade, and frame properties. Different locations are defined using meteorological data imported into the model via the Meteonorm 8.0 Software. Façade panel orientation, tilt angle, and environment albedo are also considered in the calculations.

The model considers all the relevant energy flows, which are further described in the subsequent chapters (Table 2, Figure 7). It comprehensively analyses three main types of energy transfers—radiative, convective, and conductive—as well as energy gains from solar radiation. Imported meteorological data with one-minute precision set the maximal time step for the calculation, ensuring accurate and detailed analysis.

2.3. Heat Transfer in CCF

Energy loss through CCF occurs through all three types of heat transfer—radiation, convection, and conduction (Figure 7). Among these, radiative and convective heat transfers are the most representative, while conductive heat transfer is nearly negligible [18].

Understanding heat gain, transfer, and the amount of energy in CCF is crucial, as it directly influences the temperature of each element in the façade. The simultaneous processes occurring inside the façade, such as glass deflection, gas pressure, and air exchange of the cavity, primarily depend on temperature [7]. Therefore, the thermal model of the façade forms the foundation of the entire physical and mathematical model describing the façades’ behavior.

Table 2. Energy flows observed in one time-step.

No.	Elements		Energy Transfer Type
1	Interior	Inner glass	Convective, Radiative
2	Inner glass	Air gap	Convective
3	Inner glass	Sunshade	Radiative
4	Sunshade	Air gap	Convective
5	Sunshade	Inner glass	Radiative
6	Sunshade	Middle glass	Radiative
7	Inner glass	Middle glass	Conductive (by an air gap)
8	Middle glass	Outer glass	Radiative
9	Middle glass	Outer glass	Conductive (by argon layer)
10	Outer glass	Surrounding	Convective, Radiative
11	Interior	Façade frame	Convective, Radiative
12	Façade frame	Air gap	Convective, Radiative
13	Façade frame	Surrounding	Convective, Radiative, Conductive
14	Solar radiation	Outer glass	Solar gain
15	Solar radiation	Middle glass	Solar gain (transmitted)
16	Solar radiation	Sunshade	Solar gain (transmitted)
17	Solar radiation	Inner glass	Solar gain (transmitted)
18	Middle glass	Outer glass	Solar gain (reflected)
19	Sun blinds	Middle glass	Solar gain (reflected)
20	Solar radiation	Façade frame	Solar gain

Table 2. Energy flows observed in one time-step.

No.	Elements		Energy Transfer Type
1	Interior	Inner glass	Convective, Radiative
2	Inner glass	Air gap	Convective
3	Inner glass	Sunshade	Radiative
4	Sunshade	Air gap	Convective
5	Sunshade	Inner glass	Radiative
6	Sunshade	Middle glass	Radiative
7	Inner glass	Middle glass	Conductive (by an air gap)
8	Middle glass	Outer glass	Radiative
9	Middle glass	Outer glass	Conductive (by argon layer)
10	Outer glass	Surrounding	Convective, Radiative
11	Interior	Façade frame	Convective, Radiative
12	Façade frame	Air gap	Convective, Radiative
13	Façade frame	Surrounding	Convective, Radiative, Conductive
14	Solar radiation	Outer glass	Solar gain
15	Solar radiation	Middle glass	Solar gain (transmitted)
16	Solar radiation	Sunshade	Solar gain (transmitted)
17	Solar radiation	Inner glass	Solar gain (transmitted)
18	Middle glass	Outer glass	Solar gain (reflected)
19	Sun blinds	Middle glass	Solar gain (reflected)
20	Solar radiation	Façade frame	Solar gain

Figure 7. Heat transfer scheme in a radiative, convective, and conductive manner. The numbers represent energy transfer listed in Table 2.

Figure 7. Heat transfer scheme in a radiative, convective, and conductive manner. The numbers represent energy transfer listed in Table 2.

2.4. Radiative Heat Transfer

Heat transfer by radiation in IGU predominantly occurs between the glass panels and their surroundings. The inner surfaces of the glass panels are coated with low emissivity coatings, reducing heat transfer by radiation by up to 50 times [19]. On the other hand, the outer surfaces of the IGU are of regular emissivity, making radiative heat transfer most significant at the outer surfaces of the IGU, specifically between the IGU and its surroundings [20]. Radiative heat transfer also occurs between the sun blinds inside the cavity and the glass surfaces. The amount of solar energy absorbed in the sunshade slats depends on various factors, including solar radiation, slat angle, solar azimuth, height, and surface reflectance [21,22]. Since the sunshade serves as an opaque barrier in the façade, it absorbs most of the solar radiation, leading to the excessive temperatures up to 90 °C. The blinds’ slat surface is usually made of aluminium or a polymer coating with high emissivity, which results in a substantial amount of radiative heat transfer between the blinds and the surrounding glass, as illustrated in Figure 8b [22]. In Figure 8c,d, positive values represent heat transfer direction from interior to exterior, and negative ones the other way. Both sets of data have the highest values between 02 p.m. and 06 p.m. when there are the highest temperatures inside the CCF.

Radiative heat transfer between CCF and the surrounding air is the most significant. While the sunshade becomes the hottest element in the façade, radiative heat transfer to both the inner and outer glass is reduced. This reduction is due to the smaller temperature difference between the sunshade and the middle glass, as well as the presence of a low-emissivity coating on the outer surface of the inner glass. The radiative heat transfer from a body can be written mathematically using the Stefan-Boltzmann Law (Formula (1)) [23]:

$$P=\epsilon \times \sigma \times A\times T^4, \left[\mathrm{W}\right]$$

(1)

wherein P is the radiative heat transfer rate (W), ε is surface emissivity (-), σ is the Stefan–Boltzmann constant 5.670374419...×10⁻⁸ Wm⁻² K⁻⁴, A is the surface area of the body (m²), and T is the absolute temperature of the body (K).

$$P=\epsilon \times \sigma \times A\times \left(T^4-T_0{}^4\right), \left[\frac{\mathrm{W}}{{\mathrm{m}}^2}\right]$$

(2)

Formula (2) calculates the net radiative heat transfer rate P between the body and its surroundings, considering the temperature difference to the fourth power, wherein T₀ is the absolute temperature of the ambiance (K). During the day, as the body’s temperature T increases compared to the ambient temperature T₀, the radiative heat transfer rate P will increase accordingly. Conversely, during the night, when the body’s temperature T is lower than the ambient temperature T₀, the radiative heat transfer rate P will be negative, indicating that heat is being transferred from the surroundings to the body.

The transfer between two radiating bodies can be expressed using Formula (3) [20]:

$$P=\frac{\sigma \times A\times \left(T_1^4-T_2^4\right)}{\left(\frac{1}{{\epsilon }_1}+\frac{1}{{\epsilon }_2}-1\right)}, \left[\frac{\mathrm{W}}{{\mathrm{m}}^2}\right]$$

(3)

In radiative heat transfer, energy is transferred between bodies or surroundings through infrared radiation. The radiative heat transfer rate P (W) between two bodies is calculated using Formula (3), where ε₁ is the surface emissivity of the first body, and ε₂ is the surface emissivity of the second body. It is important to note that radiative heat transfers occur only between solid bodies, as air mainly consists of oxygen and nitrogen, which do not absorb infrared radiation [24,25].

The mathematical model accounts for various radiative heat transfers between different components of the CCF. These include heat transfers between interior and inner glass, inner and middle glass (when the sunshade is opened), inner glass and the sunshade, sunshade and the middle glass, middle and outer glass, and outer glass and surrounding, as depicted in Figure 8c.

2.5. Convective Heat Transfer

Convective heat transfer includes energy transfer between solids and fluids (Figure 7). In this case, the fluid comprises inner air, outer air, and air inside the cavity, while solids are glass panels, façade frames, and sunshade slats. As the temperature difference between the solid surfaces and the surrounding air increases, a greater amount of air circulation occurs, leading to increased convection and heat transfer. Several factors influence convective heat transfer in this scenario. The properties of the fluid, such as conductive heat transfer coefficient, viscosity (cinematic and dynamic), pressure, heat capacity, temperature, and airspeed, play a crucial role. Similarly, the influencing properties of the solid elements are their temperature, size, and inclination. Convective heat transfer is characterized by Nusselt, Reynolds, Grashof, and Prandtl numbers, which are dimensionless values [26]. Newton’s law of cooling is expressed by Equation (4) used for both natural and forced convective heat transfer [27]:

$$P=h\times A\times \Delta T, \left[\frac{\mathrm{W}}{{\mathrm{m}}^2}\right]$$

(4)

where P is the rate of heat transfer between the fluid and the surface (W), A is the area of the surface in contact with the fluid, (m²), ΔT is the temperature difference between the fluid and the solid surface (K), and h is the convective heat transfer coefficient, (W/m²K). The convective heat transfer coefficient, h is calculated using Equation (5), including all the mentioned influencing factors [28].

$$h=\frac{Nu\times k}{D}, \left[\frac{\mathrm{W}}{{\mathrm{m}}^2\mathrm{K}}\right]$$

(5)

wherein D is the characteristic height of the surface (m). Nusselt’s number Nu is calculated from the Grashof, Prandtl, Rayleigh, and Reynolds numbers. Grashof number Gr can be expressed using Formula (6):

$$Gr=\frac{D^3\times {\rho }^2\times g\times \Delta T}{{\mu }^2\times T_{air}}, \left[-\right]$$

(6)

where D is the characteristic height of the surface (m), ρ is air density (kg/m³), g is gravitational constant (m/s²), ΔT is temperature difference (K), µ is air dynamic viscosity, and T_air is absolute air temperature (K). The characteristic length depends on the tilt of the surface for vertical transfer D (height of the surface), and for horizontal surface, the characteristic length is defined using Equation (7).

$$D=\frac{h\times w}{h^2+w^2}, \left[-\right]$$

(7)

where h is the height (m), and w is the width of the surface (m). Prandtl number Pr is calculated by the following Formula (8):

$$\mathrm{Pr}=\frac{\mu C_p}{k}, \left[-\right]$$

(8)

where C_p (J/(kg K)) is air-specific heat capacity, k (W/mK) is air thermal conductivity. Rayleigh number (Ra) can be written as Formula (9). For Rayleigh number values under 107, Nusselt numbers (Nu) are calculated using Formula (10), and for Rayleigh number values above 107, Nusselt numbers are calculated using Formula (11).

$$Ra=Gr\times \mathrm{Pr}, \left[-\right]$$

(9)

$$Ra<107, Nu=0.54\times R{a}^4, \left[-\right]$$

(10)

$$Ra>107, Nu=0.15\times R{a}^3, \left[-\right]$$

(11)

For forced convection, the Reynolds number (Re) is included, and the air film temperature is given as the average temperature between the surface and air (T₁ + T₂)/2, where v is wind or airspeed like shown in Formula (12).

$$\mathrm{Re}=\frac{v\times w}{\mu }, \left[-\right]$$

(12)

Nusselt number (Nu) for forced convection depends on the Reynolds number, as shown in the following Equations (13) and (14):

$$\mathrm{Re}<500000, Nu=\left(0.664\times {\mathrm{Re}}^2\right)\times \sqrt[3]{\mathrm{Pr}}, \left[-\right]$$

(13)

$$\mathrm{Re}>500000, Nu=\left(0.037\times {\mathrm{Re}}^{0.8}-871\right)\times \sqrt[3]{\mathrm{Pr}}, \left[-\right]$$

(14)

Accounted convective natural heat transfers in the model are between (1) interior and inner glass, (2) inner glass and air gap, (3) air gap and middle glass, (4) sunshade and airgap, and (5) façade frame and air gap. For airspeed > 1 m/s, convective heat transfer between the outer glass and the exterior is calculated as forced convection, as shown in Figure 8d.

2.6. Conductive Heat Transfer

In the glazed elements of the building envelope, heat is primarily transferred by radiation and convection. Conduction occurs in thin glass panes, which provide almost non-existent heat transfer resistance (

Table 3. Conductive heat transfer resistance of insulated glass unit components.

Material/ Component	Heat Transfer Coefficient, λ (W/mK)	Thickness, d (mm)	Heat Transfer Resistance, R (m2K/W)
Glass	1	5	0.005
PVB layer	0.22	0.76	0.003
Laminated glass *	-	10.76	0.013
95% Argon, 5% Air	0.025	16	0.640
Air 130 mm (simplified)	0.694	130	0.187

* 5 mm float glass + 0.76 PVB layer + 5 mm float glass.

). An Insulation Glass Unit (IGU) is filled with an inert gas like argon, krypton, or xenon [29]. In thicker gas layers, convection can take place due to the temperature difference between the inner and outer glass and gas buoyancy [30]. However, in very thin layers of inert gas, convection is not possible, and heat transfer is primarily conductive (Figure 9). As a result, the inert gas layer represents the greatest conductive heat transfer resistance in the IGU [20].

The width of the cavity and the temperature difference of elements around and inside the cavity allow for a complex airflow pattern to develop. To understand and measure such type of heat transfer, computational fluid dynamics (CFD) approach is required [18]. The heat transfer processes inside the CCF depend on ever-changing meteorological conditions such as air temperature, solar radiation, and wind speed. A very precise, iterative approach of one-minute time frame was used for meteorological data and heat transfer calculation, making the number of calculations numerous. Due to the complexity of the CFD approach, heat transfer through the air cavity in the CCF was simplified and observed as conductive.

2.7. Solar Radiation

For calculations of solar radiation, several pieces of information were required, including location data (local time zone, time zone, and geographical longitude and latitude). Atmospheric conditions at solar time, time equation, and time shift were accounted for using the date and local time. Climatic conditions obtained from Meteonorm software included solar azimuth, solar height (zenith), from which the declination of the sun, the angular incidence of the sun from the local meridian, and deviation of the earth’s orbit were calculated. Relevant input data describing the façade’s properties for further calculations included the inclination and azimuth of the façade surface. With this data, the angle of incidence of the sun on the surface can be calculated (Formula (15)).

$${\theta }_{sol;ic}=\arccos \left(\begin{array}{l}\sin (\delta )\times \sin ({\phi }_w)\times \cos ({\beta }_{ic})-\sin (\delta )\times \cos ({\phi }_w)\times \sin ({\beta }_{ic})\times \cos ({\gamma }_{ic})\\ +\times \cos (\delta )\times \cos ({\phi }_w)\times \cos ({\beta }_{ic})\times \cos ({\gamma }_{ic})+\cos (\delta )\times \sin ({\phi }_w)\times \sin ({\beta }_{ic})\times \cos ({\gamma }_{ic})\times \cos (\omega )\\ +\cos (\delta )\times \sin ({\beta }_{ic})\times \sin ({\gamma }_{ic})\times \sin (\omega )\end{array}\right), \left[°\right]$$

(15)

wherein θ_sol;ic (°) is the angle of incidence of the sun onto the façade element surface, δ (°) is the Sun declination, φ_w (°) is geographic longitude, β_ic (°) is inclination of the façade element surface, γ_ic (°) azimuth of the façade element surface, and ω is angular incidence of the Sun from the local meridian (°).

Global solar irradiance was divided into separate values, including direct solar irradiance (Formula (16)), diffuse solar irradiance (Formula (17)), albedo irradiance from the ground (Formula (18)), and circumsolar irradiance (Formula (19)). Each component’s quantity depends on various factors, such as weather conditions and the time of the day. Constants like Perez numbers were used to calculate the radiation on the inclined surface. Additionally, the clearness parameter, air mass, and sky brightness parameter were calculated using values of the direct and diffuse radiation onto the horizontal surface, the height of the Sun, the angle of incidence of the Sun onto the surface, zenith angle, and extraterrestrial irradiance [32,33,34].

$$I_{dir}=\max \left[0;G_{sol;dir}\times \cos ({\theta }_{sol;ic})\right], \left[\frac{\mathrm{W}}{{\mathrm{m}}^2}\right]$$

(16)

where I_dir (W/m²) is the direct radiation onto the surface of inclination β_ic (°), G_{sol; dir} (W/m²) is the direct radiation onto the horizontal surface and θ_sol;ic is the angle of incidence of the Sun onto the façade element surface (°). Diffuse solar irradiance is expressed by the following Formula (17):

$$I_{dif}=G_{sol;d}\left[(1-F_1)\frac{(1+(\cos ({\beta }_{ic}))}{2}+F_1\frac{a}{b}+F_2\sin ({\beta }_{ic})\right], \left[\frac{\mathrm{W}}{{\mathrm{m}}^2}\right]$$

(17)

where I_dif (W/m²) is the diffuse radiation onto the surface of inclination β_ic, while F₁ and F₂ are Perez numbers, and a and b are auxiliary Perez factors, all dimensionless values. Diffuse solar irradiance on the inclined surface due to ground reflection I_dif;grnd (W/m²) can be written mathematically using Formula (18):

$$I_{dif;grnd}=\left[G_{sol;dif}+G_{sol;dir}\times \sin ({\alpha }_{sol})\right]\times {\rho }_{sol,grnd}\frac{(1-(\cos ({\beta }_{ic}))}{2},\left[\frac{\mathrm{W}}{{\mathrm{m}}^2}\right]$$

(18)

where G_sol;dir (W/m²) is the direct radiation onto the horizontal surface, G_{sol; dif} (W/m²) is diffuse radiation onto the horizontal surface, α_sol (°) is the sun height, and ρ_sol,grnd (-) is the albedo, whose value depends on the ground surface cover. Circumsolar irradiance is calculated by Formula (19):

$$I_{circum}=G_{sol;dif}\times F_1\frac{a}{b}, \left[\frac{\mathrm{W}}{{\mathrm{m}}^2}\right]$$

(19)

Total direct solar irradiance onto the inclined surface I_dir;tot (W/m²) is a sum of a direct I_dir and circumsolar irradiance I_circum, which can be expressed by Formula (20):

$$I_{dir;tot}=I_{dir}+I_{circum}, \left[\frac{\mathrm{W}}{{\mathrm{m}}^2}\right]$$

(20)

The total diffuse solar irradiance onto the inclined surface I_dif;tot (W/m²) is the sum of the diffuse and reflected ground irradiance (ground albedo). However, circumsolar irradiance should be subtracted from the total diffuse radiation, as written in Formula (21).

$$I_{dif;tot}=I_{dif}-I_{circum}+I_{dif;grnd}, \left[\frac{\mathrm{W}}{{\mathrm{m}}^2}\right]$$

(21)

Finally, the total solar irradiance on the unshaded inclined surface is the sum of the total direct and total diffuse irradiance onto the inclined surface as shown in Formula (22).

$$I_{tot}=I_{dir;tot}+I_{dif;tot}, \left[\frac{\mathrm{W}}{{\mathrm{m}}^2}\right]$$

(22)

The last factor influencing total solar radiation on the inclined surface is the impact of the sunshade, where shaded parts of the façade lack direct radiation compared to the rest of the façade element [35,36,37,38]. Figure 8a shows different segments of global solar radiation—direct, diffuse horizontal, and extraterrestrial, along with the global horizontal irradiation as well as its part irradiated onto the façade [33].

2.8. Glazing Properties

Solar heat gain in the CCF elements depends primarily on the glazing properties and the thermal properties of the sun shading [18]. Although the sunshade is the main source of heat in the CCF, solar energy reaching the sunshade passes through one or two layers of glass. The choice of glazing properties, therefore, plays a greater role in solar gain control. The most relevant glazing properties include absorptance, reflectance, energy transmittance, and emissivity [39]. The mentioned characteristics were included in the model for each glass panel (three panels) and each side of the glass panel (six sides).

Absorptance represents a ratio of incident solar energy absorbed in the glass panel, reflectance is the ratio of reflected solar energy, transmittance ratio of the solar energy passing through the glazing and emissivity is the ratio of radiated energy compared to perfect emitter (blackbody).

2.9. CCF Elements Temperature

When the basic energy flows in the façade are recognized, the amount of energy in each component can be calculated. Known properties of the used materials, such as mass and specific heat capacity, allow for the determination of temperature. For the calculation of temperature, the specific heat formula (Formula (23)) is used [40].

$$T=\frac{Q}{cm}, \left[\mathrm{K}\right]$$

(23)

wherein Q (J) is the amount of energy in the component, c (J/kgK) is the specific heat capacity, and m (kg) is the mass of the component. The temperature was calculated for inner glass, air inside the cavity, sunshade slats, middle glass, outer glass, and façade frame (top, bottom, and sides), while the temperature of the argon layer was approximated as the arithmetic mean of the surrounding glass temperatures due to its very small thickness.

3. Results

For the validation of the thermal model of CCF, measured and calculated temperatures of the façade elements were compared. Measured data on-site included outer and inner air temperature, inner, middle, and outer glass temperature, and air gap temperature during winter southern orientation (Figure 10), winter western orientation (Figure 11), summer southern orientation (Figure 12), and summer western orientation (Figure 13). Additional meteorological data measured on site were wind speed, air pressure, and global solar radiation. Since the solar radiation in the model was calculated from direct and diffuse components, their ratio was estimated for the need of model validation.

Since only global solar radiation was measured on site, ratio between direct and diffuse solar radiation was approximated for each measurement data set that was to be validated. Measured data were imported into the model, and the results were compared for various seasons and orientations. The difference in the results was up to 3 °C which was accepted as a satisfying result. Occasional differences greater than 5 °C were attributed to the poor estimate of direct and diffuse solar radiation ratio. Approximated ratio between direct and diffuse solar radiation which gave the most desirable results, was taken as a referent one for all the days in the data set. During the period 3–6 January, the ratio between direct and diffuse solar radiation appeared to have varied greatly, what can be particularly observed during 5 and 6 January, western orientation, as shown in Figure 11d.

The results shown are for winter and summer periods and the southern and western orientations.

4. Discussion

The article presents the results of experimental and numerical simulations of the thermal behavior of passively maintained closed cavity façades during 2020–2023. Experimental model was placed in Zagreb, where measurements described in this paper were taken during 3–6 January and 18–27 June. Using simplified but unconventional methods, numerous formulae and variables, and software describing the thermal behavior of the CCF were conducted, which can be applicable to any other location, climatic conditions, and the façade’s orientation, dictating necessary façade element’s properties, respecting European standards.

Many studies in the last decade have been performed on real buildings, by monitoring parameters for longer periods on multilayered and close cavity façades. While several researchers were concerned with lowering the physical conditions of the façade such as Wüest and Luible, in 2016, who compared different glass and interspace thicknesses lowering excessive temperatures of argon from 39 °C to 34 °C only by changing the outer glass thickness. Interspace with sun blinds lowered the temperature from a non-ventilated cavity with 57 °C to 48 °C when ventilation was added [11]. Laverge, Schouwenaers, and Janssens proved in 2009, that mechanical ventilation will avoid condensation of the glass surfaces, but low airflow will not reduce excessive temperatures. They validated the thermal model of the sample by the good agreement of the new type of façade element with the European and American standard programs. Values of measured data and standard programs have the most differences in higher solar radiation, while during lower temperature rates, the differences are almost non-existing [1]. Van den Bossche and Lori made it a step further when examining Laverge’s model, creating a software which monitors actively maintained close cavity façades. Maximum error for the peak values showed in their software was in range between 5–10% [7].

Although the field of closed cavity façades is new, and there are a few researchers exploring and improving the design of CCF, the sustainability of CCF has never reached passive maintenance before. According to EN1991-1-4, a skin is considered impermeable when its permeability is lower than 0.1% of the skin surface, this façade is not fully closed. The thin capillary and outer IGU layers enable the passive functioning of the facade, using no external energy. The thermal behavior of the façade is validated both by measurements on-site and calculations in software created for this research. Maximum efficiency of the naturally ventilated façade performance is achieved during the hottest hours of the day, between 02 p.m. and 06 p.m. (Figure 8a), when the solar radiation has the highest value. Although numerous variables were taken into account, insufficient data regarding separate measuring direct and diffuse solar radiation, during cloudy days showed greater differences (5 and 6 January, western orientation, Figure 11d). Nevertheless, a comparison between the calculated and measured values of the layers’ temperatures shows the satisfying value difference of up to 3 °C, which is between 5–10% for peak values.

5. Conclusions and Future Work

This paper has focused on the mathematical model of the closed cavity façade and its validation through an experimental model. The primary advantage of this tool lies in its simplicity and ease of iteration, allowing users to enter data regarding the physical properties of the façade and verify outputs within the information regarding location, building typology, climate conditions etc. Another value lies in possible improvements to the software, introducing more complicated calculations such as FEM analysis, providing even greater accuracy.

The focus of the presented model is on thermal behavior since it represents a base for further research on the close cavity façades in terms of structural and hygric behavior. For the structural behavior, crucial is the exchange of the air between the surroundings and the façades interspace, inflating and deflating the air volume inside the cavity. With the outer air supply, moisture will be driven into the cavity as well, for which desiccant should be introduced inside the cavity to avoid excessive humidity. It is worth considering that in the future, this approach could be integrated with other engineering calculations, particularly concerning structural and hygric performance. Calculating the service life of the desiccant ensures that during this period, there will be no condensation in the cavity, i.e., the façade will fulfill its purpose. In this way, the service life expectancy can be calculated for the desiccants and, therefore, for the façades’ element.