Heat Transfer by Transmission in a Zone with a Thermally Activated Building System: An Extension of the ISO 11855 Hourly Calculation Method. Measurement and Simulation

Abstract

Water systems with pipes embedded in the horizontal concrete core slabs can be used for efficient space heating and cooling of passive and low-energy buildings. ISO 11855-4 describes the hourly simulation method of such systems while recommending to use other simulation tools to assess heat flow by transmission to the ambient environment. As it plays an important role in the thermal balance of a conditioned zone, this paper presents two calculation methods to obtain heat flow through the envelope. They were integrated with a general algorithm given in ISO 11855-4 and the simulation tool was developed. To validate the presented solution measurements were performed in a passive office building during the heating (November) and cooling (July) periods. The total heat transfer coefficient by transmission was measured and compared with the theoretical design value. Both proposed simulation algorithms provided results with very good accuracy. In the first period, the mean absolute of percentage error (MAPE) of the indoor air and floor temperatures amounted to 0.65% and 0.75%, respectively. Simulations showed that heating demand was covered mainly by the floor (28.7%), internal gains (21.7%), and ceiling (18.7%), while heat loss to the environment was mainly due to external partitions (94.0%). In the second period MAE and MAPE did not exceed 0.19 °C and 0.90%, respectively. Floor and ceiling were mainly responsible for heat gains removal (61%). Solar radiation was the main source (91%) of internal gains. The results obtained confirmed the assumptions taken. The simulation programme developed does not require the use of additional tools.

1. Introduction

Buildings are responsible for significant part of energy consumption in the economies of many countries. Therefore, numerous actions are taken to reduce it, especially in countries dependent on energy imports. This is done both politically and technically. In the first case, it takes the form of various legal regulations forcing actions towards improving energy efficiency. In the second case, the development of new technologies and technical solutions is observed.

These activities mean that the last decade has seen an increase in the number of newly constructed energy-efficient and passive buildings. Thanks to their low energy demand for space heating and cooling, they use successfully new solutions in the field of heating, ventilation and air conditioning (HVAC) systems on a large scale. These include, among others, various water-based radiant heating and cooling systems, in which pipes carrying water as a heat transfer medium are embedded in a building’s structure (floor, wall, or ceiling). They are commonly classified as Thermally Activated Building Systems (TABS) [1] and in recent decades they have gained significant interest because of favourable indoor thermal conditions and cooperation with heat pumps utilising renewable energy sources available on-site [2,3].

The design, dimensioning, installation, and control-related issues of these systems are described in the set of ISO 11855 standards. ISO 11855-2 [4] categorises them into groups labelled from A to G. An interesting solution is Type E, in the form of a concrete horizontal slab with embedded pipes and placed between two storeys, and may exchange heat with both adjacent zones: below (as a ceiling) and above (as a floor) [5].

In ISO 11855-4 [6] four sizing methods of these systems are listed. These are a rough method, simplified sizing by diagrams, simplified model based on Finite Difference Method (FDM) and dynamic building simulation programmes. This wide range of possibilities has resulted in different methods of analysing TABS systems.

An important part of studies devoted to TABS focused on thermal comfort. In [7] indoor conditions in an office room with TABS and balanced mechanical ventilation were analysed in terms of the predicted mean vote (PMV) and the predicted percent of dissatisfied (PPD) indicators were estimated based on the measurements conducted from July to November. The results showed a low variation of the indoor air temperature during working days, between 22.5 °C to 23.1 °C.

Behrendt [8] used the method from ISO 11855-4 to develop in C++ a simulation programme to obtain temperatures in a room with a radiant concrete core floor and ceiling. The object of the analysis was a virtual room (floor surface of 30 m²). The author considered many cases, differing in the duration of the numerical analysis, heat load and ceiling parameters. The results from this programme were compared with those from the commercial software IDA ICE 4.5. The differences of the internal air and ceiling surface temperatures from both tools did not exceed 0.5 °C.

Ning et al. [9] simulated of pipe-embedded radiant systems using the response factor method (RFM). The results from RFM were very close to those from the ISO 11855 standard and CFD simulations. However, the authors did not provide further details on the implementation of the ISO method.

Domínguez Lacarte and Fan [10] studied acoustic comfort in a room with TABS. Simulations in CFD were validated experimentally. Two types of free-hanging ceiling absorbers were studied and their impact on the thermal performance of the TABS was analysed. They found that sound absorbers caused cold air stagnation and reduced the cooling performance of TABS.

Some works involved other tools to study the dynamic performance of a zone with concrete core slabs. Henze et al. [11] analysed comfort issues and primary energy use of a ventilation-assisted TABS system in an office building. Then, the comparison was done in relation to a conventional all-air variable air ventilation system. Simulations were performed in TRNSYS.

Nageler et al. [12] validated four building energy simulation tools (Dymola, EnergyPlus, IDA ICE and TRNSYS) using a test box with TABS. The error analysis of internal air temperature showed RMSE from 0.87 K (IDA ICE) to 2.3 K (EnergyPlus).

Chandrashekar and Kumar [13] analysed the thermal performance and energy consumption of the thermally activated building system integrated with a chiller in an educational building. In comparison to the base system with the chiller only, the authors found that TABS with the chiller had a peak monthly energy savings of 6%.

Li et al. [14] studied the performance of a building with TABS supplied by the ground source heat pump using a thermal network model. Comparison with measurements showed the maximum relative error of 1.48% of the room air temperature.

Chen and Li [15] developed a state-space model for model predictive control (MPC) strategy of a ceiling cooling system in an experimental cell. The MAE and RMSE of the indoor air temperature amounted 0.15 °C and 0.24 °C, respectively.

Hu et al. [16] analysed an apartment in a multifamily building in Denmark with floor heating. They developed an MPC controller taking into account energy prices. Simulated in TRNSYS and measured indoor air temperature varied from 20 °C to 25 °C at a set point of 22 °C. MAE and RMSE for indoor air temperature between simulation and measurements were 0.118 °C and 0.170 °C, respectively.

Vivek and Balaji [17] analysed indoor thermal comfort in an educational building with and without TABS in cooling mode. In the second case, they noticed that floor temperature decreased by 1.5 °C and was from 29.2 °C to 29.7 °C on average.

Chandrashekar and Kumar [18] studied the impact of different types of flooring on the performance of TABS in an educational building during summer. It varied from 27.9 °C to 31.0 °C and from 27.4 °C to 29.0 °C, with a daily average of 29.9 °C and 28.5 °C for vinyl and granite floors, respectively.

Zhen et al. [19] presented measurements from the airport with floor heating in the terminal. For a chosen sunny winter day the floor surface temperature was from 18.3°C to 24.8 °C. Its uniformity, defined by the maximum difference between measuring points, was up to 3.0 °C on average.

Mugnini [20] et al. studied control techniques for heat pumps with radiant systems using MPC technique. Two validation periods were used with the resulting RMSE of the floor temperature of 0.49 °C and 0.70 °C, respectively.

Summing up the presented review it can be concluded that both simulations and experiments were presented when analysing the dynamic performance of buildings with TABS. However, experimental studies were done mainly in laboratory conditions in various test cells, experimental chambers and laboratory rooms, with insulation of external partitions to reduce impact of ambient environment [10,12,21,22,23]. Simulations [9,24] focused on design and performance aspects of these systems.

The question then becomes how simulation models will perform when studying operation of TABS in real everyday conditions. In [8] the implementation of the ISO 11855-4 hourly method in a simulation programme was described. However, it focused mainly on its detailed discussion and on a comparison of the waveforms of selected temperatures in a virtual test room over the one day. This method has neither been analysed in terms of its practical usability nor experimentally validated so far, even though this is desirable, since ISO 11855 is the basis for the sizing of TABS systems. These conclusions provide the background for the formulation of the research problems.

The first of them relates to the implementation and validation of the hourly method. While providing a general framework for dynamic simulation of a building zone with TABS, ISO 11855-4 does not impose any simulation tool, programming language or development environment. Therefore, it is up to the user to choose the method of solution.

In this study, freely available Python language was selected. It has already been used in the modelling of integrated energy systems (including TABS) [25] or control of TABS [26,27], confirming its usefulness in these applications.

The next issue concerns heat flow by transmission between the interior and exterior environments. This variable is of particular importance in passive or low-energy buildings, as it directly affects their energy performance. However, to obtain heat flow to the ambient, ISO 11855-4 recommends using other commercial software, which enables computing of the cooling loads in a zone at a constant indoor temperature. This approach is troublesome for the user, because it requires an additional tool. What is more, the constant indoor temperature is a simplification, which may result in errors.

Therefore, this paper presents two novel methods to obtain the heat flow by transmission between the analysed zone and the ambient environment within the same simulation programme without the necessity to use other tools. Each of these methods was included into the general algorithm presented in the standard. Then, results were compared with measurements according to the research plan outlined in Figure 1.

At first, measurements were conducted to estimate the heat transfer coefficient by transmission between the indoor and the outdoor environment (H_T,ie) in a real office room. This parameter was also calculated from the theoretical design parameters. Then, both values were compared and the resulting one was used in further considerations. In the next step, a simulation tool was developed and two algorithms to compute heat flow by transmission were implemented. Then, based on measurements and calculations, the error analysis was performed. Finally, concluding remarks were given.

2. Materials and Methods

2.1. Research Object and Research Algorithm

The measurement campaign was done in a passive building presented in Figure 2. It is located in Katowice in south Poland. Construction works began in August 2011, and the building was commissioned in February 2014, being the first passive office building in Poland. In May 2013, it received the Green Building award, granted by the European Commission for the most energy-efficient buildings in Europe. The total area of the building is 8100 m², and the usable area is 7500 m², including 5500 m² of offices and about 1220 m² of laboratories and a data centre. It has five floors, including the basement.

To achieve low energy demand for heating and cooling, many solutions were designed and implemented in the building. Care was taken to use daylight by the arrangement of office rooms on the perimeter of the building. Energy-efficient three-pane windows were mounted. Additionally, facade blinds stop the penetration of sunlight and protect the rooms from overheating. The external walls were insulated with a 30 cm thick layer of polystyrene.

Four air handling units (AHU) with rotary heat exchangers serve office, conference, technical and social rooms. Sanitary facilities and toilets are supplied with ventilation air from the fifth AHU, with a separate air intake and exhaust, located on the roof.

Tap water is heated by 10 vacuum solar collectors. Space heating and cooling of the building are provided mainly by TABS (in all ceilings), underfloor heating (in the basement) and auxiliary by the fan-coil units and heating and cooling coils in all air handling units. They are supplied by the set of three two-compressor water/water heat pumps with total rated heating and cooling capacities of 244 kW and 187 kW, respectively. An additional chiller was also used as a peak cold source. There is no air humidification system. The hydraulic diagram of the HVAC system is presented in Figure 3.

TABS consists of polymer pipes (internal diameter of 20 mm, the wall thickness of 2 mm, and thermal conductivity of 0.35 W/(m∙K)) which are embedded in a massive concrete layer of each ceiling (Figure 4). In the basement floor an additional bottom insulation layer was used to reduce ground heat loss.

Solar photovoltaic modules, with a total peak power of 107 kW, reduce electricity supply from the external grid. They were mounted on the roof (231 modules), on the facade (188 modules), and on three trackers in front of the building (36 modules).

The building management system (BMS) supports monitoring and energy management in the building. More details on this object can be found in [7,28].

2.2. Measurements

2.2.1. Measurement Campaign

During the research, a west-orientated office room, located on the second floor, was used (Figure 5). It was 7.95 m long, 5.80 m wide and 3.1 m high. It was located on the west side of the building.

Measurements were done in two periods: in November (10 days) and in July (6 days). In the first period the ambient temperature was from 1.9 °C (hour 204) to 17.7 °C (hour 44). The sky was cloudy. Solar irradiance incident on the external wall and entering the room was up to 96.0 W/m² and 25.1 W/m², respectively, at hour 117 (Figure 6).

During the second period typical summer conditions were observed (Figure 7). Outdoor air temperature varied from 10.9 °C (hour 122) to 24.7 °C (hour 64). Solar irradiance falling on the external wall and entering the room was up to 351 W/m² and 133.8 W/m² (hour 115), respectively.

The measuring equipment (

Table 1. The main parameters of the sensors and data logger used.

Device	Measured Variable	Measurement Range	Accuracy
Pt100 resistance sensor	Indoor air temperature	−50 °C … +150 °C	Class AA 1
Pt100 resistance sensor	Surface temperature	−50 °C … +150 °C	Class AA 1
Pt1000 resistance sensor	Ambient air temperature	−50 °C … +150 °C	Class A 1
Pt100 resistance sensor	Globe temperature	−30 °C … +120 °C	Class A 1
LP PYRA03	Solar irradiance	0 … 2000 W/m2	Spectrally Flat Class C 2
HFP01 Hukseflux	Window heat flux	−2000 … 2000 W/m2	±3% 3
HFP03 Hukseflux	Wall heat flux	−2000 … 2000 W/m2	±6% 3
Fluke 2638A data logger	Voltage input	0 … 100 mV	0.0025% MV + 0.0035% FS + 2 μV 4
Fluke 2638A data logger	Temperature input	−50 °C … +150 °C	0.038 °C at 0 °C, 0.073 °C at 300 °C

¹ According to EN 60751 [29]. ² According to ISO 9060 [30] and IEC 61724 [31]. ³ Calibration uncertainty declared by the manufacturer. ⁴ MV—measured value; FS—full scale.

) was arranged as follows. Platinum resistance sensors were used in temperature measurements. In the case of walls and windows, these were flat sensors. The globe sensor (diameter of 150 mm) was attached to a tripod, 1.5 m above the floor and placed at the centre of the room. The indoor air temperature was measured on the same tripod at three heights of 0.55 m, 1.5 m and 2.2 m above the floor. Cylindrical sensors (3 mm diameter) with low-emission shields, protecting against direct sunlight, were used. For the same reason, the sensors on the outside surface of the wall also had attached aluminium foil shields.

Heat flux was measured on the wall and the window (Figure 8). Because of low expected values of the measured heat flux in the first case, the high-sensitivity HFP03 sensor (white coloured, on the left) was used. According to the manufacturer, its declared sensitivity [32] is 500 μV/(W/m²). The second heat flux sensor, HFP01 [33] (red coloured, on the right), has a ten times lower nominal sensitivity and was used for comparative purposes. Heat flow through the window was measured with the next HFP01 sensor. To protect it from direct sunlight, an aluminium foil shield was glued to the outside surface of the window, at the height of the sensor installation.

Global solar irradiance incident on the external wall and entering the room was measured by two pyranometers. All sensors were served by the Fluke 2638A data logger.

This full set of measurements was performed only in the first period during autumn. In the second period only the most important variables were measured, namely: indoor air temperature (at three heights), globe temperature, ambient temperature, floor and ceiling temperature and solar irradiance (external and internal). They were used to compare the performance of the developed tool in different weather conditions.

2.2.2. Uncertainty Analysis

The thermal resistance of the wall (R_wall), or other partition, is obtained from an indirect measurement [34], according to the following equation:

$${\mathrm{R}}_{\mathrm{wall}}={\displaystyle \frac{{\mathrm{T}}_{\mathrm{w},\mathrm{i}}-{\mathrm{T}}_{\mathrm{w},\mathrm{e}}}{\mathrm{q}}},$$

(1)

where T_w,i and T_w,e are the temperatures of the internal and external surfaces of the wall, respectively, and q is the heat flux flowing through this wall.

If the measured physical quantity, y, is a function of independent measurements x₁, x₂, …, x_n and is given by:

$$\mathrm{y}=\mathrm{f}\left(x_1,x_2,\dots ,x_n\right),$$

(2)

then, the standard combined uncertainty (u_c) of y is calculated using the propagation model of uncertainty [35,36,37,38]:

$${\mathrm{u}}_{\mathrm{c}}\left(\mathrm{y}\right)=\sqrt{{\left({\displaystyle \frac{\partial \mathrm{y}}{\partial {\mathrm{x}}_1}}\mathrm{u}\left({\mathrm{x}}_1\right)\right)}^2+{\left({\displaystyle \frac{\partial \mathrm{y}}{\partial {\mathrm{x}}_2}}\mathrm{u}\left({\mathrm{x}}_2\right)\right)}^2+\dots +{\left({\displaystyle \frac{\partial \mathrm{y}}{\partial {\mathrm{x}}_{\mathrm{n}}}}\mathrm{u}\left({\mathrm{x}}_{\mathrm{n}}\right)\right)}^2}.$$

(3)

The expanded uncertainty of y is obtained from:

$$\mathrm{U}=\mathrm{k} ·{\mathrm{u}}_{\mathrm{c}}\left(\mathrm{y}\right)$$

(4)

where k is the coverage factor, and k = 2 for 95% confidence level of uncertainty.

In the considered case, when inserting Equation (1) into Equation (3), we get:

$${\mathrm{u}}_{\mathrm{c}}\left({\mathrm{R}}_{\mathrm{wall}}\right)=\sqrt{{\left({\displaystyle \frac{\partial {\mathrm{R}}_{\mathrm{wall}}}{\partial {\mathrm{T}}_{\mathrm{w},\mathrm{i}}}}\mathrm{u}\left({\mathrm{T}}_{\mathrm{w},\mathrm{i}}\right)\right)}^2+{\left({\displaystyle \frac{\partial {\mathrm{R}}_{\mathrm{wall}}}{\partial {\mathrm{T}}_{\mathrm{w},\mathrm{e}}}}\mathrm{u}\left({\mathrm{T}}_{\mathrm{w},\mathrm{e}}\right)\right)}^2+{\left({\displaystyle \frac{\partial {\mathrm{R}}_{\mathrm{wall}}}{\partial \mathrm{q}}}\mathrm{u}\left(\mathrm{q}\right)\right)}^2},$$

(5)

with the partial derivatives as follows:

$${\displaystyle \frac{\partial {\mathrm{R}}_{\mathrm{wall}}}{\partial {\mathrm{T}}_{\mathrm{w},\mathrm{i}}}}={\displaystyle \frac{1}{\mathrm{q}}},$$

(6)

$${\displaystyle \frac{\partial {\mathrm{R}}_{\mathrm{wall}}}{\partial {\mathrm{T}}_{\mathrm{w},\mathrm{e}}}}={\displaystyle \frac{-1}{\mathrm{q}}},$$

(7)

$${\displaystyle \frac{\partial {\mathrm{R}}_{\mathrm{wall}}}{\partial \mathrm{q}}}={\displaystyle \frac{{\mathrm{T}}_{\mathrm{w},\mathrm{e}}-{\mathrm{T}}_{\mathrm{w},\mathrm{i}}}{{\mathrm{q}}^2}}.$$

(8)

Measurement uncertainty of temperature depends on the accuracy of a temperature sensor and the accuracy of a data logger. In general, it can be written as:

$$\mathrm{u}\left(\mathrm{T}\right)=\sqrt{{\left({\mathrm{u}}_{\mathrm{sensor}}\right)}^2+{\left({\mathrm{u}}_{\mathrm{logger}}\right)}^2}.$$

(9)

Logger parameters are given in Table 1. Uncertainty of a sensor can be estimated using the data given in the IEC 60751 [29]. This standard defines tolerance of the wire wound and platinum resistors within the temperature range from −50 °C to +250 °C and from 0 °C to +150 °C, respectively, for the tolerance class AA:

$$\pm \left(0.1+0.0017\left|\mathrm{t}\right|\right).$$

(10)

where: t is the measured temperature, in °C.

In the case w of the heat flux sensors, all necessary data are given in Table 1 and the manufacturer’s manuals [32,33]. It should be noted here, that both HFP sensors operated in thermal conditions in the room close to the calibration temperature.

2.3. Simulation Model

2.3.1. General Description of the Model

The mathematical model given in ISO 11855-4 [6] includes internal walls and air in a zone, and a slab. Several assumptions were made in this model:

• The slab must be without gaps (e.g., suspended ceiling, free spaces under the floor);
• Three-dimensional heat transfer phenomena in a slab were simplified to one-dimensional. Consequently, homogeneous temperature distribution in a slab in the horizontal direction was assumed;
• Radiant heat gains are evenly distributed on all internal surfaces;
• Thermal conditions in adjacent zones are the same as in the studied one;

Applying these assumptions, the slab is divided above and below the level of the embedded pipes level into two parts, upper and lower, respectively. During the discretisation procedure the slab is subdivided into thermal nodes. The nodes are also assigned to internal walls and internal air [8]. Floor and ceiling nodes are connected with a surrounding zone by the relevant heat transfer coefficients and the slab-zone model, in the form of a resistance-capacitance thermal network, is created (Figure 9).

The standard gives the general implicit scheme to solve this network. In general, the nodal temperature in the p-th node at h-th hourly time step is given by:

$$\begin{array}{l}{\mathrm{T}}_{\mathrm{p}}^{\mathrm{h}}\\ ={\displaystyle \frac{{\mathrm{H}}_{\mathrm{A}}{\mathrm{T}}_{\mathrm{A}}^{\mathrm{h}}+{\mathrm{H}}_{\mathrm{IWS}}{\mathrm{T}}_{\mathrm{IWS}}^{\mathrm{h}}+{\mathrm{H}}_{\mathrm{F}}{\mathrm{T}}_{\mathrm{F}}^{\mathrm{h}}+{\mathrm{H}}_{\mathrm{C}}{\mathrm{T}}_{\mathrm{C}}^{\mathrm{h}}+{\mathrm{H}}_{\mathrm{Rad}}{\mathrm{Q}}_{\mathrm{Rad}}^{\mathrm{h}}+{\mathrm{H}}_{\mathrm{Conv}}{\mathrm{Q}}_{\mathrm{Conv}}^{\mathrm{h}}+{\mathrm{H}}_{\mathrm{CondUp}}{\mathrm{T}}_{\mathrm{p}-\mathrm{1}}^{\mathrm{h}}+{\mathrm{H}}_{\mathrm{CondDown}}{\mathrm{T}}_{\mathrm{p}}^{\mathrm{h}+1}+{\mathrm{H}}_{\mathrm{Inertia}}{\mathrm{T}}_{\mathrm{p}}^{\mathrm{h}-1}+{\mathrm{H}}_{\mathrm{Circuit}}{\mathrm{T}}_{\mathrm{Water}\mathrm{In},\mathrm{h}}^{\mathrm{h}}{\mathrm{f}}_{\mathrm{rm}}^{\mathrm{h}}}{{\mathrm{H}}_{\mathrm{A}}+{\mathrm{H}}_{\mathrm{IWS}}+{\mathrm{H}}_{\mathrm{F}}+{\mathrm{H}}_{\mathrm{C}}+{\mathrm{H}}_{\mathrm{CondUp}}+{\mathrm{H}}_{\mathrm{CondDown}}+{\mathrm{H}}_{\mathrm{Inertia}}+{\mathrm{H}}_{\mathrm{Circuit}}{\mathrm{f}}_{\mathrm{rm}}^{\mathrm{h}}}}\end{array}$$

(11)

The necessary relationships to obtain heat transfer coefficients (H) between nodes are given in the standard. ${\mathrm{f}}_{\mathrm{rm}}^{\mathrm{h}}$ = 1 or 0 when the system is switched on/off, respectively.

Next, the actual nodal temperatures ($T_p^h$) are compared with the ones from the previous iteration $\left({T_p^h}^{\prime }\right)$ to obtain the resulting current tolerance:

$$\mathsf{\xi }={\displaystyle \sum }_{\mathrm{p}}\left|{\mathrm{T}}_{\mathrm{p}}^{\mathrm{h}}-{{\mathrm{T}}_{\mathrm{p}}^{\mathrm{h}}}^{\prime }\right|$$

(12)

This tolerance is then compared with its maximum allowed value. If $\xi$ is lower, there are computed: heat flow through the floor ($Q_F^h$), the ceiling ($Q_C^h$) and the internal wall ($Q_{IWS}^h$) surfaces, the mean radiant temperature ($T_{MR}^h)$ and the operative temperature ($T_{Op}^h$) in a zone, and the water temperature at the coil outlet ($T_{Water,Out}^h$), and calculations follow to the next time step. If the current tolerance is too high, the next iteration is executed up to the maximum number of iterations allowed.

2.3.2. Input Conditions

As shown in Equation (11), this method requires several input parameters and variables for calculations. The heat transfer coefficients (H) depend on the type of node under consideration. They can be obtained from the thermo-physical and geometric data of all partitions in a zone. The standard provides instructions on how to calculate them. The inlet water temperature was provided by the building management system (BMS).

The next are internal heat gains in a zone due to radiation and convection. They are given by the following relationships:

$${\mathrm{Q}}_{\mathrm{Rad}}^{\mathrm{h}}=0.85\cdot {\mathrm{Q}}_{\mathrm{Transm}}^{\mathrm{h}}+{\mathrm{Q}}_{\mathrm{IntRad}}^{\mathrm{h}}+{\mathrm{Q}}_{\mathrm{Sun}}^{\mathrm{h}},$$

(13)

$${\mathrm{Q}}_{\mathrm{Conv}}^{\mathrm{h}}=0.15\cdot {\mathrm{Q}}_{\mathrm{Transm}}^{\mathrm{h}}+{\mathrm{Q}}_{\mathrm{IntConv}}^{\mathrm{h}}+{\mathrm{Q}}_{\mathrm{PrimAir}}^{\mathrm{h}}.$$

(14)

To perform calculations, the hourly values of these gains should be provided by the user. The standard does not give any guidance in this regard and it is up to the user to choose the method to determine them.

Internal gains by radiation ($Q_{IntRad}^h$) and convection ($Q_{IntConv}^h$) can be estimated based on the activity of occupants and equipment. Primary air heat gains ($Q_{PrimAir}^h$) results from the operation of the ventilation system. Solar gains can be obtained from glazing area (A_gl) and measured solar irradiance entering a zone (I_sol):

$${\mathrm{Q}}_{\mathrm{Sun}}^{\mathrm{h}}={\mathrm{I}}_{\mathrm{sol}}^{\mathrm{h}}{\mathrm{A}}_{\mathrm{gl}},$$

(15)

or from solar irradiance incident on the external surface of a window (I_sol,ext) and a solar transmittance of glazing:

$${\mathrm{Q}}_{\mathrm{Sun}}^{\mathrm{h}}={\mathrm{I}}_{\mathrm{sol},\mathrm{ext}}^{\mathrm{h}}{\mathrm{g}}_{\mathrm{gl}}.$$

(16)

2.3.3. Heat Transfer by Transmission

Heat loss from a room (heated space) by transmission, Q_Transm, can be determined along the PN-EN 12831 [39] standard. The equation to get the design heat loss, given in the standard, has the form:

$${\mathrm{Q}}_{\mathrm{Transm}}^{\mathrm{h}}=\left({\mathrm{H}}_{\mathrm{T},\mathrm{ie}}+{\mathrm{H}}_{\mathrm{T},\mathrm{iue}}+{\mathrm{H}}_{\mathrm{T},\mathrm{ig}}+{\mathrm{H}}_{\mathrm{T},\mathrm{ij}}\right)·\left({\mathrm{T}}_{\mathrm{int},\mathrm{i}}-{\mathrm{T}}_{\mathrm{e}}\right),$$

(17)

with:

H_T,ie—heat loss coefficient by transmission from the heated space to the surroundings through the building envelope, W/K,

H_T,iue—heat loss coefficient by transmission from the heated space to the surroundings through the unheated space, W/K,

H_T,ig—heat loss coefficient by transmission from the heated space to the ground in steady-state conditions, W/K,

H_T,ij—heat loss coefficient by transmission from the heated space to the adjacent heated space with a significantly different temperature, W/K,

T_e—design outdoor temperature, °C,

T_int,i—design internal temperature of the heated space, °C.

According to PN-EN 12831, heat flow in Equation (17) is positive when heat is removed from a heated zone. Following ISO 11855-4, transmission gain should inform about the amount of heat delivered to a zone, and therefore when removing heat from a room (cooling), the heat flux should have a negative value. For this reason in further considerations heat flux has the opposite sign than in Equation (17).

Because the tested room is located on the second floor, heat losses through the ground were omitted. Additionally, all rooms adjacent to the tested one are heated, and the temperature in them is similar to the temperature in the tested room, therefore there are no heat losses through unheated spaces and to adjacent heated spaces with significantly different temperatures. On this basis, it was assumed that the coefficients H_T,iue, H_T,ig and H_T,ij are equal to zero.

At the same, for consistency of the symbols used in Equation (11), T_A should be used instead of T_i. After taking into account all assumptions, Equation (17) becomes:

$${\mathrm{Q}}_{\mathrm{Transm}}^{\mathrm{h}}={\mathrm{H}}_{\mathrm{T},\mathrm{ie}}\left({\mathrm{T}}_{\mathrm{e}}^{\mathrm{h}}-{\mathrm{T}}_{\mathrm{A}}^{\mathrm{h}}\right).$$

(18)

The method to determine the heat loss coefficient by transmission to the ambient environment (H_T,ie) is also given in PN-EN 12831:

$${\mathrm{H}}_{\mathrm{T},\mathrm{ie}}={\displaystyle \sum }_{\mathrm{k}}{\mathrm{A}}_{\mathrm{k}}·{\mathrm{U}}_{\mathrm{k}}·{\mathrm{e}}_{\mathrm{k}}+{\displaystyle \sum }_{\mathrm{l}}{\mathsf{\Psi }}_{\mathrm{l}}·{\mathrm{l}}_{\mathrm{l}}·{\mathrm{e}}_{\mathrm{l}},$$

(19)

where:

A_k—area of building element k, m²,

U_k—heat transfer coefficient of partition k, W/(m²K),

Ψ_l—heat transfer coefficient of linear thermal bridge l, W/(m·K),

l_l—length of linear thermal bridge l, m,

e_k, e_l—correction factors taking into account climate influences (used when climate influences were not taken into account when calculating U_k).

Following the PN-EN 12831 standard, the thermal transmittance of windows should be calculated based on the ISO 10077-1 standard [40] from the following equation:

$${\mathrm{U}}_{\mathrm{window}}={\displaystyle \frac{{\mathrm{A}}_{\mathrm{g}}·{\mathrm{U}}_{\mathrm{g}}+{\mathrm{A}}_{\mathrm{f}}·{\mathrm{U}}_{\mathrm{f}}+{\mathrm{l}}_{\mathrm{g}}·{\mathsf{\Psi }}_{\mathrm{g}}}{{\mathrm{A}}_{\mathrm{g}}+{\mathrm{A}}_{\mathrm{f}}}},$$

(20)

where:

A_g—glazing area, m²,

A_f—frame area, m²,

U_g—thermal transmittance of glazing, W/m²K,

U_f—thermal transmittance of frame, W/m²K,

Ψ_g—heat transfer coefficient of linear thermal bridge due to the combined thermal effects of the edge of glazing, spacer and edge of frame, W/(m·K),

l_g—length of linear thermal bridge l, m,

The values of the heat transfer coefficient were calculated based on the PN-EN ISO 6946 [41] standard for the masonry part of the wall from the following relationship:

$${\mathrm{U}}_{\mathrm{wall}}={\displaystyle \frac{1}{{\mathrm{R}}_{\mathrm{si}}+{{\displaystyle \sum }}_{\mathrm{i}}{\mathrm{R}}_{\mathrm{i}}+{\mathrm{R}}_{\mathrm{se}}}},$$

(21)

with:

R_si—heat transfer resistance on the internal surface of the wall, (m²K)/W,

R_i—thermal resistance of the i-th layer of the wall, (m²K)/W,

R_se—heat transfer resistance on the external surface of the wall, (m²K)/W.

The heat transfer resistances on the inner and outer wall surfaces can be taken from EN ISO 6946. Here: R_se = 0.04 m²K/W and R_si = 0.13 m²K/W were assumed.

The properties of each layer of the external wall are presented in Table 2. The thermal resistance of the i-th layer of the wall is calculated from the formula:

$${\mathrm{R}}_{\mathrm{i}}={\displaystyle \frac{{\mathsf{\delta }}_{\mathrm{i}}}{{\mathsf{\lambda }}_{\mathrm{i}}}},$$

(22)

where:

δ_i—thickness of i-th layer, m,

λ_i—thermal conductivity coefficient of i-th layer, W/(m·K).

Table 2. Geometric and thermal properties of materials used in partitions.

Partition	Material	Thickness (mm)	Thermal Conductivity W/(m·K)
Floor	Floor covering	6	0.1875
	Cement screed	20	1.4
	Reinforced concrete	300	1.9
	Gypsum plastering	5	1.18
Internal wall	Gypsum board	25	0.24
	Mineral wool	70	0.04
	Gypsum board	25	0.24
External wall	External plaster	5	0.70
	Styrofoam	300	0.031
	Ceramic blocks	300	0.23
	Internal plaster	15	0.18

Table 2. Geometric and thermal properties of materials used in partitions.

Partition	Material	Thickness (mm)	Thermal Conductivity W/(m·K)
Floor	Floor covering	6	0.1875
	Cement screed	20	1.4
	Reinforced concrete	300	1.9
	Gypsum plastering	5	1.18
Internal wall	Gypsum board	25	0.24
	Mineral wool	70	0.04
	Gypsum board	25	0.24
External wall	External plaster	5	0.70
	Styrofoam	300	0.031
	Ceramic blocks	300	0.23
	Internal plaster	15	0.18

During in-situ heat transfer measurements, thermal conditions are usually variable over time. In our case, the temperature of the internal wall surface varied within small range, i.e., from 20.6 °C to 21.9 °C. However, as given in Figure 6, ambient temperature was from 1.9 °C to 17.7 °C. Also, the measured heat flux varied between 0.1 and 2.6 W/m².

In the case of massive building constructions, as external walls, these transient conditions may significantly influence the measurement procedure. For these reasons, it is advisable to use one of the methods suitable for the varying measurement conditions to calculate the thermal resistance of the external wall.

Such methods have been developed in recent decades [42]. Considering the ease of preparing the necessary input data and the computational cost, two dynamic methods were selected for use in this study.

The first of them is the regression model, proposed by Anderlind [43,44,45] and also known as the GullfibR Model. This model takes into account the variation of heat flux by transmission flowing through the considered partition in each, j-th time step (q_j) using three components (Equation (23)):

$${\mathrm{q}}_{\mathrm{j}}={\displaystyle \frac{{\mathrm{T}}_{\mathrm{wi},\mathrm{j}}-{\mathrm{T}}_{\mathrm{we},\mathrm{j}}}{{\mathrm{R}}_{\mathrm{wall}}}}+{\displaystyle \sum }_{\mathrm{l}=\mathrm{j}-\mathrm{p}}^{\mathrm{j}-1}{\mathrm{A}}_{\mathrm{l}}\left({\mathrm{T}}_{\mathrm{wi},\mathrm{l}+1}-{\mathrm{T}}_{\mathrm{wi},\mathrm{l}}\right)+{\displaystyle \sum }_{\mathrm{l}=\mathrm{j}-\mathrm{p}}^{\mathrm{j}-1}{\mathrm{B}}_{\mathrm{l}}\left({\mathrm{T}}_{\mathrm{we},\mathrm{l}+1}-{\mathrm{T}}_{\mathrm{we},\mathrm{l}}\right),$$

(23)

The first element on the right-hand side of this relationship describes the steady-state heat flow during each j-th measurement time-step. The second is used to account for historical temperature changes on the internal surface of the wall. The third element, in turn, models the impact of temperature changes on the external surface. These instantaneous, varying parts, are supposed to model the transient behaviour of the analysed heat flow. Thus, their separation from the total measured value shall result in the stationary part, given by the resistance R_wall.

A_l and B_l in Equation (23) are the regression coefficients and they can be obtained from the multiple regression analysis. This method is easy in application also in popular spreadsheets. The only parameter that the user should set before performing calculations is the number of historical measurement points (time steps) preceding the start of the analysis, which is given by the “p” parameter.

When the regression coefficients, A_l and B_l, are known, then they can be inserted into Equation (23) and the thermal resistance of the wall can be obtained for each time step. Then, the average R_wall and the standard deviation can be computed.

The second model, called the Pentaur Model, is an extension of the previous one and it was also proposed by Anderlind. It is base on the similar assumptions as the previous one. However, there was also added the third part, which also takes into account the variable heat flux flow in the previous time steps:

$$\begin{gathered} {\mathrm{q}}_{\mathrm{j}}={\displaystyle \frac{{\mathrm{T}}_{\mathrm{wi},\mathrm{j}}-{\mathrm{T}}_{\mathrm{we},\mathrm{j}}}{{\mathrm{R}}_{\mathrm{wall}}}}+{\displaystyle \sum }_{\mathrm{l}=\mathrm{j}-\mathrm{p}}^{\mathrm{j}-1}{\mathrm{A}}_{\mathrm{l}}\left({\mathrm{T}}_{\mathrm{wi},\mathrm{l}+1}-{\mathrm{T}}_{\mathrm{wi},\mathrm{l}}\right)+{\displaystyle \sum }_{\mathrm{l}=\mathrm{j}-\mathrm{p}}^{\mathrm{j}-1}{\mathrm{B}}_{\mathrm{l}}\left({\mathrm{T}}_{\mathrm{we},\mathrm{l}+1}-{\mathrm{T}}_{\mathrm{we},\mathrm{l}}\right) \\ +{\displaystyle \sum }_{\mathrm{l}=\mathrm{j}-\mathrm{p}}^{\mathrm{j}-1}{\mathrm{C}}_{\mathrm{l}}\left({\mathrm{q}}_{\mathrm{l}+1}-{\mathrm{q}}_{\mathrm{l}}\right), \end{gathered}$$

(24)

Further processing to obtain R_wall and the standard deviation is the same as in the previous model.

A modification of the general model given in ISO 11855-4 is the proposal to model heat transfer by transmission through an external wall using a dynamic model. For this purpose, the 2R1C thermal network model developed by Lorenz and Masy [46,47] was used. It can be solved using popular spreadsheets.

The total thermal resistance of the wall (R_wall), composed of n layers, is divided into two parts, the internal (R_i) and the external (R_e). At their junction a capacitor is connected (Figure 10). It represents the heat capacity of this partition.

R_wall is obtained from the model of Anderlind, presented above. The ratio R_i to R_e is calculated based on the physical parameters of the wall according to the following relationships:

$${\mathrm{R}}_{\mathrm{i}}=\mathsf{\theta }\times {\mathrm{R}}_{\mathrm{wall}},$$

(25)

$${\mathrm{R}}_{\mathrm{e}}=\left(1-\mathsf{\theta }\right)\times {\mathrm{R}}_{\mathrm{wall}},$$

(26)

With the accessibility factor:

$$\mathsf{\theta }=1-\left({\displaystyle \frac{{{\displaystyle \sum }}_{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{k}}{\mathrm{R}}_{\mathrm{k}}^{*}}{{\mathrm{C}}_{\mathrm{wall}}{\mathrm{R}}_{\mathrm{wall}}}}\right),$$

(27)

where C_wall is the total capacitance of the wall, C_k is the capacitance of the k-th layer, R_wall is the resistance of the wall and:

$${\mathrm{R}}_{\mathrm{k}}^{*}={\displaystyle \sum }_{\mathrm{p}=1}^{\mathrm{k}-1}\left({\mathrm{R}}_{\mathrm{p}}+{\displaystyle \frac{{\mathrm{R}}_{\mathrm{k}}}{2}}\right),$$

(28)

where ${\mathrm{R}}_{\mathrm{k}}^{*}$ is the resistance between the middle of the k-th layer and the external surface, R_p is the resistance between the p-th layer and the external surface and R_k is the resistance of the k-th layer.

2.3.4. Calculation Algorithm

As can be seen in previous sections, heat flow by transmission, given by Equation (18), depends on three input variables. The first of them, H_T,ie, can be obtained from the thermo-physical and geometric data of the used materials. The ambient temperature, if not measured, can be taken from various sources, as hourly typical meteorological years. The last variable, the indoor air temperature, results from thermal conditions in a zone at the given hour. At the design stage of the system, it can be also set according to design assumptions at a chosen constant value. Therefore, two solutions were proposed and then implemented in the simulation programme (Figure 11).

In the first method, hourly heat gains by transmission are calculated from Equation (18) based on hourly values of indoor and outdoor temperatures and then are used as the input dataset to the calculation program. The values of both temperatures can be freely set by the user depending on the needs. Q_Transm gains do not depend on the current (resulting) temperature inside the room calculated by the program. This method can be used for preliminary calculations, assuming constant values of both temperatures, when examining the effect of the internal temperature setting on the behaviour of the system, and also when using measurement data of heat losses.

In the second method, as the hourly internal temperature in a zone ($T_{int}^h$) in Equation (18) there is used the resulting current hourly air temperature ($T_A^h)$. As the ambient temperature hourly measured values of T_e is taken. Consequently, the calculation of Q_Transm is directly coupled with the running conditions in a zone and hourly heat flow by transmission is obtained automatically by the programme.

When implementing the dynamic model of the wall (Figure 10), its thermal resistance (R_wall) is taken from the method of Anderlind. To couple this 2R1C model with ambient and indoor thermal conditions, external and internal surface resistances, R_se and R_si, were assumed as in Section 2.3.3. Then, heat flow by transmission can be obtained using measured outdoor air temperature and indoor air temperature set at the required value or taken from measurements.

2.4. Statistical Analysis

To assess the models’ performance in reference to measurements of internal air temperature in a room, there were used several statistical indicators as the mean absolute error (MAE), the mean absolute of percentage error (MAPE), the mean square error (MSE) and the root mean square error (RMSE) [48,49,50].

When assuming that x_i is the reference value of the considered variable (temperature), x, taken from measurements, and ${\widehat{x}}_i$ is its value predicted by the model, and ${\overline{x}}_i$ is the average reference value of x in the analysed dataset and n is the total number of samples, then these metrics are given by the following equations:

$$\mathrm{MAE}={\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{\displaystyle \frac{\left|{\widehat{x}}_i-x_i\right|}{\mathrm{n}}},$$

(29)

$$\mathrm{MAPE}={\displaystyle \frac{1}{\mathrm{n}}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}\left|{\displaystyle \frac{{\hat{\mathrm{x}}}_{\mathrm{i}}-\mathrm{x}}{{\mathrm{x}}_{\mathrm{i}}}}\right|\times 100,$$

(30)

$$\mathrm{MSE}={\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{\displaystyle \frac{{\left({\widehat{x}}_i-x_i\right)}^2}{\mathrm{n}}},$$

(31)

$$\mathrm{RMSE}=\sqrt{{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{\displaystyle \frac{{\left({\hat{\mathrm{x}}}_{\mathrm{i}}-{\mathrm{x}}_{\mathrm{i}}\right)}^2}{\mathrm{n}}}},$$

(32)

3. Results and Discussion

3.1. Thermal Transmittance of the Wall

3.1.1. Results of Measurements

The first measurement period was used to estimate the thermal resistance of the wall, due to the favourable weather conditions and a sufficiently high temperature difference between the interior of the building and the external environment.

The external and internal surface temperatures were measured by two pairs of sensors. For further calculations and analysis, their indications were averaged. As shown in Figure 12, the internal surface temperature was stable, with variation from 20.6 °C to 21.9 °C. The external surface temperature varied more significantly, from –0.4 °C to 18.4 °C.

Despite the cloudy sky during the experiment there were also observed short periods of sunny weather. Therefore, it can be noticed that temperature on the external side was influenced by solar radiation. This resulted in disturbances in the recordings of wall heat flux (Figure 13). Hence, these periods had to be removed before further processing.

When filtering the data, the resulting average measured thermal resistance of the wall R_wall = 8.44 m²K/W. Including external and internal surface resistances from ISO 6946 the total wall resistance by transmission: R_wall,tot = 8.61 m²K/W which means heat transfer coefficient by transmission U_wall = 0.12 W/(m²K).

3.1.2. Measurement Uncertainties

In calculations the average internal and external surface temperatures of 21.2 °C and 7.3 °C, respectively, were assumed. The average heat flux was –1.68 W/m². Based on the manufacturers’ manuals and presented assumptions, there were obtained uncertainties given in

Table 3. Uncertainties in the measurement of the thermal resistance of the wall.

Uncertainty	Value	Unit
u(Twi)	0.093	K
u(Twe)	0.082	K
u(q)	0.135	W/m2
${\displaystyle \frac{\partial {\mathrm{R}}_{\mathrm{wall}}}{\partial {\mathrm{T}}_{\mathrm{w},\mathrm{i}}}}$	–0.595	W/m2K
${\displaystyle \frac{\partial {\mathrm{R}}_{\mathrm{wall}}}{\partial {\mathrm{T}}_{\mathrm{w},\mathrm{e}}}}$	0.595	W/m2K
${\displaystyle \frac{\partial {\mathrm{R}}_{\mathrm{wall}}}{\partial \mathrm{q}}}$	–4.915	W/m2K2

.

The resulting uncertainty of the thermal resistance of the wall u(R_wall) = 0.665 m²K/W. The expanded uncertainty, for the coverage factor k = 2, U(R_wall) = 1.33 m²K/W. Therefore, finally, we can write that R_wall = 8.44 ± 1.33 m²K/W.

The value obtained based on the data from Table 2 is R_wall = 11.08 m²K/W. However, it should be noted that it was calculated from the parameters declared by the manufacturers and obtained in specific test conditions, according to the relevant standards. Even a simple evaluation of the temperature effect on these materials, according to the EN ISO 10456 standard [51], assuming the average ambient temperature of 7.5 °C results in R_wall,tot = 10.88 m²K/W. Also, to get a better view of the thermal condition of this wall the measurements should be repeated during winter, when the ambient temperature is lower and measurement conditions are better due to higher indoor-outdoor temperature difference.

3.2. Thermal Transmittance of the Window

During the experiment, there was also measured the thermal transmittance of glazing. For the same reasons, as previously (Figure 14) the data had to be filtered.

The resulting average measured thermal resistance of the glazing R_glazing = 1.09 m²K/W. Including external and internal surface resistances from ISO 6946 the total wall resistance by transmission: R_glazing,tot = 1.26 m²K/W which results in heat transfer coefficient by transmission U_glazing = 0.794 W/(m²K). It is slightly better than the value of 0.8 W/(m²K) assumed at the design stage of the building. It is also very important because glazing covers a significant part of the external envelope of this building. Thus, heat loss by its envelope depends significantly on the thermal resistance of windows.

Similarly, as for the wall, the uncertainty analysis was also performed. As a result, we get: R_glazing = 1.09 ± 0.17 m²K/W.

3.3. Total Heat Transfer by Transmission

Due to simplifications and technical requirements during the building constructions, the influence of thermal bridges in the external wall on the value of the H_T,ie coefficient was omitted. This parameter was calculated in two ways:

• using measured thermal resistances of the wall and glazing;
• using declared, theoretical values of the materials parameters.

When calculating the transmittance of the window, in both cases linear thermal bridge was included (Ψ_g = 0.05 W/m·K) and the frame was also taken into account. For the considered case, from Equation (19) we obtain:

$${\mathrm{H}}_{\mathrm{T},\mathrm{ie}}={\mathrm{A}}_{\mathrm{wall}}·{\mathrm{U}}_{\mathrm{wall}}+{\mathrm{A}}_{\mathrm{window}}·{\mathrm{U}}_{\mathrm{window}}$$

(33)

The thermal transmittances of the wall (U_wall) and the window are obtained from Equations (21) and (20), respectively. Final results are given in

Table 4. Total heat transfer coefficient by transmission from measurements and theoretical data.

Parameter	Measured (Static)	Measured (Dynamic)	Theoretical	Unit
HT,ie	10.51	10.37	10.33	W/K

.

In the dynamic approach, using methods presented by Anderlind (Equations (23) and (24)), for k = 50 time steps there were obtained the values of R_wall = 9.87 ± 0.23 m²K/W and 9.96 ± 0.35 m²K/W, respectively. It means that the resulting H_T,ie is 10.37 W/K and 10.36 W/K, respectively.

Despite the aforementioned discrepancies between the measured (stationary) and theoretical values of U_wall, the difference between the resulting total heat transfer coefficients is negligible. For further simulations, as a reasonable compromise, there was assumed the value of H_T,ie = 10.42 W/K (steady state heat transfer model) and 10.37 W/K (dynamic model).

3.4. Simulations

3.4.1. First Period

The model of a room and the thermally activated slab, presented in previous sections, was implemented in Python 3.10. There were considered four simulation variants.

In the first variant, heat gains by transmission were calculated from Equation (18) based on the measured hourly values of internal (T_A) and external (T_e) air temperatures.

In the second variant, following the recommendation of ISO 11855-4, as T_A there was set constant set-point indoor temperature of 22 °C and T_e was obtained from measurements. Then, heat gains by transmission were calculated and used as input to the model.

In the third variant, the second calculation algorithm was utilised. Heat transfer through external partitions was computed automatically by the program from current outdoor and indoor conditions in the iterative procedure.

In the last, fourth, variant the total resistance of the wall was obtained from dynamic analysis of the measurement results. Then, the 2R1C model of the wall was constructed and heat flux by transmission was calculated from the measured hourly values of internal (T_A) and external (T_e) air temperatures.

The accuracy of simulations was assessed by comparing them with measurements. The first analysed variable was the indoor air temperature. It was measured at three levels, 0.55 m (T_A,0.55), 1.5 m (T_A,1.5), and 2.2 m (T_A,2.2), above the floor, showing good uniformity during the considered period (Figure 15). The hourly difference between the maximum and minimum values did not exceed 0.1 °C. Therefore, these values can be treated as reliable reference data.

Results of simulations of the indoor air temperature in the form of error analysis with regard to measurements are presented in

Table 5. Error analysis of the simulated indoor air temperature.

Error	Variant 1	Variant 2	Variant 3	Variant 4	Unit
MAE	0.13	0.13	0.14	0.24	°C
RMSE	0.17	0.16	0.17	0.28	°C
MSE	0.03	0.03	0.03	0.08	°C2
MAPE	0.65	0.61	0.65	1.15	%

.

All studied methods provided results with satisfactory accuracies. The mean absolute error, showing the average magnitude of deviations of a simulated air temperature in reference to measurements, was below 0.24 °C. The Mean Square Error (MSE) was up to 0.08 °C², which means that RMSE was no more than 0.28 °C. MAPE, treated as the average size of forecast errors during the analysed period, was 1.15% in the last variant. This means that the dynamic method involving 2R1C model of the wall showed a little worse accuracy in this case.

The next analysed variable was the operative temperature (

Table 6. Error analysis of the simulated operative temperature.

Error	Variant 1	Variant 2	Variant 3	Variant 4	Unit
MAE	0.12	0.12	0.12	0.21	°C
RMSE	0.14	0.15	0.15	0.24	°C
MSE	0.02	0.02	0.02	0.06	°C2
MAPE	0.54	0.55	0.57	0.98	%

). There were not significant differences between the studied variants. In all cases errors were low and of similar level as in the case of the indoor air temperature.

The next analysed variable was the ceiling temperature (

Table 7. Error analysis of the simulated ceiling temperature.

Error	Variant 1	Variant 2	Variant 3	Variant 4	Unit
MAE	0.18	0.19	0.19	0.12	°C
RMSE	0.22	0.23	0.23	0.15	°C
MSE	0.05	0.05	0.05	0.02	°C2
MAPE	0.81	0.87	0.85	0.57	%

). This time the results of simulations were slightly less accurate. MAE was up to 0.19 °C. RMSE was from 0.15 to 0.23 °C. More significant differences showed MAPE which was from 0.57% to 0.87%. However, these values are acceptable in all variants.

The last analysed was the temperature of the floor surface. During the measurements (Figure 16) it was from 21.2 °C to 22.6 °C. At the same it should be mentioned, that ISO 11855 refers the designer of TABS to comfort criteria given in EN-ISO 7730 [52] as design indications for TABS. This standard recommends the maximum and minimum floor temperature of 29 °C and 19 °C for heating and cooling, respectively. EN 1264 [53,54] defines the maximum temperature of a floor in the dry occupied zone at 29 °C (with 31 °C at the edge part of the floor). Therefore, this criterion was met throughout the entire analysed period.

Table 8. Error analysis of the simulated floor surface temperature.

Error	Variant 1	Variant 2	Variant 3	Variant 4	Unit
MAE	0.15	0.16	0.16	0.14	°C
RMSE	0.19	0.20	0.20	0.17	°C
MSE	0.04	0.04	0.04	0.03	°C2
MAPE	0.69	0.73	0.75	0.65	%

presents the results of the error analysis. This time the results were of the same order as previously. In the worst case, MAPE was 0.75%, which is satisfactory. The last method (with the dynamic model of the external wall) showed the best accuracy in the case of the floor and ceiling temperatures.

The last part of the performed analysis was the thermal balance of the room (Figure 17). During the studied period there was supplied to and extracted from this zone 119.9 MJ and 117.4 MJ, respectively. The difference of 2.5 MJ is about 2% and can be treated as negligible. On the other hand, it can be interpreted as the error of the method. Unfortunately, the term “error” is used in ISO 11855-4 without its definition. This problem was also noticed in other standards [55]. Therefore, this issue requires further investigations.

For better clarity heat loss by transmission was subdivided into two parts: through the window (Q_Tr, window) and the wall (Q_Tr, wall).

Heat loss was caused mainly by heat transfer by transmission to the environment (94%) with the dominating share of windows (87%) and wall (6.8%). Heating demand was covered by the floor (28.7%) and the ceiling (18.7%). However, high level of building’s insulation resulted in low demand for heating capacity of the system, which did not exceed 2.1 W/m². At the same, in such object internal gains were important. They came from internal sources (21.7%) and sun (13.9%). Simulations also showed that heat accumulation in internal walls (Q_IWS) played less important, but still a noticeable role with a share of 10.9%. Heat from ventilation air (Q_PrimAir) with a share of 6.1% can be treated as a supporting source. It is so, because the main role of the ventilation system is to bring in fresh air from outside while removing indoor air. Also, to reduce unnecessary energy consumption, ventilation was operating during the opening hours of the building.

3.4.2. Second Period

In the second period ambient conditions were significantly different (Figure 6 and Figure 7). The hourly indoor air temperature was from 21.2 °C to 24.0 °C, from 21.2 °C to 23.2 °C and from 21.2 °C to 24.0 °C, at heights of 0.2 m (T_A,0.2), 0.6 m (T_A,0.6), and 1.0 m (T_A,1.0), above the floor, respectively (Figure 18). The maximum hourly difference between these three points did not exceed 1.2 °C. At the same, the maximum difference, measured between floor and ceiling, was below 1.9 °C.

All simulations were done in the same way as in the previous period. Error analysis for indoor air, operative, ceiling and floor temperatures are presented in

Table 9. Error analysis of the simulated indoor air temperature.

Error	Variant 1	Variant 2	Variant 3	Variant 4	Unit
MAE	0.17	0.14	0.14	0.14	°C
RMSE	0.20	0.20	0.18	0.20	°C
MSE	0.04	0.04	0.03	0.04	°C2
MAPE	0.76	0.64	0.62	0.64	%

,

Table 10. Error analysis of the simulated operative temperature.

Error	Variant 1	Variant 2	Variant 3	Variant 4	Unit
MAE	0.15	0.16	0.15	0.16	°C
RMSE	0.26	0.26	0.26	0.26	°C
MSE	0.07	0.07	0.07	0.07	°C2
MAPE	0.69	0.69	0.69	0.69	%

,

Table 11. Error analysis of the simulated ceiling temperature.

Error	Variant 1	Variant 2	Variant 3	Variant 4	Unit
MAE	0.19	0.15	0.18	0.15	°C
RMSE	0.23	0.18	0.21	0.18	°C
MSE	0.05	0.03	0.05	0.03	°C2
MAPE	0.90	0.68	0.82	0.69	%

and

Table 12. Error analysis of the simulated floor surface temperature.

Error	Variant 1	Variant 2	Variant 3	Variant 4	Unit
MAE	0.11	0.10	0.08	0.10	°C
RMSE	0.13	0.13	0.10	0.13	°C
MSE	0.02	0.02	0.01	0.01	°C2
MAPE	0.50	0.46	0.38	0.46	%

, respectively.

It can be noticed that also in summer conditions the developed tool provided results with satisfactory accuracy.

During the summer period there was supplied to and extracted from this zone 122.8 MJ and 132.1 MJ, respectively (Figure 19). The difference of 9.3 MJ was about 7.5% and was larger than in the first period.

This time heat was extracted from the zone by the floor (34.3%) and ceiling (27.1%) with the maximum cooling capacity of 10.2 W/m². Important role in heat removal (16%) played windows. This was particularly evident during the night, when the ambient temperature was lower than that inside the room. Thus, without additional measures, it was possible to cool it down.

Interior walls contributed 12.9%. It should be mentioned here that they accumulated excess heat gains during the day, releasing them at night. Ventilation also played an auxiliary role (8.4%), particularly important during the day.

Heat gains resulted mainly from solar irradiance (91.1%). Further research would need to investigate the control of external blinds to reduce excess solar gains during the day, while maintaining the required light level in the room.

4. Conclusions

This study presents an attempt to model an office room with a TABS system according to the general procedure given in ISO 11855-4. Due to the lack of detailed recommendations for calculating the heat flow from the room to the external environment, two calculation methods were proposed. Measurement tests have confirmed the good accuracy of both methods.

The heat loss coefficient by transmission, between indoor and ambient environments, was calculated based on measurements and technical data of the building. In this first case both steady-state and dynamic methods were applied. All methods provided similar results.

Since the analysed object is passive, important conclusions can be drawn from this. The use of high-quality building materials and high-quality workmanship are very important, as this minimises the impact of thermal bridges and cracks. After 10 years of use of the building, the measured coefficient of heat transfer by transmission (H_T,ie) was almost equal to the value calculated based on the technical data of the materials used: 10.51 W/K and 10.33 W/K, respectively. The dynamic methods resulted in values from 10.36 W/K to 10.37 W/K.

The simulation programme, based on the general description given in ISO 11855-4, was developed. Static and dynamic methods to obtain heat flow by transmission through external wall were applied. Presented calculation algorithms were implemented and successfully validated. In all considered cases accuracies of the results were satisfactory.

In the first, autumn measurement period, MAE and RMSE of indoor air temperature were below 0.24 °C and 0.28°C, respectively. The average modelling error during the whole period, given by MAPE, was 1.15%.

The simulated ceiling temperature was slightly less accurate with MAE and RMSE up to 0.19 °C and 0.23 °C, respectively. In this case MAPE was from 0.57% to 0.87% and these values also are satisfying.

The last considered variable was the floor temperature. It was within the comfort range, defined in the relevant standards. Here RMSE was below 0.20 °C. MAPE was from 0.69% to 0.75%.

The energy analysis of the room showed that total heat loss dominates the share of heat transfer by transmission to the environment. On the other hand, almost half of the total heating demand was covered by the radiant floor and ceiling (TABS).

Satisfying accuracy of the results was also noticed in the summer period. MAE of indoor air, operative, ceiling and floor temperature was below 0.19 °C. MAPE for these variables not exceeded the value of 0.90%.

Cooling load was covered mainly by the floor and ceiling (61%) with the supporting role of heat loss through windows during nights (16%) ventilation air (8%). It confirms that in a passive building solar and internal gains play a very important role and this first element may be actively controlled by external blinds to reduce cooling loads.

In both heating and cooling modes, the floor and ceiling were the main contributors to the room’s heat load coverage. At the same, the maximum heating and cooling capacity did not exceed 2.1 W/m² and 10.2 W/m², respectively.

In the cases studied, the simulation programme was successfully validated. As the facility becomes available, further studies over a longer period of time are indicated. Under Polish conditions, measurements during the winter period are particularly desirable. This is when ambient air temperatures are low and solar gains are low, resulting in increased heating needs. Such experiments would be helpful to better understand and popularise the idea of passive buildings.