*Article* **Experimental and Theoretical Study on the Internal Convective and Radiative Heat Transfer Coefficients for a Vertical Wall in a Residential Building**

**Piotr Michalak**

Department of Power Systems and Environmental Protection Facilities, Faculty of Mechanical Engineering and Robotics, AGH University of Science and Technology, Mickiewicza 30, 30-059 Kraków, Poland; pmichal@agh.edu.pl; Tel.: +48-126-173-579

**Abstract:** Experimental studies on internal convective (CHTC) and radiative (RHTC) heat transfer coefficients are very rarely conducted in real conditions during the normal use of buildings. This study presents the results of measurements of CHTC and RHTC for a vertical wall, taken in a selected room of a single-family building during its everyday use. Measurements were performed using HFP01 heat flux plates, Pt1000 sensors for internal air and wall surface temperatures and a globe thermometer for mean radiant temperature measured in 10 min intervals. Measured average CHTC and RHTC amounted to 1.15 W/m2K and 5.45 W/m2K, compared to the 2.50 W/m2K and 5.42 W/m2K recommended by the EN ISO 6946, respectively. To compare with calculated CHTC, 14 correlations based on the temperature difference were applied. Obtained values were from 1.31 W/m2K (given by Min et al.) to 3.33 W/m2K (Wilkes and Peterson), and in all cases were greater than the 1.15 W/m2K from measurements. The average value from all models amounted to 2.02 W/m2K, and was greater than measurements by 75.6%. The quality of models was also estimated using average absolute error (AAE), average biased error (ABE), mean absolute error (MAE) and mean bias error (MBE). Based on these techniques, the model of Fohanno and Polidori was identified as the best with AAE = 68%, ABE = 52%, MAE = 0.41 W/m2K and MBE = 0.12 W/m2K.

**Keywords:** convection; radiation; heat transfer coefficient; correlation

#### **1. Introduction**

Heat transfer by convection and radiation plays an important role in numerous applications from nanoscale [1–3] to large industrial installations [4]. Among these, buildings are of special interest because of their important contribution to global energy consumption [5]. Therefore, for economic and environmental reasons, it is common for the energy performance of buildings to be legally regulated in many countries [6].

Heat transfer by convection and radiation plays important role in the internal environment of a building, influencing thermal comfort, the operation of heating, ventilation and air conditioning (HVAC) systems, and energy consumption [7–10]. Heat transfer by convection and radiation can be derived both theoretically and experimentally, however the latter method prevails when considering non-typical and more complex cases, or for proving theoretical investigations.

Energy performance assessments for buildings are usually based on the energy usage for the heating and cooling of the space [11], calculated using various computer tools. These tools employ different models to compute radiative and convective heat transfer coefficients. The nature of the physical phenomenon for a given heat transport mechanism determines the parameters necessary to compute the appropriate coefficients. Heat transfer by radiation is simple to describe, as it depends on the temperature, emissivity and position of the considered body and its surroundings. Moreover, numerous studies have proven that the difference between the air and mean radiant temperatures inside the building zone

**Citation:** Michalak, P. Experimental and Theoretical Study on the Internal Convective and Radiative Heat Transfer Coefficients for a Vertical Wall in a Residential Building. *Energies* **2021**, *14*, 5953. https:// doi.org/10.3390/en14185953

Academic Editor: Paulo Santos

Received: 31 July 2021 Accepted: 15 September 2021 Published: 19 September 2021

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**Copyright:** © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

is relatively small, typically below 2 ◦C [12], and in practice, constant values of radiative heat transfer coefficients (RHTCs) are commonly used [13,14] to simplify calculations.

In terms of heat transfer by convection, the aforementioned temperatures, flow conditions, and physical properties of the fluid (air) are needed. Hence, mathematical description is more complex than in the previous case, and analytical solutions are available mainly for relatively simple cases. For these reasons, over the past few decades, experimental correlation models of the convective heat transfer coefficients (CHTCs) have been developed. They significantly reduce computation effort by combining an unknown coefficient with the main driving physical factors in given conditions within the studied case. Usually, these models are subdivided into correlations of dimensionless numbers or temperature differences [15,16]. The first type links Nusselt with Grashof and Prandtl numbers in the form of the power equation Nu = C (Gr Pr)*<sup>n</sup>* with constant C and *n*. In the second case, an equation of the type hc =C(ΔT)*<sup>n</sup>* is used. The latter form is simpler to use because it is founded on an easily measurable quantity, which is temperature and, in certain cases, geometrical parameters (length, area or height of a surface) and air velocity.

Initially, a lack of detailed data forced the use of results from various physical experiments in buildings' thermal calculations. The similarity of buildings' partitions to typical geometric figures resulted in the use of both theoretical and experimental results, obtained mainly for horizontal, vertical, and inclined planes. Many such studies have been developed in recent decades, and different flow regimes have been studied under natural and forced convection.

Fishenden and Saunders [17–19] investigated convective heat transfer for metal (aluminium, copper, platinum, or steel) vertical plates in air, water and mercury. To study the impact of air pressure on that phenomenon, the measurements were conducted in a cylindrical steel test chamber. The temperature difference was up to 55 ◦C. Finally, the authors gave experimental correlations with dimensionless numbers, with *n* = 1/4 and *n* = 1/3 for laminar and turbulent flow, respectively.

McAdams [20] summarised the results of several previous works of various researchers on natural convection and presented correlations for vertical and horizontal plates and cylinders in the form of dimensionless equations with *n* = 1/4. In quoted experiments, metal plates were used (aluminium, copper or platinum). He also presented an alignment chart to derive a heat transfer coefficient for the turbulent flow of various gases on vertical surfaces. In the case of so-called ordinary temperatures and atmospheric pressure, simplified dimensional equations with *n* = 1/3 and *n* = 1/4 for turbulent and laminar flow, respectively, were given. The same exponent was used by Lewandowski [21,22] for copper plate in glycerine or for water [23].

Based on previous theoretical and experimental works, Churchill and Usagi [24] developed empirical dimensionless correlations with *n* = 9/4 and *n* = 8/3 under laminar free and forced convection, respectively, for an isothermal vertical plate. No information on the temperature difference range was given.

As the interest in building modelling has grown, experimental research on heat transfer phenomena in this group of objects has gained increased importance. Related studies commonly test chambers with dimensions similar to typical rooms in buildings.

Correlations for interior surfaces of buildings for vertical glazing (with no radiator under the window), vertical walls (non-heated or near the heat source) and horizontal surfaces (heated floor or cooled ceiling) were presented by Khalifa and Marshall in [25]. Using a two-zone test cell they performed nine experiments covering the most popular heating configurations in buildings under controlled steady-state conditions, as follows: forced convection on the interior surface using a fan, a cell heated by a 1 kW fan heater, a uniformly heated floor, a uniformly heated vertical wall and a uniformly heated edge using a metalized plastic foil, a cell heated by a 1.5 kW oil-filled radiator located opposite the test wall and adjacent to the test wall, and finally with a single glazed window in a test wall with a radiator located beneath and opposite the window.

The larger hot zone represented a large enclosure (such as a living room). It had one vertical wall common with a smaller cold zone built to control the outdoor temperature of the hot zone. A total of 142 tests were conducted. Vertical walls and the roof were covered with aluminium foil to minimise the impact of internal longwave radiative heat exchange on temperature and heat flux measurements, and to neglect it in heat transfer calculations. To reach the steady-state conditions between two consecutive tests with different configurations, a 24 h pre-run period was applied, and then 12 h data logging between 22:00 and 10:00 was undertaken. The CHTCs obtained for each of the different studied elements were presented as dimensional correlations against the internal air to wall surface temperature difference. For the heated vertical wall *n* = 1/4 was obtained.

Alamdari and Hammond [26] used the model of Churchill and Usagi to derive empirical correlations for buoyancy-driven convective heat transfer for the internal surfaces of naturally ventilated buildings, covering laminar, transitional, and turbulent airflows. They noticed that the experimental results related to buoyancy-driven convection given in the literature were often given for fluids, surface dimensions, and temperature differences not found in buildings. Based on Churchill and Usagi considerations and experiments presented by other researchers, they developed several correlations of different complexity and applicable for vertical and horizontal partitions.

Natural convection from heated surfaces in a room was examined by Awbi and Hatton [27]. Experiments were conducted in a test chamber and a small box. The first experiment was of a similar concept as presented in [25]. The internal layer of vertical walls, ceiling, and floor was of varnished plywood. Five heating plates (0.5 m per 2.3 m) covered with a 2 mm thick aluminium plate (to minimise radiative heat flux) were used to provide surface heating to each tested wall. To minimise the effect of internal longwave radiation, temperature sensors were shielded by small steel tubes. The small test box was a cube with sides of approximately 1 m made of the same materials. Each test was conducted for several hours to reach steady-state conditions. Obtained equations for CHTCs were in the form of power functions of temperature difference. Correlation coefficients were above 0.90, indicating strong dependences.

Delaforce et al. [28] investigated convective heat transfer at internal surfaces using a rectangular test cell with sides of approximately 2 m each. The cell was located on an exposed grassland site and experiments were conducted during winter conditions. External walls were made from bricks and an external layer of expanded polystyrene. Their internal surface had no additional covering layer, so indoor radiative heat transfer was monitored using a radiometer. Temperatures were measured at 5-min intervals. Heating was provided by a 1 kW fan heater. Continuous heating, intermittent heating, and unheated operation were examined separately for the west wall, floor, and ceiling. Before each experiment, authors used a two-day warm-up period with continuous heating. Finally, they did not provide any empirical correlations, but only averaged the numerical values of the calculated CHTCs and compared them with values recommended by the Chartered Institute of Building Service Engineers (CIBSE) Guide. The greatest discrepancies were noticed for a floor and vertical wall, where the measured CHTC was 0.5 and 1.6 W/m2K compared to 1.5 and 3.0 W/m2K from CIBSE, respectively.

Test chambers can be also used to analyse the performance of selected buildings components under various conditions. Sukamto et al. [29] used this method to investigate the thermal performance of the ventilated wall. Because of forced airflow, the obtained CHTC was about 20 W/m2K.

Furthermore, several authors studied the impact of the heating, ventilation and air conditioning systems (HVAC) on internal heat transfer in buildings. A more detailed review of embedded radiant heating/cooling systems is given in [30]. Cholewa et al. [31,32] investigated heated/cooled radiant floor and radiant ceiling using a test chamber. A similar method was used by Acikgoz and Kincay [33] for radiant cooling walls. Hydronic radiant heated walls in a climatic test chamber with a window were investigated in [34]. Guo et al., investigated convective heat transfer for night cooling with diffuse ceiling [35]

and mixing [36] ventilation in a guarded hot box used as a test chamber. To evaluate the dependence of CHTC on ventilation conditions, they used several correlations linking them with ventilation air changes per hour (ACH). Various cases of CHTCs for different building partitions in the form of empirical correlations are given in reviews [10,16,37–40]. For example, in the review by Khalifa, with CHTC correlations derived in 1, 2 and 3 dimensional enclosures [39,40], the author presented 13 studies performed for the latter case. The author concluded that large differences may arise between the presented models.

Recently, and in addition to the aforementioned methods, various computer-based techniques have been developed, and simulation tools used, in convective heat transfer analyses [41–45]. Despite their abilities to deal with complex problems that are not possible to solve analytically, they are used for proving experimental results rather than to formulate mathematical relationships to obtain convection heat transfer coefficients in building applications. For these reasons, in energy auditing, energy certification, or in other similar applications requiring relatively low computational effort and a limited number of input parameters, constant CHTC and RHTC values are commonly used [46,47]. These are given in various standards, such as EN 1264 [48], ISO 11855 [49], EN ISO 6946 [50] and guidebooks [51] providing relevant formulas to obtain CHTC or their constant values in different cases.

The presented methods and techniques more or less accurately reflect the nature of heat transfer phenomena in buildings. They allow the separation of noise, stabilisation, and control of thermal conditions, and so on [52]. On the other hand, the applicability of the theoretical models should be verified in real conditions. For this reason, the best research object seems to be a real building.

Cost, practical problems, and limitations for building users resulting from the presence of measuring equipment in the building during its everyday use are the probable reasons for so few tests being carried out in these conditions. Hence, there are few CHTCs from experiments in realistic situations [53]. Most measurements have been made in special test chambers where internal surfaces were metal coated. Therefore, it seems reasonable to address indoor convective and radiative heat transfer in buildings during their everyday use.

Wallentén [53] experimentally analysed internal convective heat transfer at an external wall with a window in a room with and without furniture, located in a single-family house built in a sandwich construction of light-weight concrete and polystyrene. The room had the dimensions of 3 × 3.6 × 2.4 m. Internal surfaces of walls and floor were painted with a mat white and light brown colour, respectively. Two electric radiators with 500 W and 1000 W heating power were used. T-type thermocouples were used to measure indoor air and partition temperatures. Longwave radiation was calculated from diffuse grey surface temperatures. A total of 14 different experiments (radiator placement, presence of furniture, wall and window analysis, ventilation schedule), each lasting approximately one week, were performed across four years. A sampling time of 1 min was assumed and then measurements were averaged in 4, 10, 30 and 60 min intervals. The author did not provide any conclusions on the impact of the averaging time on final results. Several graphs with CHTC depending on the surface (vertical wall or window) and indoor air temperature were presented, but no correlations were given. As far as the wall is concerned, the data from measurements were quite scattered and showed little correlation. The spatial distribution of internal air temperature was also presented. An impact from the radiator or ventilation was visible, but the maximum difference between the floor and ceiling did not exceed 3 ◦C. Only near the window, at a distance of several cm, it reached 5 ◦C.

The review presented here shows several important outcomes. Thermal conditions in climatic chambers during the aforementioned tests were controlled according to the requirements necessary to perform the assumed tests. Noises could be easily limited using selected construction materials or HVAC system components. On the other hand, heat transfer in buildings during everyday use is influenced by different environmental factors, such as occupation, solar radiation, air ventilation, and so on. The very low number

of studies dedicated to this issue makes it worthy to compare the models developed theoretically or on a basis of experiments in test chambers, with those derived from real conditions in a building during its normal operation. The identification of the most significant limitations during such an experiment is also a possibility, and could indicate possible improvements to similar studies in the future.

The environmental conditions during measurements and experimental setup are given in the next section. Calculation procedures to obtain CHTC and RHTC and uncertainty analysis are then presented. For comparative purposes, 14 theoretical and empirical correlations to obtain CHTC for a vertical wall under free convection are used. The results are then presented and compared with relevant studies. Finally, concluding remarks are given.

#### **2. Materials and Methods**

#### *2.1. Introduction*

The research was conducted from 30 January to 10 February 2021 in a residential building located in southern Poland, equipped with a hydronic heating system.

During the studied period the outdoor air temperature (Figure 1) varied from −15.1 ◦C (at 3:50 on 1 February) to 10.1 ◦C (at 9:30 on 4 February). Global solar irradiance (Figure 2) incident on the external surface of the considered wall was from 0 to 113.3 W/m2 (at 13:20 on 31 January).

**Figure 1.** External air temperature during measurements.

**Figure 2.** Global solar irradiance on the west-facing wall during measurements.

#### *2.2. Experimental Setup*

The west-oriented wall (strictly—its north part) in a living room was chosen for the measurements (Figure 3). Also located in this corner of the room are wooden stairs. The room is on a rectangular plane of 6 m × 4 m and has a height of 2.50 m.

**Figure 3.** (**a**) View of the room selected for experiment; (**b**) View of the measurement equipment during assembly.

The measured data was recorded in 10 min intervals with a MS6D Comet data logger. Periods from 4 to 60 min have typically been used in other studies [25,28,53].

In the experiment, two HFP01 heat flux sensors and three Pt1000 temperature sensors were mounted on the internal surface of the wall, while two Pt1000 internal air temperature sensors, one sensor of indoor radiant (ø 150 mm TP875.1 globe thermometer with the Pt100 sensor) and one Pt1000 of outdoor air temperature (mounted in a radiation shield) and a CMP11pyranometer for global solar irradiance incident were placed on the external wall. Their main parameters are given in Table 1.

**Table 1.** The main metrological parameters of the measuring sensors used.


<sup>1</sup> According to EN 60751. <sup>2</sup> According to ISO 9060 and IEC 61724.

#### *2.3. Calculation Procedure*

The conduction heat flux flowing through the wall (qw) and measured by the heat flux sensors (Figure 1) is the sum of radiative (qr) and convective (qc) fluxes entering an interior space of the room:

$$\mathfrak{q}\_{\rm W} = \mathfrak{q}\_{\rm I} + \mathfrak{q}\_{\rm C}.\tag{1}$$

The radiative heat flux is given by the relationship:

$$\mathbf{q}\_{\rm r} = \mathbf{h}\_{\rm r} (\mathbf{T}\_{\rm s} - \mathbf{T}\_{\rm r})\_{\rm \prime} \tag{2}$$

Within the building enclosure, the considered wall at temperature Ts is exposed to remaining walls and heat transfer by radiation occurs between them. The mean radiant temperature of the given wall surroundings is measured by the globe thermometer [28–30]. Then, based on the Stefan–Boltzmann law and at the given emissivity of the wall, ε, the following equation can be written [31]:

$$\mathbf{q}\_{\mathbf{r}} = \varepsilon \sigma \left( \mathbf{T}\_{\mathrm{s}}^{4} - \mathbf{T}\_{\mathrm{r}}^{4} \right). \tag{3}$$

Comparing Equations (2) and (3) we obtain:

$$\mathbf{h\_{r}} = \varepsilon \sigma \frac{\mathbf{T\_{s}^{4}} - \mathbf{T\_{r}^{4}}}{\mathbf{T\_{s}} - \mathbf{T\_{r}}} = \varepsilon \sigma (\mathbf{T\_{s}} + \mathbf{T\_{r}}) \left(\mathbf{T\_{s}^{2}} + \mathbf{T\_{r}^{2}}\right). \tag{4}$$

Equation (4) shows that hr can be obtained without direct measurement of radiative heat flux.

Convective heat flux is given by:

$$\mathbf{q}\_{\mathbf{c}} = \mathbf{h}\_{\mathbf{c}} (\mathbf{T}\_{\mathbf{s}} - \mathbf{T}\_{\mathbf{i}}).\tag{5}$$

With qr known, qc can be obtained from Equation (1), and then, from Equation (5) an unknown convective heat transfer coefficient can be computed.

#### *2.4. Uncertainty Analysis*

Experimental investigations inevitably have associated measurement errors. Both RHTC and CHTC are computed in the present study indirectly from formulas given by Equations (4) and (5), respectively.

In cases where the indirect measurement of a physical quantity, y, is a function of independent measurements *x*1, *x*2,... , *x*n:

$$\mathbf{y} = \mathbf{f}(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n) \tag{6}$$

the standard combined uncertainty uc of y can be calculated applying the propagation model of uncertainty from the formula [54–59]:

$$\mathbf{u}\_{\mathbf{c}}(\mathbf{y}) = \sqrt{\left(\frac{\partial \mathbf{y}}{\partial \mathbf{x}\_{1}} \mathbf{u}(\mathbf{x}\_{1})\right)^{2} + \left(\frac{\partial \mathbf{y}}{\partial \mathbf{x}\_{2}} \mathbf{u}(\mathbf{x}\_{2})\right)^{2} + \dots + \left(\frac{\partial \mathbf{y}}{\partial \mathbf{x}\_{n}} \mathbf{u}(\mathbf{x}\_{n})\right)^{2}}.\tag{7}$$

In the next step, the expanded uncertainty of the measured quantity is calculated from the equation:

$$\mathbf{U} = \mathbf{k} \,\mathrm{u}\_{\mathrm{c}}(\mathbf{y}) \tag{8}$$

where k—coverage factor; k = 2 for 95% confidence level of the uncertainty.

Following the calculation procedure presented in the previous section, the uncertainty of RHTC is derived. From Equation (4) we obtain:

$$\mathbf{h\_{r}} = \varepsilon \sigma \left( \mathbf{T\_{s}^{3}} + \mathbf{T\_{s}} \mathbf{T\_{r}^{2}} + \mathbf{T\_{r}} \mathbf{T\_{s}^{2}} + \mathbf{T\_{r}^{3}} \right). \tag{9}$$

Inserting Equation (9) into Equation (7) we get an expression to obtain the standard combined uncertainty of the measured RHTC. Three independent variables can be distinguished, namely: ε, Tr and Ts. Hence, Equation (7) can be written as:

$$\mathbf{u}\_{\rm c}(\mathbf{h}\_{\rm r}) = \sqrt{\left(\frac{\partial(\mathbf{h}\_{\rm r})}{\partial \varepsilon}\mathbf{u}(\varepsilon)\right)^{2} + \left(\frac{\partial(\mathbf{h}\_{\rm r})}{\partial \mathbf{T}\_{\rm r}}\mathbf{u}(\mathbf{T}\_{\rm r})\right)^{2} + \left(\frac{\partial(\mathbf{h}\_{\rm r})}{\partial \mathbf{T}\_{\rm s}}\mathbf{u}(\mathbf{T}\_{\rm s})\right)^{2}}.\tag{10}$$

Then, partial derivatives in Equation (10) are computed, as follows:

$$\frac{\partial \mathbf{h\_{r}}}{\partial \varepsilon} = \sigma \left( \mathbf{T\_{s}^{3}} + \mathbf{T\_{s}} \mathbf{T\_{r}^{2}} + \mathbf{T\_{r}} \mathbf{T\_{s}^{2}} + \mathbf{T\_{r}^{3}} \right) \tag{11}$$

$$\frac{\partial \mathbf{h\_{r}}}{\partial \mathbf{T\_{r}}} = \varepsilon \sigma \left( \mathbf{2T\_{s}T\_{r} + T\_{s}^{2} + 3T\_{r}^{2}} \right) \tag{12}$$

$$\frac{\partial \mathbf{h\_{r}}}{\partial \mathbf{T\_{s}}} = \varepsilon \sigma \left( \mathbf{3T\_{s}^{2}} + \mathbf{T\_{r}^{2}} + \mathbf{2T\_{s}T\_{r}} \right). \tag{13}$$

The uncertainty associated with the measurement of emissivity, mean radiant temperature and surface temperature can be derived based on sensors and measurement equipment data presented in Section 2.2.

The estimation of emissivity uncertainty is a very difficult task. Several authors have presented methods to address this issue, involving different approaches. Ficker [60] used two precise black body etalons of different emissivities heated to the same temperature to develop a comparative method to estimate virtual emissivity of infrared (IR) thermometers. Chen and Chen [61] determined the emissivity and temperature of construction materials by using an IR thermometer and two contact thermometers. For this purpose, they developed empirical regression equations. Höser et al. [62] found unknown emissivities of rock samples by comparing the temperature of the tested samples with a reference sample at a known temperature. They concluded that the relative uncertainty in rock emissivity is proportional to the relative uncertainty in the rock temperature (T) in the absolute sale. It can be written as:

$$\mathbf{u}(\varepsilon) = \left| -4\frac{\mathbf{u}(\mathbf{T})}{\mathbf{T}}\varepsilon \right|,\tag{14}$$

This method seems to be the most appropriate in the considered case and was applied here.

The remaining uncertainties of two temperatures, Tr and Ts, are both comprised of two elements. The first results from the accuracy of a temperature sensor. The latter is the effect of the data logging device accuracy [55,63]. Hence, we obtain:

$$\mathbf{u}(\mathbf{T}\_{\mathbf{r}}) = \sqrt{(\mathbf{u}\_{\text{sens,r}})^2 + (\mathbf{u}\_{\text{dev,r}})^2} \tag{15}$$

and:

$$\mathbf{u}(\mathbf{T}\_s) = \sqrt{(\mathbf{u}\_{\text{sens},s})^2 + (\mathbf{u}\_{\text{dev},s})^2} \tag{16}$$

where "sens" and "dev" subscripts refer to sensor and measuring device, respectively. Subscripts after a comma, r and s, refer to radiant temperature and surface temperature, respectively.

Similar considerations should be made to obtain CHTC uncertainty. Firstly, Equation (5) is rearranged to the form:

$$\mathbf{h\_{c}} = \frac{\mathbf{q\_{c}}}{\mathbf{T\_{s}} - \mathbf{T\_{i}}}.\tag{17}$$

In the above formula, three independent variables can be distinguished that influence measured CHTC, i.e., qc, Ti and Ts. Then, the uncertainty of CHTC is given by:

$$\mathbf{u}\_{\mathrm{c}}(\mathbf{h}\_{\mathrm{c}}) = \sqrt{\left(\frac{\partial(\mathbf{h}\_{\mathrm{c}})}{\partial\mathbf{q}\_{\mathrm{c}}}\mathbf{u}(\mathbf{q}\_{\mathrm{c}})\right)^{2} + \left(\frac{\partial(\mathbf{h}\_{\mathrm{c}})}{\partial\mathbf{T}\_{\mathrm{i}}}\mathbf{u}(\mathbf{T}\_{\mathrm{i}})\right)^{2} + \left(\frac{\partial(\mathbf{h}\_{\mathrm{c}})}{\partial\mathbf{T}\_{\mathrm{s}}}\mathbf{u}(\mathbf{T}\_{\mathrm{s}})\right)^{2}}.\tag{18}$$

Subsequent partial derivatives are as follows:

$$\frac{\partial \mathbf{h}\_{\mathbf{c}}}{\partial \mathbf{q}\_{\mathbf{c}}} = \frac{1}{\mathbf{T}\_{\mathbf{s}} - \mathbf{T}\_{\mathbf{i}}} \,\tag{19}$$

$$\frac{\partial \mathbf{h\_{c}}}{\partial \mathbf{T\_{s}}} = \frac{-\mathbf{q\_{c}}}{\left(\mathbf{T\_{s}} - \mathbf{T\_{i}}\right)^{2}} \tag{20}$$

$$\frac{\partial \mathbf{h\_{c}}}{\partial \mathbf{T\_{i}}} = \frac{\mathbf{q\_{c}}}{\left(\mathbf{T\_{s}} - \mathbf{T\_{i}}\right)^{2}}.\tag{21}$$

The uncertainty associated with the measurement of indoor air temperature u(Ti) and surface temperature (Ts) can be derived similarly, as in the previous case on the basis of Equations (15) and (16). However, a measurement of heat flux density requires some additional explanation. In this case, according to the manufacturer's manual [64], measurement uncertainty consists of several components. These are calibration uncertainty, the difference between measurement and reference calibration conditions, the duration of sensor employment, and application errors resulting from working conditions. They can be included in the relationship:

$$\mathbf{u}(\mathbf{q}\_c) = \sqrt{\sum\_{i=1}^n \mathbf{u}\_{i\prime}^2} \tag{22}$$

where n is the number of uncertainty components and ui is the i-th component.

#### *2.5. Selected Correlations for CHTC Calculations*

During the test, there was no measured air velocity near the wall. A gravity ventilation system is present in the building, but the ventilation register in the room, however, is located far from the studied wall and was closed, and it was assumed that it has no impact on the flow regime near the wall. Hence, for further consideration, 13 correlations were chosen to obtain convective heat transfer coefficients for vertical walls under natural convection [10,16,37,65,66]. In all following relationships, it is assumed that:

$$
\Delta \mathbf{T} = \mathbf{T}\_{\mathbf{s}} - \mathbf{T}\_{\mathbf{i}}.\tag{23}
$$

The model of Wilkes and Peterson (quoted from [16]) is given by:

$$\mathbf{h}\_{\mathrm{c}} = \mathbf{3.05} (\boldsymbol{\Delta T})^{0.12}. \tag{24}$$

In [19] it was stated that that model was developed on the basis of a test for temperature difference between 4.5 to 15.5 ◦C with two heated plates 2.4 × 0.8 m with 0.1 m air space. The resulting correlation was derived on the three data points only.

Khalifa in his review [39] quoted the relationship developed originally by Hottinger:

$$\mathbf{h}\_{\mathrm{c}} = 2.50 \,\mathrm{(\Delta T)}^{1/4}.\tag{25}$$

Min et al. [40] determined natural convection coefficients by using three different sized rectangular test chambers. Hence, they are not applicable in the considered case. For the vertical wall, they obtained correlations in the case of the heated ceiling and heated floor. They also referred to other correlations from various other studies. The first of these, presented here, was developed using a 0.60 m square plate, the temperature difference up to 555 ◦C and the height H. For laminar flow:

$$\mathbf{h}\_{\mathrm{C}} = 1.368 \left( \frac{\Delta \mathbf{T}}{\mathrm{H}} \right)^{1/4} \,. \tag{26}$$

The second correlation was developed during tests with a 1.2 m square plate. Apart from the temperature difference up to 100 ◦C, no additional information on the application range was given. The model was given by two equations, as follows:

$$\mathbf{h}\_{\mathbb{C}} = 1.776 \left(\Delta \mathbf{T}\right)^{1/4} \tag{27}$$

and for the turbulent flow:

$$\mathbf{h}\_{\mathrm{c}} = 1.97 \Im(\Delta \mathbf{T})^{1/4}.\tag{28}$$

The third is the correlation from King. Unfortunately, no data related to the test details or applicability ranges were provided. It is given by:

$$\mathbf{h\_{c}} = 1.517(\Delta \mathbf{T})^{1/3}.\tag{29}$$

Alamdari and Hammond [26] derived correlations for convective heat transfer from the internal surfaces in naturally ventilated buildings. Using data from other studies, and applying the mathematical model of Churchill and Usagi [24], they proposed the following relationship:

$$\mathbf{h\_{\mathcal{C}}} = \left\{ \left[ 1.5 \left( \frac{\Delta \mathbf{T}}{\mathcal{L}} \right)^{1/4} \right]^{6} + \left[ 1.23 (\Delta \mathbf{T})^{1/3} \right]^{6} \right\}^{1/6} \,. \tag{30}$$

In addition, they provided a simplified formula valid within the limited range of temperatures and applicable to naturally ventilated buildings, given by the expression:

$$\mathbf{h}\_{\mathrm{c}} = 0.134 \mathbf{L}^{-1/2} + 1.11 (\boldsymbol{\Delta T})^{1/6} \text{\AA} \tag{31}$$

where L is a hydraulic diameter expressed as:

$$\mathbf{L} = 4\mathbf{A}/\mathbf{P} \tag{32}$$

Fohanno and Polidori [67] analysed laminar and turbulent heat transfer modelling at uniformly heated internal vertical building surfaces. The resulting correlations for CHTC were derived theoretically and compared with other models. Their model is given by the equation:

$$\mathbf{h}\_{\mathrm{C}} = 1.332 \left( \frac{\Delta \mathbf{T}}{\mathrm{H}} \right)^{1/4} \,\mathrm{.}\tag{33}$$

Musy et al. [68], quoting the work of Allard, presented the following equation for walls with natural convection:

$$\mathbf{h\_{c}} = 1.5(\Delta \mathbf{T})^{1/3}.\tag{34}$$

Churchill and Chu [65] investigated uniformly heated and cooled vertical plates. Their correlation is given by:

$$\mathbf{h}\_{\mathrm{c}} = \frac{0.0257}{\mathrm{H}} \left( 0.825 + 7.01 (\Delta \mathrm{T})^{1/6} \mathrm{H}^{3/6} \right)^{2} . \tag{35}$$

Khalifa and Marshall [25] investigated internal convection in a real-sized indoor test cell. Various combinations of relative positions of a heater, radiant panels and wall were studied. Two cases similar to that studied in the present paper were chosen. The first one is the case with a room heated by a radiator located adjacent to the test wall:

$$\mathbf{h}\_{\mathbb{C}} = 2.20(\Delta \mathbf{T})^{0.21}.\tag{36}$$

The second is a room heated by an oil-filled radiator located under a window:

$$\mathbf{h}\_{\mathbb{C}} = 2.35(\boldsymbol{\Delta T})^{0.21}.\tag{37}$$

Rogers and Mayhew [69] presented the correlation for vertical plates under laminar or transitional flow conditions:

$$\mathbf{h}\_{\mathrm{C}} = 1.42 \left( \frac{\Delta \mathbf{T}}{\mathrm{H}} \right)^{1/4}. \tag{38}$$

#### *2.6. Statistical Analysis*

Additionally, various statistical measures can be used to perform error analysis of the presented models against measured CHTC [70]. Among them, there is an average absolute error (AAE), average biased error (ABE), mean absolute error (MAE) and mean bias error (MBE) [71,72]. Assuming that hm,i is a i-th measured value of CHTC, hp,i is a i-th value of CHTC predicted by the model and m is a total number of conducted measurements, they are given by the following equations:

$$\text{AAE} = \frac{1}{\text{m}} \sum\_{\text{i=1}}^{\text{m}} \frac{\left| \mathbf{h}\_{\text{P},\text{i}} - \mathbf{h}\_{\text{m},\text{i}} \right|}{\mathbf{h}\_{\text{m},\text{i}}} \times 100\text{\%},\tag{39}$$

$$\text{ABE} = \frac{1}{\text{m}} \sum\_{\text{i=1}}^{\text{m}} \frac{\left(\mathbf{h}\_{\text{p},\text{i}} - \mathbf{h}\_{\text{m},\text{i}}\right)}{\mathbf{h}\_{\text{m},\text{i}}} \times 100\text{\%} \tag{40}$$

$$\text{MAE} = \frac{1}{\text{m}} \sum\_{i=1}^{\text{m}} |\mathbf{h}\_{\text{P},i} - \mathbf{h}\_{\text{m},i}| \,\tag{41}$$

$$\text{MBE} = \frac{1}{\text{m}} \sum\_{\text{i=1}}^{\text{m}} (\mathbf{h}\_{\text{p},\text{i}} - \mathbf{h}\_{\text{m},\text{i}}). \tag{42}$$

The average absolute error (AAE) is the average of all the absolute errors calculated for the consecutive measurements in a given dataset, and indicates the average error of a correlation. The average biased error (ABE) shows the degree of overestimation (ABE > 0) or underestimation (ABE < 0) of the considered correlation. The mean absolute error (MAE) shows the average magnitude of deviations of a modelled variable against the reference values. Low MAE indicates the high accuracy of a model. MBE is used to determine the overall bias of the correlation. Positive MBE means the overestimation of the model.

#### **3. Results and Discussion**

#### *3.1. Radiative Heat Transfer Coefficient*

#### 3.1.1. Introduction

The radiative heat transfer coefficient was computed from Equation (4). During the whole period, the radiant temperature was higher than that of the wall surface (Figure 4). This difference varied from 1.54 ◦C (at 13:10 on 7 February) to 0.28 ◦C (at 6:50 on 6 February), with an average of 0.91 ◦C.

**Figure 4.** Average wall surface temperature (Ts) and radiative temperature (Tr).

RHTC, calculated under the aforementioned conditions, varied during the measurement period (Figure 5) from 5.373 W/m2K (at 16:40 on 3 February) to 5.516 W/m2K (at 21:00 on 31 January), with an average of 5.445 W/m2K. The EN ISO 6946 standard [50] gives the value hr = 5.7 W/m2K for the blackbody at 20 ◦C. Consequently, for ε = 0.95 resulting hr = 5.415 W/m2K is very close to that average value.

**Figure 5.** Radiative heat transfer coefficient from measurements.

Assuming that:

$$\mathbf{T\_m} = \mathbf{0.5} \left( \mathbf{T\_r} + \mathbf{T\_s} \right) \tag{43}$$

For cases with small Tm values, Equation (4) can be linearised [73,74] to the form:

$$\mathbf{h\_{f}} = 4\varepsilon\sigma \mathbf{T\_{m}^{3}}.\tag{44}$$

Despite the nonlinear dependence of the RHTC on Tm, given by Equation (44), hr almost linearly changes with Ts (Figure 6). This occurs primarily because of the relatively narrow range of Ts variation within the range below 2.5 ◦C. Similar observations were reported by Evangelisti et al. [65].

**Figure 6.** Radiative heat transfer coefficient from measurements.

#### 3.1.2. Uncertainty Analysis

Before the calculations were made, the emissivity of the internal walls was assumed at ε = 0.95. This value was confirmed using a DIT-130 IR-thermometer (declared accuracy ±1.5% of a measured value + 2 ◦C) and the reference DFT-700 thermometer with a K-type thermocouple probe (accuracy ±0.2% of a full scale). Because of its low variation under the conditions considered here, it was reasonable to assume a constant emissivity value of the wall surfaces [75].

Based on the manufacturers' manuals, and assuming a temperature of 25 ◦C, relevant uncertainties were computed (Table 2). The resultant uncertainty of RHTC measurements is uc(hr) = 0.18046 W/m2K. Applying coverage factork=2 (see Equation (8)) we obtained the expanded uncertainty U(hr) = 0.36 W/m2K. Assuming a temperature of 20 ◦C the value of U(hr) = 0.34 W/m2K was obtained.


*3.2. Convective Heat Transfer Coefficient*

#### 3.2.1. Introduction

The measured internal air temperature varied from 20.44 ◦C (at 16:40 on 3 February) to 23.40 ◦C (at 22:10 on 30 January), with an average of 21.98 ◦C. It was higher than that of the wall surface (Figure 7). This difference varied from 1.54 ◦C (at 13:10 on 7 February) to 0.28 ◦C (at 6:50 on 6 February), with an average of 0.91 ◦C.

**Figure 7.** Wall surface temperature (Ts), internal air temperature (Ti) and their difference.

The calculated CHTC varied throughout the whole period (Figure 8) from 0.069 W/m2K (at 6:30 on 5 February) to 3.027 W/m2K (at 14:10 on 2 February), with an average of 1.153 W/m2K. The EN ISO 6946 standard gives the value hc = 2.5 W/m2K for the internal surface of the vertical wall. The recommended internal surface resistance in such a case is Rsi = 0.13 m2K/W. Taking the measured average hr = 5.445 W/m2K we obtain Rsi = 0.15 m2K/W.

**Figure 8.** Convective heat transfer coefficient from measurements.

#### 3.2.2. Uncertainty Analysis

At first, the uncertainty of the heat flux density measurement should be determined (Equation (22)). In the present study, two HFP sensors were used. Several authors have presented various experimental studies on the thermal performance of buildings using that sensor [76–78], but none of them discussed the problem of heat flux density measurement uncertainty. In some cases, the measurement accuracy of the HFP01 heat flux plate was assumed from 5% [79] to 20% [80]. For this reason, this issue is presented here in detail.

Following recommendations given by the manufacturer [64], the following factors were distinguished:


The calibration, non-stability and temperature errors are relatively easy to estimate because of clear criteria given by the manufacturer. The first is given as expanded uncertainty (see Equation (8)), and for uncertainty budget calculation it was assumed as 1.5%. The latter should be estimated for the worst conditions during measurement, i.e., the greatest temperature measured during the test: Ti = 23.40 ◦C.

The resistance error is related to the influence of the sensor thermal resistance added to the resistance of a given partition on the resultant heat flux density. While the manufacturer does not recommend the use of thermal paste because it tends to dry out, it was

applied here because of the relatively short measurement period and moderate temperature. Assuming an average thickness of this paste of 0.1 mm and a thermal conductivity of 1.5 W/m·<sup>K</sup> it gives the additional thermal resistance of 6.67 × <sup>10</sup>−<sup>6</sup> m2K/W in comparison to <sup>71</sup> × <sup>10</sup>−<sup>4</sup> m2K/W of the sensor specified by the manufacturer. Hence, the resulting resistance is 0.93%.

Deflection error (called operational error in ISO9869) appears as a result of the difference between the thermal conductivity of the surrounding environment and that of a sensor. The sensor thermal conductivity is 0.76 W/m·K and is in the same order as values typically met in buildings materials: 0.3–0.4 W/m·K for ceramic blocks and 0.7–0.9 W/m·K for cement-lime plaster. Assuming a thermal resistance of the wall of approximately 4 m2K/W, we can see that the deflection error is at a negligible level. The average measured heat flux density of 7.42 W/m2 calculated u(qc) = 0.1526 W/m2.

The derivative components, given by Equations (19)–(21), were computed for the most unfavourable conditions during the experiment, i.e., the smallest Ts—Ti, which amounted to 1.23 ◦C for qc = 2.54 W/m2. Results are given in Table 3. Calculated uc(hc) = 0.475218 W/m2. Taking k = 2 (see Equation (8)) we get a resulting U(hc) = 0.95 W/m2K. On the other hand, for the maximum temperature difference of 2.77 ◦C and qc = 2.11 W/m<sup>2</sup> the resulting uncertainty is U(hc) = 0.16 W/m2K. Other authors estimated the accuracy of experimentally measured CHTCs from approximately 20% [22,26] to 30% or even 35% at ΔT ≈ 1 ◦C [53].

**Table 3.** Uncertainties in CHTC measurement.


#### 3.2.3. Comparison with Other Models

It can be noticed that fluctuations of the CHTC were connected with variations in the difference between internal air and wall surface temperature (Figure 9), but this relationship was rather weak (calculated coefficient of determination R<sup>2</sup> = 0.2498). Some authors recommended the use of nonlinear correlations [81,82], but they did not improve the quality of the fit. In case of the logarithmic, exponential, quadratic function and 4-th order polynomial, R2 = 0.2252, 0.2459, 0.2308, and 0.2323, respectively.

**Figure 9.** Convective heat transfer coefficient versus indoor air temperature.

There are several reasons for this. In the experimental studies performed in test chambers [7,25,27,39,40], thermal conditions during the measurements were stable. As noted in Section 2.5, the ventilation register in the room was closed, however, and located far from the studied wall, and it was assumed that it has no impact on the flow regime near the wall. The hydronic heater was located approximately 4 m from the measurement equipment. Solar irradiance incident on the external wall was very low (Figure 2), and that falling on internal surfaces could be neglected without any doubt, due to its reduction through windows and curtains. On the other hand, because of the everyday use of the building during the experiment, there could be disturbances in the airflow due to the movement of people in the building, however, indoor air temperature variation was below 3 ◦C. Hence, to find a solution to this problem, technical factors should be taken into account. Considerations presented in Section 2.4 indicate that an error of CHTC estimation results from uncertainty of heat flux density and air and surface temperature measurement. In the present case, the most important influence was that of the Ts—Ti temperature difference (Table 3). Both temperatures were measured by accurate sensors (Table 1). However, their difference is in the denominator of the fraction to calculate their uncertainty (Equations (19)–(21)). Hence, not only is the uncertainty of temperature measurement important, but the uncertainty of the temperature difference is as well. In the present study, ΔT varied from 1.23 ◦C to 2.77 ◦C. These were low values, difficult to measure with high accuracy and which, in certain cases, may lead to unacceptable uncertainties [83]. Numerous studies have pointed out this problem [84–86], and a common practice is to filter out the data when ΔT was lower by several Celsius degrees (typically 2–5 ◦C). This problem should be, however, investigated in more detail in the next experiment.

Several studies have presented the results of experiments on convection at vertical walls at temperature differences below 3 ◦C. In [26], the heat transfer coefficient for buoyancy-induced airflow near vertical surfaces for ΔT of 1 to 10 ◦C was between 1.5 to 4 W/m2K. Khalifa [40] presented the results of numerous experimental and theoretical works on convection at various surfaces. For a vertical wall at ΔT=2 ◦C, CHTC was between 1.2 to 3.0 W/m2K. CHTC in a wall with an intermittently heated room [28] was between about 0.2 to 1.5 W/m2K during the non-heating period. At continuous heating, the average CHTC was 1.6 W/m2K, with a variation between maximum-minimum values of about 1.5 W/m2K. CHTC for the temperature difference below 2 K was between 0.2 to 2.5 W/m2K. The author did not provide any empirical correlation for hc, and only compared averaged values with that recommended by the relevant standards. This significant dispersion of experimental results in a real building was also confirmed in [53]. CHTC was measured in that study at a wall with a radiator at the back wall and no ventilation, and varied from approximately 0.3 to 5.0 W/m2K at ΔT<2 ◦C. The dominant results for a wall temperature measured at half of its height ranged between 0.5 and 2.0 W/m2K.

CHTC calculated from the 14 correlations presented in Section 2.5 varied from 1.305 W/m2K (Equation (26)) to 3.328 W/m2K (Equation (24)). In all cases, it was greater than measurements by 1.153 W/m2K (Figure 10). The average value from all models was hc = 2.024W/m2K and was greater than measurements by 75.6%. The results closest to the measured value were obtained for models given by the Equations (26), (31), (33) and (38) and they were greater than measurements by 13.4%, 14.9%, 10.2% and 17.5%, respectively.

Six correlations, given by Equations (26), (30), (31), (33), (35) and (38), included geometrical parameters of the considered wall (hydraulic diameter or height). The relevant columns in Figure 9 were filled with hatching. CHTC in this group of models varied from 1.271 W/m2K (Equation (33)) to 1.829 W/m2K (Equation (35)), with an average of 1.313 W/m2K. The remaining eight models produced results from 1.912 W/m2K (Equation (34)) to 3.328 W/m2K (Equation (24)), with an average of 2.446 W/m2K, i.e., more than twice the measured CHTC. The results presented here indicate that better compliance with measurements was obtained for models that take into account the geometry of the wall.

**Figure 10.** Average CHTCfor different models for measurement conditions.

The theoretical calculation of CHTC revealed that various models provide similar results for ΔT up to 10 K (Figure 11). These can be gathered into several groups. The first consists of models given by Equations (26), (31), (33) and (38). All except for Equation (31) have an (ΔT/H)1/4 element. In Equation (31), (ΔT)1/6 was used along with L−1/2 and resulted in a slightly lower rise in CHTC with ΔT. Similarities can also distinguished between models given by Equations (27) and (28), by (36) and (37), and by (29), (30), (34) and (35).

**Figure 11.** CHTC for different models.

All the models studied here had low coefficients of determination against measurements (Table 4). Their value in the case of the 10 min sampling time did not exceed R<sup>2</sup> = 0.0437 for the linear regression equation. In the next step, hourly averaged values of CHTCs (the typical length of the simulation time step in the building simulation tools) were computed from 10 min variables. The R<sup>2</sup> increase was negligible, in the third decimal place.


**Table 4.** Convective heat transfer coefficients from presented models and coefficients of the determination referred to measured values.

The results presented here show significant differences between measurements and models. The best degree of convergence between predicted and measured results, given by AAE, was obtained in models given by Equations (26), (31), (33) and (38). MAE ranged from 0.41 W/m2K (Equation (33)) to 2.18 W/m2K (Equation (24)), showing relatively high deviations from the modelled CHTC versus measurements. The average bias in the models given by MBE varied from 0.12 W/m2K (Equation (33)) to 2.18 W/m2K (Equation (24)). From these results, it can be stated that the best match with measurements was obtained by the model of Fohanno and Polidori given by Equation (33).

Such discrepancies were also reported in other studies. For example, Kalema and Haapala [7] analysed the impact of interior heat transfer coefficient models on the thermal dynamics of a two-room test cell with radiator heating. Experimental results from the tests under steady-state and dynamic conditions were compared with that simulated in the thermal analysis program TASE. ASHRAE, Alamdari and Hammond, and Khalifa and Marshall correlation equations were used in the calculations of convective heat transfer coefficients. Measured air and surface temperatures were within the simulated minimummaximum range of 4 ◦C. Only the window surface temperature was 0.7 ◦C greater than the calculated maximum temperature. All calculated heat fluxes were within the 10% error band of the measured values.

The discrepancies presented here may have an impact on simulation results when using computer tools for the dynamic simulation of buildings [10]. Commonly in such cases, constant values of internal CHTC and RHTC are assumed. If vertical walls are of interest, in the first case the values within the range from 2.0 W/m2K to 5.0 W/m2K [87–89] are met.

In the energy auditing of buildings, when monthly calculation methods are applied, constant values of total combined convective and radiative heat transfer coefficient (or internal surface resistance) are normally used [90–92].

In contrast to measured RHTC, it is not easy to present any recommendations on CHTC values to be used based on this study. Further detailed analyses are needed to find and minimise measurement uncertainties. The presented analysis shows that the most important contribution to CHTC uncertainty is the measurement of the temperature difference between internal air and wall surface.

#### **4. Conclusions**

In the present study, convective (CHTC) and radiative (RHTC) heat transfer coefficients for the internal surface of a vertical wall were calculated from measurements. For

comparative purposes, in the first case, 14 correlations were used to obtain CHTC from temperature difference and, in several cases, the height of a wall.

The results showed significant differences between measurements and mathematical models. This was likely because in the presente study measurements were taken during the everyday use of a building. Hence, it was not possible to obtain the stable thermal conditions possible in the test chambers used in other experimental studies. Moreover, the internal surfaces of the test chambers were covered with materials different from those used in the building. For example, in the study of Khalifa and Marshall [10] authors used aluminium foil on the internal and external surfaces of the external partitions of their climatic chamber. This was employed to minimise the effect of longwave radiation exchange on the temperature and heat flux measurements, and meant that only convective heat transfer was studied. In the present study, combined convective and radiative heat transfer was considered.

It should also be noted that aluminium foil has different physical properties (especially roughness influencing the flow of air) than the cement-lime plaster used as the surface layers of ceramic walls. The flow of air could be also influenced because of stairs located near the sensors. The stairs are about 2 cm away from the wall and their construction allows air to flow, but its presence cannot be excluded from the measurement results.

Model quality was estimated using four statistical goodness-of-fit criteria: AAE, ABE, MAE and MBE. On this basis, the model of Fohanno and Polidori was chosen as the best, with AAE = 68%, ABE = 52%, MAE = 0.41 W/m2K and MBE = 0.12 W/m2K. The resulting hc = 1.217 W/m2K. The worst model was that of Wilkes and Peterson, for which AAE = 298%, ABE = 298%, MAE = 2.18 W/m2K, MBE = 2.18 W/m2K and hc = 3.328 W/m2K.

The experiment could be repeated over an extended period to avoid possible temporary disruptions and with the addition of airflow velocity measurement near the wall. Moreover, a shorter sampling time could be used and then time averaging in longer periods could be applied for comparison with hourly values calculated from other models.

The study showed, however, that calculated total surface resistance is close to that recommended by EN ISO 6946 standard, commonly used in energy auditing and certification of buildings. However, it is a rather crude estimation applied in less accurate, annual or monthly methods.

Because of the noticeable impact of a temperature difference between internal air and wall surface on CHTC uncertainty, in future studies, emphasis should be given to this issue.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Symbols**



#### **References**


**Sultan Kobeyev 1, Serik Tokbolat 2,\* and Serdar Durdyev 3,\***


**Abstract:** In times of unprecedented climate change and energy scarcity, the design and delivery of energy-efficient and sustainable buildings are of utmost importance. This study aimed to design a hotel building for hot and dry climate conditions and perform its energy performance analysis using energy simulation tools. The model of the hotel building was constructed by a graphical tool OpenStudio and EnergyPlus following the ASHRAE Standard 90.1. To reduce the energy demand of the hotel, parametric analysis was conducted and building envelope parameters such as the thickness of insulation layer in the exterior wall and the roof, thermal conductivity of insulation layer, rate of infiltration, U-factor of windows, and thermal resistance of air gap in the interior walls (R-value), window-to-wall ratio, and orientation of the building were tested and the impact on the energy use of the building was analyzed. It was found that most of the design assumptions based on the ASHRAE standard were already optimal for the considered locality, however, were still optimized further to reach the highest efficiency level. Apart from this, three sustainable technologies—thermochromic windows, phase change materials, and solar panels—were incorporated into the building and their energy consumption reduction potential was estimated by energy simulations. Cumulatively, these sustainable technologies were able to reduce the total energy use from 2417 GJ to 1593 GJ (i.e., by 824 GJ or 34%). Calculation of payback period and return on investments showed that thermochromic windows and solar panels have relatively short payback periods and high return on investments, whereas PCM was found to be economically nonviable. The findings of this study are deemed to be useful for designing a sustainable and energy-efficient hotel building in a sub-tropical climate. However, the overall design and energy performance analysis algorithm could be used for various buildings with varying climate conditions.

**Keywords:** energy performance; design parameters; energy simulation; building envelope

#### **1. Introduction**

Globally, buildings account for about one-third of the final energy use, and the major part of that energy is consumed, among others, by heating, ventilation, and air conditioning systems [1,2]. Thus, buildings offer a significant potential to reduce the energy consumed by the building sector and thus mitigate the effects of climate change and the ongoing energy crisis [3]. Therefore, it is imperative for all the new buildings to satisfy the highest standards of energy efficiency and to operate at the lowest possible energy consumption (EC) levels. The up-to-date research findings refer to a combination of measures that could increase the energy efficiency of the buildings such as, for example, design optimization and integration of sustainable energy technologies and sustainable materials [4]. Generally, solutions for achieving net-zero energy buildings (NZEB) can be broadly categorized into

**Citation:** Kobeyev, S.; Tokbolat, S.; Durdyev, S. Design and Energy Performance Analysis of a Hotel Building in a Hot and Dry Climate: A Case Study. *Energies* **2021**, *14*, 5502. https://doi.org/10.3390/en14175502

Academic Editor: Paulo Santos

Received: 12 August 2021 Accepted: 1 September 2021 Published: 3 September 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

three groups—passive efficiency strategies, integration of renewables, and active energy technologies [5,6].

Design optimization includes addressing such measures as the window-to-wall ratio (WWR), building orientation, shadings, cool roofs, modifications to buildings' envelope and thermal mass, etc. There is a wide consensus among the scientific community that early actions taken with optimizing the above-mentioned parameters yield substantial improvements in the efficiency of buildings, which will eventually impact the sizing of HVAC equipment and help to achieve a higher-performing building [7–9]. In this study, the following design parameters were considered: building envelope optimization (thickness and conductivity of insulation layers, infiltration rate, U-factor of windows, and thermal resistance of interior walls), WWR, and building orientation.

Several studies have been conducted by researchers to determine the most energyefficient options for optimization of building design at a preliminary stage. Tuhus-Dubrow and Krarti [10] used a genetic algorithm to determine optimum building shapes, orientation, wall and roof construction, window type, window area, foundation, infiltration rate, insulation thickness, and shading of residential buildings. Ascione et al. [11] optimize the annual energy consumption of buildings in the Mediterranean climate zone by using optical properties of external plastering, wall insulations properties, and composition of outer walls as design parameters. Wang and Wei [12], on the other hand, focused on the composition of walls, number, type, dimensions, and material of windows to optimize energy consumption and cost of office buildings using a quantum genetic algorithm. Similarly, Carlucci et al. [13] used multi-objective optimization by the non-dominated sorting genetic algorithm to enhance the thermal comfort of a building with U-values of walls, roofs, and floors as well as U-value and visible transmittance of windows as design variables.

Apart from design optimization, the overall energy efficiency and general sustainability levels of buildings were found to be improved by integrating various sustainable energy technologies and materials. Since the hotel buildings tend to have a great number of windows due to the nature of the room allocation as well as the potential impact on solar gain and overheating, the study paid significant attention to the selection of window types. The study also considered integrating phase change materials (PCMs) and solar energy technologies.

There is a wide consensus among the scientific community and industry practitioners that windows and skylights are building elements with the lowest energy efficiency [14–16]. Owing to their relatively smaller thickness compared to exterior walls and roofs, windows have a much higher rate of thermal heat transfer [16]. As there are already a number of heat generators inside the building, such as equipment and people, extra heat gain through windows is often unwanted, especially in moderate climates.

Thermochromic windows are one of the promising technologies intended to address this issue that has drawn particular attention among researchers in recent years [17]. These are a type of smart windows with a special glazing that can passively modulate infrared transmittance from sunlight and alter its optical and thermal properties depending on the intensity of solar heat gain [16]. By reflecting infrared and near-infrared lights while still transmitting visible light, thermochromic windows can reduce cooling energy demand by up to 81.7% and maintain healthy indoor conditions with visual and thermal comfort [16].

Thermochromic windows can be manufactured from various materials, including mercury (II) iodide, chromium (III) oxide, and lead (II) oxide, but so far, the most researched material with thermochromic properties is vanadium dioxide (VO2) [14,16,18]. Traditionally, pure VO2-based coatings were used for manufacturing thermochromic windows as this material provides adequate luminous transmittance and solar modulation properties. However, it has a number of shortcomings, such as relatively low stability of properties, cost inefficiency, and color alterations [16]. To resolve these issues, VO2 is usually doped with additional chemical elements or various organic and inorganic nanomaterials.

Phase change materials (PCMs) are a group of materials that could be considered as sustainable materials that help with energy savings in buildings. PCMs have a high latent heat that can passively store and release thermal energy by undergoing a solid-liquid phase transition. By doing so, PCMs can decrease heating and cooling energy consumption of buildings, reduce indoor temperature fluctuation, shift peak times of energy demand and thus reduce grid load and assist intermittent renewable energy sources, and maintain comfortable indoor temperature conditions [19]. The model of PCM in EnergyPlus was analytically validated by Tabares-Velasco et al. [20], who states that for the accuracy of the model, (1) Conduction Finite Difference algorithm should be used, (2) the number of timesteps per hour should be higher than 20, and (3) space discretization value should be set to at least 3.

Usually, due to partial charging and discharging, PCMs are not exploited to their full potential. Moreover, PCMs cause an additional problem by discharging heat into indoor spaces during cooler times of the day [21]. Therefore, various means of assisting discharge of PCMs, such as natural ventilation, night ventilation, mechanical ventilation, were actively investigated in recent years. One of them is temperature-controlled natural ventilation demonstrated by Prabhakar et al. [21] and Pisello et al. [22], which is reported to significantly improve the energy efficiency of PCMs.

Solar panels or photovoltaic PV panels are devices made of semiconductors that convert energy from photons into electricity by displacing electrons from the semiconductor material's atoms. It has such advantages as carbon neutrality, low maintenance cost, and relatively low environmental footprint among others. However, it is intermittent and significantly dependent on ambient weather conditions, set aside such challenges as large space requirements and the necessity of energy storage systems. Nevertheless, solar energy technologies are generously subsidized in many parts of the world, which, together with technological advancements, help lower their capital costs and foster their widespread installation [23]. For instance, in the United States, prices of solar panels have dropped by half since 2008, while the capacity of installed panels increased by 35 times [24].

As it can be seen from the literature review, various studies address the impacts of design parameters on the energy use of buildings in general [10–13]. Some studies separately study the impacts of the integration of renewable energy technologies [5]. However, it was also noted that most articles are concerned about residential buildings and tend to consider only impacts of either design parameters or integration of sustainable energy technologies as standalone solutions. This study, however, is aimed to adopt a systems-based approach and to provide a combined energy performance analysis of both aspects at the same time.

For the purposes of this research, a hotel building was selected. Hotel buildings are quite common around the world and a significant number of them are located near holiday destinations which are known by their tropical and sub-tropical climate conditions [25]. They usually have high comfort levels and, thus, heavily rely on HVAC systems. This motivated the authors of the research to investigate the hotel buildings as their energy optimization and energy efficiency could potentially make a significant contribution to the overall sustainability of the built environment. Thus, this study conducted an energy performance analysis of a hotel building. The primary objectives of this study are:


The results of the study would contribute to the existing body of knowledge at least in two important ways. First of all, it will provide practical insights into the effect of envelope optimization and the performance of the above-mentioned sustainable technologies on the energy efficiency of a hotel building in the subtropical climate zone, which, to the best of the authors' knowledge, has not been well studied yet. Moreover, the following paper would suggest a methodology for optimization of building envelopes and green technologies at the preliminary stages of building design. Although a case study was shown for a high-rise hotel building only, the application of the methodological flow can readily be extended to other building types as well. A simulation model of a theoretical hotel building was created using graphical and energy simulation tools, and then six potential energy performance improvement measures were considered. Subsequently, a basic cost-benefit analysis was performed to evaluate the economic viability and financial attractiveness of the building optimization options. The findings of the study were collated to a set of recommendations for designing and building an energy-efficient hotel building in a sub-tropical hot and dry climate. The steps undertaken in this study are presented in Figure 1.

**Figure 1.** The flow of the research work.

#### **2. Methodology**

The adopted methodology in this study is based on energy performance simulation using a combination of graphical and energy simulation tools which were built based on the relevant standards and codes used in the United States. Building energy efficiency levels have to comply with International Energy Conservation Code. All the analysis presented in this section will be in accordance with ASHRAE [26]. To optimize building envelope characteristics, window-to-wall ratio, building orientation, and characteristics of three sustainable technologies considered in the following study, a simple parametric analysis will be performed and the optimization will be carried out sequentially considering each variable separately. The advantage of adopting this method is that it is simple and quick to perform, which may come at the cost of possibly overlooked interactions between some parameters, such as properties of the building envelope, building orientation, and PCM melting temperature. To account for this kind of interdependencies between various parameters, one may want to use more mathematically rigorous optimization methods, such as the genetic algorithm or stochastic gradient descent.

#### *Development of the Building Energy Model*

The energy model of the building was created using the OpenStudio plugin in SketchUp and improved in OpenStudio. The energy simulation of the model was then performed separately in EnergyPlus v9.3. For the validity of the PCM model, the number of timesteps per hour was set to 20 and the Conduction Finite Difference (CondFD) algorithm was used following the recommendations of Tabares-Velasco et al. [20].

Each floor was divided into separate spaces (for guest rooms, bathrooms, lobby, corridors, kitchens, etc.) and each room was set to be a distinct thermal zone. The material composition of walls, slabs, roof, and windows in the initial baseline model was assigned by using templates provided by ASHRAE Standard 90.1 [26] taking into account that the hotel is to be located in climate zone 3B according to the IECC Climate Zone Map [27]. The material composition of exterior walls and roof are shown below in Table 1.


**Table 1.** Material composition of exterior walls and roof.

Moreover, space types and schedules for occupancy loads, lighting, and electric equipment were also assigned referring to ASHRAE Standard 90.1. For heating, cooling, and air conditioning of spaces in the hotel Packaged Terminal Air Conditioning (PTAC) system with direct expansion cooling and electric heating was selected due to the following reasons:


Thermostat set points of different space types have some variations, but for guest rooms, setpoints were selected to be 21 ◦C and 24 ◦C of mean air temperature for heating and cooling, respectfully, all days of the week and round-the-clock.

The model was initially built having all 24 floors (see Figure 2), but later, to reduce the simulation time, the number of residential floors was reduced and a zone multiplier of 19 was set for the eleventh floor. Floor and roof surfaces exposed because of this were set to be adiabatic. For simulation, the Conduction Transfer Function algorithm with 6 timesteps per hour was used for all retrofits except for PCMs.

In general, there is no exact scientific reason for choosing this particular type of building other than an attempt to make it representative of common hotel buildings. The hotel is rectangular with an aspect ratio of 2 with rooms arranged along its perimeter and an atrium at the center, which is a common practice for accommodating hotel floors with as many rooms as possible, while also ensuring that indoor areas are well lit.

**Figure 2.** Energy simulation model of a hotel building.

#### **3. Results and Discussions**

#### *3.1. Building Envelope*

As was mentioned earlier, the baseline composition of exterior walls and roof were assigned based on the type of building and the climate zone. With this initial configuration, the building consumes an annual heating and cooling energy of 2557.82 GJ. To reduce this value, a parametric analysis was conducted to optimize some key parameters of the building envelope, namely thickness of insulation layer in the exterior wall and the roof (Di, wall and Di, and roof, respectively), the thermal conductivity of the insulation layer (ki), rate of infiltration (I), U-factor of windows, and thermal resistance of air gap in the interior walls (R-value). All six variables control the rate of heat transfer through the envelope of a building and significantly influence its energy demand. Table 2 summarizes the range and increments of these parameters used in the parametric analyses:

**Table 2.** Parametric analyses for optimization of the building envelope.


To reduce the number of simulations required while still accounting for interdependencies between these parameters, two cycles of "multi-objective" optimizations were conducted first involving only parameters of insulation layers, followed by the inclusion of infiltration, U-values of windows, and thermal resistance of air gap.

From Figure 3, it can be noticed that the original building envelope has already been quite energy-efficient and that in most cases, further alterations of building envelope properties tend to cause negative energy savings. The parametric analysis also reveals that for all six properties, higher values are preferable and the bottom line for all the analyses performed in this section is—the building should be well insulated yet adequately ventilated. The magnitude of energy consumption is the most sensitive to the U-factor of windows and thermal resistance (R-value). Ultimately, the building envelope was optimized as follows, Table 3.

**Figure 3.** Parametric analysis of building envelope properties.


Here, energy savings (ES) and the percentage-based savings (%savings) refer to energy savings relative to the baseline model expressed respectively in terms of absolute magnitude and percentage.

#### *3.2. Window-to-Wall Ratio*

Windows and skylights are usually components of building envelopes that have the least thermal resistance, therefore the window-to-wall ratio (WWR) plays a crucial role in improving buildings' energy efficiency. In the original design, the WWR of different sides of the building are non-uniform, but for the sake of simplicity, optimization will be carried out assuming that WWR will be the same for all sides of the hotel. Table 4 shows the original window-to-wall ratios:

> **West (225**◦**–315**◦**)**


**Table 4.** Original window-to-wall ratios of the building.

WWR was varied from 5% to 70% with an increment of 5%. Figure 4 below shows how the energy consumption of the building is affected by these changes (Note: percent savings indicated on the graph are relative to the baseline model):

Gross Wall Area, m<sup>2</sup> 17,197.80 5712.45 2886.45 5712.45 2886.45 Window Area, m<sup>2</sup> 2400.80 882.40 265.20 988.00 265.20 Gross WWR, % 13.96 15.45 9.19 17.30 9.19

**Figure 4.** Effect of window-to-wall ratio on building energy consumption.

As it is shown in Figure 4, the higher WWR, the higher the energy demand of the building. However, having very small windows would be impractical for hotel buildings both from the point of view of aesthetics and the impact of insufficient daylighting on the physical and mental well-being of tenants. Therefore, considering these factors, it was decided to keep the WWR close to the original value, i.e., 15%.

#### *3.3. Building Orientation*

Since the building has an aspect ratio of 2, the orientation of the building has a significant influence on the energy consumption of the building mostly owing to the amount of solar irradiation (refer to Figure 2, to see how the building is originally orientated). In fact, from the annual sun path diagram in Figure 5, it is evidently clear that throughout the year, the northern face of the building is exposed to the direct solar irradiation less compared to other sides of the building:

**Figure 5.** Sun path diagram for the city of Los Angeles, USA [28].

To find how the energy demand required for heating and cooling of the hotel is affected by the orientation, the orientation of the hotel relative to the north direction was varied from 0◦ to 355◦ with an increment of 5◦. The result of these simulations is provided in Figure 6:

**Figure 6.** Effect of building orientation on heating and cooling energy demand.

As can be seen in Figure 6, the energy demand of the hotel is the lowest when its longer side is oriented perpendicular to the North direction. In terms of energy required for air conditioning, there is not much difference between orienting it at 0◦ or 180◦ relative to the north. Therefore, architectural considerations would prevail in this case: since an open-air restaurant is to be located on the top floor, it is preferable to have it oriented towards the

southern direction—where several landmark buildings are located—to offer guests a better view. Thus, the building will be oriented at 180◦ relative to the North direction.

#### *3.4. Thermochromic Windows*

For the purposes of the hotel project, properties of double-layer VO2 glazing covered by aluminum-doped zinc oxide (ZnO:Al) described by Kang et al. [17] were used for modeling of thermochromic windows. Kang et al. [17] and Kamalisarvestani et al. [14] reported properties of this material at 20 ◦C and 90 ◦C, therefore, referring to Giovannini et al. [18], it was assumed that optical and thermal properties vary linearly between these two temperatures (Figure 7).

**Figure 7.** *Cont.*

**Figure 7.** Optical properties of the thermochromic windows [17].

Figure 8 shows changes in energy consumed by cooling, heating, and fan of the original building and energy consumption before and after installation of thermochromic windows. As can be seen from this figure, heating energy demand decreased by 160 GJ (11.9%), and cooling energy demand increased by 33 GJ (22.4%) due to the installation of thermochromic windows. The fan energy changes proportionally to the total cooling and heating energy consumption, therefore its share in total energy consumed by the HVAC system remains practically unchanged.

**Figure 8.** Energy consumption of the hotel in its original design and before and after installation of thermochromic windows.

#### *3.5. PCM with Temperature-Controlled Natural Ventilation*

The energy model of the hotel building and heat balance algorithms were configured following recommendations in the literature described in the introduction section [19,20]. A homogenous PCM layer with a 0.020 m thickness was installed into the exterior wall between the insulation layer and gypsum board (refer to Table 5). Density (ρ), conductivity (k), latent heat (Hf), and specific heat (Cp) of organic PCM A28 manufactured by PlusICE® [29] will be taken as a basis for thermophysical properties:

**Properties Values** ρ [kg/m3] 789 k [W/(m·K)] 0.21 Hf [kJ/kg] 265 Cp [J/(kg·K)] 2220 Tm [ ◦C] 20–30

**Table 5.** Properties of PCM A28 (PlusICE®) [29].

To find the optimum PCM, melting temperature (Tm) will be varied from 20 ◦C to 30 ◦C and enthalpy curves will be generated using Feustel's model [30]:

$$h(T) = \mathbb{C}\_{p, \text{const}} T + \frac{H\_f}{2} \left\{ 1 + \tanh\left[\frac{2\beta}{\tau} (T - T\_m) \right] \right\}$$

where *T* is the temperature (◦C), *β* is the inclination of enthalpy curve (–), *τ* is the width of the phase transition zone (◦C). Figure 9 shows enthalpy curves of PCMs for the entire search domain used in optimization, which were generated using Feustel's model:

**Figure 9.** Enthalpy-temperature curves of PCMs for search domain used in optimization.

For the building energy analysis of the hotel, the temperature-controlled natural ventilation described by Prabhakar et al. [21] and Jacobson and Jadhav [31] was adopted with some minor modifications. To model this type of natural ventilation, the airflow network model (AFN) in EnergyPlus was used. In this model, algorithms can take into account wind, buoyancy, and pressure differences created by natural and forced movement of air for calculations of airflow [32]. In case of temperature-controlled natural ventilation, windows were opened by 50% at any time during the day and nighttime, if (1) outdoor temperature will be above HVAC heating setpoints and (2) temperature difference between indoor and outdoor temperatures were more than 1.5 ◦C (originally, in the above two papers, 3 ◦C temperature difference was used). Airflow network control was set to multizone without distribution and wind pressure coefficients were calculated by surface averages. Figure 10 below demonstrates the results of simulations:

**Figure 10.** Optimization of PCM melting temperature.

As can be seen in this graph, the highest savings are achieved by PCM with a melting temperature of 24 ◦C, which yields 222 GJ (10.03%) of energy consumption reductions.

#### *3.6. Photovoltaic Panels*

Several photovoltaic (PV) panels shall be installed on the rooftop of two kitchens and bathrooms on the 24th floor. According to Jacobson and Jadhav [31], in Bakersfield (Kern County, CA, USA), a city located 112 km from Los Angeles, the optimal tilt angle of solar panels is 29◦. Using this tilt angle and taking one solar panel with the unit area, the effect of panels' orientation relative to the North was checked. The results of these simulations are presented in Figure 11:

**Figure 11.** Effect of orientation of PV on electricity generation potential.

As can be seen from this figure, the energy generation potential of photovoltaic panels is maximum if they are oriented at 190◦ relative to the north direction, i.e., when the panel is facing towards the southern direction. Therefore, a total of ten PV panels were fit onto the rooftop with dimensions and interspacing as shown in Figures 12 and 13 (the total area of all solar panels is 273.0 m2). Their annual electrical energy generation capacity was estimated via energy simulation to be 397.86 GJ.

**Figure 12.** Dimensions and tilt angle of photovoltaic panels.

**Figure 13.** General overview of solar panels.

#### **4. Cost-Benefit Analysis**

To summarize the results of previous sections, due to the installation of thermochromic windows and PCMs, energy savings of 204 GJ and 222 GJ were obtained. Additionally, 398 GJ of electricity is to be generated by solar panels installed on the rooftop. Collectively, these three technologies are estimated to reduce the total energy consumption from 2417 GJ to 1593 GJ (i.e., by 824 GJ or 34%). In this section, a dynamic payback period, net present value (NPV), and a rough estimate of return on investments (ROI) due to the installation of these three sustainable energy technologies are provided. Optimization of building envelope properties, window-to-wall ratio, and building orientation were not considered during economic analysis since all these energy analyses are being considered at the preliminary stages of building's design, and as such investments required for their implementation are virtually nonexistent. However, it is still important to consider these parameters to improve the energy efficiency of the design and to reduce the operational costs required for heating and cooling the building during its lifetime.

Unit prices of thermochromic windows, PCMs, and solar panels are obtained from manufacturers. They are shown in Table 6 together with material costs, their design life, and energy-saving potential over their lifetime. For simplicity, transportation and installation costs were ignored at this stage.

**Table 6.** Material quantities, costs, and energy savings by thermochromic windows, PCMs, and solar panels.


<sup>a</sup> [33], <sup>b</sup> [29], <sup>c</sup> [34].

According to the U.S. Bureau of Labor Statistics [35], the average electricity price in Los Angeles from January to September in 2020 was 54.44 USD/GJ. Assuming that the electricity prices in Los Angeles will follow the national average over the last 20 years, which, according to statista.com [36], is 2.30% per year, and taking MARR to be 6%, dynamic payback period, NPV, ROI, and revenue per available room (RevPAR) can be calculated for each of these three technologies, Table 7:

**Table 7.** Payback period and return on investments for TC window, PCM, and solar panels.


Here, dynamic payback period, *NPV*, *ROI*, and *RevPAR* were calculated using the following formulas:

$$\begin{aligned} \text{Dynamic payback period} &= \frac{\log\left\{ \left[ 1 - \frac{\text{Initial Investments} \times MRRR}{\text{Annual Cost} \times \text{Sales} - \text{Maintenance Cost}} \right]^{-1} \right\}}{\log\{1 + MARR\}} \\ NPV &= (Annual \text{Cost Savings} - \text{Maintenance Cost}) \left[ \frac{(1 + MARR)^{design\\_lft} - 1}{MARR(1 + MARR)^{design\\_lft}} \right] \\ &- \text{Initial Investments} \\ ROI &= \frac{NPV}{\text{Initial Investments}} \times 100\% \end{aligned}$$

*RevPAR* = *Total Room Revenue Number of Available Rooms*

As the above results show, to finance TC windows and PCMs, 4.9 USD and 254.9 USD have to be allocated annually from each room.

As can be seen from Table 7, TC windows and solar panels have relatively short payback periods and high returns on investments, whereas PCM seems to be not economically viable at all. Overall, among the considered technologies, only PV panels have positive NPV, meaning that only this technology yields profits by the end of its lifetime. Nevertheless, it should be taken into account that these calculations did not take into account various other overhead costs and the fact that these technologies may significantly deteriorate over their lifetime and thus have an annual energy saving potential that is cardinally different from what was estimated in this section. In addition, 25 years is a long enough period for significant climate change events to occur, therefore energy performance analysis of PCMs and solar panels under future weather conditions is advised for higher accuracy.

#### **5. Conclusions**

The study aimed to identify ways of improvement of the building energy efficiency of a newly designed hotel building developed in accordance with ASHRAE Standard 90.1. To minimize heating and cooling energy demand of the model building, building envelope properties (thickness and thermal conductivity of insulation layers, U-factor of windows, thermal resistance of air gaps, and infiltration rate), window-to-wall ratio, and building orientation were optimized. Moreover, the energy-saving potential of thermochromic windows, phase change materials, and photovoltaic panels were estimated along with the economic feasibility of their installation. Based on the performed parametric analysis, the study found that while some of the assumptions made in the design model were already optimal, other design considerations could be improved. Specifically, the parametric analysis indicated that all 6 properties were optimal in terms of energy efficiency which indicates that the ASHRAE Standard 90.1-2019 is quite strict and allows achieving good results in terms of the building envelope. The parametric analysis also indicated that the energy consumption is the most sensitive to the U-factor of windows and thermal resistance (R-value) of building envelope components. Therefore, the optimal set of values for designing hotel buildings for the climate conditions of the considered locality was provided. In addition, the energy performance analysis also showed that the window-towall ratio and orientation of the hotel building had an impact on the energy use levels. It was found that the higher the WWR is, the higher the energy demand of the building was. Keeping WWR close to the original value, i.e., as 15% was also the most optimal solution. In terms of orientation, the energy demand of the hotel was found to be the lowest when its longer side was oriented perpendicular to the North direction.

The integration of thermochromic windows, phase change materials, and photovoltaic panels allowed to increase the energy efficiency of the building. After installing the thermographic windows heating energy demand decreased by 160 GJ (11.9%) and cooling energy demand increased by 33 GJ (22.4%) due to the installation of thermochromic windows. The energy-saving is significant. It was also the case with the PCMs as the highest savings were achieved with a melting temperature of 24 ◦C, which yielded 222 GJ (10.03%) of energy consumption reductions. Finally, the solar panels were able to generate 398 GJ of electricity. Cumulatively, these three technologies were able to reduce the total energy use from 2417 GJ to 1593 GJ (i.e., by 824 GJ or 34%). These findings indicate that the initial design considerations were optimal after minor adjustments but adding extra sustainable energy solutions brought notable energy use reductions. Since design parameters stayed close to the original values, the cost-benefit analysis of these three technologies only was carried out. It was found that windows and solar panels have relatively short payback periods and high returns on investments, whereas PCM was found to be economically nonviable in the given climate zone. Therefore, the designers and relevant stakeholders should weigh the benefits and costs of the sustainable measures and adopt only feasible ones.

These analyses may be extended further to include HVAC with model predictive control and analyses of the future energy demand of the building.

**Author Contributions:** Conceptualization, S.K.; methodology, S.K.; software, S.K.; formal analysis, S.K. and S.T.; resources, S.K., S.T. and S.D.; data curation, S.K.; writing—original draft preparation, S.K.; writing—review and editing, S.K., S.T. and S.D.; supervision, S.K., S.T. and S.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** Research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviation**


#### **References**

