1 Introduction

The HTC is a very important factor affecting the energy performance of a building. Various studies have been undertaken to investigate the HTC effect [1,2,3,4]. The calculation of the HTC on external building walls is complex and is influenced by a wide range of parameters, including building geometry, building environment, wind speed, wind direction, position on the building façad. Therefore, it is necessary to consider the development of cities and towns in different countries (the height and density of buildings have changed). Deb et al. [5] presented an international survey that targets several stakeholders from 14 countries in Europe to get their perspectives on HTC measurements in reducing the performance gap in buildings. As a result, they provide interesting insights on different aspects (measurement duration and the acceptable error). Sami et al. [6] investigated the impact of external convective heat transfer coefficient on the energy demand of building for different geometry parameters (height and width) by using different models. The results provided significant different predictions for the building energy demand. In the same goals, Evangelisti et al. [7] examined the total external heat transfer coefficients (convective and radiative) for different buildings placed in Rome and Perugia. The analysis of the total heat transfer coefficients shows that the most significant differences are based on the quantification of convective heat transfer phenomena. Furthermore, Bruno et al. [8] provided an experimental and theoretical evaluation on the global heat transfer coefficient by using the thermo-reflective materials in non-ventilated air gaps. It can be deduced that by growing emissivity, the convective heat transfer coefficients show an optimal thickness. Nevertheless, the radiative heat flux decreases with the reduction of the emissivity. Furthermore, in their other experimental work realized at the University of Calabria (Italy), a comparison in terms of surface temperatures and transmitted thermal fluxes was conducted. As a results, it can be showed a large discrepancy by adopting steady and combined heat transfer coefficients, especially for the inner surfaces [9]. Palyvos [10] gives in his work a literature review about different correlations of wind external convection coefficient for building envelope to help designers make better use of the available correlations. Matthew et al. [11] presented an estimation study of building heat transfer coefficients from in-use data. They illustrate in their work the scale of the impacts unmetered heat flows on the accuracy with which a building’s thermal performance may be inferred for two homes at opposite ends of the energy performance scale. The results of the analysis suggest that simple quasi-steady state heat balance methods may be appropriate for dwellings with very high heat loss some researchers analyzed the impact of external heat transfer on building integrated photovoltaic facades for performance evaluation [12,13,14,15]. In fact, wind direction, building geometry, and the interaction between naturel/forced convection on the exterior surfaces have a significant effect on temperature variations in cell temperature.

Seohoon et al. [16] attempted to measure thermal heat losses in a building envelope as an important parameter of the energy performance analysis. Correspondingly, the U value of the building envelope was measured using the combination of the heat flow meter (HFM) method and co heating test. Zhang and Yang [17] proposed two measures to enhance the air layer’s thermal insulation performance in building envelopes. The results showed that the radiation occupies more than 60% of the total heat transfer rate, thus playing the dominate role. In addition to that, reducing the surface emissivity enjoys a great potential for the thermal performance improvement of insulation air layers. Wang et al. [18] compared in their paper convective and radiative heating systems for intermittent heating. The results indicated that convective heating systems were more comfortable and more efficiency for intermittent operation than radiative heating systems. Yang et al. [19] carried out a simulation study for assessing the thermal performance of building external wall. Furthermore Srhayri and Bah [20, 21] discussed the effect of using a U-shape exterior walls on heating and cooling demand of buildings, the results show that using a U-shape wall can reduce or increase the total hearing and cooling demand of the building depending on the climate zone. The current research provides an innovative method for choosing the reasonable structure and constitution of external wall in building with discontinuous radiant heating.

As mentioned earlier, it seems that there are only a limited number of researches which investigate the HTC effect on building performances. The aim objective of the current study is to investigate the effect of U-shape external wall with different scenarios (A 0.25 m, B 0.75 m, and C 1.25 m) on HTC as well as building performances. Based on simulation study and numerical analysis, a proposed administrative building is tested in the six climate zones of Morocco under the climatic ambient conditions of a typical summer day of July 15th, 2021.

2 Methodology

2.1 Building description

The simulation study developed in the present work focused on convective and radiative heat transfer coefficients effects on building performances (U-shape exterior walls effects on convective and radiative heat transfer coefficients of a building). With this view, a simple administrative building [19] is tested in six climatic zones of Morocco (Rabat, Meknes, Marrakech, Errachidia, Tangier, and Ifran). The building consists of 27m2 of surface area and the measured overall heat transfer coefficient of external walls is U = 1.43 W/m2.k (Fig. 1). Furthermore, the geometry of the current model is modified and developed at the external wall oriented south direction as mentioned in Fig. 2 in order to evaluate the effect of this parameter on HTC of building. Hence, three cases are tested (case A 0.25 m of U-shape depth; case B 0.75 m of depth and case C 1.25 m) to be compared with the reference case. A typical summer day of July 15th, 2021 has been chosen, were the meteorological data (solar radiation, ambient temperature, and wind velocity) are shown in Fig. 5.

Fig. 1
figure 1

a 3D building model. b 2D building model [22]

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Fig. 2
figure 2

a U-shape scenarios at south orientation of building. b Walls studies for the U-shape model

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The heat transfer coefficients are studied for each scenario in the three walls of the U-shape (Fig. 2b), in order to evaluate the impact of different orientations on HTC.

2.2 Formulation of the problem

The building models described above, were used to evaluate the convective and radiative heat transfer coefficients for different scenarios (Fig. 2). A dynamic 3D model is developed in this work taking into account the following assumptions: thermo-physical properties are taken constant, all component of the system are in perfect contact.

Accordingly, two parameters were defined in the present study:

2.2.1 Convective heat transfer coefficient

The convective heat transfer coefficient (Fig. 3a) can be defined through the ratio [23]:

$${h}_{conv}=\frac{q_{cv}}{\left({T}_{\mathit{\operatorname{ext}}. wall}-{T}_a(t)\right)}$$
(1)
$${h}_{conv}=\left\{\begin{array}{l}2\frac{k}{L}.\frac{0.3387.{\Pr}^{1/3}{\operatorname{Re}}^{1/2}}{{\left(1+{\left(\frac{0.0468}{\Pr}\right)}^{2/3}\right)}^{1/4}}\kern2.04em if\operatorname{Re}\le {5.10}^5\\ {}2\frac{k}{L}.{\Pr}^{1/3}\left(0.037{\operatorname{Re}}^{4/5}-871\right)\kern0.96em if\operatorname{Re}>{5.10}^5\end{array}\right.\kern0em$$
(2)
Fig. 3
figure 3

Convective and radiative heat flux in the external wall of the studied building

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Where qcv is the heat flux transferred by convection (W), Text. wall is the external wall temperature (°C) and Ta(t)the ambient temperature (°C) of the considered site, k is the thermal conductivity (W/m.K), L is the length of the external wall (m). Re is Reynolds number and Pr is the Prandtl number.

2.2.2 Radiative heat transfer coefficient

The radiative heat transfer coefficient (Fig. 3b) is defined as [7]

$${q}_{rad}= A\varepsilon \sigma \left({T^4}_{\mathit{\operatorname{ext}}. wall}-{T^4}_{sky}(t)\right)$$
(3)
$${h}_{rad}=\frac{\varepsilon \sigma \left({T^4}_{\mathit{\operatorname{ext}}. wall}-{T^4}_{sky}(t)\right)}{\left({T}_{\mathit{\operatorname{ext}}. wall}-{T}_{sky}(t)\right)}$$
(4)

Where qrad is the radiative heat flux (W), ε is the surface emissivity, A is the area of the external wall (m2), σ is the constant of Stefan-Boltzmann (W/m2.K4) and Tsky(t) sky temperature(°C) [24].

With

$${T}_{sky}(t)={0.0552}^{\ast }{\left({T}_a(t)\right)}^{3/2}$$
(5)

2.3 Initial and boundary conditions

  • Initially (t = 0), all the domains of the system are at constant temperature (T0 = 21 °C).

  • All system is adiabatic except the external wall oriented at south direction.

  • The external wall of the building exchanged energy by convection and radiation with the ambient area.

2.4 Mesh generation

COMSOL Multiphysics is a commercial solver and simulation software, based on the finite element method, which has the possibility to generate the suitable mesh. In fact, the most adequate mesh used in this study consists of 56,041 free tetrahedral elements (Fig. 4).

Fig. 4
figure 4

Mesh was used for the numerical simulation of the studied building (scenario B)

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3 Results and discussion

The thermal performance of the studied building is affected by the heat transfer processes occurred on the external modified wall for different cases (A, B, and C). Thus, an analysis is required in this study to evaluate the convective and the radiative heat transfer coefficients, which in actual conditions vary dynamically. In this context, the present section summarizes the main results of the developed work.

3.1 Climatic conditions

To carry out this study, it is fundamental to have weather data of the typical day (July 15th, 2021) for the six climatic zones of Morocco (Errachidia, Marrakech, Meknes, Rabat, Tangier, and Ifran). Hence, Fig. 5a–c shows respectively the solar radiation intensity, ambient temperature, and wind velocity for the six climatic zones [25]. All measurements were carried out during 24 h (from 7am to 7am of the next day). It is clear from the Fig. 5a that maximum solar intensity variation exposures in the range of 925 W/m2 to 1023 W/m2 which indicates the same solar conditions. However, it can see from Fig. 5b that the ambient temperature present great fluctuations between the different concerned cities with a maximum range of 25.92 °C to 41.13 °C. Although, in Fig. 5c the wind velocity gives a maximum range of 4.12 m/s to 10.01 m/s which effect closely on the heat transfer coefficients.

Fig. 5
figure 5

Climatic conditions of a typical day 15 July 2021. a Solar radiation. b Ambient temperature. c Wind velocity

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3.2 Effect of the U shape on HTC

The present section consists on calculating the HTC by COMSOL software for different U-shape scenarios discussed above. The results will be analyzed for each scenario and compared to the heating and cooling consumption calculated by the TRNSYS software in the next section. Furthermore, this was done in order to assess the effect of the U-shape on the HTC variations and subsequently on the total consumption of the building.

At the first step, it is essential to present the reference case were the U-shape = 0 m. Hence, Fig. 6 indicates the convective and radiative heat transfer variations occurred in the vertical external wall under the climatic conditions of Rabat City (15 July 2021 at 1pm). Figure 6 also confirms that the convective and radiative heat transfer coefficients (hconv and hrad) exhibit constant behavior along the south external vertical wall (3.6 m). The “scenario A” to “scenario C” sections describe in details the results of three cases considered in the present simulation (0.25, 0.75, and 1.25 m of depth). It can be clear that changing a form of the external wall affect the HTC in the three walls of the U-shape for all climate zones of Morocco. Hence, we have more details in sections below for Rabat City (details of other zones are given in “Appendices 1, 2, 3, 4 and 5” section).

Fig. 6
figure 6

HTC variations of the reference case in Rabat City

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3.2.1 Scenario A

Figure 7 presents the first case: scenario “A” with variation of the HTC (convection and radiation) according to the U-shape depth of 0.25 m in Rabat City. We can observe that the HTC decrease over length particularly for wall 1 and wall 3. Furthermore, a part of the curves (Fig. 7a, c, d, and f) is related to the thickness of the external wall (0.35 m) after that a significant decrease is remarked. Hence, depending on the time of day the HTC can change according to the climatic conditions. Thereby, the suitable time for our tests is considered at 1pm with a maximum solar intensity. From Fig. 7b, e, the results showed small deviations for the measured convective and radiative HTC respectively for the wall 2. In other words, the curve contains two parts: the first concerns the east-facing half of the wall and the other west-facing half affected by the shadow area which has a significant impact on cooling and heating consumption. More importantly, it should be noted that the influence of wind speed, interaction between naturel/forced convection on the exterior surfaces of the building have more effect on heating demand.

Fig. 7
figure 7

HTC variations for scenario A (0.25 m): convective (a–c/radiative (d–f) heat transfer coefficient for different walls

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3.2.2 Scenario B

In this subsection, the figures below present variation of the HTC according to the U-shape depth of 0.75 m in Rabat City. We observe that increasing the depth form 0.25 m to 0.75 m the present results are marginally improved (Fig. 8). Although, the trend of convective and radiative heat coefficients is similar to the previous curves. Then, wall1 oriented on West direction receive more solar radiation compared to wall 3 which is oriented at East and affected by shadow which explain the low value of convective HTC (Fig. 8a, c). Therefore, important values of radiative HTC are observed (Fig. 8d, f). Specifically, the highest enhancement in the HTC over the depth wall (form 0.25 m to 0.75 m) is found to be more than 33% for the maximum radiative coefficient in wall 2 (Fig. 8e). However, a small loss of 0.16% is remarked for maximum convective coefficient in the same wall2 (Fig. 8b).

Fig. 8
figure 8

HTC variations for scenario B (0.75m): convective (a–c)/radiative (d–f) heat transfer coefficient for different walls

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3.2.3 Scenario C

For the concerned building, the variation of the HTC (convective and radiative) according to the U-shape depth of 1.25 m and depending on ambient conditions of Rabat City is illustrated in Fig. 9. Furthermore, it is necessary to reminisce that a typical day of 15 July 2021 at 1pm is considered for the simulation process. It can be clear from the results of scenario C that a similar behavior of scenario B is observed a long the depth, the maximum radiative HTC register an important increase of 64.62% for wall 3 compared to 12.33% for wall 1 (Fig. 9d, f). Also, it can be noted that the convective coefficient decrease progressively according to the U-shape wall (Fig. 9a–c).

Fig. 9
figure 9

HTC variations for scenario C (1.25 m): convective (a–c)/radiative (d–f) heat transfer coefficient for different walls

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In the next step of this analysis, it was investigated the HTC effect on energy consumption for the administrative building under the climatic conditions of Rabat City.

3.3 Effect of the HTC variations on energy consumption

The final step of the present analysis focused on the effect of HTC on building energy consumption. It can be clear from the cooling analysis (Table 1) that the building achieves tangible and gradual reduction in the cooling load over U-shape according to our previous work [21]. However, an increase is observed for heating load. Hence, we can say that in temperate and humid climates, the total energy consumption decreases as well as the HTC decrease (Tables 1 and 2). Also, a gradual increase is observed in thermal total loads when using a depth of more than 0.25 m. Generally, it can be noted that the optimal depth coincides with the best value of thermal total loads according to the climatic conditions of Rabat City.

Table 1 Energy consumption for different scenarios (reference case, A 0.25 m, B 0.75 m, and C 1.25 m)
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Table 2 HTC average for different scenarios (reference case, A 0.25 m, B 0.75 m, and C 1.25 m)
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In all climates of the other zones, we can observe that the convective and radiative HTC keep the same trend as results of Rabat City (see Appendices 1, 2, 3, 4, and 5). Furthermore, each area is characterized by an optimal U-shape for lowest thermal loads. Table 3 shows the general analysis of thermal performances for different scenarios at six climatic conditions (Rabat, Errachidia, Marrakech, Ifran, Meknes, and Tangier).

Table 3 General analysis of thermal performances for different scenarios at six climatic zones
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4 Conclusion

This work present a robust analysis of the convective and radiative heat transfer coefficient s effect on energy performance of a building for the six zones climates of Morocco according to different U-shape of the south external wall. Three cases are discussed in the present paper, (case A 0.25 m of U-shape depth; case B 0.75 m of depth and case C 1.25 m). Indeed, a numerical modeling is developed using COMSOL Multiphysics to evaluate the convective and radiative heat transfer coefficients. Hence, the obtained results can be summarized in the following points:

  • The shading effect has a decisive role on reducing the sunny area as well as energy consumption.

  • The radiative HTC increase by 33% and 64.62% by increasing the U-shape from 0.25 to 0.75 m and from 0.75 m to 1.25 m respectively

  • A maximum reduction of total energy consumption is achieved as well as the HTC decrease.

  • The optimal depth coincides with the best value of thermal total loads.

  • This research provides a guide for optimized depth for thermal performances of building envelope.

The developed study was founded to be of real benefit to many communities with same climatic conditions given their importance on the wind flow around buildings.