2.1

CHTC identification

There are three main methods for identifying CHTC: field measurements, wind-tunnel experiments, numerical simulation.

Field measurements were performed in earlier studies which focused on modelled and real urban blocks3-8. Previous study9 measured the CHTC distribution on the external surfaces of a residential district, and they examined the relationship of CHTC with the wall-adjacent wind speed. Previous study10 also measured CHTC of a window of a full-scale building by using the filter-paper evaporation method. However, field measurements are limited to measurements at specific points, it is also difficult to obtain all values of CHTC and sensible heat flux on an entire building’s external surfaces with complex geometries.

Wind-tunnel experiments also have been carried out in earlier studies which focused on modified urban blocks11. Previous study12 performed wind-tunnel experiments with urban canopy models and the filter-paper evaporation method, and then measure CHTC by using the similarity between CHTC and convective mass transfer coefficient (CMTC). They then discussed the dependency of prevailing wind direction on CHTC. However, the scales of the windtunnel experiments are different from those of heat and mass transfer in real-building surfaces, and it is difficult to obtain similarities between CHTC and CMTC on a realistic scale. Numerical simulation is considered to be an appropriate method to understand not only the properties of flow fields around buildings, but also to determine the CHTC of entire surfaces on a consistent scale (Figure 2).

Figure 2.

Diagram of sensible heat flux from external surfaces of buildings to the surrounding atmosphere.

There have been several studies which aimed to identify CHTC using numerical simulation13-16, and some turbulence models have been used. The high-Reynolds-number k–ε model (the High-Re model, where k is the turbulent kinetic energy, ε is the turbulent dissipation) is a widely used Reynolds-averaged Navier–Stokes (RANS) model which applies wall functions to cells with adjacent walls (wall-adjacent cells), so the results will not account for the behaviour inside the boundary layer, which will influence the value of CHTC. Meanwhile, CHTC can be reproduced by the wall-function in view of engineering usage in the High-Re model (Figure 3). Previous study17 used the High-Re model and predicted the CHTC of a building’s external surfaces and the outdoor thermal environment in a high-rise campus block with 10- story buildings.

Figure 3.

Wall boundary layer and the cell division near the surfaces of the Low-Re model.

The Jürges equation18, shown below, is one of the most commonly used wall functions in the field of urban and built environment:

where v is the wind velocity above the surface boundary layer. These equations were obtained from a single copper plate in a wind-tunnel experiment (Appendix B). Therefore, verification is necessary regardless of whether the equations are appropriate for entire building surfaces. The High-Re model has a lower computational cost and is considered to be appropriate for the prediction of sensible heat flux on the scale of an urban block in a built environment.

The low-Reynolds-number k–ε model (the Low-Re model) is another method to predict the value of CHTC, and it can also be used to simulate the wind flow and temperature fields inside the boundary layer with very fine grids. The difference between the High-Re model and the Low-Re model is a modelling level of the boundary layer. The Low-Re model is under the no-slip condition for the layer, and CHTC can be calculated by the following equation:

where q_c is the sensible heat flux, W/m²; λ is the thermal conductivity of air, W/(m·K); dT/dy is the temperature gradient inside the viscous sublayer, K/m; T_s is the surface temperature of walls, K; and T_ref is the air temperature above the thermal boundary layer, K.

Using the Low-Re model, the behaviour of heat and momentum transfer inside the boundary layer, which affect the value CHTC, can be reproduced (Figure 4). To ensure that at least one of the wall-adjacent cells is inside the vicious sublayer, this study examined the wall distance y⁺ with value below 5 at every surface point. The wall distance y⁺ is defined as

Figure 4.

Wall boundary layer and the cell division near the surfaces of the High-Re model.

where U_τ is the frictional velocity, y_p is the distance from the centre point ‘p’ of the wall-adjacent cell to the wall, and ν is the kinematic viscosity of air.

Since this model is based on the theory of heat-transfer, the accuracy of sensible heat flux prediction is high. Meanwhile, when the model is applied on the urban-block scale, it requires high computational cost because of the enormous number of meshes. Previous study19 used the Low-Re model to predict the CHTC distribution on the surfaces of a small building. However, the detailed geometries of the building were not reproduced, and they quantitatively analysed only the distribution of CHTC on the windward façade with a flat surface. In addition, the differences in prediction accuracy of sensible heat flux on the building external surfaces between the Low-Re model and the High-Re model have not been concretely confirmed in reality. Moreover, it is necessary to examine if it is possible to derive the CHTC-predicting equation for the High-Re model from the Low-Re model.

As suggested by the subjects mentioned above, the goal of this study was to quantitatively confirm the effectiveness of the turbulence model and spatial geometry in predict the sensible heat flux. It was further examined whether it was possible to derive a new CHTC-predicting equation that can be applied for the High-Re model from the Low-Re model. Because the forced convection field is dominant in outdoor urban spaces, except under the condition of low wind speed, in which case natural convection prevails, this study calculated CHTC under the condition of forced convection field.

2.2

Prediction of sensible heat flux

2.2.2

Calculation Methods of Heat Balance and Coupled Analysis.

This study used the 3D CAD-based thermal environment simulator to calculate surface temperature. The simulator generated cells on the external surfaces of building, and for each of the cells, this tool calculated the heat balance as follows:

where R_S is the net short-wave radiation, R_L is the net long-wave radiation, Q_H is the sensible heat flux, Q_E: is the latent heat flux, and Q_G is the conductive heat flux (see Chapter 5 for details).

The 3D CAD-based thermal environment simulator considers wind and temperature distribution in urban canopy as constants, so it cannot calculate the surface-temperature distribution by taking wind- and air-temperature distribution into account. To resolve the issue, this study coupled the simulator with computational fluid dynamics (CFD), which is called the coupled analysis. The flow chart of the coupled analysis is shown in Figure 5. First, this study performed isothermal air flow simulation by the CFD simulation and estimated the CHTC distribution on the entire building’s surfaces. This study used the value of CHTC calculated from the forced convection field for the boundary condition of the heat-balance simulation. Then, this study performed heat balance simulation (Appendix A) and calculate the distribution of surface-temperature, taking the CHTC distribution into account. Finally, this study calculated and analysed the sensible heat flux from the CHTC and surface temperature distribution.

Figure 5.

Flow chart of the coupled analysis.

2.2.3

Simulation Conditions.

At first, the turbulence model of the CFD simulation was used as the Low-Re model, the prediction accuracy of which is shown in Section 6.1. The High-Re model was then applied to confirm similarities and differences with the Low-Re model, and applicability to sensible heat flux prediction. Before the comparison, the relationship between wall-adjacent wind speed and CHTC is examined in the Low-Re model to derive a prediction equation which can be apply to the prediction of CHTC for the High-Re model.

Table 1 shous the simulation conditions. The cell-division section of the Low-Re model with detailed geometries is shown in Figure 6, along with an enlarged figure of the veranda. This study generated the cells near the surface with smaller size than that in the space. In terms of the cells close to the surface, this study refined it with 8 layers. This study confirmed that even at vertical partitions of the verandas, the generated cells were very fine and closely fitted to the surface. The results of wind speed obtained by the CFD simulation are shown in Figure 7. It can be confirmed that the air circulation behind the building was reproduced, and also wind speed was reduced in the veranda spaces.

Figure 6.

Simulation of the target building with detailed geometries and cell division near the surface of the Low-Re model.

Figure 7.

Distribution of wind velocity vector in the horizontal (6 m above ground level) and vertical (around the centre of the building) directions.

Table 1.

Simulation and weather conditions.

	Case 1	Case 2
Turbulence model	Low-Re number k-ε model (Lounder-Sharma model)
Advection scheme	Upwind difference scheme
CFD simulation tool	OpenFOAM-2.0
Vertical profile	v = 2.17(Z/30)0.2
Inlet boundary	Power law
Outlet boundary	Pressure at the interface is 0 (Pa)
Cell numbers on building surfaces	About 4 million	About 25 million
Cell size near the surface	6.1e-5 m
Boundary condition (ground, wall)	No-slip condition
Calculation method of CHTC	Definition equation:
qc=λ*dT/dy
Weather condition	Nov. 5th 2004 12:00:00, Temp. [].2℃, Sunny, Wind speed at 1.8 times the building height: 2.0m/s, South wind

3.1

Effect of a building’s detailed geometries on the CHTC prediction

In order to confirm the effect of detailed geometries such as verandas and eaves, which can affect the wind flow near the walls and surface temperatures, this part of study compared the case without detailed geometries (Case 1) and the case with detailed geometries (Case 2).

Figure 8 shows the distribution of CHTC on the windward facade. In both cases, CHTC distribution shows the same tendency of increasing from the bottom of the building centre to the surrounding area. Previous study19 had already reported this distribution tendency for a flat building facade. In addition, by reproducing the detailed geometries Case 2), the wind speed in the veranda space becomes small, and then CHTC of the wall becomes small owing to the obstacles in the form of eaves and vertical partitions of verandas. At the vertical partitions of verandas, there are some places on the edge where CHTC is larger than that at the centre. Case 2 shows not only the simple tendency for CHTC which increase from the bottom to the top and also increase from the centre to the sides, but also variations caused by the detailed geometries. The surface-temperature distribution in Figure 9 confirms that the surface temperature shows a tendency opposite to that of the CHTC distribution, because high CHTC takes away large convective heat transfer from the surface, decreasing the surface temperature as a result. Meanwhile, in Case 2, the surfaces of the space inside the verandas show low surface temperatures because of solar shading. As shown by the sensible heat flux in Figure 10, even though the distribution of CHTC and surface temperature caused by the air flow near the surfaces have opposite tendencies except inside the veranda space, the distributions of sensible heat flux and CHTC show almost the same trends. On the surfaces with the same solar radiation, CHTC plays a more important role than the surface temperature on the sensible heat flux. The histogram shows sensible heat flux in Figure 11. It shows that both cases exhibit similar general trends, while the sensible heat flux of Case 1 is 20% higher than that of Case 2. Based on these findings, it shows that the detailed geometries could affect the prediction of sensible heat flux by up to 20%.

Figure 8.

CHTC distribution on the windward facade of Case 1 (top) and Case 2 (bottom).

Figure 9.

Surface-temperature distribution on facade the windward facade of Case 1 (top) and Case 2 (bottom).

Figure 10.

Sensible heat flux distribution on the windward facade of Case 1 (top) and Case 2 (bottom).

Figure 11.

Histogram of sensible heat flux of Case 1 and Case 2.

3.2

Relationship between CHTC and wind speed near the surface

The relationship between CHTC and wind velocity at a distance of 10 cm above the surface U (where in CASE1 is shown in Figure 12. In order to obtain higher correlations, this study added the turbulence kinetic energy k, which represents the strength of disturbance to the average wind speed to create another parameter U_k20:

Figure 12.

Relationship between CHTC and U (Case 1).

where U_x, U_y, and and U_z are the x, y, and z component, respectively, of the wind velocity, and k is the turbulence kinetic energy.

The relationship between CHTC and U_k was also examined, and the result is shown in Figure 13. Compared to the coefficient of determination R² of 0.52 between U and CHTC, the R² between CHTC and U_k is 0.69. It shows that CHTC has a stronger correlation with U_k because in some area where the average wind speed is small but the turbulence is large, resulting in a larger CHTC. At these places, U_k is a better index than U for predicting CHTC. The correlation between CHTC and U_k for Case 2 is also shown in Figure 14. The R² is 0.77. In both cases, CHTC shows strong correlation to U_k for both linear and power approximation. While Case 2 shows many more values of CHTC under 4 W/(m²·K) owing to the detailed geometries. In any case, it can be concluded that it is possible to approximately calculate CHTC using the equation of U_k.

Figure 13.

Relationship between CHTC and Uk (Case 1).

Figure 14.

Relationship between CHTC and Uk (Case 2).

3.3

Difference between sensible heat flux of the High-Re model and the Low-Re model

This part of the study aims to clarify the effect of selection of the turbulence model on sensible heat flux prediction. By applying the approximation equation (called the prediction equation here) derived in the previous section to the high-Re model, it can confirm that the prediction accuracy of the prediction equation by comparing it to the Low-Re model. The prediction equation applied is CHTC = 5.7U_k.

This study performed the coupled analysis for the High-Re model, as well as the Low-Re model, for the same target building with the detailed geometry. Hereafter, the High-Re model calculated by Jürges equation is referred to as Case 3, and the High-Re model calculated by the prediction equation is referred to as Case 4. This study compared Case 2, Case 3, and Case 4, and the results are shown in Figure 15. Overall, Case 4 shows results that are closer to those of Case 2 than Case 3. However, the edge effects on the edge of verandas cannot be confirmed from Case 4 unlike it could be confirmed in Case 2. The windward facade is divided into four areas (shown in Figure 15) and Table 2 shows the average sensible heat flux for each area, as well as the average sensible heat flux from the entire windward external surface. This study discovered that in general, for the sensible heat flux from the entire windward external surface, Case 4 also shows a result that is quantitatively closer to that of Case 2 than Case 3. Case 3 shows a -20% difference from Case 2, while Case 4 shows a lower difference of <15%. It can thus conclude that the predicting accuracy of sensible heat flux is increased by applying the prediction equation instead of the Jürges equation to the High-Re model.

Figure 15.

Distribution of sensible heat flux on the windward facade (from top to bottom for Case 2, Case 3, and Case 4).

Table 2.

Average sensible heat flux from the windward façade of the building model.

The results of Case 4 still show differences from those of Case 2 (the Low-Re model). These differences originate from the modelling with wall function such as the prediction equation and the Jürges equation, and these wall functions have limitations when predicting heat and mass transfer on detailed surfaces with complex geometries. By investigating the effective wind speed U_k in the flow field of both the Low-Re model and the High-Re model, this study discovered that the average value of U_k of entire surfaces shows different values for the Low-Re model and the High-Re model. As a result, the different values of CHTC are predicted in each case. However, the surface-temperature distribution, obtained by taking into consideration the CHTC distribution, shows almost the same results. Therefore, the sensible heat flux calculated by the prediction equation (Case 4) shows slightly different results compared with those of the Low-Re model (Case 2). Nevertheless, the prediction equation shows more reasonable values than those obtained Jürges equation in sensible-heat-flux prediction in this modelling.