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Modeling Thermal Interactions between Buildings in an Urban Context^{ †}

^{1}

^{2}

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Thermal Interactions between Buildings

#### 2.1. Modeling Longwave Radiant Heat Exchange between Buildings in EnergyPlus

- $\mathsf{\epsilon}=\mathrm{longwave}\mathrm{emittance}\mathrm{of}\mathrm{the}\mathrm{surface},$
- $\mathsf{\sigma}=\mathrm{Stefan}-\mathrm{Boltzmann}\mathrm{constant}\left(\mathrm{W}\cdot {\mathrm{m}}^{-2}\cdot {\mathrm{K}}^{-4}\right),$
- ${\mathrm{T}}_{\mathrm{surf}}=\mathrm{Temperature}\mathrm{of}\mathrm{the}\mathrm{exterior}\mathrm{surface}\left(\mathrm{K}\right),$
- ${\mathrm{T}}_{\mathrm{sky}}=\mathrm{Sky}\mathrm{temperature}\left(\mathrm{K}\right),$
- ${\mathrm{F}}_{\mathrm{sky}}=\mathrm{View}\mathrm{factor}\mathrm{of}\mathrm{the}\mathrm{sky},$
- ${\mathrm{T}}_{\mathrm{g}}=\mathrm{Ground}\mathrm{temperature}\left(\mathrm{K}\right),$
- ${\mathrm{F}}_{\mathrm{g}}=\mathrm{View}\mathrm{factor}\mathrm{of}\mathrm{the}\mathrm{ground},$
- ${\mathrm{T}}_{\mathrm{a}}=\mathrm{Temperature}\mathrm{of}\mathrm{the}\mathrm{outdoor}\mathrm{air}\left(\mathrm{K}\right),$
- ${\mathrm{F}}_{\mathrm{a}}=\mathrm{View}\mathrm{factor}\mathrm{of}\mathrm{the}\mathrm{outdoor}\mathrm{air}.$

- ${\mathrm{T}}_{\mathrm{si}}=\mathrm{Temperature}\mathrm{of}\mathrm{surrounding}\mathrm{surface}\mathrm{i}\left(\mathrm{K}\right),$
- ${\mathrm{F}}_{\mathrm{si}}=\mathrm{View}\mathrm{factor}\mathrm{of}\mathrm{surrounding}\mathrm{surface}\mathrm{i}.$

_{si}at the previous time step, assuming one timestep lag. This simplification may lose some fidelity (if a larger time step of 60 min is used in simulation) but significantly improves computing performance.

#### 2.2. Modeling Urban Buildings and Their Surrounding Surfaces

#### 2.3. View Factor Calculation

_{1}& A

_{2}), and the fundamental expression is written as:

_{1}and A

_{2}(m

_{2}) are the areas of surfaces 1 and 2; θ

_{1}and θ

_{2}(°) are the angles between the norms to surface differential elements dA

_{1}and dA

_{2}and the vector between those differential elements; r is the length of that vector [26]. This method is used in the View3D tool developed for evaluating radiation view factors between two 3D surfaces of interior surfaces in EnergyPlus. When a third surface is added between two planar surfaces, View3D calculates the blockage factor by tracing the rays between each two differential elements dA

_{1}and dA

_{2}, represented as the shadowed area in Figure 3 [27]. The adaptation to the partial obstruction is less efficient in computing, and it cannot be used to calculate concave and curve surfaces as well.

_{1}to a target surface A

_{2}can be then deduced as a sum of the $\frac{\mathrm{cos}{\mathsf{\theta}}_{1}\mathrm{cos}{\mathsf{\theta}}_{2}}{\mathsf{\pi}{\mathrm{r}}^{2}}$ terms over all surviving rays subject to proper normalization. Compared with the traditional analytical solutions and geometry analogy algorithms introduced in the literature review, the view factor tracer we developed carries the following advantages:

## 3. Case Study

#### 3.1. Simulation Settings

#### 3.2. View Factors to the Surrounding Surfaces

#### 3.3. Thermal Behaviors of Building Exterior Surfaces

#### 3.4. Energy Impact Analysis

#### 3.5. Environmental Impact Analysis

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Case study urban district in downtown Chicago. (

**a**) Site view of the studied district; (

**b**) Footprints and building types of the studied district.

**Figure 7.**Monthly average exterior surface temperature of different facades. (

**a**) Walls; (

**b**) Windows.

**Figure 8.**Daily exterior wall temperature of different façades—21 January. (

**a**) East and West; (

**b**) South and North.

**Figure 9.**Daily exterior wall temperature of different façades—21 July. (

**a**) East and West; (

**b**) South and North.

**Figure 10.**Temperature differences between facades and their surrounding surface—building 16. (

**a**) 21 January; (

**b**) 21 July.

**Figure 11.**Energy impact of the LWR for the studied buildings—Annual heating and cooling energy use difference in percentage.

**Figure 12.**Surrounding buildings’ LWR received by exterior walls and roofs of building 16. (

**a**) January 21

^{st}; (

**b**) 21 July.

**Figure 14.**LWR impact on the exterior surface temperature of different façade and roof on 21 January. (

**a**) Building 13—21 January; (

**b**) Building 13—21 July; (

**c**) Building 16—21 January; (

**d**) Building 16—21 July.

**Figure 15.**LWR impact on the exterior surface temperature of different façades and roofs—21 July. (

**a**) Building 13—21 January; (

**b**) Building 13—21 July; (

**c**) Building 16—21 January; (

**d**) Building 16—21 July.

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**MDPI and ACS Style**

Luo, X.; Hong, T.; Tang, Y.-H.
Modeling Thermal Interactions between Buildings in an Urban Context. *Energies* **2020**, *13*, 2382.
https://doi.org/10.3390/en13092382

**AMA Style**

Luo X, Hong T, Tang Y-H.
Modeling Thermal Interactions between Buildings in an Urban Context. *Energies*. 2020; 13(9):2382.
https://doi.org/10.3390/en13092382

**Chicago/Turabian Style**

Luo, Xuan, Tianzhen Hong, and Yu-Hang Tang.
2020. "Modeling Thermal Interactions between Buildings in an Urban Context" *Energies* 13, no. 9: 2382.
https://doi.org/10.3390/en13092382