# Insights about Modelling Environmental Spatiotemporal Actions in Thermal Analysis of Concrete Dams: A Case Study

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dam Description

## 3. Governing Equations

^{2}K

^{4}). When $T$ and ${T}_{a}$ are close, which is the case in civil engineering structures, it is possible to rewrite (6) in a quasi-linear form

## 4. Finite Element Model

#### 4.1. Finite Element Formulation

#### 4.2. Finite Element Mesh

## 5. Thermal Properties

^{3}was adopted for the density.

## 6. Convection Heat Transfer

^{3}, and 1.8 × 10

^{−5}kg/(m s), ${V}_{w}$ is the average wind speed in m/s, and $L$ represents the size of the considered flat surface, for which Silveira [16] adopted the value of 0.60 m. Then, considering an average wind speed ${V}_{w}$= 2.8 m/s (corresponding to 10 km/h), the convection coefficient results in ${h}_{c}$ = 14.6 W/(m

^{2}K).

^{2}K). Therefore, a constant value for the total thermal transmission coefficient ${h}_{t}$ = 20 W/(m

^{2}K) was applied to the whole model.

## 7. Solar Radiation

#### 7.1. The Path of the Sun across the Celestial Sphere

#### 7.2. Solar Position

#### 7.3. Angle of Incidence

#### 7.4. Solar Radiation Components

#### 7.5. Solar Irradiance

^{2}).

#### 7.6. Shading of Beam Radiation

## 8. Reservoir Water Temperature

## 9. Analysis and Results

## 10. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Downstream view of Aguieira Dam [12].

**Figure 6.**Comparison of the solar irradiance on the downstream face of the dam at 12 h (solar time) during the summer solstice, the winter solstice, and the equinox.

**Figure 8.**Schematic summer thermal stratification of the reservoir showing epilimnion, metalimnion, and hypolimnion and associated thermal profile.

**Figure 9.**Mean observed values and empirical water temperature variations of ${T}_{m}\left(y\right)$, $A\left(y\right)$ and ${d}_{o}\left(y\right)$ for the Aguieira dam reservoir.

**Figure 12.**Comparison of the predicted and monitored temperatures at thermometers T27, T29, and T31.

**Figure 13.**Comparison of the predicted and monitored temperatures at thermometers T58, T60, and T62.

**Figure 14.**Comparison of the predicted and monitored temperatures at thermometers T15, T17, and T77.

**Figure 15.**Comparison of the predicted and monitored temperatures at thermometers T10, T12, and T14.

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**MDPI and ACS Style**

Leitão, N.S.; Oliveira, S.
Insights about Modelling Environmental Spatiotemporal Actions in Thermal Analysis of Concrete Dams: A Case Study. *Thermo* **2023**, *3*, 605-624.
https://doi.org/10.3390/thermo3040036

**AMA Style**

Leitão NS, Oliveira S.
Insights about Modelling Environmental Spatiotemporal Actions in Thermal Analysis of Concrete Dams: A Case Study. *Thermo*. 2023; 3(4):605-624.
https://doi.org/10.3390/thermo3040036

**Chicago/Turabian Style**

Leitão, Noemi Schclar, and Sérgio Oliveira.
2023. "Insights about Modelling Environmental Spatiotemporal Actions in Thermal Analysis of Concrete Dams: A Case Study" *Thermo* 3, no. 4: 605-624.
https://doi.org/10.3390/thermo3040036