# Investigation of Temperature Dynamics in Small and Shallow Reservoirs, Case Study: Lake Binaba, Upper East Region of Ghana

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Water Bodies Modeling

## 3. Description of Study Site and Data Collection

^{−2}for most of days. The measured values of relative humidity (RH) over the water surface are shown in Figure 2c. The wind speed and directions, are shown in Figure 2d with south-western direction being the most dominant direction with a maximum speed of 4.0 m/s. Since the wind speed values have been averaged over 30-min intervals (as for the other parameters), instantaneous wind speed may be larger. The variation of atmospheric pressure during the study period was very small and could be ignored. Therefore, the pressure was taken to be a constant 102 kPa for all of the simulations.

## 4. Mathematical Model

#### 4.1. Governing Equations

^{2}); ${\mathsf{\nu}}_{eff}={\mathsf{\nu}}_{0}+{\mathsf{\nu}}_{t}$ is the effective kinematic viscosity (m

^{2}/s), with ${\mathsf{\nu}}_{0}$ and ${\mathsf{\nu}}_{t}$ denoting molecular and turbulent viscosity, respectively; $\mathsf{\beta}$ the coefficient of expansion with temperature of the fluid (for water ≈ 0.207 × 10

^{−3}J·kg

^{−1}·K

^{−1}); ${T}_{ref}$ reference temperature (= 293.15 K); and $\mathsf{\delta}$ is the delta of Kronecker (dimensionless). The Boussinesq approximation is valid under the assumption that density differences are sufficiently small to be neglected, except where they appear in the term multiplied by ${g}_{i}$ [35]. In the model, for incompressible flows the density is considered as effective (driving) kinematic density and calculated as a linear function of temperature as

^{−3}).

^{2}/s) and ${S}_{T}$ is the heat source term in lake due to the penetrating solar radiation (K/s). Heat transfer conductivity can be given by:

^{−6}m

^{2}s

^{−1}), ${\mathsf{\nu}}_{t}$ turbulent kinematic viscosity (m

^{2}/s), ${C}_{p}$ specific heat of water (≈ 4.1818 × 10

^{3}J·kg

^{−1}·K

^{−1}), $Pr$ is Prandtl number (≈ 7.07), $P{r}_{t}$ turbulent prandtl number (≈ 0.85) and ${\mathsf{\alpha}}_{eff}$ is effective thermal conductivity (m

^{2}/s). Changes in temperature in water body might occur mainly due to heat exchange across the air–water interface. Accurate estimation of heat fluxes is extremely crucial in the simulation of temperature dynamics in the water body [16]. Atmospheric heat fluxes include incoming short-wave (solar) and long-wave (atmosphere) radiations, outgoing long-wave radiation, conductive heat at the free surface and evaporative heat flux. Computationally, all these terms, except for incoming short-wave radiation, are considered at the water surface as boundary conditions.

^{*}within the water (W/m

^{2}), ${Q}_{Rs}^{0}$ is the net solar radiation at the air–water interface (W/m

^{2}), ${f}_{i}$ is the fraction of energy contained in the i

^{th}bandwidth (dimensionless), and ${\mathsf{\eta}}_{i}$ represents the composite attenuation coefficient of the i

^{th}bandwidth (m

^{−1}) [39,40]. The values of ${f}_{i}$ and ${\mathsf{\eta}}_{i}$ are presented in Table 1. The attenuation coefficient (light extinction coefficient) for visible light theoretically is a function of wave length, temperature and water turbidity [16,41] and typically ranges from 0.02 to 31.6 for inland shallow waters [16,41,42,43,44]. Usually a linear relationship is applied to calculate the extinction coefficient value from observed Secchi depth in inland water bodies [16,45]. For this study, the attenuation coefficient is assumed to be constant in the whole water body ($\mathsf{\eta}=3.0$ m

^{−1}). This value was computed from the turbidity measurements. By applying the approach proposed by Williams et al. (1981) [46] and using the available measurements for Secchi depth values during the simulated period, the attenuation coefficient was calculated. The attenuation coefficient value was estimated only in one point and therefore, it is assumed that the distribution of the attenuation coefficient in the water body is homogeneous. The temperature profiles in a shallow lake are significantly sensitive to the attenuation coefficient and, hence, this parameter should be considered carefully in the simulation [29].

^{2}) and ${r}_{ws}$ is the reflection coefficient of solar radiation from water surface (≈ 0.08) [48].

#### 4.2. Turbulence Modelling

^{2}/s

^{2}) and the turbulent dissipation rate ($\mathsf{\epsilon}$ in m

^{2}/s

^{3}) are obtained from:

^{2}/s), ${G}_{k}$ are the production of turbulent kinetic energy by the mean velocity gradient (m

^{2}/s

^{3}), ${G}_{b}$ is the production of turbulent kinetic energy by the buoyancy (m

^{2}/s

^{3}), and ${S}_{k}$ (m

^{2}/s

^{3}) and ${S}_{\mathsf{\epsilon}}$ (m

^{2}/s

^{4}) are the source terms which include the effects of wind on $k$ and $\mathsf{\epsilon}$ equations respectively. The parameter ${C}_{\mathsf{\epsilon}3}$ (dimensionless) is not constant and depends on the flow conditions and is a function of the ratio of the velocity components in the vertical and longitudinal directions [53]:

## 5. Numerical Simulation

#### 5.1. Numerical Grid

#### 5.2. Numerical Setup

## 6. Boundary Conditions

#### 6.1. Temperature

^{−8}Wm

^{−2}K

^{−1}), and ${T}_{air}$ is absolute air temperature in K. Similarly, long-wave radiation from the water surface was estimated by [1,11,43]:

^{2}(positive if it flows from the atmosphere into the water surface) and ${h}_{s}$ is the convective heat transfer coefficient (Wm

^{−2}·K

^{−1}), which relates the convective heat flux normal to the water surface to the difference between the water surface temperature (${T}_{ws}$) and surrounding air temperature (${T}_{air}$). The convective heat transfer coefficient can be estimated by [64]:

^{−2}, ${X}_{air}$ and ${X}_{ws}$ are the water vapor mixing ratio of air and water surface, respectively (kg(water)/kg(dryair)), ${\mathsf{\rho}}_{a}$ is the air density, assumed constant in this study ≈ 1.186 kg m

^{−3}, the coefficient $(24\times 3600)$ is used for converting the latent heat flux to mm/day and the coefficient of $(28.4)$ converts it to W/m

^{2}to be consistent with the rest heat flux components over the water surface, and ${h}_{m}$ is the mass transfer coefficient given by [64]:

^{−1}and ${U}_{2}$ in ms

^{−1}and the respective constant values in the sensible and latent heat flux equations specific to the studied lake obtained from CFD-Evap model [64], ${X}_{air}$ and ${X}_{ws}$, were calculated by:

#### 6.2. Velocity

^{3}), ${\mathsf{\tau}}_{sx}$ and ${\mathsf{\tau}}_{sy}$ are horizontal shear stress components over the water surface (kg m

^{−1}s

^{−2}), ${u}_{w}$ and ${v}_{w}$ are horizontal components of the mean wind speed over the water surface (m/s), and ${\mathsf{\nu}}_{eff}$ is the effective kinematic viscosity (m

^{2}/s). The empirical dimensionless drag coefficient (${C}_{D}$) depends to large extent only on wind speed and the state of wave development or wave age. For small shallow lakes, wind speed is generally low U

_{10}< 5 ms

^{−1}, where U

_{10}is wind speed at height 10 m above the water surface (Figure 2d) and measurements of the drag coefficient are relatively scarce. Confusingly, in the literature, the values of ${C}_{D}$ vary over a wide range and it is associated with large scatter [3,43,68]. In this study, the following empirical relationship for low wind speeds measured at a height of 10 m is used [68,69]:

## 7. Numerical Results and Discussion

- 1)
- Estimating heat fluxes over the water surface as boundary condition is very uncertain especially for latent heat flux. The location, climate, shape, depth, bathymetry, atmospheric stability conditions, etc. make it difficult to estimate evaporation accurately from the water surface.
- 2)
- There are no measurements for some important parameters that can affect the flow field and temperature in the water body, such as turbidity, and heat fluxes at the bottom and side walls where using simplified temperature boundary conditions could be considered as a source of error.
- 3)
- The measurements were taken only at one point. This means that the distribution of parameters over the water surface was assumed homogeneous. For shallow and small lakes with limited fetch, this assumption could produce a large error in the results.
- 4)
- Coupling the turbulent flow and heat transfer in a shallow water body is complex and computational issues such as numerical errors, mesh dependency and residuals control should be considered.
- 5)
- Errors in field measurements on the water surface especially for water surface temperature or heat fluxes.

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The shape of Lake Binaba and its surroundings (Google earth). Location of the thermistor chain is shown by a filled square over the lake.

**Figure 2.**Measured meteorological parameters used in the simulation: (

**a**) air and water surface temperature; (

**b**) short-wave radiation; (

**c**) relative humidity; and (

**d**) wind speed and its direction.

**Figure 3.**(

**a**) The generated geometry of Lake Binaba with depth distributions; and (

**b**) computational grid in the water body used in the simulation (vertical exaggerated by 100).

**Figure 4.**Simulated water temperature (S.) and observed values (M.) at different depths (at water surface as well as depths of 0.100, 0.200, 0.500, 1.100, 1.550, 1.850, 2.150, 2.800 and 3.465 m).

**Figure 5.**Analysis of the simulated (S.) and observed (M.) temperature values on the selected day (24 November 2012). (

**a**) Amplitude of the daily temperature variations as a function of depth. Amplitude is defined as the half-temperature fluctuations (the difference between the maximum and minimum temperature values in each depth); (

**b**) The phase shift with respect to the maximum short-wave radiation value as a function of depth where t and t

_{ref}are the times of the maximum temperature and short-wave radiation respectively.

**Figure 6.**Simulated vertical temperature profiles (S.) and observed values (M.) at selected time frames.

**Figure 7.**Simulated velocity vectors and their magnitudes over the water surface at t = 1:00 p.m. where U

_{2}= 1.0 ms

^{−1}. Line A-A is the vertical section illustrated in Figure 13.

**Figure 8.**Simulated velocity vectors and theirs magnitudes at 1 meter beneath the water surface at t = 1:00 p.m. where U

_{2}= 1.0 ms

^{−1}.

**Figure 9.**Simulated vertical distribution of velocity components in the water body where U

_{2}= 3.0 ms

^{−1}.

**Figure 10.**Simulated vertical distribution of velocity components in the water body where U

_{2}= 0.7 ms

^{−1}.

**Figure 11.**Simulated velocity field (stream lines) at 1 m beneath the water surface at t = 9:00 a.m. where U

_{2}= 3.8 ms

^{−1}.

**Figure 12.**Simulated Temperature field (values and contours) at 1 m beneath the water surface at t = 1:00 p.m.

**Figure 13.**Simulated Temperature field (values and contours) in a vertical section of the lake shown as line A-A in Figure 7 at t = 1:00 p.m. (vertical exaggerated by 100).

**Table 1.**Short-wave radiation bandwidth fractions of the total energy ($f$) and composite attenuation coefficients ($\mathsf{\eta}$) (adopted from [39]).

Wavelength (nm) | $f$ | $\mathsf{\eta}$ [m^{−1}] |
---|---|---|

<400 (UV) | 0.046 | assume same as VIS |

400–700 (VIS) | 0.430 | 3.0 (assumed) |

700–910 | 0.214 | 2.92 |

910–950 | 0.020 | 20.4 |

950–1090 | 0.089 | 29.5 |

1090–1350 | 0.092 | 98.4 |

>1350 | 0.109 | 2880 |

**Table 2.**Calculated metrics of model performance (MAE: mean absolute error; RMSE: root mean square error; ME: mean error; RME: relative mean error) for simulated temporal temperatures at different depths.

Depth (m) | MAE (°C) | RMSE (°C) | Mean Error (°C) | RME(%) | ||||
---|---|---|---|---|---|---|---|---|

max | min | ave | max | min | ave | |||

0.0 | 0.029 | 0.043 | 0.2053 | −0.1985 | 0.0032 | 0.6556 | 0.0017 | 0.0969 |

0.1 | 0.079 | 0.110 | 0.0650 | −0.4645 | −0.0773 | 1.4922 | 0.0000 | 0.2659 |

0.2 | 0.117 | 0.172 | 0.0023 | −0.7523 | −0.1166 | 2.4385 | 0.0076 | 0.3960 |

0.5 | 0.169 | 0.297 | 0.0760 | −1.4065 | −0.1402 | 4.6780 | 0.1038 | 0.5764 |

1.10 | 0.258 | 0.442 | 0.0335 | −1.6010 | −0.2541 | 5.4021 | 0.0009 | 0.8870 |

1.55 | 0.282 | 0.407 | 0.0340 | −1.5325 | −0.2816 | 5.2681 | 0.0075 | 0.9770 |

1.85 | 0.298 | 0.415 | 0.1960 | −1.2730 | −0.2931 | 4.3723 | 0.0094 | 1.0320 |

2.15 | 0.253 | 0.360 | 0.2560 | −1.0185 | −0.2277 | 3.4922 | 0.0009 | 0.8750 |

2.80 | 0.283 | 0.374 | 0.1123 | −1.0593 | −0.2780 | 3.6444 | 0.0009 | 0.9800 |

3.465 | 0.308 | 0.385 | 0.0425 | −1.0400 | −0.3057 | 3.5659 | 0.0043 | 1.0690 |

Total | 0.208 | 0.329 | 0.2560 | −1.6010 | −0.1972 | 5.4021 | 0.0000 | 0.7162 |

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**MDPI and ACS Style**

Abbasi, A.; Annor, F.O.; Van de Giesen, N.
Investigation of Temperature Dynamics in Small and Shallow Reservoirs, Case Study: Lake Binaba, Upper East Region of Ghana. *Water* **2016**, *8*, 84.
https://doi.org/10.3390/w8030084

**AMA Style**

Abbasi A, Annor FO, Van de Giesen N.
Investigation of Temperature Dynamics in Small and Shallow Reservoirs, Case Study: Lake Binaba, Upper East Region of Ghana. *Water*. 2016; 8(3):84.
https://doi.org/10.3390/w8030084

**Chicago/Turabian Style**

Abbasi, Ali, Frank Ohene Annor, and Nick Van de Giesen.
2016. "Investigation of Temperature Dynamics in Small and Shallow Reservoirs, Case Study: Lake Binaba, Upper East Region of Ghana" *Water* 8, no. 3: 84.
https://doi.org/10.3390/w8030084