**5.2 Influence of the discharge rates**

In **Figure 7**, we present the influence of different discharge rates on the thermal plume area. For low rates, we observe a high temperature far from the discharge area. However, for high rates, the temperature seems to be high near the emission zone and small far from the discharge zone. In this last case, the polluted area is more extended.


**Table 1** *Physical parameters of the river.*

#### **Figure 5.**

*Temperature contours. (a) Temperature contour after 0 second, (b) Temperature contour after 10 second, (c) Temperature contour after 20 second, (d) Temperature contour after 30 second, (e) Temperature contour after 40 second, and (f) Temperature contour after 50 second.*

**5.3 Heat emission optimal control**

dispersion simulation *U* ¼ 0°*C:s*

*f* <sup>1</sup> ¼ ð Þ 0*;* �10 *m:s*

*Optimal Control of Thermal Pollution Emitted by Power Plants*

*DOI: http://dx.doi.org/10.5772/intechopen.88646*

by **Figure 8**. At the initial step, we assume that *U* ¼ �1°*C:s*

*Cost functional after each iteration of the steepest descent algorithm.*

initialized to *U* ¼ �1°*C:s*

**Figure 12**.

**Figure 8.**

**33**

**Figure 7.**

*(d) vmax = 50.*

Similarly to the first case, a source term *f* is applied on Ω1. The control on Ω<sup>2</sup> is

The cost function value according to the optimization iteration is represented

In **Figure 10**, the stopping criteria ∣∇~*J U<sup>m</sup>* ð Þ∣*=*∣∇~*J U*<sup>0</sup> <sup>∣</sup> in terms of the number of iterations are reported. At initial step, the stopping criteria is equal to 1 and then decreases to reach 9*:*56428 ∗ 10�<sup>3</sup> after 10 iterations. From this observation, it can be deduced that the control sequence *U<sup>m</sup>* converges to the optimal control rate.

**Figure 11** compares the temperature evolution in Ω*OBS* for the thermal plume

�1, the initial step *<sup>U</sup>* ¼ �1°*C:<sup>s</sup>*

�1

*Temperature contours for different discharge rates. (a) vmax = 20, (b) vmax = 30, (c) vmax = 40, and*

function *J T*ð Þ¼ *; <sup>U</sup>* <sup>1</sup>*:*<sup>37274</sup> <sup>∗</sup> <sup>10</sup>�<sup>2</sup> (**Figure 9**). The optimal solution is obtained after 10 iterations, for an optimal control rate *U t*ð Þ of order �10�<sup>2</sup> illustrated by

�1. The velocity sources on Ω<sup>1</sup> and Ω<sup>2</sup> are, respectively:

�1

*:* (41)

�1, thus obtaining a cost

�1, and the optimal

*, f* <sup>2</sup> ¼ ð Þ 0*;* �10 *m:s*

#### **Figure 6.**

*Velocity fields. (a) Velocity field after 0 second, (b) Velocity field after 10 second, (c) Velocity field after 20 second, (d) Velocity field after 30 second, (e) Velocity field after 40 second, and (f) Velocity field after 50 second.*

*Optimal Control of Thermal Pollution Emitted by Power Plants DOI: http://dx.doi.org/10.5772/intechopen.88646*

#### **Figure 7.**

**Figure 5.**

**Figure 6.**

*second.* **32**

*Temperature contours. (a) Temperature contour after 0 second, (b) Temperature contour after 10 second, (c) Temperature contour after 20 second, (d) Temperature contour after 30 second, (e) Temperature contour*

*Velocity fields. (a) Velocity field after 0 second, (b) Velocity field after 10 second, (c) Velocity field after 20 second, (d) Velocity field after 30 second, (e) Velocity field after 40 second, and (f) Velocity field after 50*

*after 40 second, and (f) Temperature contour after 50 second.*

*Numerical Modeling and Computer Simulation*

*Temperature contours for different discharge rates. (a) vmax = 20, (b) vmax = 30, (c) vmax = 40, and (d) vmax = 50.*

#### **5.3 Heat emission optimal control**

Similarly to the first case, a source term *f* is applied on Ω1. The control on Ω<sup>2</sup> is initialized to *U* ¼ �1°*C:s* �1. The velocity sources on Ω<sup>1</sup> and Ω<sup>2</sup> are, respectively:

$$f\_1 = (\mathbf{0}, -\mathbf{10}) \, m \, \text{s}^{-1}, \quad f\_2 = (\mathbf{0}, -\mathbf{10}) \, m \, \text{s}^{-1}.\tag{41}$$

The cost function value according to the optimization iteration is represented by **Figure 8**. At the initial step, we assume that *U* ¼ �1°*C:s* �1, thus obtaining a cost function *J T*ð Þ¼ *; <sup>U</sup>* <sup>1</sup>*:*<sup>37274</sup> <sup>∗</sup> <sup>10</sup>�<sup>2</sup> (**Figure 9**). The optimal solution is obtained after 10 iterations, for an optimal control rate *U t*ð Þ of order �10�<sup>2</sup> illustrated by **Figure 12**.

In **Figure 10**, the stopping criteria ∣∇~*J U<sup>m</sup>* ð Þ∣*=*∣∇~*J U*<sup>0</sup> <sup>∣</sup> in terms of the number of iterations are reported. At initial step, the stopping criteria is equal to 1 and then decreases to reach 9*:*56428 ∗ 10�<sup>3</sup> after 10 iterations. From this observation, it can be deduced that the control sequence *U<sup>m</sup>* converges to the optimal control rate.

**Figure 11** compares the temperature evolution in Ω*OBS* for the thermal plume dispersion simulation *U* ¼ 0°*C:s* �1, the initial step *<sup>U</sup>* ¼ �1°*C:<sup>s</sup>* �1, and the optimal

**Figure 8.** *Cost functional after each iteration of the steepest descent algorithm.*

**Figure 9.** *Control rate after iterations 0, 4, and 10.*

#### **Figure 10.**

*Stopping criteria according to the number of iterations.*

solution, the temperature is >30°*C* but does not exceed 30.08°*C*. Moreover for all the time, it is closer to the optimal value than for the thermal dispersion case. According to these remarks, it can be concluded that the computed optimal rate allows to maintain the temperature in Ω*OBS* at a value close to the desired

*Optimal control: temperature field (in °C) at time intervals of 1 (a), 5 (b), 10 (c), 30 (d), 45 (e),*

*Optimal Control of Thermal Pollution Emitted by Power Plants*

*DOI: http://dx.doi.org/10.5772/intechopen.88646*

**Figure 12** illustrates the temperature distributions at times 1, 5, 10, 30, 45, and 59.9 s. A reduction of the thermal pollution is observed, due to the cold water

Numerical models are essential to predict the thermal effluent impacts on natural systems. This work is of particular relevance for the coastal area managements,

threshold 30°*C*.

**Figure 12.**

*and 59.9 s (f).*

source in Ω2.

**6. Conclusion**

**35**

#### **Figure 11.**

*Time series of water mean temperature in the observation zone ΩOBS. In blue, only hot water discharge in Ω*<sup>1</sup> *is considered. In green, simulations are carried out by using the initial control U* ¼ �1°*C:s* �<sup>1</sup>*. In red, temperature evolution is corresponding to the optimal control.*

case. For the thermal dispersion, it can be noticed that after 8 s the temperature increases to reach a maximum of 30.15°*C*. For the initial step *U* ¼ �1°*C:s* �1, the temperature is lower than 30°*C* and reaches a minimum of 29.87. For the optimal

*Optimal Control of Thermal Pollution Emitted by Power Plants DOI: http://dx.doi.org/10.5772/intechopen.88646*

#### **Figure 12.**

*Optimal control: temperature field (in °C) at time intervals of 1 (a), 5 (b), 10 (c), 30 (d), 45 (e), and 59.9 s (f).*

solution, the temperature is >30°*C* but does not exceed 30.08°*C*. Moreover for all the time, it is closer to the optimal value than for the thermal dispersion case. According to these remarks, it can be concluded that the computed optimal rate allows to maintain the temperature in Ω*OBS* at a value close to the desired threshold 30°*C*.

**Figure 12** illustrates the temperature distributions at times 1, 5, 10, 30, 45, and 59.9 s. A reduction of the thermal pollution is observed, due to the cold water source in Ω2.

## **6. Conclusion**

Numerical models are essential to predict the thermal effluent impacts on natural systems. This work is of particular relevance for the coastal area managements,

case. For the thermal dispersion, it can be noticed that after 8 s the temperature

*Time series of water mean temperature in the observation zone ΩOBS. In blue, only hot water discharge in Ω*<sup>1</sup> *is*

temperature is lower than 30°*C* and reaches a minimum of 29.87. For the optimal

�1, the

�<sup>1</sup>*. In red, temperature*

increases to reach a maximum of 30.15°*C*. For the initial step *U* ¼ �1°*C:s*

*considered. In green, simulations are carried out by using the initial control U* ¼ �1°*C:s*

**Figure 9.**

**Figure 10.**

**Figure 11.**

**34**

*Control rate after iterations 0, 4, and 10.*

*Numerical Modeling and Computer Simulation*

*Stopping criteria according to the number of iterations.*

*evolution is corresponding to the optimal control.*
