**1. Introduction**

Thermal pollution is defined as the degradation of water quality by any process that changes ambient water temperature [1]. Coastal areas are often subject to thermal effluents originating from the cooling processes in industrial plants (nuclear reactors, electric power plants, petroleum refineries, steel melting factories, etc.) [2]. The industries collect water from lakes, rivers, or ocean, for cooling purpose, and return it in the environment at a high temperature. The hot water affects aquatic life, causes the substitution of aquatic fauna and flora, increases the mortality of certain species, and has indirect effects including bacterial development. More precisely, increasing the water temperature often increases the susceptibility of organisms to toxic substances (which are undoubtedly present in contaminated water) [3–6].

Studying the thermal effluents in receiving environments can contribute to efficiently manage the discharges, reducing environmental and economic impacts. Hence the reduction of thermal pollution must be included in the installation of cooling systems. Moreover distance between inlet and outlet must be carefully determined to avoid a decrease of the power plant efficiency.

By the middle of the 1960s, there were many research projects concentrating on thermal discharges, where major publications focus on the environmental impacts of power plant thermal discharges. Early mathematical models took place with works of [7]. The first treatments addressed the equilibrium iso-contour of elevated temperature within the receiving waters. Slightly later more advanced models allow the analysis of thermal plumes across extensive data in relation to seasonal and climate change fluctuations [3, 8–11].

*2.2.1 Temperature*

By hypothesis, three processes influence the temperature evolution: the thermal conduction, the convection, and the internal reactions. The thermal conduction translates the fact that the heat flux is proportional to the temperature gradient. The convection expresses the temperature transfer by the fluid velocity. The internal reactions are represented by the different sources of temperature and the industrial plant discharges in this situation. By taking into account these processes, for a time

*Tf* . 0, the temperature dynamic in Ω��0*, Tf* ½ is described by the equation

þ *u:*∇*T* |ffl{zffl} convection

*T x*ð Þ *; t* represents the fluid temperature at position *x*∈ Ω and time *t*∈ �0*, Tf* ½. *k* stands for the thermal diffusion coefficient. *u x*ð Þ *; t* is the fluid velocity inducing the advection process. The velocity is obtained by solving the Navier-Stokes system, described below. *ψ*1ð Þ *x* and *ψ*2ð Þ *x* are, respectively, Ω<sup>1</sup> and Ω<sup>2</sup> characteristic functions. They allow to localize the source term *f t*ð Þ and control *U t*ð Þ, respectively, in the subdomains Ω<sup>1</sup> and Ω2. The source term *f t*ð Þ is given, while the control *U t*ð Þ must be computed as a solution of an optimal control problem, described in the

On the impermeable boundary, no heat flux boundary condition is considered:

mal vector. On the outflux boundary, the heat flux is proportional to the velocity

where *α<sup>T</sup>* . 0 is a constant. The boundary condition allows us, as we will see in the sequel in Subsection 3.2, to obtain an explicit formula for the cost function gradient. These boundary conditions for the temperature are summarized in

¼ *fψ*<sup>1</sup> |{z} discharges þ *Uψ*<sup>2</sup> |ffl{zffl} control

*T* ¼ *Tin* on Γ*IN*��0*, T*½*:* (3)

! <sup>¼</sup> 0 on <sup>Γ</sup>*N*��0*,T ,*<sup>½</sup> (4)

! � �*<sup>T</sup>* <sup>¼</sup> 0 on <sup>Γ</sup>*OUT*��0*,T ,*<sup>½</sup> (5)

! defined on the boundary constitutes the outward unit nor-

*:* (2)

*∂T ∂t* |{z} variation

� *k*Δ*T* |ffl{zffl} diffusion

*Optimal Control of Thermal Pollution Emitted by Power Plants*

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

sequel. *Tin*ð Þ *x; t* is the temperature distribution in the inlet border:

�*k ∂T ∂n*

! <sup>þ</sup> *<sup>α</sup><sup>T</sup> <sup>u</sup>: <sup>n</sup>*

�*<sup>k</sup> <sup>∂</sup><sup>T</sup> ∂n*

where the vector *n*

and the temperature:

**Figure 2**.

**Figure 2.**

**23**

*Boundary conditions for temperature.*

#### **Figure 1.**

*Water domain Ω, industrial plants Ω*1*, control zone Ω*2*, observation zone ΩOBS, impermeable boundary ΓN, inflow boundary ΓIN, and outflow boundary ΓOUT.*

In this research, thermal pollution due to industrial activities was modeled by a system of partial differential equations, and optimal control is applied to reduce the associated thermal pollution. The location of the understudy area is illustrated by **Figure 1**.

The paper is organized as follows. First, the thermal pollution is modeled by a coupling of Navier-Stokes and heat equations. The cold water injection rate is minimum of a cost function, in order to reduce the temperature variation and the energy required to refresh injected water. Afterward, the well-posedness of this problem is investigated. It follows a numerical resolution of the optimal control by means of a gradient descent algorithm. Finally, numerical simulations are performed to illustrate our approach.
