**2. CFD modeling of the thermal dispersion**

### **2.1 Geometry representation**

We are interested in the evolution of the system in space and time. Then, we denote *x* and *t*, respectively, as the space and time variables. Ω is the domain occupied by the water. Its boundary is denoted as *∂Ω* and is divided into three disjoint subborders. It is written

$$
\partial \mathfrak{Q} = \Gamma\_N \cup \Gamma\_{IN} \cup \Gamma\_{OUT},
\tag{1}
$$

where Γ*IN* is the entering border, Γ*OUT* is the outflow boundary, and Γ*<sup>N</sup>* is the impermeable part. Ω contains three subdomains Ω1, Ω2, and Ω*OBS*. Ω<sup>1</sup> stands for the industrial plants, where the source of pollution modeled by *f x*ð Þ *; t* is defined. In Ω2, cold water is injected at a rate *U* in order to control the temperature in Ω*OBS*. The objective consists in finding the optimal rate *Uopt* so that the temperature in Ω*OBS* will be as close as possible to a desired value *Td*. *Td* can be the temperature favorable to the survival of the ecosystem. The geometric domain is illustrated by **Figure 1**.

## **2.2 Mathematical model**

We present the system of partial differential equations representing the evolution of the river parameters (temperature, velocity, and pressure). Then, the cost function to be minimized in order to reduce the thermal pollution is described.

#### *2.2.1 Temperature*

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

*Water domain Ω, industrial plants Ω*1*, control zone Ω*2*, observation zone ΩOBS, impermeable boundary ΓN,*

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

We are interested in the evolution of the system in space and time. Then, we denote *x* and *t*, respectively, as the space and time variables. Ω is the domain occupied by the water. Its boundary is denoted as *∂Ω* and is divided into three

where Γ*IN* is the entering border, Γ*OUT* is the outflow boundary, and Γ*<sup>N</sup>* is the impermeable part. Ω contains three subdomains Ω1, Ω2, and Ω*OBS*. Ω<sup>1</sup> stands for the industrial plants, where the source of pollution modeled by *f x*ð Þ *; t* is defined. In Ω2, cold water is injected at a rate *U* in order to control the temperature in Ω*OBS*. The objective consists in finding the optimal rate *Uopt* so that the temperature in Ω*OBS* will be as close as possible to a desired value *Td*. *Td* can be the temperature favorable to the survival of the ecosystem. The geometric domain is illustrated

We present the system of partial differential equations representing the evolution of the river parameters (temperature, velocity, and pressure). Then, the cost function to be minimized in order to reduce the thermal pollution is

*<sup>∂</sup><sup>Ω</sup>* <sup>¼</sup> <sup>Γ</sup>*<sup>N</sup>* <sup>∪</sup> <sup>Γ</sup>*IN* <sup>∪</sup> <sup>Γ</sup>*OUT,* (1)

means of a gradient descent algorithm. Finally, numerical simulations are

**Figure 1**.

**Figure 1.**

performed to illustrate our approach.

*inflow boundary ΓIN, and outflow boundary ΓOUT.*

*Numerical Modeling and Computer Simulation*

**2.1 Geometry representation**

disjoint subborders. It is written

by **Figure 1**.

described.

**22**

**2.2 Mathematical model**

**2. CFD modeling of the thermal dispersion**

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

$$\underbrace{\frac{\partial T}{\partial t}}\_{\text{variation}} - \underbrace{k \Delta T}\_{\text{diffusion}} + \underbrace{\mu . \nabla T}\_{\text{convection}} = \underbrace{f \mu\_1}\_{\text{discharge}} + \underbrace{U \mu\_2}\_{\text{control}}.\tag{2}$$

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

$$T = T\_{\rm in} \quad \text{on} \quad \Gamma\_{\rm IN} \times [0, T]. \tag{3}$$

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

$$-k\frac{\partial T}{\partial \overrightarrow{n}} = \mathbf{0} \quad \text{on} \quad \Gamma\_N \times ]\mathbf{0}, T[. \tag{4}$$

where the vector *n* ! defined on the boundary constitutes the outward unit normal vector. On the outflux boundary, the heat flux is proportional to the velocity and the temperature:

$$-k\frac{\partial T}{\partial \vec{n}} + a\_T \begin{pmatrix} u \ \vec{n} \end{pmatrix} T = \mathbf{0} \quad \text{on} \quad \Gamma\_{OUT} \times ]\mathbf{0}, T[.\tag{5}$$

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 **Figure 2**.

**Figure 2.** *Boundary conditions for temperature.*

Function *T*0ð Þ *x* represents the distribution of the temperature at the initial time:

$$T(\mathbf{0}) = T\_0 \text{ on } \Omega. \tag{6}$$

*J T*ð Þ¼ *; <sup>U</sup>* <sup>1</sup>

**3. Optimal control**

**3.1 Cost functional**

where <sup>U</sup>*ad* <sup>¼</sup> *<sup>L</sup>*<sup>2</sup>

the cost function is written

**3.2 Directional derivative**

*3.2.1 Sequences convergence*

following one:

**25**

First, for a fixed *h*∈*L*<sup>2</sup>

are considered, for all λ . 0.

*∂w*<sup>λ</sup>

to *Td*.

2

ð *Tf*

0

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

B@

ð

*Optimal Control of Thermal Pollution Emitted by Power Plants*

ð Þ *T* � *Td*

2

The aim is to find an optimal control *U* minimizing the cost function:

*dxdt* þ *β*

where *Ud* is the ideal control rate and *β* . 0 is the cost-efficiency ratio. More *β* is great; more energy must be provided to maintain the temperature in Ω*OBS* close

ð *Tf*

ð

ð Þ *U* � *Ud*

<sup>~</sup>*J v*ð Þ¼ *JTv* ð Þ ð Þ*; <sup>v</sup> ,* <sup>∀</sup>*v*<sup>∈</sup> <sup>U</sup>*ad,* (13)

*F v*ð Þ¼, *Td* � *T*0*,T v*ð Þ� *T*<sup>0</sup> . *,* (15)

<sup>2</sup> *<sup>π</sup>*ð Þþ *<sup>v</sup>; <sup>v</sup>* <sup>∥</sup>*Td* � *<sup>T</sup>*0∥<sup>2</sup> � � � *F v*ð Þ*:* (16)

*T*<sup>λ</sup> ¼ *T U*ð Þ þ λ *h , w*<sup>λ</sup> ¼ *T*<sup>λ</sup> � *T* (17)

ð Þ Ω<sup>2</sup> is the admissible function space. By considering the

*π*ð Þ¼ *s; v* , *T s*ð Þ� *T*0*,T v*ð Þ� *T*<sup>0</sup> . þ *β* , *s* � *Ud, v* � *Ud* . (14)

We are in the framework of Theorem 16.1 in [12] that establishes the existence

ð�0*, Tf* ½ Þ *;* U*ad* , two function sequences

The difference between equations satisfied by *T*<sup>λ</sup> and *T* results to the

� *<sup>k</sup> <sup>∂</sup>w*<sup>λ</sup> *∂n*

*<sup>∂</sup><sup>t</sup>* � *<sup>k</sup>*Δ*w*<sup>λ</sup> <sup>þ</sup> *<sup>u</sup>:*∇*w*<sup>λ</sup> <sup>¼</sup> <sup>λ</sup>*hψ*<sup>2</sup> in <sup>Ω</sup>��0*, Tf* <sup>½</sup>*,*

*w*<sup>λ</sup> ¼ 0 on Γ*IN*��0*, Tf* ½*,*

(18)

! <sup>¼</sup> <sup>0</sup> on <sup>Γ</sup>*N*��0*, Tf* <sup>½</sup>*,*

2 *dxdt*

1

CA*,* (12)

Ω<sup>2</sup>

0

Ω*OBS*

0

symmetric, continuous, coercive bilinear form

for all *s, v*∈ *Uad*, and the linear bounded functional

<sup>~</sup>*J v*ð Þ¼ <sup>1</sup>

and uniqueness of solution to the minimization problem.

#### *2.2.2 Velocity and pressure*

The fluid velocity *u* is obtained by solving, in Ω��0*, Tf* ½, the incompressible Navier-Stokes system:

$$\frac{\partial u}{\partial t} - \nu \Delta u + (u.\nabla)u + \nabla p = f\_1 \nu\_1 + f\_2 \nu\_2,\tag{7}$$

$$d\dot{v}(u) = 0,$$

where *p x*ð Þ *; t* is the water pressure; *ν* . 0 is the kinematic viscosity; *f* <sup>1</sup>ð Þ*t* and *f* <sup>2</sup>ð Þ*t* are, respectively, the velocity sources in Ω<sup>1</sup> and Ω2. At the inlet, the velocity is known and given by a function *uin*ð Þ *x; t* . It is written

$$
\mu = \mu\_{\text{int}} \text{ on } \Gamma\_{IN} \times ]0, \, T\_f[.\tag{8}
$$

On the impermeable boundary, it is assumed that the velocity is equal to zeros due to the viscosity:

$$
\mu = \mathbf{0} \text{ on } \Gamma\_N \times ]\mathbf{0}, T\_f[.\tag{9}
$$

On the outflow boundary, the pressure is equal to zeros:

$$p = \mathbf{0} \text{ on } \Gamma\_{OUT} \times [\mathbf{0}, T\_f],\tag{10}$$

The boundary conditions applied to the velocity and the pressure are summarized in **Figure 3**.

The system is also completed by the initial condition for the velocity:

$$
\mu(\mathbf{0}) = \mu\_0 \text{ on } \overline{\Omega}. \tag{11}
$$

#### *2.2.3 Cost functional*

In order to reduce the pollution in an arbitrary area Ω*OBS*, a freshwater is introduced in the subdomain Ω2. We are seeking the optimal rate *U* at which the freshwater is introduced, such that the temperature in Ω*OBS* must be as closed as possible to a prescribed threshold denoted *Td*. This optimal control must be the minimum of the cost function:

**Figure 3.** *Boundary conditions for the velocity and pressure.*

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

$$J(T, U) = \frac{1}{2} \left( \int\_0^{T\_f} \int\_{\Omega\_{\text{OLS}}} (T - T\_d)^2 dxdt + \beta \int\_0^{T\_f} \int\_{\Omega\_{\text{\}}} (U - U\_d)^2 dxdt \right), \tag{12}$$

where *Ud* is the ideal control rate and *β* . 0 is the cost-efficiency ratio. More *β* is great; more energy must be provided to maintain the temperature in Ω*OBS* close to *Td*.
