The parabolic trough is one of the main technologies currently used in solar thermal power plants for electricity production (operating temperature between 300˚C and 400˚C) and also for heat production for industry (operating temperature between 100˚C and 250˚C) [1] .

A lot of research has been done in recent years on the types of solar collectors. While some studies aimed at cost reduction and performance improvement, others focused on improving the reliability of the receiver tube (absorber), the performance of the selective coating and new measurement methods to test the parabolic trough system [2] - [12] . The majority of models are focused on simulation studies. The solar flux and flow rate are generally assumed to be uniform, and many correlations used in the models are also based on a constant temperature. However, due to the non-uniform solar flux on the outer surface of the inner absorber tube, the flow is heated asymmetrically and thus non-uniformly [13] [14] . Balbir et al. [2] conducted a sizing study of a parabolic trough concentrator. The study aimed to improve the efficiency and reduce the cost of the solar collector operating with synthetic oil and water. Campo [15] developed a simple algebraic method to improve the heat transfer in turbulent regime in a cylindrical tube for different gases. Grald [16] conducted a parametric study to determine the influence of thermo-physical properties and the performance of the parabolic trough solar collector. Thus, this study aims to model the heat transfer phenomenon in the different compartments of the parabolic trough concentrator. Here, we determine the evolution of the temperature as a function of time by establishing a thermal balance that takes into account the different thermal exchanges, followed by a numerical model based on the following steps [17] :

It is presented as a module having a reflector of parabolic shape arranged cylindrically according to Figure 1 .

Heat exchanges here take place between three elements which are: Heat transfer fluid, absorber and glass, as shown in Figure 2 .

The thermal power emitted by sun and received by concentrator is [23] :

$q_{ab}=S_e\cdot R_d\left(\beta \right)$ (31)

$S_e$: effective area of a reflector of the parabolic trough;

$R_d\left(\beta \right)$: sunlight or direct radiation.

Since the surface of absorber $S_a$ receives the same power as $S_e$, then the concentration has the effect of increasing power per unit area at the level of $S_a$; but due to optical losses around transparent envelope (glass) and absorber, the power per unit length of absorber is:

$q_{ab}={\tau }_v\cdot {\alpha }_{ab}\cdot {\rho }_m\cdot S_e\cdot R_d\left(\beta \right)\cdot k\left(\theta \right)$ (32)

where:

${\rho }_m$: Mirror reflectance factor;

${\alpha }_{ab}$: Absorption coefficient of absorber tube;

${\tau }_g$: Coverage transmission coefficient (glass);

$k\left(\theta \right)$: Angle of incidence modify.

For cylindro-parabolic collectors, the following equation [24] , gives coefficient $k\left(\theta \right)$ as a function of angle of incidence:

$k\left(\theta \right)=1-b_0\left(\frac{1}{\cos \left(\theta \right)}\right)$ (33)

where:

$b_0$: coefficient of modified angle less than zero for a cylindro-parabolic concentrator.

The coefficient of thermal losses $U_L$ is expressed by [25] :

$\begin{array}{c}{U}_L={\left(\frac{1}{C_1{\left[\left(T_a-T_{amb}\right)/\left(1+\lambda \right)\right]}^{0.25}}+\frac{D_{ai}}{D_{ae}}\times \frac{1}{h_V}\right)}^{-1}\\ +\frac{\sigma \left(T_a^2+T_{amb}^2\right)\left(T_a+T_{amb}\right)}{{\left[{\epsilon }_a-0.04\left(1-{\epsilon }_a\right)\left(T_a/450\right)\right]}^{-1}-\left[\left(D_{ai}/D_{ae}\right)\left(1/{\epsilon }_g\right)+\lambda /{\epsilon }_g\right]}\end{array}$ (34)

where:

$\lambda$: wind loss coefficient, obtained by equation (35):

$\lambda =\frac{D_{ai}}{D_{ai}^{1.4}}\left(0.61+1.3{\epsilon }_a\right)h_g^{-0.9}\exp \left[0.00325\left(T_a-273\right)\right]$ (35)

$C_1=\frac{1.45+0.96{\left({\epsilon }_a-0.5\right)}^2}{D_{ai}{\left(1/D_{ai}^{0.6}+D_{ae}^{0.6}\right)}^{1.25}}$ (36)

$h_V=4V_V^{0.58}{D}_{ae}^{-0.42}$ (37)

The finite difference method we use for our problem resolution, consists in approximating derivatives of equations obtained above, by means of Taylor expansions and is deduced directly from definition of derivative.

Discretization of one-dimensional equations (1D)

The space is discretized into N + 1 nodes of coordinates $z_j$ ( $j$ varying from 0 to N) regularly spaced. Note $\Delta z$ the space step. Time is discretized in constant step intervals $\Delta t$.

Let us note $T_j^n$ the temperature at the node $z_j=j\Delta z$ and at time $t=n\Delta t$.

We use an explicit before order 1 scheme to evaluate temporal and spatial derivative:

${\left(\frac{\partial T}{\partial t}\right)}_j^n=\frac{T_j^n-T_j^{n-1}}{\Delta t}$ (38)

${\left(\frac{\partial T}{\partial z}\right)}_j^n=\frac{T_j^n-T_{j-1}^n}{\Delta z}$ (39)

For the fluid

Equation (6) gives:

${\rho }_f\cdot C_f\cdot S_{ai}\cdot \frac{T_{f,j}^n-T_{f,j}^{n-1}}{\Delta t}=-{\rho }_f\cdot C_f\cdot \stackrel{\dot{}}{V}\cdot \frac{T_{f,j}^n-T_{f,j-1}^n}{\Delta z}+q_u\left(z,t\right)$

By replacing $q_u\left(z,t\right)$ by its expression and by expressing $T_{f,j}^n$ depending on other parameters, we get:

$\begin{array}{l}\left(1+\frac{\Delta t\cdot \stackrel{\dot{}}{V}}{\Delta z\cdot S_{ai}}+\frac{\Delta t}{{\rho }_f\left(j\right)C_f\left(j\right)S_{ai}}{h}_f\cdot S_{ai}\right)T_{f,j}^n+\frac{\Delta t}{{\rho }_f\left(j\right)C_f\left(j\right)S_{ai}}{h}_f\cdot S_{ai}{T}_{a,j}^n\\ =\frac{C_f\left(j-1\right){\rho }_f\left(j-1\right)\Delta t\cdot \stackrel{\dot{}}{V}}{C_f\left(j\right){\rho }_f\left(j\right)\Delta z\cdot S_{ai}}{T}_{f,j-1}^n+T_{f,j}^{n-1}\end{array}$ (40)

For the absorber

Equation (9) gives: ${\rho }_a\cdot C_a\cdot S_{ae}\cdot \frac{T_{a,j}^n-T_{a,j}^{n-1}}{\Delta t}=q_{ab}\left(t\right)-q_{a,g}\left(t\right)-q_u\left(t\right)$.

By replacing $q_{ab}\left(t\right)$, $q_{a,g}\left(t\right)$ and $q_u\left(t\right)$ by their expressions, we get:

$\begin{array}{l}\frac{\Delta t\cdot h_f\cdot S_{ai}}{{\rho }_a{C}_a{S}_{ae}}{T}_{f,j}^n\\ +\left(1-\frac{\Delta t}{{\rho }_a{C}_a{S}_{ae}}\left(\frac{2\pi \cdot k_{eff,air}}{\ln \left(D_{gi}/D_{ae}\right)}+h_f\cdot S_{ai}+\frac{\sigma \cdot S_{ae}\left({\left(T_{a,j}^2\right)}^n+{\left(T_{g,j}^2\right)}^n\right)\left(T_{a,j}^n+T_{g,j}^n\right)}{\frac{1}{{\epsilon }_a}+\frac{1-{\epsilon }_g}{{\epsilon }_g}\left(\frac{D_{ae}}{D_{gi}}\right)}\right)\right)T_{a,j}^n\\ +\frac{\Delta t}{{\rho }_a{C}_a{S}_{ae}}\left(\frac{2\pi \cdot k_{eff,air}}{\ln \left(D_{gi}/D_{ae}\right)}+\frac{\sigma \cdot S_{ae}\left({\left(T_{a,j}^2\right)}^n+{\left(T_{g,j}^2\right)}^n\right)\left(T_{a,j}^n+T_{g,j}^n\right)}{\frac{1}{{\epsilon }_a}+\frac{1-{\epsilon }_g}{{\epsilon }_g}\left(\frac{D_{ae}}{D_{gi}}\right)}\right)T_{g,j}^n\\ =T_{a,j}^{n-1}+\frac{\Delta t}{{\rho }_a{C}_a{S}_{ae}}{\tau }_g\cdot {\alpha }_{ab}\cdot {\rho }_m\cdot S_e\cdot R_d\left(\beta \right)\cdot k\left(\theta \right)\end{array}$

(41)

For the glass

Equation (10) gives: ${\rho }_g\cdot C_g\cdot S_g\cdot \frac{T_{g,j}^n-T_{g,j}^{n-1}}{\Delta t}=q_{a,g}\left(t\right)-q_{g,amb}\left(t\right)$.

By replacing $q_{a,g}\left(t\right)$ and $q_{g,amb}\left(t\right)$ by their expressions, we get:

$\begin{array}{l}-\frac{\Delta t}{{\rho }_g{C}_g{S}_g}\left(\frac{2\pi \cdot k_{eff,air}}{\ln \left(D_{gi}/D_{ae}\right)}+\frac{\sigma \cdot S_{ae}\left({\left(T_{a,j}^2\right)}^n+{\left(T_{g,j}^2\right)}^n\right)\left(T_{a,j}^n+T_{g,j}^n\right)}{\frac{1}{{\epsilon }_a}+\frac{1-{\epsilon }_g}{{\epsilon }_g}\left(\frac{D_{ae}}{D_{gi}}\right)}\right)T_{a,j}^n\\ +(1+\frac{\Delta t}{{\rho }_g{C}_g{S}_g}(\frac{2\pi \cdot k_{eff,air}}{\ln \left(D_{gi}/D_{ae}\right)}+\frac{\sigma \cdot S_{ae}\left({\left(T_{a,j}^2\right)}^n+{\left(T_{g,j}^2\right)}^n\right)\left(T_{a,j}^n+T_{g,j}^n\right)}{\frac{1}{{\epsilon }_a}+\frac{1-{\epsilon }_g}{{\epsilon }_g}\left(\frac{D_{ae}}{D_{gi}}\right)}\\ \begin{array}{c}\\ \\ \\ \end{array}+h_V\cdot S_{ge}+{\epsilon }_g\cdot \sigma \cdot S_{ge}\left({\left(T_{g,j}^2\right)}^n+{\left(T_{amb,j}^2\right)}^n\right)\left(T_{g,j}^n+T_{amb,j}^n\right)))T_{g,j}^n\\ =T_{g,j}^{n-1}+\frac{\Delta t}{{\rho }_g{C}_g{S}_g}\left(h_V\cdot S_{ge}+{\epsilon }_g\cdot \sigma \cdot S_{ge}\left({\left(T_{g,j}^2\right)}^n+{\left(T_{amb,j}^2\right)}^n\right)\left(T_{g,j}^n+T_{amb,j}^n\right)\right)T_{amb,j}^n\end{array}$ (42)

Equations (40), (41) and (42) form a system of equations with three unknowns to be solved.

It is also important to know that time step is one hour [26] .

This system can be in the following matrix form:

$\left[\begin{array}{ccc}{b}_1& c_1& 0\\ a_2& b_2& c_2\\ 0& a_3& b_3\end{array}\right]{\left[\begin{array}{c}{T}_{f,j}\\ T_{a,j}\\ T_{v,j}\end{array}\right]}^n=\left[\begin{array}{c}d\\ e\\ g\end{array}\right]$

where:

$b_1=1+\frac{\Delta t\cdot \stackrel{\dot{}}{V}}{\Delta z\cdot S_{ai}}+\frac{\Delta t}{{\rho }_f\left(j\right)C_f\left(j\right)S_{ai}}{h}_f\cdot S_{ai}$

$c_1=\frac{\Delta t}{{\rho }_f\left(j\right)C_f\left(j\right)S_{ai}}{h}_f\cdot S_{ai}$

$d=\frac{C_f\left(j-1\right){\rho }_f\left(j-1\right)\Delta t\cdot \stackrel{\dot{}}{V}}{C_f\left(j\right){\rho }_f\left(j\right)\Delta z\cdot S_{ai}}{T}_{f,j-1}^n+T_{f,j}^{n-1}$

$a_2=\frac{\Delta t\cdot h_f\cdot S_{ai}}{{\rho }_a{C}_a{S}_{ae}}$

$b_2=1-\frac{\Delta t}{{\rho }_a{C}_a{S}_{ae}}\left(\frac{2\pi \cdot k_{eff,air}}{\ln \left(D_{gi}/D_{ae}\right)}+h_f\cdot S_{ai}+\frac{\sigma \cdot S_{ae}\left({\left(T_{a,j}^2\right)}^n+{\left(T_{g,j}^2\right)}^n\right)\left(T_{a,j}^n+T_{g,j}^n\right)}{\frac{1}{{\epsilon }_a}+\frac{1-{\epsilon }_g}{{\epsilon }_g}\left(\frac{D_{ae}}{D_{gi}}\right)}\right)$

$c_2=\frac{\Delta t}{{\rho }_a{C}_a{S}_{ae}}\left(\frac{2\pi \cdot k_{eff,air}}{\ln \left(D_{gi}/D_{ae}\right)}+\frac{\sigma \cdot S_{ae}\left({\left(T_{a,j}^2\right)}^n+{\left(T_{g,j}^2\right)}^n\right)\left(T_{a,j}^n+T_{g,j}^n\right)}{\frac{1}{{\epsilon }_a}+\frac{1-{\epsilon }_g}{{\epsilon }_g}\left(\frac{D_{ae}}{D_{gi}}\right)}\right)$

$e=T_{a,j}^{n-1}+\frac{\Delta t}{{\rho }_a{C}_a{S}_{ae}}{\tau }_g\cdot {\alpha }_{ab}\cdot {\rho }_m\cdot S_e\cdot R_d\left(\beta \right)\cdot k\left(\theta \right)$

$a_3=-\frac{\Delta t}{{\rho }_g{C}_g{S}_g}\left(\frac{2\pi \cdot k_{eff,air}}{\ln \left(D_{gi}/D_{ae}\right)}+\frac{\sigma \cdot S_{ae}\left({\left(T_{a,j}^2\right)}^n+{\left(T_{g,j}^2\right)}^n\right)\left(T_{a,j}^n+T_{g,j}^n\right)}{\frac{1}{{\epsilon }_a}+\frac{1-{\epsilon }_g}{{\epsilon }_g}\left(\frac{D_{ae}}{D_{gi}}\right)}\right)$

$\begin{array}{c}{b}_3=1+\frac{\Delta t}{{\rho }_g{C}_g{S}_g}(\frac{2\pi \cdot k_{eff,air}}{\ln \left(D_{gi}/D_{ae}\right)}+\frac{\sigma \cdot S_{ae}\left({\left(T_{a,j}^2\right)}^n+{\left(T_{g,j}^2\right)}^n\right)\left(T_{a,j}^n+T_{g,j}^n\right)}{\frac{1}{{\epsilon }_a}+\frac{1-{\epsilon }_g}{{\epsilon }_g}\left(\frac{D_{ae}}{D_{gi}}\right)}\\ \begin{array}{c}\stackrel{}{}\\ \\ \end{array}+h_V\cdot S_{ge}+{\epsilon }_g\cdot \sigma \cdot S_{ge}\left({\left(T_{g,j}^2\right)}^n+{\left(T_{amb,j}^2\right)}^n\right)\left(T_{g,j}^n+T_{amb,j}^n\right))\end{array}$

$g=T_{g,j}^{n-1}+\frac{\Delta t}{{\rho }_g{C}_g{S}_g}\left(h_V\cdot S_{ge}+{\epsilon }_g\cdot \sigma \cdot S_{ge}\left({\left(T_{g,j}^2\right)}^n+{\left(T_{amb,j}^2\right)}^n\right)\left(T_{g,j}^n+T_{amb,j}^n\right)\right)T_{amb,j}^n$

Which give: $\left[A\right]\left[T\right]=\left[B\right]$.

Or $\left[A\right]=\left[\begin{array}{ccc}{b}_1& c_1& 0\\ a_2& b_2& c_2\\ 0& a_3& b_3\end{array}\right]$; $\left[T\right]=\left[\begin{array}{c}{T}_{f,j}\\ T_{a,j}\\ T_{v,j}\end{array}\right]$; $\left[B\right]=\left[\begin{array}{c}d\\ e\\ g\end{array}\right]$.

The matrix $A$ of system being tridiagonal, we will use Gaussian pivot method for resolution of our problem.

Computation is carried out in two stages: triangularization and determination of unknowns.

Which give:

$T_{a,j}^n=\frac{b_1\left(e{b}_3-c_{2}g\right)-b_3{a}_{2}d}{b_1\left(b_2{b}_3-c_1{a}_2\right)-b_2{c}_2{a}_3}$

$T_{f,j}^n=\frac{d{b}_1\left(b_2{b}_3-c_1{a}_2\right)-d{b}_2{c}_2{a}_3-c_1{b}_1\left(e{b}_3-c_{2}g\right)+c_1{b}_3{a}_{2}d}{b_1^2\left(b_2{b}_3-c_1{a}_2\right)-b_1{b}_2{c}_2{a}_3}$

$T_{g,j}^n=\frac{g{b}_1\left(b_2{b}_3-c_1{a}_2\right)-g{b}_2{c}_2{a}_3-a_3{b}_1\left(e{b}_3-c_{2}g\right)+a_3{b}_3{a}_{2}d}{b_3{b}_1\left(b_2{b}_3-c_1{a}_2\right)-b_3{b}_2{c}_2{a}_3}$

Table 1 and Table 2 group together the geometric and optical characteristics of our cylindro-parabolic concentrator.

A computer program written in Matlab language has been developed to simulate outlet temperature of heat transfer fluid, absorber and glass of cylindro-parabolic solar concentrator.

In this part, we present results of numerical simulation, as well as the effect of some geometric and climatic parameters [12] on evolution of outlet temperature T_f.

The aim of our work was to model a parabolic trough concentrator, to numerically solve the resulting equation and to simulate the heating process of the heat transfer fluid.

Thus, a heat balance was established for the heat transfer fluid, the absorber and the glass. This allowed us to establish a system of equations whose resolution was done by the finite difference method. Then, a computer program was developed to simulate the temperatures of the heat transfer fluid, the absorber tube and the glass as a function of time and space.

The simulation of the fluid heating process was done in one-hour time steps, from 6 am to 6 pm. The results obtained show that the temperature difference between the inlet and outlet of the sensor is very large, thus making it possible to increase the power and efficiency of the parabolic trough concentrator.