The following section was carried out using the Fourier law, related to heat conduction through a material, given by eq. (1)

where T is the temperature (K), т is the instantaneous time (s), kp is the conduction coefficient of the absorber (W/mK), a is its thermal diffusivity (m2/s), p is its density (kg/m3), cp is its specific heat value (J/kg K), and qv is the internal heat generation in the absorber (W/m3).

1.1. — Mathematical model

To carry out the steady state study, the absorber surface was divided, using a mesh, in multiple similar units, all of them with same dimensions, x and y, so the problem could be considered discrete. Then, Eq.(1) was approximated by finite differences. For each (i, j) pair, derivatives from x (as well as from y) depend on the increment Ax [Ketkar, 1999]:

A sample network is shown in Fig. 1. Because pipes distribution was considered to be the same over the absorber plate, only a portion of it will be considered. The plate presented 15 equal tubes, symmetrical to the y-axis, dividing it in 30 equal zones that will present the same results. The boundary line is considered adiabatic, due to the presence of an isolator. Boundary losses are around 3% of the captured heat [Alaiz, 1981], so they could be neglected. Flow distributions estimation along the risers is the same, unless the regime is laminar [Weitbrecht et al., 2002].

After using eq. (1) and (2) in the previously mentioned mesh (see Fig. 1), it can be seen that there is a relationship between temperatures obtained form energy balance in each point (see eq. (3)). This expression will experience some changes in the boundary line, due to the previously mentioned adiabatic considerations.

Tj-ij + Tj+1,] + Tj, ]_1 + Tj, ],1 — 4Tj, j =- q AxAy

Internal heat generation, qv, includes solar radiation and heat losses by re-radiation, conduction and convection through the isolators and the crystal. Also, it is important to notice the losses of available heat that is transferred to the water that is going to be heated:

Then, unitary heat release by radiation q1 matches S (W/m3):

q1 = S = IT (та)! e (5)

where It is the incident global radiation over the inclined absorber (W/m2), та is the product transmittance-absorbance of the crystal-absorber, and e is the thickness of the plate (m). On the other hand, the unitary losses of the plate, q2, are obtained from the global losses coefficient, Ul (W/m2K), and finally, the temperature difference is calculated between the point and the environment.

• (T. — T )

?2 = Ul 1 con UL = Ub + Ut (6)

e

where Ub represents the losses behind the collector, due to the heat conduction through the isolator and the casing, and the convection to the atmosphere. Its values depend on the materials properties and wind velocity. In this way, Ut is related to front losses, and is originated from solar radiation and re-radiation, system inclination and orientation. Once calculated, both coefficients will be considered as constants in all the network points (no shadow will be considered to take place). The ambient temperature is Ta.

Finally, q3 provides the available heat that passes through the working fluid in that precise point. It can be calculated by means of the global transmission coefficient, Uc, which takes into account the heat convection between the internal wall of the pipe and the water. This available heat could be calculated as shown in eq. (7).

where h (W/m2 K) is the film coefficient (convection coefficient), R is the internal radius of the pipe (m), eq is the pipe thickness, and Kuc is a proportional coefficient depending on the pipe geometry, plate weld and position. The sub index m refers to the limit over the pipe. According to this, eq. (3) will appear as follows (eq. 8.a does not consider heat transfer to the fluid, while eq. 8.b does).

general form (8.a):

The sub index m+1 refers to the fluid at each point, related to the dimension y. Let consider m the number of distributed points along the x-axis, and n the number of points distributed over the y-axis. For each of them, there will be an equation, as eq. (8), depending on their position. This equation could be modified if adiabatic boundary is considered. Then, m x n equations will appear, for (m+1) x n unknown quantities. So, to solve this system m additional equations are needed.

These additional equations can be obtained after calculating the heat balance of the working fluid in the y direction. The working fluid will receive a certain quantity of heat from point y=j to point y=j+1, which depends on the medium temperatures of riser, Tp, and fluid, Tf, between the extremes j and j+1. The fluid flow mass, mf, the specific heat value of

water, Of, and the increment of temperature of water, AT, will also be considered (see eq. (9) and (10)).

Film coefficient, h, is calculated by means of an initial value ho, obtained from empirical relations of heat transfer theory, multiplied by a proportionality coefficient, Kh, which depends on the same factors as KUc.

Equations (8) and (11) constitute a set of (m+1) x n linear equations with (m+1) x n unknown data. In this case, however, the input fluid flow, Tff, is known, thus facilitating the equations set resolution. Moreover, this temperature serves to stabilize the system calculus, thus promoting the finding of feasible results.