EL DERECHO AL AGUA EN LA CONSTITUCIÓN BOLIVIANA:

As derived in Section 1.2, the focus in this work is placed on non-concentrating, low-temperature ST and photovoltaic module (PV) systems. A novel high concentration photovoltaic and thermal system (HCPVT) system [86–88] co-generating electricity and low-temperature heat (<100 °C) is also compared to traditional technologies. In the subsequent sections the static performance equations of the three solar systems are presented.

2.3.1.1 Solar irradiation on inclined surfaces

Global horizontal radiation The global solar radiation incident on a horizontal plane is composed

partly of direct and partly of diffuse radiation reflected from the ground, clouds and the atmosphere. As mentioned in the meteorological data section, a common measurement provided by weather stations around the globe is the global horizontal irradiation (GHI, gh, [W/m2]) as well as the direct normal irradiation (DNI, bn, [W/m2]) or beam radiation. The global horizontal irradiation (GHI) is derived from the diffuse and the direct radiation incident on a plane [128].

gh = dh+ bh

= dh+ bn· cos (θs) (2.1)

Where dh[W/m2] is the diffuse horizontal and bh[W/m2] is the direct horizontal radiation,θs[°] is the solar zenith angle indicated in Figure 2.1 (in grey) as the angle between the earth surface normal (zenith, z) and the sun.

Global radiation on inclined surface The global radiation present on an inclined surface, i, is

derived from the solar beam on the slope and the present diffuse radiation. The direct beam on an inclined surface is calculated by the product of the direct horizontal radiation and the cosine of the

Figure 2.1 – Angles of the sun towards earth normal and inclined surface.

incidence angle between beam and slope. Calculation of the diffuse component on a slope is not trivial since the diffuse sky radiation is anisotropic meaning that it is not uniformly distributed over the hemisphere. Perez et al. [21, 22] offer correlations for modeling the anisotropic component of the diffuse radiation. In order to keep the problem at reasonable complexity the isotropic diffuse model proposed by Lui and Jordan [129], reprinted in Duffie and Beckman [26], is used in this work. It is divided into three components: the beam, isotropic diffuse, and global radiation diffusely reflected from the ground (Eq. 2.15.1 Duffie and Beckman [26]).

gi = bn· cos (λis) | {z } bi + dh·µ 1 + cos(θi) 2 ¶ | {z } di + gh·ρ_g·µ 1 − cos( θ_i) 2 ¶ | {z } ggr,i = bi+ di+ ggr,i (2.2)

Where bi[W/m2] is the direct beam, di[W/m2] is the sky diffuse and ggr,i[W/m2] is the ground reflected diffuse radiation present on the surface i,θi[°] is the slope inclination angle. The ground

reflectivity,ρg[-], is fixed at 0.154 as given by Ineichen et al. [130].

Under the assumption that the solar angles are known, the incidence angle of the solar beam with respect to an inclined surface,λis[°], can be calculated (Eq. 1.6.3 Duffie and Beckman [26]) as follows.

cos (λis) = cos (θs) · cos(θi) + sin(θs) · sin(θi) · cos¡γ_s−γ_i¢

= sin (αs) · cos(θi) + cos(αs) · sin(θi) · cos

¡

γ_s−γ_i¢

(2.3)

Whereθs[°] is the solar zenith angle (vector),θi [°] surface inclination angle,γs [°] is the solar azimuth angle (vector), andγi[°] is the surface, i, azimuth angle.

2.3.1.2 Solar (non-concentrating) thermal system (ST): flat plate thermal collector (FP)

The efficiency of solar thermal collectors at steady state conditions is commonly described by a quadratic performance curve that depends on the operating temperature, TFPm= 0.5·¡TinFP+ ToutFP¢ [°C],

the incoming radiation intensity during that period, g_i,p [W/m2], the ambient temperature during period p, Ta,p[°C], the conversion factor,η0FP[-], and two experimental parameters accounting for

convection and radiation heat losses, a1FP[W/Km2] and aFP2 [W/K2m2].

The general conversion factor,η0[-], is defined by the absorber material, thickness, and heat transfer

fluid flow characteristics. The first experimental coefficient is usually related to the collector convec- tive heat losses, and the latter is influenced by the collector re-radiation losses. The temperature dependent efficiency Equation 2.4 is then written as follows (Duffie and Beckman [26] Eq. 6.17.7, [60]). ηFP_th,p=ηFP₀ − aFP₁ · TFP_m − Ta,p g_i,p − a FP 2 · gi,p· Ã TFP m− Ta,p g_i,p !2 ∀p ∈ P (2.4)

The formula accounts for a reduction in efficiency for operating temperatures higher than the ambient (due to thermal losses) and for reduction in efficiency due to reduced radiation intensities at normal incidence. However, since panels are installed at a fixed position, an incidence angle modifier, fIAM[-], is introduced in order to account for optical losses related to the angle of the

incident radiation. By definition, it is set to one at 0 ° incidence and is usually provided at 50 °. In order to find other data points a cosine law is traditionally suggested (e.g. 6.17.10 [26] also in ASHRAE 93-2003), which however cannot be evaluated for angles close to 90 °. Therefore an Ambrosetti-type equation [131] is used here which can be evaluated up to 90 °.

fIAMFP (λ) = 1 − tana

µλ 2 ¶

(2.5)

Where the coefficient a is usually derived from known data at a certain inclination (e.g. 50 °); For beam radiation, the incidence angle, λ [°], is equivalent to the solar angle of incidence on the slope,λis[°]. For the diffuse and ground reflected component, the incidence angle is found from Equation 2.6 based on the slope inclination angle,θFPi [°], (Duffie and Beckman [26], Figure 5.4.1, Eq.

5.4.1, 5.4.2). λid = 90 − 0.5788 ·θFPi + 0.002693 · ¡ θFP_i ¢2 λigr = 59.7 − 0.1388 ·θFPi + 0.001497 ·¡θFPi ¢ 2 (2.6)

With this, the time-dependent thermal energy production, ˙QFPp [kW], of the solar flat plate collectors

FP can be formulated.

˙

QFP_p =η_thFP_,p· fFP_field·

³

bi,p· fFP_IAM,ib,p+ di,p· fFP_IAM,id+ ggr,i,p· fFPIAM,ir

´

· AFP ∀p ∈ P (2.7)

Where each type of radiation (direct beam, bp,i[W/m2], sky diffuse, dp,i[W/m2], ground reflected

diffuse, g_p,gr,i[W/m2]) is multiplied with the respective incidence angle modifier from Equation 2.5 and Equation 2.6, the collector area, AFP[m2], the thermal field loss factor, ffieldFP [-], and the efficiency

from Equation 2.4.

2.3.1.3 Photovoltaic module (PV)

As mentioned before, the two main parameters influencing the PV performance is the cell tempera- ture and the irradiation intensity. The cell temperature can be determined by correlations found in the literature (Eq. 23.3.4 Duffie and Beckman [26]).

Tc,pPV= Ta,p+ g_i,p g_NOCT· 9.5 5.7 + 3.8 · va,p · " 1 −η PV el (τα) # ·¡TNOCTPV − Ta,NOCT¢ ∀p ∈ P (2.8)

where Ta,p [°C] is the (time dependent) ambient temperature during period p, va,p [m/s] is the

ambient wind speed during period p, and g_i,pis the global incident radiation on an inclined sur- face during period p (see 2.3.1.1). The irradiation, gNOCT, of 800 W/m2and ambient temperature ,

Ta,NOCT, of 20 °C during nominal cell operating temperature (NOCT) conditions are pre-set parame-

ters. The nominal cell operating temperature, TNOCTPV [°C], is a parameter measured and provided

by the module manufacturer. The ambient temperature and wind speed are provided from the meteorological data described in Section 3.2.3.2.

A factor accounting for the influence of the incident radiation intensity is calculated by linear interpolation between the standard testing conditions (STC), defined at a radiation, gSTC, of 1000

W/m2and a cell temperature, TSTC, of 25 °C, and the certified indication at g200200 W/m2.

fg,pPV= f200+ ³ g_i,p− g200 ´ · 1 − f200 g_STC− g200 ∀p ∈ P (2.9)

23.2.16 [26]).

˙

EPVp =η_elPV· fPV_gen· fg,pPV·£1 − fPVT ·¡Tc,pPV− TSTC¢¤ · gi,p· APV ∀p ∈ P (2.10)

where APV[m2] is the module area,η_elPV [-] is the module reference electrical efficiency, fTPV[-] is the

temperature reduction factor, fgenPV[-] is the generator electrical conversion efficiency factor. 2.3.1.4 Photovoltaic and thermal (HCPVT)

The high concentration photovoltaic and thermal system (HCPVT) system is novel technology under development by Airlight Energy Holding SA [86, 87, 132]. It consists of a multi-facet concentrating dish (reaching concentrations up to 2000).

High concentration devices only convert direct beam radiation. In the focal point, a multi-cell receiver is placed, which is equipped with PV cells, efficiently cooled by a micro-channel cooling system. A secondary concentrator is placed around the receiver which also needs to be cooled. Thermal energy is harvested from the micro-channel (primary cooling), and the secondary con- centrator (secondary cooling). Due to the high concentration ratio, and based on the results from dynamic modeling [112], the efficiency is assumed to be independent of the incident radiation intensity and due to the two axis tracking the angle of incidence is always zero. The time-dependent electricity, ˙EHCPVT_p [kW], and thermal energy production, ˙QHCPVT_p [kW], is formulated according to the following. ˙ EHCPVT_p = fHCPVTgen ·ηHCPVTel · bn,p· A HCPVT ∀p ∈ P ˙ QHCPVTp = QHCPVTprim,p ¯ ¯ ¯ THCPVT_out,prim THCPVT in,prim + QHCPVTsec,p ¯ ¯ ¯ THCPVTout,sec THCPVT out,prim

= fHCPVT_field ·³ηHCPVT_{th, prim}+ηHCPVT_{th, sec}

´

· bn,p· AHCPVT ∀p ∈ P

(2.11)

where bp,nis the direct beam normal radiation in period p andηelHCPVTis the PV electrical conversion

efficiency, and fHCPVT_field is the thermal field loss factor. The primary efficiency,ηHCPVT_th,prim [-], stems

from the PV cell cooling while the secondary efficiency,ηHCPVT_th,sec [-], is derived from the cooling of

the secondary concentrators positioned immediately prior to the receiver. The PV cell cooling is constrained by the cell temperature which should not exceed 100°C; and assuming a minimum temperature difference in the heat exchanger ofΔTminof 5K, the cooling stream cannot reach

temperatures higher than 95°C. The secondary cooling, contrary to the restriction imposed for the primary PV cooling, can reach any temperature. Therefore, two thermal streams are produced which are between the three temperatures THCPVT_in,prim, THCPVT_out,prim, and THCPVT_out,sec.