**4. Solar panel thermal model**

What said above clearly highlights the need of a thermal model of the solar panel taking into consideration the heat exchange on both sides of it in case of a deployable one; or usually considering the rear side as adiabatic in case of the panel is body mounted. The panel is considered as rigid, with honeycomb structure on which the solar cells are applied; the following table reports the components recognizable in solar panel cross-section:


Table 1. Solar panel composition

The panel temperature is computed taking into account the direct sun radiation, the albedo radiation, the irradiation to deep space, and irradiation between the earth surface and the panel itself. The sun illumination is variable during the year and considering only missions around the earth it may range between 1315.0 (summer solstice) and 1426 W/m2 (winter solstice), while the albedo of the earth surface is about 30% of the incident sun illumination. The panel exchanges heat with the deep space and this is seen as a black body at 3°K, as well as the earth irradiates as a black body at 250°K. The following simplifying assumptions can been made; a deployed solar panel does not exchange heat with the outer surfaces of the satellite body, a body mounted solar panel is adiabatically isolated from the rest of the satellite body and finally the panel surface temperature is considered as uniform. The conduction across the panel also plays an important role, and it has to be taken into account in case of a deployed solar panel. At a first glance, the in-plane conductivity may be neglected, this because under the hypothesis of uniform temperature over the panel, the heat exchange between adjacent cells is basically zero.

The thermal equilibrium is computed by solving the differential equation which takes into account the different heat exchange modalities.

$$\mathbf{C} \cdot \frac{dT}{dt} = Q\_{Rad} + Q\_{Alb} + Q\_{Earth} + Q\_{Space} + Q\_{cond} \tag{5}$$

Where:

$$Q\_{Rad} = \alpha \left(1 - \eta\right) \cos \theta \cdot f \tag{6}$$

What said above clearly highlights the need of a thermal model of the solar panel taking into consideration the heat exchange on both sides of it in case of a deployable one; or usually considering the rear side as adiabatic in case of the panel is body mounted. The panel is considered as rigid, with honeycomb structure on which the solar cells are applied; the

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The panel temperature is computed taking into account the direct sun radiation, the albedo radiation, the irradiation to deep space, and irradiation between the earth surface and the panel itself. The sun illumination is variable during the year and considering only missions around the earth it may range between 1315.0 (summer solstice) and 1426 W/m2 (winter solstice), while the albedo of the earth surface is about 30% of the incident sun illumination. The panel exchanges heat with the deep space and this is seen as a black body at 3°K, as well as the earth irradiates as a black body at 250°K. The following simplifying assumptions can been made; a deployed solar panel does not exchange heat with the outer surfaces of the satellite body, a body mounted solar panel is adiabatically isolated from the rest of the satellite body and finally the panel surface temperature is considered as uniform. The conduction across the panel also plays an important role, and it has to be taken into account in case of a deployed solar panel. At a first glance, the in-plane conductivity may be neglected, this because under the hypothesis of uniform temperature over the panel, the

The thermal equilibrium is computed by solving the differential equation which takes into

*dT C QQQ Q Q dt*

*Q J Rad* 

 1 cos 

*Rad Alb Earth Space cond*

 

(5)

(6)

Thickness, μm

following table reports the components recognizable in solar panel cross-section:

Components from front to rear side

**4. Solar panel thermal model** 

Table 1. Solar panel composition

heat exchange between adjacent cells is basically zero.

account the different heat exchange modalities.

Where:

It is the contribution of the direct sun radiation J that is not converted into electrical power by the photovoltaic cell;

$$Q\_{\rm alb} = \alpha \left(1 - \eta \right) \cdot F \cdot A \boldsymbol{l} \cdot F\_{1-2} \cdot \boldsymbol{l} \tag{7}$$

It is the contribution of the albedo radiation;

$$Q\_{Earth} = \sigma \cdot \varepsilon \cdot F\_{12} \cdot \left(T\_E^4 - T^4\right) \tag{8}$$

It is the heat exchanged with the Earth surface.

$$Q\_{\text{Space}} = \sigma \cdot \varepsilon \cdot \left(1 - F\_{12}\right) \cdot T^4 \tag{9}$$

It is the heat released to the deep space.

$$Q\_{cond} = \frac{\left(T\_{front} - T\_{rear}\right)}{\Delta \mathbf{x}} \cdot k \tag{10}$$

Is the heat transmitted by conduction between the front and rear faces of the panel at Tfront and Trear temperature respectively.

The parameters appearing in these equations have the following meanings:

C = thermal capacitance of the panel per unit area, main contribution is provided the honeycomb structure;

= solar cell absorptivity;

= solar cell emissivity;

Al = albedo coefficient, about 0.3 for earth;

F = albedo visibility factor;

F12 = View factor between radiating surface and planet

= Stephan-Boltzmann, constant: 5.67210-8 W/(m2°K4);

TE = Black body equivalent temperature of the earth;

 = incidence angle of the sunlight on the panel; 

k = panel transverse thermal conductivity;

Δx =panel thickness;

The radiating view factor of a flat surface with respect to the Earth surface is function of the altitude h, earth radium R and the angle λ between the nadir and the normal to the panel. The albedo view factor is computed according to the following formulas:

$$F\_{alb} = \left[1 - \cos\left(\beta\_{app}\right)\right] \cdot \cos\left(\beta\right) \tag{11}$$

Where β is the angle between the nadir and the earth-sun direction:

$$\mathcal{B}\_{\text{app}} = \arcsin\left(\frac{R\_{\oplus}}{h + R\_{\oplus}}\right) \tag{12}$$

The integration in the time domain of the equation (5) gives the actual temperature of the panel along the propagation of the orbit in eclipse and sunlight, taking into account the orientation of the panel itself with respect to the earth and the sun. The thermal model

Architectural Design Criteria for Spacecraft Solar Arrays 169

solids which are of interest to the solar array designer are ionisation and atomic

Ionisation occurs when orbital electrons are removed from an atom or molecule in gases, liquids, or solids. The measure of the intensity of ionising radiation is the roentgen. The measure of the absorbed dose in any material of interest is usually defined in terms of absorbed energy per unit mass. The accepted unit of absorbed dose is the rad (100 erg/g or 0.01 J/kg). For electrons, the absorbed dose may be computed from the incident fluence *Φ* (in cm-2) as: Dose (rad) = 1.6x10-8 *dE*/*dx Φ*, where *dE*/*dx* (in MeV cm2 g-1) is the electron stopping power in the material of interest. In this manner, the effects of an exposure to fluxes of trapped electrons of various energies in space can be reduced to an absorbed dose. By the concept of absorbed dose, various radiation exposures can be reduced to absorbed dose units which reflect the degree of ionisation damage in the material of interest. This concept can be applied to electron, gamma, and X-ray radiation of all energies. Several ionisation related effects may degrade the solar cell assemblies. The reduction of

transmittance in solar cell cover glasses is an important effect of ionising radiation.

irradiation can be described as shown for *I*sc in the following case:

which are in some way related to the minority carrier diffusion length.

And for the maximum power;

The basis for solar cells damage is the displacement of semiconductor atoms from their lattice sites by fast particles in the crystalline absorber. The displaced atoms and their associated vacancies after various processes form stable defects producing changes in the equilibrium of carrier concentrations and in the minority carrier lifetime. Such displacements require a certain minimum energy similar to that of other atomic movements. Seitz and Koehler [1956] estimated the displacement energy is roughly four times the sublimation energy. Electron threshold energies up to 145 keV have been reported. Particles below this threshold energy cannot produce displacement damage, therefore the space environment energy spectra are cut off below this value. The basic solar cell equations (1) may be used to describe the changes which occur during irradiation. This method would require data regarding the changes in the light generated current, series resistance, shunt resistance, but most investigations have not reported enough data to determine the variations in the above parameters. The usual practice is then to reduce the experimental data in terms of changes in the cell short circuit current (*I*sc), open circuit voltage (*V*oc), and maximum power (*P*max). The variation of common solar cell output parameters during

 *I*sc = *I*sc0 - *C* log (1 + *Φ* / *Φ*x) (13) Where *Φ*x represents the radiation fluence at which *I*sc starts to change to a linear function of the logarithm of the fluence. The constant *C* represents the decrease in *I*sc per decade in radiation fluence in the logarithmic region. In a similar way, for the Voc it can be written;

 *V*oc = *V*oc0 - *C'* log (1 + *Φ* / *Φ*x). (14)

 *P*max = *P*max0 - *C*'' log (1 + *Φ*/*Φ*x). (15) In the space environment a wide range of electron and proton energies is present; therefore some method for describing the effects of various types of radiation is needed in order to get a radiation environment which can be reproduced in laboratory. It is possible to determine an equivalent damage due to irradiation based upon the changes in solar cell parameters

displacement.

exposed so far is sufficient for the design of a solar array for space application at system level.

Fig. 7. View Factors F12 as function of H=h/R, parameter:

Fig. 8. Albedo Visibility factor F as function of h for different β value
