Contents
4.2 Float Phase Analyses
4.2.3 Analytical analyses
the same criteria as in [15]. They represent points with an equal probability of finding points with a higher albedo and OLR for the hot case and vice versa for the cold case.
These points cause a higher or lower temperature on the system for the hot and cold case respectively. The 2D distributions for the lowest and highest SZA in the region and epoch of study are shown in Figure 4.10a and Figure 4.10b respectively.
(a) (b)
Figure 4.10: Albedo to OLR correlation for a SZA of (a) 45.5° and (b) 87.7°.
4. Planetary thermal environment for stratospheric balloon thermal design 59
Figure 4.11: Sketch of the analytical model used to estimate main radiative fluxes.
distance between the gondola and balloon is dG−B, 60 m and the altitude of the balloon is hB, 37.5 km, values of reference that are taken from SUNRISE I flight.
The balloon skin has been modelled as a three-layer, zero pressure balloon of 34 MCF (0.96 million m3, and 65 m radius at float altitude). It has been modelled as a semi-transparent surface with equivalent thermo-optical properties taken from Ref. [65]. The estimation of the successive reflections inside the balloon have been calculated following Ref. [66], by means of the effective reflectance defined as ρ′B =ρB+ρ2B+ρ3B, for each band, solar (superscript S) and infrared (superscript IR). The transmittance τ is calculated from α+ρ+τ = 1.
To identify the two (a, OLR) pairs that give the worst hot and cold temperatures for each ratio α/ε, the energy balance equation in steady-state conditions has been written for each thermal node, balloon B, and gondola G:
Q˙neti = 0, (4.1)
where ˙Qneti is the net heat transfer onto the nodei, which depends on its temperature, and on the environmental thermal loads, which in turn depend on its relative position with respect to the Sun and Earth.
Based on the relative sizes of balloon and gondola, the balloon surface is almost 4 orders of magnitude higher than the gondola surface. For this reason, the heat radiation received by the balloon surface from the gondola is negligible when compared to other radiation exchanges, whereas the heat received by the gondola from the balloon has to be taken into account. Both surfaces have a view of the Earth, its infrared radiation is calculated from the OLR value, and
view of the cold space, considered at this altitude to be at 3 K, and therefore, negligible in terms of infrared radiation.
The model includes only radiative thermal interactions. Convection has been considered negligible since the air at the floating altitude is very rarefied, with a pressure of only some 300 Pa. Heat conduction between balloon and gondola has also been neglected.
Based on the previous assumptions, the balloon film temperature can be obtained from the energy balance equation, which can be written as:
Q˙SB + ˙QaB + ˙QIREB+ ˙QIRBint − Q˙IRB = 0. (4.2) The terms in Equation (4.2) are calculated in the same way as in [66]:
Solar radiation absorbed by the balloon film
Q˙SB =αBABpGSh1 +τBS1 +ρSB′i, (4.3) where ABp is the area of the balloon projected in the Sun’s direction, GS is the solar constant, and the thermo-optical properties, absorptanceα, transmittance τ, and effective reflectance ρ′, are those corresponding to short wavelengths.
Albedo radiation absorbed by the balloon Q˙aB = 1
4αBaFBEABGScosθh1 +τBS1 +ρSB′i, (4.4) where FBE is the view factor between two spheres, from the balloon to Earth, and θ, the SZA.
Earth infrared radiation absorbed by the balloon
Q˙IREB =εBABFBEqEh1 +τBIR1 +ρIRB ′i, (4.5) where qE is the radiative infrared flux emitted by the Earth (OLR).
Balloon inner infrared radiation absorbed by itself
Q˙IRBint =ε2BABσTB41 +ρIRB ′. (4.6) This term is present because the balloon film is semi-transparent to infrared radiation.
Therefore, not all the radiation emitted by the internal surface of the balloon is absorbed by itself. To account properly for this interaction, both the internal radiation emitted and that absorbed by the balloon are considered in the calculations.
Balloon film emitted infrared radiation
Q˙IRB = 2εBABσTB4, (4.7)
4. Planetary thermal environment for stratospheric balloon thermal design 61
where the factor 2 accounts for both sides of the balloon surface.
Successive reflections on the gondola, the balloon or the Earth contribute to heat fluxes in an amount smaller than 0.03 %. For this reason they have been neglected.
Substituting Equation (4.3) to (4.7) in Equation (4.2), it is possible to obtain the balloon film temperature directly as a function of the environmental conditions:
solar constant GS, albedo a and Earth infrared flux OLR,
TB =
Q˙SB+ ˙QaB+ ˙QIREB εBσAB[2−εB(1 +ρIRB ′)]
!1/4
. (4.8)
On the other hand, the gondola-payload thermal balance equation, can be written as:
Q˙SG+ ˙QaG+ ˙QIREG+ ˙QIRBG−Q˙IRG = 0. (4.9) The terms in Equation 4.9 are calculated in a similar way as done for the balloon.
Solar radiation absorbed by the gondola Q˙SG =αGAGpGS+ 1
4αGAGFGBGS
τBS21 +ρSB′, (4.10) where AGp is the area of the gondola projected in the Sun direction. The second term accounts for the sun radiation passing through the balloon surface that finally reaches the gondola.
Albedo radiation absorbed by the gondola Q˙aG = 1
4αGaGSAGsinθ(FGE +FGBFBE[ρSB+τBS2ρSB′]), (4.11) where both direct albedo and the albedo reflected on the balloon surface have been considered.
Earth infrared radiation absorbed by the gondola
Q˙IREG=εGAGFGEqE. (4.12) Balloon infrared radiation absorbed by the gondola
Q˙IRBG =εGεBABFBGσTB4h1 +τBIR1 +ρIRB ′i. (4.13) Gondola emitted infrared radiation
Q˙IRG =εGAGσTG4. (4.14)
By substituting Equation 4.10 to 4.14 in Equation 4.9, it is possible to obtain the temperature of the gondola as a function of the environmental conditions, the ratio α/ε and the balloon temperature, as follows,
TG= Q˙SG+ ˙QaG+ ˙QIREG+ ˙QIRBG εGσAG
!1/4
. (4.15)
The results presented here correspond to a value of the solar constant GS = 1370 W/m2 and a Solar Zenith Angle of 65°. For a given mission, a further study would be necessary to identify the worst SZA for each case hot and cold.
For each value of the ratioα/εof the gondola, its temperature has been obtained along the regression curves obtained in Section 4.2.2, and shown in Figure 4.8.
Firstly, the maximum temperature of the gondola with a fixed ratio α/ε= 0.26, representative of SUNRISE white paint is shown in Figure 4.12. In this graph, the relevance of choosing the adequate pair (a, OLR) can be observed. For a given value ofα/ε, temperatures can vary up to 10 K (7 K in the case of example shown), even considering only the (a, OLR) pairs leading to the potential worst hot cases. As can be seen, in this case, the pair (a, OLR) that gives the maximum temperature is the first pair in the regression line, that is,a = 0.12 and OLR = 290 W/m2.
Then, following the same procedure, the values of pairs (a, OLR) leading to the maximum and minimum temperatures, respectively, have been identified for each value of the ratio α/ε of the gondola.
Figure 4.12: Temperature of the gondola for every thermal environment defined by the hot case regression curve (α/ε= 0.26).
The value of the pair leading to the maximum temperature is not always the first pair of the regression line, and it moves towards higher albedo values as the ratio
4. Planetary thermal environment for stratospheric balloon thermal design 63
α/ε increases. The results obtained, that is, the pairs (a, OLR), that maximize the temperature of the instrument for each ratio α/ε are represented in Figure 4.13. It can be observed that there is a critical value of α/ε (α/ε = 0.5 for the case under study) where there is a sudden change in the conditions leading to the maximum temperature. Below that α/ε critical value, the worst hot case is driven by the OLR, whereas above it, the worst hot case is driven by the albedo coefficient a.
This is a quite relevant result that has to be considered when selecting the extreme environmental parameters: the worst environmental parameters cannot be selected without taking into account the ratio α/ε of the system under study. Furthermore, the system behaves in a similar way in cold conditions, as it is explained below.
This provides an additional critical value to consider. Therefore, in order to have a robust design, it would be advisable either to choose a value of α/ε far enough from those critical values or to carry out a study in depth around such points.
Figure 4.13: OLR and albedo coefficient maximizing gondola steady-state temperature as a function of the ratio α/ε.
Following the same reasoning and procedure with the potential minimum regression curve leading to worst cold temperatures, the results show similar behaviour, as can be seen in Figure 4.14. The critical value of α/ε, 1.0 in this case, is not the same as the one found for the hot case.
This parametric sweep has been carried out for an analytical model based on a sphere where the influence of the relative position of the considered instrument cannot be analyzed. Sometimes, the SZA that maximizes the temperature of an instrument on board a satellite or a balloon-borne telescope is not the minimum that can be found during the flight. This happens due to the radiation received as a function of its relative position with respect to the Earth and the balloon film. In
(a) (b)
Figure 4.14: (a) Temperature of the gondola for every thermal environment defined by the cold case regression curve (α/ε= 0.26). (b) OLR and albedo coefficient minimizing gondola steady-state temperature as a function of the ratioα/ε.
order to determine also the SZA that corresponds to the hottest case, and also to define an analytical model which simulates the behavior of the analyzed instrument, a more complex analytical model could be analysed or more complex analysis could be performed with a dedicated software such as ESATAN-TMS.