Hot Worst-Case

Focusing the analysis on the range where the maximum temperature is obtained for the instantaneous heat flux, a correlation between the low characteristic time system and the heat flux can be observed as shown in Figure 5.21a. This fact makes it possible to identify the environmental conditions that would lead to the maximum temperatures, just focusing on the maximum peak of the heat flux variations. However, if the system has a higher characteristic time (Figure 5.21b), the maximum peak of the heat flux may or may not correspond to the maximum

5. Low Earth Orbit thermal environment for space thermal design 159

Figure 5.19: Location of the maximum temperature point for a satellite with a characteristic time of 112 s together with its heat loads.

temperature response. In order to identify the maximum contribution to the temperature system response, the exponential weighted moving average of the intantaneous heat flux variation corresponding to a window, w, of value t_c has been obtained based on the following equation:

y(t) = x(t) + (1−α)x(t−1) + (1−α)²x(t−2) +...

1 + (1−α) + (1−α)²+...

α = 2

w+ 1

. (5.14)

As shown in Figure 5.21b, the result is a heat flux profile where its maximum value corresponds to the maximum peak temperature.

Once the peaks have been identified, it is necessary to establish an equivalence between the maximum temperature response to the instantaneous heat flux and the temperature response of the system to a tuned standard peak input that would be used for the extreme case analysis. A step excitation of a magnitude equal to the instantaneous peak corresponding to the maximum temperature has been selected in order to obtain the same response on the system. Knowing the ∆T generated in the system, the required duration of the step function for reaching this value is calculated as shown in Figure 5.22 where the characteristic time response has been also represented.

Figure 5.20: Location of the maximum temperature point for a satellite with a characteristic time of 896 s together with its heat loads.

(a) (b)

Figure 5.21: Temperature response of the systems with (a)t_c = 87 s and (b) t_c = 1733 s, to the resid heat load of albedo and OLR where SZA is lower than 60^◦.

In the case under study, each system reaches its maximum temperature at different environmental conditions. This is because the different peak widths.

Systems with lower characteristic times would reach their maximum for shorter but greater peaks. In contrast, systems with higher characteristic times would need more time to increase their temperature and their maximum would correspond to wider peaks. Values for the albedo, OLR and the duration of the tuned peak are shown in Table 5.4.

Once the residuals of the series are treated and the step function that would lead

5. Low Earth Orbit thermal environment for space thermal design 161

Figure 5.22: Variation of temperature with time of the step function response defined to obtain an equivalent ∆T for the Hot Case.

Table 5.4: Albedo and OLR peak step function characteristics for the Hot Case.

Albedo [-] OLR [W/m²/K]

Width [s]

t_c = 87 s 0.19 74 196

t_c = 1733 s 0.20 66 359

to the maximum temperature response extracted, it is possible to build the albedo coefficient and OLR profile that would be used as the hottest extreme environmental conditions for the analysed orbit. This profile is composed of the following elements:

• Maximum values of the trends. The partial correlation of albedo and OLR, as shown in Figure 5.23, should be considered depending on the thermo-optical properties of the system.

• Seasonal component of the albedo and OLR series.

• Equivalent tuned peak step function at the orbital point where maximum or minimum temperatures are found.

The resultant profiles for both series and different characteristic times are shown in Figure 5.24. The step function can be observed centred in the point where the SZA is minimum, and the associated planetary heat load is maximum. Values for the albedo and OLR step functions at t = 5200 s correspond to the peak of the total planetary heat flux that maximizes the temperature of the system only considering the planetary heat flux. If the thermal behaviour previously shown in Figure 5.20 were considered, the peak location of the 1733 s time constant system would be

Figure 5.23: Albedo and OLR trend components value distribution.

the end of the illumination period. However, just to show the characteristics of the proposed methodology, only the planetary heat load is to be considered from now on.

Figure 5.24: Albedo and OLR worst hot case profiles.

From the analysis of the time response of the system to these albedo and OLR profiles, it is possible to obtain the behaviour of the system with the thermal environment variations when reaching the cyclic solution starting from a given initial condition. Results shown in Figure 5.25 only considers the external heat flux corresponding to the Earth. The effect of direct solar radiation has not been computed.

Comparing the temperatures obtained for the instantaneous heat flux with regard to the computed temperatures using the developed profiles, some conclusions can be drawn:

5. Low Earth Orbit thermal environment for space thermal design 163

• The proposed profiles get the temperature response to overpass the real behaviour providing a maximum temperature above the real maximum temperature.

• The variable OLR and albedo coefficient profiles allow us to obtain a temper- ature response that follows the orbital trend of the real temperatures.

In order to evaluate the advantages or disadvantages of this new methodology, it is necessary to compare the results obtained with the existing criteria for defining the worst-case thermal environment parameters. They are typically defined based on the criteria established by NASA in 1994 [14] using the ERBE data from ERBS and NOAA satellites. Although this methodology is being successfully and widely used in space missions, there are some elements which could limit its applicability to some systems.

Table 5.5: NASA provided extreme worst hot cases.

Average Time

16 s 896 s

Albedo [-] OLR [W/m²] Albedo [-] OLR [W/m²]

Alb 0.50 180 0.35 202

Comb 0.32 263 0.28 259

OLR 0.22 332 0.20 294

The temperature response of both, the low and high characteristic time systems, have been represented in Figure 5.25a and Figure 5.25b, respectively. NASA worst- cases defined in Table 5.5 have also been represented in both figures. The Simple Thermal Environmental Model [17] selects as the hot worst-case for each system the pair of values (albedo and OLR) which maximizes the temperature response for given thermo-optical properties. Three potential cases are provided for the thermal engineer in order to check which of them correspond to the worst-case. These values are established based on statistics. Real data retrieved from ERBE is processed by averaging the albedo and OLR values across the satellite trajectory. The appropriate average time to be used should be between a quarter of the characteristic time and the characteristic time of the analysed system [17].

This methodology does not aim at replacing the NASA developed criteria [14].

As it is shown, NASA worst-case values allow for the definition of the maximum higher limit of the temperature response. However, when the system tends towards small values of characteristic times, it sometimes does not appropriately represent the temperature variations during an orbital period, which could be of great

(a) (b)

Figure 5.25: Comparison between the real response, the hot extreme case, and NASA proposed worst-cases for a)t_c = 87 s and b)t_c = 1733 s.

interest for the system design. If the characteristic time of the system increases, its predicted response amplitude becomes more similar to the real one, but the maximum temperature reached could be overestimated.