Contents
4.3 Ascent Phase Analyses
4.3.2 Analytical thermal model
Solar irradiance
When studying the solar irradiance reaching different layers of the atmosphere, the solar spectrum and the atmosphere opacity to some wavelengths due to its composition explain its dependence with the altitude and the SZA since "the amount of atmosphere" to pass through decreases with it.
For altitudes above the TOA, it can be considered that the solar irradiance is the solar constant, GS, and is not affected by the atmosphere. Its value varies through the year due to the variable distance between the Earth and the Sun.
Moreover, the solar cycle modifies its value, but the variations are below 1%, so they will be considered negligible. For altitudes below the TOA, CERES data has been statistically treated obtaining the mean values for each altitude depending on the SZA, as shown in Figure 4.33.
Figure 4.33: Mean solar irradiance heatmap.
Sky temperature
As has been shown, the sky temperature, which has been derived from the incoming longwave flux, is strongly dependent on the altitude but not on the SZA. Computing the mean values for each altitude, the heatmap shown in Fig- ure 4.34 is obtained.
At the top of the atmosphere and in higher altitudes, the sky temperature could be considered as the deep space equivalent temperature since there is no significant amount of atmosphere above it. Values of 2.7 K can be used for thermal calculations.
4. Planetary thermal environment for stratospheric balloon thermal design 87
Figure 4.34: Mean sky temperature heatmap.
are mainly centred on what happen to the balloon film, without focusing on the payloads. In order to see the influence of the different parameters involved, the mathematical approach of a generic node of a payload is considered here.
The payload geometrical model should represent not only its thermo-optical properties but also the relative position of the subsystem, structure or instrument object of study with respect to other nearby elements, the Earth and the Sun. This is why the payload is represented by a single node tilted plate with an area Ai.
Figure 4.35: Sketch of the angles pitch Ω and yaw φwhich define the relative position of the plate.
The relative position of such a node has been defined by the two rotation angles shown in Figure 4.35, where Ω = 0 corresponds to the plate in vertical position with respect to the Earth and φ= 0 corresponds to the plate’s face 1 normal pointing to the Sun’s rays projection on the horizontal plane. Only the thermal loads on face 1 are going to be analysed as face 2 is considered to be insulated.
The normal vector to the plate is defined as a function of Ω and φ in the reference frame shown in Figure 4.35 by the rotation matrix,
n=
cos Ω cosφ(t) cos Ω sinφ(t)
−sin Ω
. (4.19)
Once the normal vector to the surface in the reference frame is known, the angle γ with respect to the Sun direction can be obtained with the dot product of both vectors. The node thermal balance equation, can be written as:
Q˙S,i+ ˙Qa,i+ ˙QIRE,i+QIRSky,i+ ˙QIRRad,i+ ˙QIRi + ˙QDis,i+ ˙QCond,i+ ˙QConv,i =micidTi
dt . (4.20) The terms in Equation 4.20 are explained as follows.
Absorbed solar radiation flux, ˙QS,i
The total amount of solar radiation absorbed by the system depends not only on its geometry and its solar absorptance, αSi, but also on its orientation with respect to the Sun,
cosγ(t) = cosφ(t) cos Ω sinθ(td)−sin Ω cosθ(td), (4.21) where t is the elapsed time of the flight. In addition,γ is the angle between n and the Sun direction defined by the SZA,θ, which is considered to be in XZ plane.
The absorbed solar flux by the node i can be expressed as, Q˙S,i =ξiαS i AiGS(h) cosγi(t) +
n
X
j=0
RSj−i. (4.22)
The factor ξi is 1 if cosγi > 0 and 0 if cosγi < 0. The second term accounts for the Sun radiation reflected by the other nearby elements, represented by node j, which finally reach the node i.
Absorbed albedo radiation flux, ˙Qa,i
The absorbed albedo flux is the amount of the shortwave radiation which is reflected by the Earth and absorbed by the node i. It can be expressed as,
Q˙a,i =αSia(td, h)GS(td, h)Aicosθ(td)Fi−E(h) +
n
X
j=0
Raj−i, (4.23) where the first term refers to direct albedo, and the second term to the albedo reflected by other nodes, j. Fi−E(h) is the view factor from the nodei to the Earth which depends on the altitude of the system.
Absorbed Earth infrared radiation flux, ˙QIRE,i
The absorbed infrared radiation coming from the Earth depends on the node infrared emissivity, εi,
4. Planetary thermal environment for stratospheric balloon thermal design 89
Q˙IRE,i =εiAiFi−E(h)qE(td, h) +
n
X
j=0
REj−i, (4.24)
where the first term refers to the direct Earth Infrared radiation and the second term to the reflected one by other nodes, j.
Emitted infrared radiation flux, ˙QIRi
The thermal radiation emitted in the infrared spectrum by the node i, can be written as a function of its temperature, Ti,
Q˙IRi =−εiAiσTi4. (4.25) The negative sign of this expression refers to the fact that it is an outgoing flux from the node.
Absorbed infrared radiation flux emitted by other elements j,Q˙IRRad,i
All matter emits thermal radiation due to the fact of being at a certain temperature,Tj. Part of this emitted radiation in the infrared spectrum is absorbed by the node i. This flux is expressed as,
Q˙IRRad,i =
n
X
j=0
εiAiBi−jσTj4, (4.26) where Bi−j is called the REF or Gebhart Factor from nodei to the nodej [87]. The Gebhart factor can be calculated by solving the following equation (iteratively),
Bi−j =Fi−jεj +
n
X
k=0
Fi−k(1−εk)Bk−j, (4.27) where Fi−j is the view factor from node i to the node j.
Absorbed infrared radiation flux from the sky, ˙QIRSky,i
This flux takes into account the infrared radiation coming downward, and its value decreases with the altitude up to the top of atmosphere. Due to the low value of the sky equivalent temperature, it acts as a heat sink. The absorbed flux depends on the infrared emissivity of node i and it is calculated as,
Q˙IRSky,i =εiAiFi−SkyσTSky4 (h), (4.28) where Fi−Sky is the view factor from node i to the Sky.
Internal dissipated heat flux, ˙QDis,i
A given node can represent an internal heat dissipation power corresponding to some electronics, heaters, etc. This internal flux would be dissipated through either or both the radiative interface or the conductive one, or produce an increment of the node temperature. This flux is expressed as ˙QDis,i.
Conductive heat flux from other elementsj, Q˙Cond,i
The conductive heat transfer is proportional to the temperature difference between all model’s nodes and the considered node i. The incoming flux would be positive due to a higher temperature of node j and the outgoing flux would be negative due to the opposite effect. This conductive heat flux from j to i is calculated as,
Q˙Cond,i =
n
X
j=0
GLij(Tj −Ti), (4.29) where GLijis the linear conductor between the node i and j.
Convective heat flux to the atmospheric air, ˙QConv,i
The convective heat transfer rate can be expressed as,
Q˙Conv,i =−¯hL(h)Aef f,i(Ti−Tair(h)), (4.30) whereAef f,i is the effective area exposed to the air flow and ¯hL(h) is the average convective heat transfer coefficient for the entire surface. The convective coefficient depends on the fluid properties such as density, viscosity, thermal conductivity and specific heat, and on the surface geometry and the flow conditions. Convection heat transfer is determined by the boundary layer which appears on the surface and it is strongly dependent on whether it is laminar or turbulent and its calculation is not straight-forward [73].
Analysing the ascent phase of these kinds of flights, an external flow appears over the system’ surfaces with its associated convection heat transfer. It is forced convection which depends on the relative motion between the fluid and the surface.
Moreover, natural convection could also appear due to temperature gradients in the fluid inducing a relative motion by buoyancy forces.