2. Heat transfer mechanisms

24

2. Heat transfer mechanisms

Heat transfer in space is mainly achieved in two ways: conduction and radiation. The presence of convection is usually confined at particular cases, for example, when active TCS is required by making use of heat pipes or for manned missions. The latter is the case of capsules, where an air-conditioned compartment or Environmental Control and Life-Support Subsystem (ECLSS) is appropriately sized to be a sort of “home away from home” for space travelers, [13].

Ablation, which is a of particular interest when speaking of re-entry vehicles, is not another transfer mechanism, but the result of a phase transition that a materials undergoes when the heat load it receives overcomes its latent heat of vaporization, [14]. The produced thin film of ablated fluid phase is then removed due to the continuous external convection and substituted by a new film, and this continuous process protects internal layers of the structure from melting. In order to describe ablation, a completely different approach is required and is out of the objectives of the work.

2.1 Conduction

Conduction is very important in space applications since it allows balancing the internal powers between the components and permits to evacuate the dissipated power through external radiative surfaces. Inside a spacecraft, it is the main mode of heat transfer in the range of temperatures [-50 °C, 100 °C], [4].

Physically speaking, it is the result of the presence of a thermal gradient inside a solid medium which, according to the 2

law of Thermodynamics, tends to make its temperature as uniform as possible by making heat flowing from the higher- temperature to the lower-temperature region. The resulting conductive heat flux q is due to energy exchanges at molecular level (without motion) and is a measure of the rate at which heat is transferred by conduction. The following Fourier’s law, for a homogeneous medium, describes this phenomenon in its most general form:

𝜌𝑐

𝜕𝑡

= 𝑘∇

𝑇 + 𝑄

(2.1)

25 where (unless specified, the following quantities are properties of the solid medium)

 𝜌 is the density, [kg/m

],

 c

is the specific heat at constant pressure, [J/(kg*K)], weak function of T,

 k is the thermal conductivity, [W/(m*K)], weak function of T,

 Q

is a possible heat volumetric flux generated internally, [W/m

], and the “squared-Nabla” indicates the Laplacian operator.

Under the assumptions of steady T and 1-D flux and selecting average values of k and c

in the appropriate range of T, the previous equation in Cartesian coordinates rewrites as:

𝑞 = −𝑘𝐴

𝜕𝑥

(2.2) where A is the area of the solid section perpendicular to q, [W], the conductive heat flow rate. This is the starting expression considered for thermal modelling of conductive conductors (see Chapter 6).

Table 2.1 - Values of k at various T for common materials of space application

Values of k in the figure above are extracted by semi-empirical formulae, as stated

in [15]. About c

, it must be distinguished by c

(specific heat at constant volume) if

referred to gases, while for both solid and liquids the quantities can be considered

almost equal.

26 Table 2.2 - Mean values of c

at 298 K for common materials in space application

Polynomial approximations are available also for c

and should be considered if the analyzed range of T is very large, but constant values available in literature, [7], are sufficient for these purposes.

A specific situation in which conduction is particular interesting is the case of two adjacent solids of different materials in contact. In this case, not only there is conduction inside each of them, but there is also a parameter called “contact thermal resistance”, [7], which accounts for the surface effects of the two solids and usually results in a local temperature leap.

Figure 2.1 - Heat conduction through contacting points and relative temperature

leap at the interface, [7]

27 This phenomenon is very hard to model because of its dependence on several quantities (e.g. surface roughness). Although the case of two adjacent solids has been taken into consideration, this kind of conduction has been neglected.

2.2 Convection

Heat transfer by means of convection actually consists of two mechanisms operating simultaneously. The first is the energy transfer due to molecular motion, that is, the conductive mode (this would be the only mode, ideally, at the interface between a standing fluid and a solid). Superimposed upon this mode is energy transfer by macroscopic motion of fluid parcels. The fluid motion is a result of parcels of fluid, each consisting of a large number of particles, moving by an external force, [15].

This force may be produced by:

1. a density gradient  natural convection,

2. a pressure difference artificially created  forced convection.

Figure 2.2 - Schematics of natural (left) and forced (right) convection, [15]

The main difference is that in forced convection the velocity far from the surface

approaches the free-stream value imposed by an external force, while in natural

convection the velocity reaches a maximum at an intermediate point between the

body and the freestream conditions. The reason of this behavior is that the action of

viscosity diminishes rather rapidly with distance from the surface, while the density

difference decreases more slowly, [15]. The role of velocity is fundamental in

convective exchanges: it is externally imposed in forced convection, but, in natural

28 convection, it also depends on the temperature difference between the body surface and the fluid. An in-depth analysis of the question would require facing a fluid- dynamic problem. Despite this, the solid-fluid convective exchange can be analytically expressed as:

𝑞

= ℎ

𝐴(𝑇

− 𝑇

) (2.3) where:

 q

is the convective heat flow rate, [W],

 h

is the convection heat transfer coefficient, [W/(m

*K)], and accounts on average for all the complex phenomena involved in convection,

 A is the heat transfer area, [m

],

 T

– T

is the difference between surface temperature of the solid and freestream temperature of the fluid, [K].

It is intuitive that h

depends on the geometry of the surface, the velocity, the physical properties of the fluid and sometimes even on T

– T

. The table below lists some values that should be only used for a first-order analysis of convection.

Table 2.3 - Values of h

for common fluids, [15]

Vacuum in space does not leave any possibility for natural convection, therefore the

only way to transport heat from one point of the spacecraft to the other, if conduction

cannot be exploited, is by means of “pumps” such as the Pumped Fluid Loops (PFL),

which are rather complex and expensive. Their working principles and sizing criteria

are extensively described in [7].

29

2.3 Radiation

Thermal radiation is the only heat transfer mechanism that does not require contact among the parts involved. The “energy carriers” in radiation are the electromagnetic waves, which are emitted by all existing bodies independently of their state of matter and have an energy content that varies with the wavelength, as shown by the Herschell’s experiment, [6].

2.3.1 Definitions and laws

A “blackbody” is an ideal body capable of absorbing and emitting energy at any wavelength, so it is also called a “perfect radiator”, [15]. When kept at a constant T, its emission at a specific wavelength λ is described by the Planck’s law:

𝐸 _𝜆 = ^2ℎ𝑐

𝜆

(𝑒

−1) (2.4) where:

 E

, [W/(m

*μm)], is the energy emitted per surface unit at wavelength λ, [μm], by a blackbody, also called hemispherical (i.e., through all directions) spectral emissive power, [16], or spectral emittance,[6],

 h ≈ 6.626*10

, [Js], is the Planck’s constant,

 c ≈ 299 792 458, [m/s], is the speed of light,

 k

≈ 1.386488*10

, [J/K], is the Boltzmann constant.

E

undergoes a maximum at a certain values of λ. By differentiation of Eq. (2.4), the Wien’s law is obtained:

𝜆

≅

𝑇

(2.5)

The highest emissive power takes place in the visible and IR spectrum (λ ≈ 0.1 - 200

μm), [6], and the expression “thermal radiation” commonly refers to this range of

wavelengths.

30 Figure 2.3 - Thermal radiation in the electromagnetic spectrum

The total power content inside the spectrum (the total emittance) is given by integration of Eq. (2.4), expressed by Stefan-Boltzmann law:

𝐸 = 𝜎𝑇

(2.6) where σ = 5.67*10

is the Stefan-Boltzmann constant, [W/(m

*K

)]. Another important property of a blackbody is that it emits and absorbs equally in all directions.

This assumption is retained even for real bodies, since considering directional properties would add a complication that is not justified at this level of analysis.

Because of its definition, the blackbody has:

 α

= α =1, (total) absorptivity or absorption coefficient,

 ε

= ε = 1, (total) emissivity or emission coefficient.

Real bodies do not meet the specifications of an ideal radiator, because they emit radiation at lower rate, while part of the received heat can be not only absorbed, but either transmitted or reflected, according to the following energy balance, [15]:

𝛼

+ 𝜌

+ 𝜏

= 1 (2.7) where:

 𝜌

is the total reflectivity or reflection coefficient,

 𝜏

is the total transmissivity or transmission coefficient.

and which is also valid for the relative spectral properties.

31 The total emittance of a real body at a given T can be always compared to the same quantity referred to a blackbody at the same T: this ratio defines the emissivity of a real body:

𝜀 = ^𝐸(𝑇)

𝜎𝑇

(2.8) where both numerator and denominator of the right member of Eq. (2.8) are expressed by integrals, as Eq. (9.31) of [15] shows, since also ε varies with λ.

Absorptivity is defined by a similar equation (see Eq. (9.33) of [15]), with the difference that there is a dependence on λ even after the integration, unlike what happens for emissivity.

Integrals in previous equation simplify when one introduces the hypothesis of “grey body”, a body whose α and ε do not depend on λ. Even though real surfaces do not meet this specification exactly, it is often possible to choose suitable average values for the emissivity and absorptivity to make the grey body assumption acceptable for engineering analysis.

Figure 2.4 - Comparison of hemispherical monochromatic emission for a black, a

grey (ε = 0.6), and a real surface, [15]

32 Kirchhoff law states that, at thermal equilibrium and at any fixed wavelength:

𝛼

= 𝜀

(2.9) which easily translates in “α equals ε” for grey-bodies. However, for spacecraft thermal-analysis purposes, α and ε are chosen as the mean values of visible and IR spectrum, respectively. This is because the Sun, which is the main source of heating in space, emits mainly in the visible spectrum, while a satellite generally experiences temperatures for which maximum emission is in the IR spectrum, like the Earth itself.

Table 2.4 - Optical properties of common spacecraft materials

Both α and ε are less than 1 for real bodies. In this dissertation, these quantities are intended as the only surface optical properties of the materials in order to represent passive thermal control components such as paints and coatings, typically used in spacecraft design.

2.3.2 Grey-body factors and view factors

In most practical problems involving radiation, the intensity of thermal radiation

passing between surfaces is not appreciably affected by the presence of intervening

33 media because, unless the temperature is so high as to cause ionization or dissociation, monatomic and most diatomic gases as well as air are transparent, [15]. This is perfectly true in space, where any surfaces not in contact can only exchange heat by radiation. In this case, the net heat transfer by radiation between two grey bodies 1 (emitter) and 2 (absorber), respectively at temperatures T

and T

(< T

), is given by:

𝑞

= 𝜎𝜀

𝐴

𝐵

(𝑇

− 𝑇

) (2.10) where:

 q

is the heat flow rate, [W], exchanged by radiation and plays the same role as q and q

previously defined,

 ε

is the emissivity of body 1,

 A

is the area of the radiative surface of 1, [m

],

 B

is a non-dimensional parameter called “grey-body factor”, [18], or Radiative Exchange Factor (REF), [17], between the two surfaces.

The computation of the “grey-body factors” is the heart of radiation problems, since they are function of the relative position and geometry of the bodies, their optical properties and are affected by possible occlusions or reflections from other bodies.

Grey-body factors must not be confused with the “view factors”, which instead represent the purely geometric part of the computation. A good distinction is the following one, [18]:

 A view factor quantifies the fraction of energy emitted from one surface that arrives at another surface directly; in other words, it describes “how well”

surface 1 sees surface 2 via a direct link and is indicated with F

;

 A grey-body factor quantifies the fraction of energy leaving surface 1 that is absorbed by surface 2 through all possible paths.

Nevertheless, the view factor computation is the first step in evaluating the more

useful grey-body factors. The only case in which they coincide is the radiative

34 exchange between two blackbodies of finite dimensions, for which Eq. (2.10) stands with B

= F

but has no practical interest.

The view factor between two finite surfaces 1 and 2 of area A

and A

is given by:

𝐹 ₁₋₂ = ¹

𝐴

∫ ∫ ^cos(𝜃

_𝜋𝑟 ^{) cos(𝜗}

⁾

122

𝐴

𝐴

𝑑𝐴 ₂ 𝑑𝐴 ₁ (2.11) where all the geometrical quantities are represented in the figure below.

Figure 2.5 - Geometry for view-factor computation between two finite surfaces. θ

and θ

are the line-of-sight inclination angles, while r

is the distance between the centroids of the surfaces

The previous double integral returns an analytic closed-form solution only for very simple geometries. In practical cases, numerical integration is required and, as anticipated in Chapter 1, the VFS adopts the Monte Carlo ray-tracing method as the best solution for the triangular-meshed models coming out from the Mesher. Despite that, the following view-factor main properties turned out to be effective in the computation of the grey-body factors and, in the end, the radiative couplings in the thermal model:

1. 0 ≤ 𝐹

≤ 1 (2.12) where 0 is the case of two parallel adjacent flat plates

and 1 is the case of a body 1 at the centre of an enclosure 2;

2 Value 0 also stands for two completely occluded surfaces, as it turns out from the view-factor computation in [17], where a “line-of-sight occultation factor” X1-2 is considered

35 2. 𝐴

𝐹

= 𝐴

𝐹

(2.13)

is the “reciprocity law”, easily deduced from symmetry of Eq. (2.11);

3. ∑

𝐹

= 1 (2.14) is the “sum law” and is a direct consequence of the conservation of energy, [17].

Starting from these simple properties, a complete view-factor algebra is derived and many problems can be solved without complex computation. The work described in [6] mainly concerns with methods for computing view factors in spacecraft.

Once view factors are obtained, the grey-body factors can be determined as described in Chapter 6. In doing so, the following assumption about the radiative properties of the surfaces is retained, [17]: any radiative surface emits as if it were a

“perfect diffuser”, thus following the Lambert’s cosine law:

𝐸

= 𝜋𝐼

(2.15) where I

, [W/m

*sr], is the power emitted per unit area of emitting surface projected along a fixed direction into a given solid angle. Subscript b stands for “blackbody”: in fact, only a blackbody is a “perfect radiator” but grey bodies can inherit this property since radiation from industrial rough surfaces well approaches diffuse characteristics, [15]. In other words, the intensity of radiation I is uniform in all angular directions and the total emittance is proportional to it, although the hemispheric one goes as the cosine of the angle between the normal to the surface and the direction considered. Such surfaces are therefore called “Lambertian”.

Figure 2.6 - Lambert's law and the dependence of hemispherical emittance on

direction