A-to-Z Guide to Thermodynamics,
Heat & Mass Transfer, and Fluids Engineering
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Heat protection is a system of measures taken for preventing or decreasing the heat flow from the surrounding medium to the surface of the body being protected. Six basic techniques of heat removel or absorption are currently known:

• by thermal conductivity and heat capacity of a heat- absorbing layer of a substance,

• by releasing (re-radiating) absorbed heat to the surrounding medium,

• by convection (bleeding) of a coolant through a system of ducts in the subsurface layer,

• by coolant percolation through permeable wall with subsequent injection into the surrounding medium (see Transpiration Cooling

• by sacrificial ablation of the surface layer (see Ablation

• by producing an electromagnetic effect on the surrounding medium.

Under some operating conditions a combination of two or more techniques may be used for heat removal and absorption. The mechanisms and limits of applicability of methods most widespread in practice are set forth below.

### Thermal conductivity/heat capacity

Heat protection systems based on thermal conductivity and heat capacity or, in other words, on accumulation of thermal energy are in essence low-temperature ones. They do not permit overheating of the surface layer above the temperature of physicochemical transformation. The capability of transferring energy depends on the thermal conductivity λ of the surface layer. Thus, by Fourier's law:

where is the heat flux into the surface layer and (∂T/∂n)n=0 is the temperature gradient at the surface (n = 0). The maximum quantity of heat that can be absorbed by this system in time τ is determined by the expression

where M is the mass of a substance, c and are the true and the mean (over the temperature interval) heat capacities, Tp and T0 are the limiting and the initial temperatures of the substance, As is the area of the heated surface.

The efficiency of this cooling technique is the higher, the higher the heat capacity of a material , its thermal conductivity λ and the limiting temperature (or the temperature of physicochemical transformation) Tp.

Table 1 presents the thermal characteristics of copper, tungsten, and graphite at T0 = 20°C, while the melting point Tp = Tm is taken as a limiting temperature.

Table 1.  Heat-absorbing materials properties

In order to determine the time of functioning of the heat-absorbing system τT and the mass needed M, it is necessary to solve the problem of heat propagation in the solid with a given distribution of heat flux . We assume that the dependence can be approximated by the second-degree polynomial = b2τ2 + b2τ + b0, where b0, b1, and b2 are constant. Then the temperature of the heated surface will vary with time as

Hence, it follows, in particular, that for a time-constant heat flux the temperature of the body surface Tw reaches the limiting value Tp in the time interval

The operating range results from a linear variation of heat flux .

In this time interval, the heat wave will propagate in the coating to the depth δT. We assume for definiteness that the depth of the heated layer δT corresponds to the distance from an outer (heated) surface to the isotherm T = Tδ, where θδ = (Tδ – T0)/(Tw – T0) is a preset small value, for instance θδ = 0.05. In this case, the temperature of the outer surface veries monotonically, δT = with k .

Since τ ≤ τT, δT or the dimensionless number Bi = cannot grow indefinitely.

It can be shown that it is not advisable to use heat-absorbing coatings, even metals, with a thickness over 25 to 50 mm. It is constraints imposed on the time τT and the needed thickness δT or mass M of the heat-absorbing layer that establish the applicability range of the thermal conductivity/heat capacity heat protection technique in the coordinate (Figure 1).

Systems with partial release of the thermal energy supplied are also known as radiation cooling systems. The maximum quantity of energy that can be radiated by a body at a given temperature on a given wavelength is governed by the Planck law. Figure 2 depicts the spectral density of radiation E(Tw) for different temperatures of a black body surface. It is apparent that the maximum radiation shifts toward the region of short wavelengths in accordance with Wien's displacement law

The distribution of relative monochromatic radiation density is shown in Figure 3, where it can be seen that 80 per cent of radiation lies in the λT region varying from 2000 to 8000 μm K.

The total radiation energy emitted per unit time from a unit surface is

This relation is known as the Stefan-Boltzmann law for a black body. Here the constant σ = 5.710−8W/(m3K4).

Real substances are not black radiators. They radiate on any wavelength only a part of maximum possible energy E(Tw) that is equal to σλE(Tw) . The coefficient σλ is called spectral emissivity.

σλ should be distinguished from the integral emissivity

The spectral emissivity factor of solids depends slightly on temperature Tw of the radiating surface, but only changes greatly with wavelength λ. The result is that the integral emissivity factor σ depends substantially on temperature, because the maximum in E shifts, with growing Tw , toward the region of the short wavelength (Wein's displacement law).

The shape of σλ versus λ curves (Figure 4) fundamentally differs for polished metals or, in general case, conductors (curves 1 and 2) and oxides, or dielectrics (curves 4 and 5). At the room temperatures metals are characterized by a low integral emissivity factor, but surface contamination (oxidation, roughness) may equalize emissivity factors of metals and dielectrics (Figure 4, curve 3). With increasing temperature the integral emissivity grows for metals and drops for dielectrics (Figure 5).

The idea of equalizing the input convective heat flux q0 and the surface-radiated heat flux is the basis of the radiation cooling method. If , i.e., the substrate is perfectly heat-insulated, the surface temperature acquires an equilibrium value

The radiation cooling method is used in particular on a shuttle spacecraft. Since Tw cannot exceed the temperature of phase or physicochemical transformation Tp, the applicability of this method is bounded by the heat flux per unit area (Figure 1).

In principle, protection from radiation heat fluxes is also possible. Thus, in deep space, temperature-controlling coatings are used that possess a low emittance, or bulk absorption coefficient, in the visible spectral region in which solar radiation band largely lies and a high σλ in the IR region, where the coating radiates spontaneously (bearing in mind the low temperature of its surfaces).

### Convection cooling systems

Convective cooling systems have gained wide acceptance in engineering (Figure 6). Heat from a hot surface 3 is transferred to cooling liquid or gas pumped from a reservoir 1 by a special feeding system 2. Convective cooling systems may be classified as "closed" or "open": in the open system, the coolant is discharged to the surroundings. In the closed system, the coolant flows in a closed circuit. A necessary component of closed systems is a heat exchanger (4) in which heat from the coolant is transferred to another heat transfer medium. In this case, the amount of coolant needed in the closed circuit does not depend on the duration of operation. On the whole, the applicability of this method is limited by a number of factors among which is the limiting temperature of the outer wall Tw < Tp. Under a stationary operating regime, the temperature of the outer heated wall is calculated using the set of equations

where Ti and As are the temperature and the area of the inner surface of the heated wall, the heat flux from the outside, δ the wall thickess, c and T0 the coolant heat capacity and the temperature at the system outlet, and the coolant flow rate. Among gaseous coolants the highest heat capacity (c = 14.5kJ(kg)) is characteristic of hydrogen, while amongst liquid coolants, fluids such as water and alcohol are the most widely used. Sodium or lithium melts can be used for cooling at high wall temperatures.

Extension of the range of permissible heat fluxes is possible by either increasing the coolant flow rate or by raising its temperature Ti. The latter may bring about the boiling regime on the inner surface with additional heat being removed as latent heat of vaporization. A voluminous amount of literature discusses these problems, in particular as applied to cooling of units at thermal power stations and nuclear power plants.

Liquid-propellant rocket engines commonly employ a convective cooling system in which the fuel is used as a coolant before it enters the combustion chamber and there reacts with oxidizers.

### Injection cooling systems

Systems with a permeable wall permeated by the coolant can be implemented as a film, transpiration, or jet curtain cooling (Figure 7). Injection of a gas or a liquid directly into the near-wall layer of a high-temperature free stream makes this layer thicker. The free stream is displaced from the surface protected which results in diminishing of heat transfer rate. This cooling technique is sometimes called active cooling. It is distinguished from ablation, first, by maintaining the geometry of the body being protected and, second, by the opportunity of controlling the coolant, flow rate to maintain the surface temperature at the desired level.

Transpiration cooling is the easiest to realize. Its mechanism consists of two physical processes:

1. internal heat transfer during which the gaseous coolant is filtered through a porous (permeable) matrix (wall) and absorbs part of thermal energy delivered from the outside (see Porous Medium

2. external heat transfer when the gaseous coolant leaves the wall and penetrates into the boundary layer diluting and displacing the high-temperature free stream fluid from the surface.

The latter process strongly depends on the regime of flow around the body, but it makes transpiration cooling highly efficient in relation to convective cooling. Sometimes this technique of heat protection is referred to as mass transfer cooling, which also emphasizes the role the transfer processes play in the boundary layer.

In a simplified formulation, external heat transfer is reduced by making an allowance for the injection effect. A gaseous coolant penetrating into the boundary layer is heated from the surface temperature Tw to a temperature close to gas temperature Te on the outer layer boundary.

If the gas injected and the free stream are of the same composition, then it can be assumed, by analogy with thermal conductivity of solids, that absorption of heat in the boundary layer corresponds to the equation

where and are the heat fluxes to impermeable and porous surfaces, respectively, Gw the specific flow rate (mass flow per unit area of surface) of the coolant, he and hw the stagnation gas enthalpies on the outer boundary of the boundary layer and on the wall, respectively.

The parameter γ, commonly referred to as the injection coefficient, must take account of incomplete heat absorption by the coolant due to the drift by the free stream at a temperature lower than Te. In other words, this is the coefficient allowing for fluidity of the coolant due to convective and diffusion heat transfer in the boundary layer.

Making use of the traditional representation for the heat flux

yields for the injection effect a linear relation (Figure 8), where is the dimensionless flow rate of the coolant.

Some merely qualitative estimates indicate that the injection coefficient γ can be modified such that different physical properties of the coolant and the free stream are taken into account. To this end it is sufficient to introduce into it the ratio of the injected to the free gas molecular masses

Table 2 gives the values of injection coefficient for various gases for laminar regime flow in the vicinity of a blunt body stagnation point. These data can be represented approximately as:

However, the range of applicability of a linear approximation for the injection effect is, obviously, limited. First, even injection of homogeneous gases such as carbon dioxide gives rise to a substantial deviation from a linear dependence when the heat flux attains the level (qw/q0) ≤ 0.5 . Therefore, another approximation is required for calculating intense injection heat transfer. Preferable among many others is a quadratic dependence on the dimensionless coolant flow rate (Figure 8)

in which the injection coefficient γ is related, just as in the linear approximation, to the ratio of molecular masses / .

In contrast to blowing of homogeneous chemically inert gases, the real processes may involve changes in the thermodynamic and transfer properties, oxidizer suction from the outer flow, and its interaction with the coolant components. Figure 8 presents the results of numerical solution of the equation for the laminar boundary layer on a glass-reinforced plastic ablative surface and a linear and a quadratic approximations of the injection effect are plotted.

It is important to note that an analogy of heat and mass transfer is also satisfactorily retained in injection, therefore, the ratio of mass transfer coefficients βw0 can be estimated using the same approximating relations as the ratio , however, the injection coefficient is

Table 2.  Injection coefficients for various gases

In order to account for the effect of coolant injection in a turbulent boundary layer the formula

is recommended. Proceeding from the analysis of experimental data on the injection effect on heat transfer in the turbulent boundary layer on a plane plate, we can also recommend a linear approximation of the type used for the laminar flow. But , i.e., the blowing efficiency is reduced twofold.

If a perforated surface is to be used choice of hole diameter and concentration of holes depends on the boundary layer thickness. It is required that the hole diameter be no more than the boundary layer thickness δ while the spacing between the neighboring holes be less than 5δ. As the experiments have shown, under the turbulent flow regime in the boundary layer perforation cooling is equivalent to transpiration cooling only at relative flow rates ≤ 0.5. As the flow rate increases, or if the concentration of holes is low, perforation cooling is less efficient than transpiration cooling.

Let us compare the efficiency of transpiration and convective cooling. We assume an ideal situation in which the temperature difference in the wall being cooled is absent and the heat transfer coefficient between the coolant and the wall is infinitely large. Thus, we assume that the coolant temperature is equal to that of the outer surface. The heat balance in the convective cooling system can be reduced to

Here Gc = /As is the flow rate of the coolant per unit area of the protected surface. If, however, the transpiration cooling is used under the same heat conditions, then

Here a linear approximation is used for estimating the injection effect. With consideration for equality of heat fluxes and the temperatures of the walls being cooled we obtain

The higher the temperature of the free stream Te in relation to the wall protected Tw, the more efficient is transpiration cooling compared with convective cooling. The range of applicability of transpiration cooling is not in fact limited. As is shown in Figure 1, in the variables all the regimes to the right of or above the limiting curves can be realized only for permeable and ablative coatings. The last, sixth, method of heat protection based on an exposure of the surrounding medium to electromagnetic force is so far only within the scope of laboratory tests.