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Melting is the process of changing a solid substance into a liquid accompanied by heat absorption. The reverse of melting, solidification, occurs, in equilibrium conditions, at the same temperature Tm and with the evolution of the same amount of heat ΔQm which is absorbed in melting. Some of the substances with a complicated molecular structure such as glass, resin and polymers, which are supercooled liquids at room temperature, soften gradually on heating and have no definite melting temperature Tm. Melting is only associated with a certain weakening of interatomic bonds and therefore ΔQm is only three to four per cent heat of substance sublimation ΔQυ, which corresponds to a complete break of the bonds.

The heat of melting is related to melting temperature. The molar entropy of melting for metals is ΔHm/Tm = ΔSm 9.7 kJ/(k) for monovalent interatomic bonds, but for covalent crystals changing to a metal state, this ratio can increase three times.

The model of melting crystalline substances in a high-temperature gas flow is widely used to establish general laws of conjugated heat transfer problems, in particular, as applied to problems of heat protection (see Heat Protection, Ablation).

If the parameters of a gas flow (pressure, velocity and temperature) do not change significantly with time, then we can clearly distinguish two periods in the character of heating melting substances (Figure 1). At first, for time τ ≤ τT, the temperature of the heated surface grows from the initial value T0 to the phase transition (melting) temperature Tm = Td where Td is the temperature at which surface destruction begins.

Change of temperature with time in the melting and destruction of a material.

Figure 1. Change of temperature with time in the melting and destruction of a material.

Beginning at τ =τT, a melt film is formed on the surface of the crystalline substance, its viscosity, however, is so small that it may be carried away instantaneously by the gas flow under the action of shear stresses (friction, pressure gradient). By the term "instantaneously" we mean that the temperature drop over the melted film is always less than the temperature drop (Tm — T0) over the heated layer of the solid, therefore the contribution of sublimation is negligibly small and the entire ablation occurs in a molten state.

In what follows, a model is presented which aims to represent all the relevant parameters and mechanisms (including melting) in the ablation of a surface under the influence of a gas flow at constant (high) temperature. Generalization of the results of the theoretical analysis is obtained by the use of dimensionless coordinates. One of the dimensionless groups used is a parameter describing the thermal efficiency of ablation or destruction:

where C is the specific heat capacity of the solid material and ΔQm is the heat of phase transitions. Presented schematically in Figure 2 is the heat destruction and heating of the body at time τ > τT. Defining S(τ) as the thickness of the layer of the substance already removed from the surface, z is a dimensioniess space coordinate with respect to the moving interface between the gas and liquid (melted) phases:

where is the thermal diffusivity of the solid material (= λ/ρc where λ is the thermal conductivity and ρ the density). The value of λT may be calculated from:

where is the heat flux to the surface. We can define a dimensionless time coordinate as:

Figure 3 shows how coordinate z changes with t for the isotherm θb = (Tδ − T0)/(Tm − T0) = 0.1 for various values of m. By changing the given value of θb = const, we obtain the temperature field inside the destructing body.

Temperature profile in solid material during melting.

Figure 2. Temperature profile in solid material during melting.

Variation of z with for a given dimensionless temperature isotherm q.

Figure 3. Variation of z with Variation of z with for a given dimensionless temperature isotherm q. for a given dimensionless temperature isotherm q.

It is of interest to correlate the process of development of the temperature field with the change of the dimensionless carried-away layer thickness S(τ)/ . Figure 4 gives the coordinates of three isotherms for a fixed value of the parameter m (= 0.1) and also gives dimensionless thickness of the carried away layer, all as a function of dimensional time t. We notice that with an accuracy of 10-15% the coordinate of the corresponding isotherm approaches its set (stationary) state at the moment of its intersection with the curve S(τ)/ . We may choose the onset time tδ of the "quasi-stationary" value of depth of the heated layer to correspond to δT = z (θb = 0.1).

Comparison isotherms and melt layer thickness.

Figure 4. Comparison isotherms and melt layer thickness.

The stationary temperature profile is an exponential function of the coordinate z and the parameter m:

Under these quasi-stationary conditions, there is a constant value of ablation rate :

The stationary temperature profile makes possible the establishment of a simple relationship for the amount of heat required for heating the subsurface layers of the destructing body, this amount of heat playing an important role in the energy balance on the surface of destruction:

Within the framework of this simple one-parameter model of destruction we can obtain the dependence on m of the times for the onset of quasi-stationary values of ablation velocity tυ and of the heated layer depth, tυ and tυ respectively (Figure 5). Here, we define a quasi-stationary value of ablation velocity (dS/dτ) as one that differs from the stationary one by no more than 10%. The coordinate of the isotherm θb = 0.1 is defined as the depth of heated layer.

Variation of times for the onset of a quasi-stationary ablation rate (tυ and quasi-stationary depth of heated layer (tδ) with m).

Figure 5. Variation of times for the onset of a quasi-stationary ablation rate (tυ and quasi-stationary depth of heated layer (tδ) with m).

In contrast with crystalline substances which have a specific melting temperature Tm and a heat of phase transition ΔQm, amorphous substances change from a solid to a liquid state gradually without practically an additional heat absorption. They have a high melt viscosity which depends exponentially on temperature:

Therefore, not only is the temperature of transition from a solid to a liquid state unknown in advance but also the temperature of the outer surface. The fraction Г of the heat load which goes into sublimation (gasification) of the solid material depends on the thermal characteristics of the body over which the hot gas flows. The resultant characteristics of ablation of amorphous materials is the sum of the action of two opposite processes: the higher the density or pressure of the incident flow the faster the melt film is carried away, but also the heat flux also grows which brings about an increase in the outer surface temperature and in the evaporation rate.

A careful analysis of the equation of the melt film motion over the surface of amorphous bodies allows us to reveal an interesting rule. It turns out that the ratio of the film mass loss to the evaporated substance flow rate depends mainly on the parameter pe/(α/cp)o2, where pe is the stagnation pressure, a the heat transfer coefficient and cp the gas specific heat capacity. The body dimensions, the enthalpy of the stagnated flow and other determining quantities, together affect the ablation characteristics of amorphous substance of the quartz glass type by not more than ±30%.

In a wide range of incident flow pressures (from 1 kPa to 1 MPa) almost identical characteristics of heat ablation efficiency are realized for a laminar flow in the boundary layer (Figure 6), since (α/cp) . In a turbulent boundary layer in which (α/cp) , a noticeable redistribution of ablation in the direction of sublimation will occur, and the share of melt on the surface will decrease the more substantially the higher pe (Figure 6) (see also Ablation).

Figure 6. 

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