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Sublimation, or volatization, is the process of changing from a solid phase to a gaseous one, without first forming a liquid. Sublimation is one type of vaporization (see Vapor-liquid equilibrium). As with evaporation, sublimation is possible within the whole range of temperatures T and pressures p over which the solid and gaseous phases coexist. Figure 1 presents a typical phase diagram in p-T coordinates (a. water, b. carbon dioxide). It is well known that any substance can exist in one of the three states of aggregation: solid, liquid or gas. Two phase conditions can correspond to the solid state: crystal and amorphous; therefore, the notion "phase condition" is broader than the "aggregate" one. Below, however, the term "phase transition" implies exactly the change of the state of aggregation.

Phase diagrams for water (a) and carbon dioxide (b).

Figure 1. Phase diagrams for water (a) and carbon dioxide (b).

The curves of phase equilibrium on the p-T plane intersect at the triple point, where all three states of aggregation of the substance (solid, liquid and gas) take place simultaneously. The change from a solid state to a liquid state is called "melting": the process of changing from a solid state to a gaseous one is called "sublimation" and from a liquid to a gaseous one is called "evaporation". The reverse process to evaporation and sublimation is called "condensation". The pressure at which the gaseous and condensed (liquid or solid) phases coexist is called the "saturated vapor pressure". For any substance relation between and T is close to exponential (the Clapeyron-CIausius Equation):

where ΔQv is the heat of sublimation, vapor molecular mass, R the universal gas constant, and k is the experimentally defined constant. The heat of sublimation depends weakly on the temperature Tw.

According to the molecular-kinetic concept, sublimation and evaporation are continuous processes of molecular emission from the interface between the gas and condensed phases, the rate of emission being governed by the thermal motion of molecules. The velocity of the reverse process (condensation) is proportional to the number of molecules per unit volume, i.e., to the partial pressure pv of the molecular species condensing on the interface. In sublimation (evaporation), a state of dynamic eqilibrium is established in a closed cavity when the condensation rate is equal to the sublimation rate. The appropriate, partial pressure is called the saturated vapor pressure, pv = (T).

According to this model, the mass flow rate of substance during sublimation is the result for two counter processes, i.e, it is defined by the difference between the saturated vapor pressure which applies at the interface, and the partial pressure in the bulk vapor, pv the interface temperature


This relation is known as the Knudsen-Langmuir equation. The factor a is called the evaporation coefficient (see Accommodation Coefficient). More accurate investigations based on the methods of the molecular-kinetic theory of gases, show that in Eq. (1) the coefficient before the brackets is in the form 2a/(2 − a). This takes into account the transverse constituent of the mass velocity in the function of distribution of gas molecule velocity near the evaporation surface.

With gas flow around bodies, the process of sublimation of their surfaces is nonequilibrium. This is due to the diffusional and convective entrainment of sublimation products into the external flow (Figure 2). To predict the partial pressure pi of the ith component of the gas mixture at the surface of the body, one should consider the mass balance and allow for convective and diffusional transfer. If the mass loss rate per unit area of the body surface is Gw, and if the fraction of the ith component in the subliming material is φi, then

Here, β is the mass transfer coefficient and ci is the concentration of the ith component in the boundary layer of the incoming flow. The indices w and e refer to the body surface and the boundary layer external limit.

Sublimation from the surface of a body in crossflow.

Figure 2. Sublimation from the surface of a body in crossflow.

According to the heat transfer analogy β = (α/cp)w. To a first approximation the heat transfer coefficient (α/cp) on the sublimating smface is related to (α/cp)0 on a nonpermeable (heated) surface by the following relation:

where = Gw/(α/cp)0. If we assume that in the external (oncoming) flow the products of sublimation are absent ci,e = 0, and that the sublimating body does not contain extraneous admixtures φi = 1, then we obtain the following equation for the mass loss rate:


This equation takes into account that the partial pressure pv is related with mass concentration cv by the relationship

wherein pe, M are the pressure and molecular mass of the gas mixture.

Solving Eqs. (1) and (2) simultaneously, one can reach a number of interesting conclusions. Thus, eliminating the mass loss rate Gw we obtain the relationship for estimating the degree of non-equilibrium of the sublimation process, i.e., the relation between the partial pressure pv and the saturated vapour pressure :

The larger the ratio ( /pv), the further the process departs from equilibrium. When the temperature and heat transfer coefficient increase, and when the pressure pe decreases, the departure from equilibrium becomes more significant. During sublimation in a vacuum, as numerous investigations of intense evaporation show, in the steady state case the maximum flow rate Gw,max = (Tw)/ is not obtained, since a portion of outgoing molecules, even in the case of evaporating in a vacuum, return to the surface as a result of intermolecular collisions. It may be shown that in this case the following expression can be used (assuming a = 1):

The higher the partial pressure pe of the subliming material, the closer the sublimation regime is to an equilibrium one, and closes the vapor pressure is to the saturation pressure (Tw). In this case, it can be easily shown at large values of the mass loss rate, the sublimating surface temperature asymptotically tends to a limiting value which depends only on the partial pressure in the gas flow (pe) and is defined from the equation:

Introduction of the limiting temperature Tw,max is very important for estimating the possibilities of evaporative cooling (see Evaporative Cooling).

Under actual conditions, there can be a mixture of condensed media on the sublimation surface. In this case, the equation of the Knudsen-Langmuir kind is applied to each component separately. The mass loss rate is determined by summation of sublimation velocities of all components: Gw = ∑iGwi.

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