Accounting for the Influence of Convective Heat Transfer during Sublimation of Substances

V. K. Safonov

In the process of sublimation, to convert substances from the solid state to the vapor state, radiant energy supply, energy supply by radio waves, ultrasonic action, etc., have recently been used. In this case the intensity of the evaporation rate is many times greater than the evaporation rate when energy is supplied by the conductive or convective method. But regardless of the type of energy used, the processes of convective transfer in the region of medium vacuum play an essential role.

As experiments show [1], at a radiant-energy flux density \(q_u = 698 \div 1070\ \mathrm{W/m^2}\) and at a constant pressure in the sublimator \(P_c = 66.6\ \mathrm{N/m^2}\), the share of the convective component of \(q_u\) varies, respectively, within the limits \(0 \div 6.5\%\). In this case the convective component may be neglected.

An entirely different picture is observed at \(P_c > 66.6\ \mathrm{N/m^2}\). According to the investigations of E. A. Ermakova [2], at a low radiant-energy density (the temperature of the sublimator walls \(t_{\mathrm{st}} = 24^\circ\mathrm{C}\) and \(P_c = 133.3\ \mathrm{N/m^2}\)), the share of the convective component of \(q_u\) is equal to \((31 \div 40)\%\). On this basis it may be considered that, under conditions of a viscous and molecular-viscous regime, the intensity of the heat- and mass-transfer process cannot be ensured solely by the molecular transfer mechanism.

For an analytical accounting of the convective component, a detailed study is necessary of the hydrodynamic conditions that arise during evaporation from the solid state. The most important in this case may be the study of the velocity field near the specimen. However, measurement of the velocity under the conditions of interest to us is associated with great difficulties.

The distribution of the pressure, density, and temperature fields in vacuum is also characteristic. At present there are satisfactory data on the distribution of density in the boundary layer of the sublimating body [3] and on the distribution of temperatures in the volume of the sublimator [2]; [4], but as yet there are no reliable methods for determining pressure fields. A preliminary consideration of the phenomenon does not lead to its detailed model and, consequently, we cannot derive the basic equations. If, however, the equations cannot be derived, the investigation has to be constructed on the basis of relations that are much less specific. Let us establish the relation between the heat-transfer coefficient, the hydrodynamics of mass transfer

and the thermodynamic properties of the vapor–gas medium, proceeding from the following assumptions.

In the boundary layer of a sublimating body, inertial forces cannot be neglected. As the distance from the sublimation surface increases, these forces decrease so rapidly that at a distance of \(0.015\ m\) they are practically equal to zero. In this case it becomes possible to construct the Péclet complex, which rapidly degenerates with distance from the surface of the sublimating body. The Péclet complex (\(\mathrm{Pe}\)) includes the flow velocity, but in sublimation it is impossible to specify any velocity of circulation of the vapor–gas medium at the surface of the specimen. According to the physical situation [5] that develops in the boundary layer, the vapor–gas mixture circulates repeatedly. There are descending and ascending flows, and most likely this resembles the free motion of a gas at a heated horizontal surface. In this case, in order to exclude the velocity, we shall use one more complex, \(\pi_{GF}\), which represents a measure of the ratio of the force of gravity to the force of internal friction. Then the product \(\mathrm{Pe}\cdot\pi_{GF}=Gr\cdot Pr\), as we see, gives us the product of two new complexes, one of which is the Grashof criterion and the other the Prandtl criterion. These criteria also contain quantities that are determined from the condition.

Let us now consider which complexes characterize the thermodynamic properties of the vapor–gas medium in the boundary layer. During sublimation of a substance from a free surface, the smallest particles of the substance enter the boundary layer of the vapor–gas mixture. In this case a heat sink appears in this layer, caused by the evaporation of the particles. This expression for the sink can be taken into account by the Gukhman complex (\(\mathrm{Gu}\)). The Gukhman criterion is a variable quantity, since the temperature of the vapor–gas mixture changes across and along the boundary layer. However, at a constant sublimation intensity for a given temperature interval, an averaged criterion \(\mathrm{Gu}\) can be introduced. Thus, we have determined the most essential complexes affecting convective heat transfer under the condition that the main amount of energy is supplied by radiation. Since the sought quantity is the heat-transfer coefficient, the dependence between the similarity criteria must be represented in the form

\[ \mathrm{Nu}=f(Gr, Pr, Gu), \]

where \(\mathrm{Nu}\) is the Nusselt number.

LITERATURE

1. D. P. Lebedev. Kinetics of the process of ice sublimation under vacuum with radiant heat supply. In the collection Heat and Mass Transfer, vol. 10, Minsk, “Nauka i tekhnika” Publishing House, p. 204, 1968.

2. E. A. Ermakova. Investigation of temperature distributions during evaporation from a solid state under vacuum. IFZh, vol. 7, No. 5, p. 39, Minsk, 1964.

3. V. A. Solov’ev. Investigation of the density of the vapor–gas medium in the boundary layer of a sublimating body. IFZh, vol. 15, No. 5, p. 804, Minsk, 1968.

4. A. V. Lykov. Heat and Mass Transfer in Drying Processes. Gosenergoizdat, 1956.

5. A. A. Gukhman. Application of Similarity Theory to the Study of Heat- and Mass-Transfer Processes. Moscow, “Vysshaya shkola,” p. 283, 1967.