Thermogravitational convection

Statement of the problem. Let us consider the motion of a viscous fluid in an infinite layer of constant thickness 2h. The force of gravity is directed normally to the layer. The lower plane is a hard surface on which a constant temperature gradient is maintained. The nonuniformity of the temperature field results in two effects that can bring about the motion of the fluid, namely, the thermogravitational effect related to the heat expansion of the fluid and the appearance of Archimedes forces, and the thermocapillary effect (if the second surface is free) produced by tangential stresses on the interface due to the temperature dependence of the surface tension coefficient. 

To describe the two-dimensional problem, we use the rectangular coordinate system X, Y, where the X-axis is directed oppositely to the temperature gradient on the lower surface and the Y-axis is directed vertically upward. The origin is chosen to be in the middle of the layer therefore, -h Y h. The velocity and temperature fields are described by the equations [142, 143] 

Here P is the pressure (taking into account the gravity potential), x is the thermal diffusivity coefficient, g is the gravitational acceleration, and 7 is the thermal expansion coefficient. 

The thermogravitational motion is described in the Boussinesq approximation in which the variable density in the equations of motion (5.9.1)—(5.9.3) and in the convective heat conduction equation (5.9.4) is taken into account only in the Archimedes term (the last term in (5.9.2)). This term is proportional to the temperature deviation T from the mean value. The thermocapillary motion 

Mass and Heat Transfer Under Complicating Factors 

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For example, the convective motion of a liquid in a vessel whose opposite walls are maintained at different temperatures is due to the fact that the fluid density is normally a decreasing function of temperature. The lighter liquid near the heated wall tends to rise, whereas the heavier liquid near the opposite wall tends to lower. This is one of the examples in which the so-called gravitational (in this case, thermogravitational) convection manifests itself. 

First, let us consider the case of purely thermogravitational convection. It corresponds to the case in which heat is supplied through both boundaries and a constant temperature gradient is maintained on both boundaries. The boundary conditions can be written in the form... 

Density is one of the most basic thermophysical properties. As shown in Eq. (4.3), the temperature derivative of density corresponds to the volumetric thermal expansion coefficient P, which corresponds to the driving force for thermogravitational convection. 

Thermal conductivity is one of the most demanding thermophysical properties but difficult to obtain experimentally, because on Earth thermogravitational convection exerts a major effect on heat transfer and it is almost impossible to suppress this effect. There are four methods to obtain the thermal conductivity of molten silicon. Historically, thermal conductivity has been estimated from the measurement of electrical conductivity x and applying the Wiedemann-Franz law, as shown in Eq. (4.2) [5, 8, 52, 53]. Thermal diffusivity was measured also by a laser flash method [7, 24, 54] and is converted into thermal conductivity using density p and mass heat capacity Although transient hot-wire and hot-disk methods assure direct... 

The basic construction of a horizontal thermal diffusion cell is sketched in Figure 19.14(a). When gases are to be separated, the distance between the plates can be several mm for liquids it is a fraction of a mm. The separation effects of thermal diffusion and convection currents are superimposed in the equipment of Figure 19.14(b), which is called a thermogravitational or Clusius-Dickel column after the inventors in 1938. A commercially available column used for analytical purposes is in Figure 19.14(c). Several such columns in series are needed for a high degree of separation. 

In some cases, natural convective flow plays an integral role in separations. For example, thermogravitational (TG) columns rely on a combination of thermal convective flow and relative (selective) displacement by thermal diffusion. 

Convective flow is used in both the thermogravitational (Clusius-Dickel) column and in electrodecantation. In the thermogravitational system, one wall of the channel is heated or, alternatively, a hot wire is placed along the axis of the channel. The fluid at the cold surface then tends to sink relative to that at the hot surface. Simultaneously, thermal diffusion (Section 8.8) causes different levels of enrichment in the hot and cold regions of the channel. The enriched solutes then move up and down the channel at a rate depending upon their distribution between hot and cold regions. In binary... 

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