Forces thermophoretic

In a discussion of thermal-force theories it is necessary to address the problem of experimental data for the following reason. It has been known for some time that experimental thermal-force data determined by the Millikan-cell method [2.97, 98,137-139] differed from the data obtained by measuring the velocity of particle motion due to the thermal force (thermophoretic velocity) in various flow systems with different configurations [2.121,140,141]. 

The basic operations in dust collection by any device are (1) separation of the gas-borne particles from the gas stream by deposition on a collecting surface (2) retention of the deposit on the surface and (3) removal of the deposit from the surface for recovery or disposal. The separation step requires (1) application of a force that produces a differential motion of a particle relative to the gas and (2) a gas retention time sufficient for the particle to migrate to the coUecting surface. The principal mechanisms of aerosol deposition that are apphed in dust collectors are (1) gravitational deposition, (2) flow-line interception, (3) inertial deposition, (4) diffusional deposition, and (5) electrostatic deposition. Thermal deposition is only a minor factor in practical dust-collectiou equipment because the thermophoretic force is small. Table 17-2 lists these six mechanisms and presents the characteristic... 

The efficiencies of electrostatic precipitators, bag filters, and scrubbers are given in Figure 16-3. The costs for installing and operating these devices are given in reference 20. This source also describes some exotic methods, such as the use of thermophoretic or diffusiophorctic forces and sonic agglomeration, that have not yet been commercialized. 

Air movement indoors is much slower than outdoors, but it is usually enough to ensure that concentrations are fairly uniform in a room. Convection from heating appliances gives air speeds typically in the range 0.05-0.5 m s-1 (Daws, 1967). However, to undergo deposition, vapour molecules or particles must be transported across the boundary layer, typically a few millimetres thick, of almost stagnant air over surfaces. This may be achieved by sedimentation, molecular or Brownian diffusion, or under the action of electrostatic or thermophoretic forces. 

Deposition other than in rain is termed dry deposition, and this includes sedimentation of particles, molecular and Brownian diffusion to surfaces, impaction on roughness elements and deposition under electrical or thermophoretic forces. The velocity of deposition is defined... 

Thermophoretic forces can be used in sampling aerosols the particles are passed through the dark space surrounding a hot body and are collected with nearly 100 percent efficiency on a cold surface placed nearby. To date, however, there has been no successful utilization of thermal forces for large-scale air cleaning. 

Thermophoretic forces produce very obvious effects near areas of significant temperature gradients. For instance, one can often observe a black deposit on the wall just above a hot-water radiator or pipe. Convection currents conduct the warm gas and particles over the radiator, but since the cooler surfaces nearer the radiator are not protected by a dust-free space, deposition takes place. On a ceiling or on walls of rooms heated by convection, one can often see a replica of the construction behind the plaster formed by deposited particles. Again, the dust is deposited on the cooler portions of the surface on spaces between the laths if the laths are poor heat conductors and directly opposite the laths if they are good conductors. In a room that is heated by direct radiation, such as by an open fire, the walls and furniture of the room are warmer than the air, so that particles suspended in the air are not deposited by thermal forces (Lodge, 1883 Gibbs, 1924). 

To determine the thermophoretic velocity, Stokes law can be utilized by assuming that the Cunningham or slip correction factor (Eq. 5.3) is applicable for cases where Kn > 1. Thermophoretic velocity will be independent of particle diameter since Cc Kn(A + Q) when Kn > 1. Then, equating the thermal force (Eq. 11.8) with the resisting force (Stokes law) and solving for the thermophoretic velocity vT give (Talbot et al., 1980)... 

To determine thermophoretic velocity, the Stokes resisting force is equated with the thermal force. Then... 

Using Brock s equation, determine the thermophoretic force on a 1-p.m-diameter glycerol particle. For this calculation use C = 1.147, C, = 2.20, and Cm = 1.146. How does this estimate of thermal force compare with the estimate made by using Epstein s equation (Eq. 11.11) ... 

For illustrative purposes, we exhibit temperature and vapor density profiles along the line connecting sphere centers for two water drops growing imder a supersaturation ratio of 1.05 at atmospheric pressure and 27°C ambient temperature these profiles are shown in Figures 1 and 2. Also in Figures 3 and 4, the profiles for a water drop and an inert sphere are shown. It may be of some interest to use the present generalized formalism to treat thermophoretic and diffusiophoretic forces between spheres. 

The change of momentum for a particle in the disperse phase is typically due to body forces and fluid-particle interaction forces. Among body forces, gravity is probably the most important. However, because body forces act on each phase individually, they do not result in momentum transfer between phases. In contrast, fluid-particle forces result in momentum transfer between the continuous phase and the disperse phase. The most important of these are the buoyancy and drag forces, which, for reasons that will become clearer below, must be defined in a consistent manner. However, as detailed in the work of Maxey Riley (1983), additional forces affect the motion of a particle in the disperse phase, such as the added-mass or virtual-mass force (Auton et al., 1988), the Saffman lift force (Saffman, 1965), the Basset history term, and the Brownian and thermophoretic forces. All these forces will be discussed in the following sections, and the equations for their quantification will be presented and discussed. 

In summary, the Boussinesq-Basset, Brownian, and thermophoretic forces are rarely used in disperse multiphase flow simulations for different reasons. The Boussinesq-Basset force is neglected because it is needed only for rapidly accelerating particles and because its form makes its simulation difficult to implement. The Brownian and thermophoretic forces are important for very small particles, which usually implies that the particle Stokes number is near zero. For such particles, it is not necessary to solve transport equations for the disperse-phase momentum density. Instead, the Brownian and thermophoretic forces generate real-space diffusion terms in the particle-concentration transport equation (which is coupled to the fluid-phase momentum equation). 

Deposition of nanoparticles was investigated in the free molecular regime approximation for thermophoretic force and the Brownian motion. The analytical solution was obtained by the Galerkin method for the heat transfer between gas flow and substrates and convective diffusion. Relative roles of two channels of nanoparticle deposition are discussed. 

Emphasize that for the higher modes, obtained by the Galerkin method, characteristic lengths are much shorter. In turn, if there is no significant temperature gradient, we can neglect the influence of thermophoretic force on deposition of nanoparticles. 

---