Thermophoretic velocity

The light-scattering experiment showed that the particle size was on the order of 50 nm, significantly smaller than the gas mean-free path. In this limit the thermophoretic velocity is [136]... 

Postprocess the solution, either with a short special purpose program or in a spreadsheet, to calculate the thermophoretic velocity at each point. (This velocity has the opposite sign of the convective velocity from the flow simulation.)... 

Find (or interpolate) the position at which the convective and thermophoretic velocities sum to zero (i.e., the predicted position of the particles above the surface). 

Figure 11.4 Thermophoretic velocity for particles of two different thermal conductivities.

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)... 

It is also possible to derive an equation for the thermophoretic velocity by considering that the suspended particles are a dilute suspension of giant molecules mixed with a much greater number of smaller molecules. This was done by Mason and Chapman (1962) who found essentially the same form for FT as that given in Eq. 11.9. 

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

Talbot et al. (1980) have shown that thermophoretic velocity determined from Brock s equation (Eq. 11.14) degenerates into thermophoretic velocity determined from Waldmann s equation (Eq. 11.8) when the limit of Jd — (except for the multiplication factor CJCm 1). They point out that there appears to be no theoretical justification for this result except that it appears to fit the available experimental data quite well. 

Figure 11.3 is a plot of reduced thermophoretic velocity as a function of Knudsen number showing some experimental data along with curves for Brock s and Derjaguin and Yalamov s equations. It can be seen that although these equations all predict the form of the data set, there appears to be still much room for improvement in both data analysis and theory. 

A schematic plot of thermophoretic velocity (Eq. 11.21) as a function of particle diameter for air at normal temperature and pressure is shown in Fig. 11.4. It can be seen that the thermophoretic velocity decreases from a high value at small particle sizes to a somewhat lower constant value for large particle sizes. The range of the region of changing vT is approximately 0.01 < d p,m < 40, and the thermal conductivity effect of a particle begins to become apparent above about d 0.2 p,m. 

Friedlander (1977) gives for thermophoretic velocity at large Knudsen numbers the equation... 

For dp p the mechanism of particle transport in a temperature gradient is easy to understand Particles are bombarded by higher-energy molecules on their hot" side and thus driven toward the tower temperature zone. Their thermophoretic velocity can be calculated from the kinetic theory of gases (Waldmann and Schmitt. 1966) ... 

It is more difficult to explain the motion of particles that are larger than the mean free path. The explanation Is based on the tangential slip velocity that develops at the surface of a particle in a temperature gradient (Kennard, 1938). This creep velocity is directed toward the high-temperature side, propelling the particle in the direction of lower temperature. An expression for the thermophoretic velocity based on the continuum equations of fluid mechanics with slip-corrected boundary conditions was derived by Brock (1962). Talbot et al. (1980) proposed an interpolation formula for the thermophoretic velocity... 

Values of the dimensionles.s thermophoretic velocity are. shown in Fig. 2.9 as a function of the Knudsen number with kg/kp as a parameter. For Knudsen numbers larger than unity, the dependence of the dimensionless thermophoretic velocity on particle size and chemical nature Is small. Particle sampling by theimophoresis in this range offers the advantage that particles are not selectively deposited according to size. 

Thermophoretic velocities have been measured for single particles suspended in a Millikan-lype cell with controlled electrical potential and temperature gradients. Particle diameters are usually larger than about 0.8 pm, for convenient optical observation. 

Figure 2.9 Dimensionless thermophoretic velocity eakuiated fTOin (2.56), an interpolation formula that closely approaches theoretical limits for large and small Kn. For Kn > 1 (panicles smaller tliun the mean free path), the velocity becomes nearly independent of the particle material.

For a pseudo-steady state, the thermophoretic velocity pth can be obtained using (8.76),... 

Note that the thermophoretic velocity is proportional to the heat flux kgVT and that it goes to zero as A /j —> oo. 

For the two main processes determining aerosol stability, sedimentation was discussed already in Section 2.5.1, and aggregation is discussed further in Section 5.5. For quite large particles, and Kn<, Stokes law describes the sedimentation process. Thermophoretic velocity is not strongly influenced by interactions between the aerosol particles and molecules of the gas. 

The thermophoretic velocity u is defined as u = — DtVT, the coefficient Dj- is the thermal diffusion coefficient, and VT is the temperature gradient either imposed externally or induced internally by some means. The determinatiOTi of thermal diffusion coefficient is the starting point of any thermophoresis study. 

Different from the afore-described three methods, the fluorescence detection method allows direct visualizatitm of the motion of fluorescent-tagged particles suspended in solution, without the necessity of specifically optical fixtures or designed test cells. Indeed, this method appears to be the only method directly measuring the thermophoretic velocity of dilute particles. However, its practical applications are limited by its fluorescent particles requirement. 

It would go too far to derive the complete set of equations here so we suffice with the thermophoretic velocity given by Brenner [6]. The thermophoretic velocity t/ of a sphere is... 

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]. 

DERJAGUIN and co-workers have offered the following explanation for the differences between their experimental data for the measured thermophoretic velocity and Millikan-cell measurements of thermal force. The values of thermophoretic velocity determined by DERJAGUIN and co-workers are usually from two to four times the corresponding values obtained by the Millikan-cell method. DERJAGUIN and coworkers have claimed that there are uncontrolled convection currents in the Millikan cell and cite the experiments of PARANJPE [2.142] who observed convection currents in the central portion between two parallel plates when the ratio of plate diameter to separation between the plates was less than 5 or 6. There are several reasons to discount this explanation. GIESEKE [2.139] has published experimental... 

It is still necessary to explain the large differences between the Millikancell thermal-force data and the thermophoretic-velocity measurements, principally of DERJAGUIN and co-workers. One possible explanation, which might explain some of the difference, lies in the theory and experiment of PHILLIPS [2.128]. PHILLIPS... 

---