Thermal transpiration

As shown previously (Section 14.7), thermal transpiration or thermomole-cular flow arises when gas is contained in two vessels at equal pressures but different temperatures and a connection is made between the vessels. A vacuum microbalance represents just such a system. The long tube surrounding the hang-down wire separates the sample, immersed in the coolant bath from the warm upper portion of the apparatus. With vacuum microbalances the effects of thermal transpiration are further exacerbated by the temperature gradient along the hang-down wire. 

With nitrogen the data required for BET analysis or pore volumes lies above the thermal transpiration region. 

When using microbalances for adsorption measurements, those adsorbates which do not require thermal transpiration corrections are the most susceptible to buoyancy errors while those adsorbates not requiring buoyancy corrections, such as krypton, because of its low vapor pressure, are most susceptible to thermal transpiration errors. 

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Thermal transpiration and thermal diffusion effects have been neglected in developing the dusty gas model, and will be neglected throughout the rest of the text. The physics of these phenomena and the justification for neglecting them are discussed in some detail in Appendix I. 

Thermal transpiration and thermal diffusion will not be considered here, but it would be incorrect to assume that their influence is negligible, or even small in all circumstances. Recent results of Wong et al. [843 indi cate that they may Influence computed values of the effectiveness factor iby as much as 30. An account of thermal transpiration and thermal diffu-Ision is given in Appendix I. 

Equations (12.13) and (12.14) are themselves approximations, since in Chapter 3 we eliminated thermal transpiration and diffusion from consideration before constructing the dusty gas model flux relations. See Appendix 1. 

The phenomenon of thermal transpiration was discovered by Osborne Reynolds [82], who gave a clear and detailed description of his experiments, together with a theoretical analysis, in a long memoir read before the Royal Society in February of 1879. He experimented with porous plates of stucco, ceramic and meerschaum and, in the absence of pressure gradients, found that gas passes through the plates from the colder to the hotter side. His experimental findings were summarized in the following "laws" of thermal transpiration. 

In the second part of hla memoir Reynolds gave a theoretical account of thermal transpiration, based on the kinetic theory of gases, and was able CO account for Che above "laws", Chough he was not able to calculate Che actual value of the pressure difference required Co prevent flow over Che whole range of densities. ... 

At very low densities It Is quite easy Co give a theoretical description of thermal transpiration, alnce the classical theory of Knudsen screaming 9] can be extended to account for Che Influence of temperature gradients. For Isothermal flow through a straight capillary of circular cross-section, a well known calculation [9] gives the molar flux per unit cross-sectional area, N, In the form... 

Now in the present case i/a 1, so it follows from equation (A.1.7) that G/a 1, provided f differs significantly from zero. Thus che first term on the right hand side of (A.L.8) is a close approximation to the familiar Poisoiille flux. The second term, on the other hand, represents thermal transpiration. In particular, setting N 0, we find... 

This point was taken up by Reynolds in a letter addressed to G. G. Stokes, in the latter s capacity as Secretary of the Royal Society [83]. Reynolds pointed out that Maxwell s theory evaluated the effects of thermal transpiration only in circumstances where they were too small to be measured, and complained that Maxwell had misrepresented his own theoretical treat ment of the phenomenon. However, this incipient controversy never developed... 

It is also interesting to examine the relative importance of thermal transpiration and thermal diffusion in the two limiting cases. From equations (A. 1.12) and (A. 1.13)... 

Finally, let us return to the question of the practical importance of thermal diffusion and thermal transpiration in modeling reactive catalyst... 

Tables 2,3, and 4 outline many of the physical and thermodynamic properties ofpara- and normal hydrogen in the sohd, hquid, and gaseous states, respectively. Extensive tabulations of all the thermodynamic and transport properties hsted in these tables from the triple point to 3000 K and at 0.01—100 MPa (1—14,500 psi) are available (5,39). Additional properties, including accommodation coefficients, thermal diffusivity, virial coefficients, index of refraction, Joule-Thorns on coefficients, Prandti numbers, vapor pressures, infrared absorption, and heat transfer and thermal transpiration parameters are also available (5,40). Thermodynamic properties for hydrogen at 300—20,000 K and 10 Pa to 10.4 MPa (lO " -103 atm) (41) and transport properties at 1,000—30,000 K and 0.1—3.0 MPa (1—30 atm) (42) have been compiled. Enthalpy—entropy tabulations for hydrogen over the range 3—100,000 K and 0.001—101.3 MPa (0.01—1000 atm) have been made (43). Many physical properties for the other isotopes of hydrogen (deuterium and tritium) have also been compiled (44).

Ochoa, F., Eastwood, C., Ronney, P. D., and Dunn, B., (2003) Thermal Transpiration Based Microscale Propulsion and Power Generation Devices, 7th International Workshop on Microgravity Combustion and Chemically Reacting Systems, NASA/CP-2003-21376, 2003. 

When using adsorbates with low vapor pressure no correction for nonideality is necessary. However, another type of correction is required due to thermal transpiration which occurs if two parts of a system contain the same gas at the same pressure but at different temperatures. Thermal transpiration can be understood by considering the number of collisions n per square centimeter per second which a gas makes with the walls of its container... 

For the effects of thermal transpiration to be observed it is necessary that the mean free molecular path be greater than the tube diameter. At higher pressures and smaller mean free paths, molecular collisions destroy the effect. 

With krypton, the ability to use larger samples of low-area powders facilitates measuring low surface areas because larger signals are generated in the absence of thermal diffusion. Also, as is true for nitrogen, krypton measurements do not require void volume, or ideality corrections, nor is thermal transpiration a factor as in the volumetric measurements. 

When using beam microbalances the effects due to buoyancy and thermal transpiration must be considered. The gas displaced by the sample produces a buoyancy force that reduces the weight by exactly the mass of... 

Both the volumetric and gravimetric apparatus are subject to this effect. Data acquired at low pressures must be corrected for thermal transpiration. The continuous flow method is not subject to this phenomenon. 

Crowell and Young (15) compare their differential heats of adsorption with those of Jura and Criddle (38) and conclude that there were errors in the low surface coverage region of the latter investigation due to the effect of thermal transpiration on pressure measurement. 

As a consequence of molecular considerations, when two systems are connected for transfer of mass without significant transfer of energy, such as two containers at different temperatures connected by a capillary tube, we have the relation of thermal transpiration. 

B. Thermal Transpiration. In most tensimetry, it is correctly assumed that the pressure at the manometer is equal to the pressure in the sample container to... 

Thermal transpiration is discussed in many books on vacuum technology. See, for example S. Dushman, Scientific Foundations oj Vacuum Technique, 2nd ed.t rev. by J. M. Laffcrty et al. (New York Wiley, 1962). 

We see that establishing a temperature difference between the chambers has resulted in a density and pressure difference The movement of material due to a temperature differential is known as thermal diffusion, and in the special case where the flow is molecular, it is known as thermal transpiration. 

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