Mean thermal molecular velocity

The thermal conductivity coefficient has been derived from Browniah motion theory by Irving and Kirkwood33 in terms of the equilibrium singlet and pair distribution functions °/a) and °/Thermal conduction under a macroscopic temperature difference involves a gradient in the mean square molecular velocity rather than in the mean molecular velocity. The steady-state radial distribution function then remains spherically symmetric except for a small correction arising from the number density variation with the temperature. As the analysis introduces no new assumptions and is somewhat lengthy, it will not be reproduced here. The resulting equation for the thermal conductivity coefficient x is... 

The translational energy of one mole of gas is given by 3/2RT, which corresponds to an average thermal molecular velocity v (the root mean square velocity), while the most probable velocity v = /(0.67) v. 

Here kg is the Boltzmann constant (1.38 x 10" Joule/K/molecule), T is in K, and m is the molecular mass. The thermal molecular speed increases with temperature and decreases with molecular mass. To have a feel of how fast a molecule moves, we take the case of helium at ambient temperature (T = 298 K), the mean thermal molecular speed is 1250 m/s, showing the very high random thermal velocity of molecules. 

From the technology of combustion we move to the molecular mechanism of flame propagation. We shall give a molecular-kinetic expression for the heat release rate by calculating the frequency v of collisions of fuel molecules with other molecules (v is proportional to the molecular velocity and inversely proportional to the mean free path), further taking into account that only a small (1/j/) part of all collisions are effective. The quantity 1/v—the probability of reaction taken with respect to a single collision— depends on the activation heat of an elementary reaction event, as well as on the fraction of all molecules comprised of those radicals or atoms by means of which the reaction occurs. The molecular-kinetic expression for the coefficient of thermal conductivity follows from formulas (1.2.4) and (1.2.3). 

Inasmuch as mean molecular velocity is given by U = Jlc, Eq. 3.14, the velocity induced by thermal diffusion is simply... 

In this section we shall be concerned with a molecular theory of the transport properties of gases. The molecules of a gas collide with each other frequently, and the velocity of a given molecule is usually changed by each collision that the molecule undergoes. However, when a one-component gas is in thermal and statistical equilibrium, there is a definite distribution of molecular velocities—the well-known Maxwellian distribution. Figure 1 shows how the molecular velocities are distributed in such a gas. This distribution is isotropic (the same in all directions) and can be characterized by a root-mean-square (rm speed u, which is given by... 

Calculating the mean free path requires assumptions about intermolecular collisions. First, the gas is assumed to be a dilute gas, i.e., a gas for which the mean molecular diameter of the molecules is small compared with the mean molecular spacing. This condition is verified with a good precision for all usual gases under standard pressure and temperature conditions. It involves binary collisions between molecules, and the mean free path may be expressed as X = c j = sJ%RTl%j, ratio of the mean thermal velocity c to the collision rate v. The mean thermal velocity depends on the temperature T and on the specific gas cmistant R. 

We can write the tangential momentum flux on a surface s located near the wall as equal to Here, is the number density of molecules crossing surface Y m is the molecular mass is the tangential (slip) velocity on the surface and Vg is the mean thermal speed of the molecule. 

From this formula it is seen that to calculate h we need to determine the mean molecular speed ( ci )a/- For real systems the average molecular speed is difficult to determine. Assuming that the system is sufficiently close to equilibrium the velocity distribution may be taken to be Maxwellian. For molecules in the absolute Maxwellian state the peculiar velocity equals the microscopic molecular velocity, i.e.. Cl = Cl, because the macroscopic velocity is zero vi = 0, hence it follows that the speed of the microscopic molecular velocity equals the thermal speed C )m = ( Ci )m = c )m = ( ci )m-... 

The methods used to estimate gas mixture thermal conductivity are less accurate as compared to empirical data than those used to estimate gas mixture viscosity. Thermal conductivity is more susceptible to variations in the size of the molecules, variations in mass, concentration dependence, and temperature dependence (Singh, Dham, and Gupta, 1992). Additionally, the kinetic theory assumes a single average of the mean molecular velocities deviations from actual thermal conductivity may be accounted for in the wide spectrum of molecular velocities in a mixture (Poling, Prausnitz, and O Connell, 2001). Therefore, there is a greater expected error in HeXe gas mixture thermal conductivities than in gas mixture viscosities. 

Extracted from U.S. Standard Atmosphere, 1976, National Oceanic and Atmospheric Administration, National Aeronautics and Space Administration and tte U.S. Air Force, Washington, 1976. Z = geometric altitude, T = temperature, P = pressure, g = acceleration of gravity, M = molecular weight, a = velocity of sound, i = viscosity, k = thermal conductivity, X = mean free path, p = density, and H = geopotential altitude. The notation 1.79.—5 signifies 1.79 X 10 . ... 

In these equations x and y denote independent spatial coordinates T, the temperature Tib, the mass fraction of the species p, the pressure u and v the tangential and the transverse components of the velocity, respectively p, the mass density Wk, the molecular weight of the species W, the mean molecular weight of the mixture R, the universal gas constant A, the thermal conductivity of the mixture Cp, the constant pressure heat capacity of the mixture Cp, the constant pressure heat capacity of the species Wk, the molar rate of production of the k species per unit volume hk, the speciflc enthalpy of the species p the viscosity of the mixture and the diffusion velocity of the A species in the y direction. The free stream tangential and transverse velocities at the edge of the boundaiy layer are given by = ax and Vg = —ay, respectively, where a is the strain rate. The strain rate is a measure of the stretch in the flame due to the imposed flow. The form of the chemical production rates and the diffusion velocities can be found in (7-8). 

For the other extreme of the free molecular regime where Kn - oc, the particle radius is small compared to the mean free path. In this case, the thermal velocity distribution of the gas is not distorted by uptake at the surface. In effect, the gas molecules do not see the small particles. For this case, Fuchs and Sutugin (1970, 1971) show that for diffusion to a spherical particle of radius a... 

Random motion is ubiquitous. At the molecular level, the thermal motions of atoms and molecules are random. Further, motions in macroscopic systems are often described by random processes. For example, the motion of stirred coffee is a turbulent flow that can be characterized by random velocity components. Randomness means that the movement of an individual portion of the medium (i.e., a molecule, a water parcel, etc.) cannot be described deterministically. However, if we analyze the average effect of many individual random motions, we often end up with a simple macroscopic law that depicts the mean motion of the random system (see Box 18.1). 

The kinetic theory of gases was briefly discussed. It enables the mean or thermal velocity (c) of gas molecules at a given temperature to be obtained and gas flux to be calculated. From the latter, effusion rates, area-related condensation rates and conductances under molecular flow can be determined (see Examples 1.5 and 1.7-1.10). Calculation of collision frequency (obtained from c, n and the collision cross-section of molecules), enables the mean free path (f) of particles to be determined. The easily obtained expression for Ip is a convenient way of stating the variation of / withp (Examples 1.11-1.15). 

The Prandtl number via has been found to be the parameter which relates the relative thicknesses of the hydrodynamic and thermal boundary layers. The kinematic viscosity of a fluid conveys information about the rate at which momentum may diffuse through the fluid because of molecular motion. The thermal diffusivity tells us the same thing in regard to the diffusion of heat in the fluid. Thus the ratio of these two quantities should express the relative magnitudes of diffusion of momentum and heat in the fluid. But these diffusion rates are precisely the quantities that determine how thick the boundary layers will be for a given external flow field large diffusivities mean that the viscous or temperature influence is felt farther out in the flow field. The Prandtl number is thus the connecting link between the velocity field and the temperature field. 

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