# Rarefied vapor self-diffusion model

The discussions so far are limited to the case where the density of the fluid is sufficiently high to permit the continuum assumption. There are cases in which the assumption of continuum is not valid (${\rm{Kn}} \ge 0.001$). For example, in the early stage of heat pipe startup from the frozen state, the vapor pressure and density are very small in the heat pipe core. Because of the low density, the vapor in the rarefied state is somewhat different from the conventional continuum state. Also, the vapor density gradient is very large along the axial direction of the heat pipe. The vapor flow along the axial direction is mainly caused by the density gradient via vapor molecular diffusion. The low-density vapor state that has partly lost its continuum characteristics is referred to as rarefied vapor.

Neglecting the presence of noncondensable gases, the rarefied vapor flow can be simulated by a self-diffusion model. The term self-diffusion here means the interdiffusion of particles of the same mass due to a gradient in density. The governing equations for startup of a heat pipe from the frozen state were derived by applying the principles of the conservation of mass and energy in a differential cylindrical control volume in conjunction with the definition of mass flux (Cao and Faghri, 1993).

The mass self-diffusion equation is

$\frac{{\partial \rho }}{{\partial t}} - \frac{\partial }{{\partial z}}\left( {{D_v}\frac{{\partial \rho }}{{\partial z}}} \right) - \frac{1}{r}\frac{\partial }{{\partial z}}\left( {r{D_v}\frac{{\partial \rho }}{{\partial r}}} \right) = 0 \qquad \qquad(1)$

and the energy equation is

$\frac{{\partial (\rho {c_v}T)}}{{\partial t}} + \frac{1}{r}\frac{\partial }{{\partial r}}({\dot m''_r}{c_p}rT) + \frac{\partial }{{\partial r}}({\dot m''_z}{c_p}T) = \frac{1}{r}\frac{\partial }{{\partial z}}\left( {r{k_v}\frac{{\partial T}}{{\partial r}}} \right) + \frac{\partial }{{\partial z}}\left( {{D_v}\frac{{\partial T}}{{\partial z}}} \right) \qquad \qquad(2)$

where the mass fluxes ${\dot m''_r}$ and ${\dot m''_z}$ are

${\dot m''_r} = \rho v = - {D_v}\frac{{\partial \rho }}{{\partial r}} \qquad \qquad(3)$
${\dot m''_z} = \rho w = - {D_v}\frac{{\partial \rho }}{{\partial z}} \qquad \qquad(4)$

where Dv is the self-diffusion coefficient, and kv is the vapor molecular conductivity. The evaluation of low-density properties such as Dv is carried out using the kinetic theory of gases. The coefficient of self-diffusion is obtained from the relation based on the Chapman-Enskog kinetic theory (Hirschfelder et al., 1966):

${D_v} = 2.628 \times {10^{ - 7}}\frac{{\sqrt {{T^3}/MW} }}{{p{\sigma ^2}{\Omega _D}({k_b}T/\varepsilon )}} \qquad \qquad(5)$

where p is the pressure in atmospheres, σ is the collision diameter in Å, $\varepsilon$ is the maximum energy of attraction between a pair of molecules, kb is the Boltzmann constant, and ΩD is the collision integral for mass diffusion. For sodium, both σ and $\varepsilon /{k_b}$ can be found from the table for constants of the Lennard-Jones potential model (Edwards et al., 1979), which gives σ= 3.567 Å, and $\varepsilon /{k_b}$= 1375 K. The value of the collision integral ΩD can also be found from the same reference, which is listed as a function of $\varepsilon /{k_b}$.

## References

Cao, Y., and Faghri, A., 1993, “Simulation of the Early Startup Period of High Temperature Heat Pipes From the Frozen State by a Rarefied Vapor Self-Diffusion Model,” ASME Journal of Heat Transfer, Vol. 115, pp. 239-246.

Edwards, D.K., Denny, V.E., and Mills, A.F., 1979, Transfer Process, Hemisphere, New York.

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Hirschfelder, J.O., Curtiss, C.F., and Bird, R.B., 1966, Molecular Theory of Gases and Liquids, Wiley, New York.