Continuum flow limitations

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The transport phenomena are usually modeled in continuum states for most applications – the materials are assumed to be continuous and the fact that matter is made of atoms is ignored. Recent development in fabrication and utilization of nanotechnology, micro devices, and microelectromechanical systems (MEMS) requires noncontinuum modeling of transport phenomena in nano- and microchannels. When the characteristic dimension, $L$ , is small compared to the molecular mean free path, λ, which is defined as average distance between collisions for a molecule, the traditional Navier-Stokes equation and the energy equation based on the continuum assumption have failed to provide accurate results. The continuum assumption also fails when the gas is at very low pressure (rarefied).

The continuum assumption may also not be valid in conventional sized systems – for example, the early stages of high-temperature heat pipe startup from a frozen state (Cao and Faghri, 1993) and microscale heat pipes (Cao and Faghri, 1994). During the early stage of startup of high-temperature heat pipes, the vapor density in the heat pipe core is very low and partly loses its continuum characteristics. The vapor flow in this condition is usually referred to as rarefied vapor flow. 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 caused mainly by the density gradient via vapor molecular diffusion.

The validity of the continuum assumption can also be violated in micro heat pipes. As the size of the heat pipe decreases, the vapor in the heat pipe may lose its continuum characteristics. The heat transport capability of a heat pipe operating under noncontinuum vapor flow conditions is very limited, and a large temperature gradient exists along the heat pipe length. This is especially true for miniature or micro heat pipes, whose dimensions may be extremely small.

The continuum criterion is usually expressed in terms of the Knudsen number

${\rm{Kn}} = \frac{\lambda }{L} \qquad\qquad(1)$

Based on the degree of rarefaction of gas or the gases size, the flow regimes in various devices can be classified into four regimes:

Continuum regime ( $Kn < 0.001$ ). The Navier-Stokes and energy equations are valid (with no-slip/no jump boundary conditions).
Slip flow regime ( $0.001 < Kn < 0.1$ ). The Navier-Stokes and energy equations can be used with the application of slip or jump boundary conditions, i.e., allowing non-zero axial fluid velocity near the wall of the object.
Transition regime ( $0.1 < Kn < 10$ ). The Navier-Stokes equation is not valid, and the flow must be solved using molecular based models such as the Boltzmann equation or Direct Simulation Monte Carlo (DSMC).
Free molecular flow regime ( $Kn > 10$ ). The collision between molecules can be neglected and a collisionless Boltzmann equation can be used.

In the slip flow regime, the slip boundary condition refers to circumstance when the tangential velocity of the fluid at the wall is not the same as the wall velocity. Temperature jump is similarly defined as when the temperature of the fluid next to the wall is not the same as the wall temperature.

The mean free path for dilute gases based on the kinetic theory can be rewritten in terms of temperature and pressure

$\lambda = \frac{{1.051{k_b}T}}{{\sqrt 2 \pi {\sigma ^2}p}} \qquad\qquad(2)$

The transition density under which the continuum assumption is invalid can be obtained by combining eqs. (1) and (2) and $Kn = 0.001$ i.e.,

${\rho _{tr}} = \frac{{1.051{k_b}}}{{\sqrt 2 \pi {\sigma ^2}{R_g}D{\rm{Kn}}}} \qquad\qquad(3)$

where the ideal gas equation of state, $p = ρ R g T$ was used. Assuming that the vapor is in the saturation state, the transition vapor temperature $T t r$ corresponding to the transition density can be obtained by using the Clausius-Clapeyron equation (see Chapter 2) combined with the equation of state:

${T_{tr}} = \frac{{{p_{sat}}}}{{\rho {R_g}}}\exp \left[ { - \frac{{{h_{\ell v}}}}{{{R_g}}}\left( {\frac{1}{{{T_{tr}}}} - \frac{1}{{{T_{sat}}}}} \right)} \right] \qquad\qquad(4)$

where $P s a t$ and $T s a t$ are the saturation pressure and temperature, ${h_{\ell v}}$ is the latent heat of vaporization, and the vapor density $P$ is given by eq. (3). Equation (4) can be rewritten as

$\ln \left( {\frac{{{T_{tr}}\rho {R_g}}}{{{p_{sat}}}}} \right) + \frac{{{h_{\ell v}}}}{{{R_g}}}\left( {\frac{1}{{{T_{tr}}}} - \frac{1}{{{T_{sat}}}}} \right) = 0 \qquad\qquad(5)$

and solved iteratively for $T t r$ using the Newton-Raphson/secant method. The transition vapor temperature is the boundary between the continuum and noncontinuum regimes.

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.

Cao, Y., and Faghri, A., 1994, “Micro/Miniature Heat Pipes and Operating Limitations,” Journal of Enhanced Heat Transfer, Vol. 1, pp. 265-274.

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.

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Continuum flow limitations

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