# Interfacial resistance in vaporization and condensation

High-heat transfer coefficients, typically associated with evaporation and condensation processes in heat transfer devices, are restricted by interfacial resistance. When condensation occurs at an interface, the flux of vapor molecules into the liquid must exceed the flux of liquid molecules escaping to the vapor phase. When evaporation occurs, on the other hand, the flux of liquid molecules escaping to the vapor phase must exceed the flux of vapor molecules into the liquid. Schrage (1953) used the kinetic theory of gases to describe condensation and evaporation processes and considered separately the fluxes of condensing and vaporizing molecules in each direction.

From kinetic theory, the mass flow rate (of molecules) passing in either direction (right or left) through an imagined plane is given by

 $\left| {{{\dot{m}}}_{\delta }} \right|={{\left( \frac{{{M}_{v}}}{2\pi {{R}_{u}}} \right)}^{1/2}}\frac{p}{{{T}^{1/2}}}$ (1)

where ${{\dot{{m}''}}_{\delta }}$ is the flux of molecules, Ru is the universal gas constant, and Mv is the molecular mass of the vapor. The net molecular flux through an interface is

 ${{\dot{{m}''}}_{\delta }}={{\dot{{m}''}}_{\delta +}}-{{\dot{{m}''}}_{\delta -}}$ (2)

Actually, the Boltzmann transport equation should be solved with appropriate boundary conditions of thermal equilibrium at several mean free path distances from the interface. However, a reasonable approximation can be obtained by means of correction factors if it is assumed that the interaction between molecules leaving the interface and those approaching was at equilibrium. This obtains the following relation for the net mass flux at the interface:

 ${{{\dot{m}}''}_{\delta }}=\frac{{{{{q}''}}_{\delta }}}{{{h}_{\ell v}}}=\alpha \sqrt{\frac{{{M}_{v}}}{2\pi {{R}_{u}}}}\left( \frac{\Gamma {{p}_{v}}}{\sqrt{{{T}_{v}}}}-\frac{{{p}_{\ell }}}{\sqrt{{{T}_{\ell }}}} \right)$ (3)

where α is the accommodation coefficient ($\alpha \le 1$), and the function is given by Schrage (1953):

 $\Gamma \left( a \right)=\exp \left( {{a}^{2}} \right)+a\sqrt{\pi }\left[ 1+\text{erf}\left( a \right) \right]$ (4)

 $\Gamma \left( -a \right)=\exp \left( {{a}^{2}} \right)-a\sqrt{\pi }\left[ 1-\text{erf}\left( a \right) \right]$ (5)

where

 $a=\frac{{{{{q}''}}_{\delta }}}{{{\rho }_{v}}{{h}_{\ell v}}}\sqrt{\frac{{{M}_{v}}}{2{{R}_{u}}{{T}_{v}}}}$ (6)

and

 $\text{erf}\left( a \right)=\frac{2}{\pi }\int_{0}^{a}{{{e}^{-{{x}^{2}}}}dx}$ (7)

is the Gaussian error function.

The heat flux to the interface is equal to the net mass flux multiplied by the latent heat $\left( {{{{q}''}}_{\delta }}={{{\dot{{m}''}}}_{\delta }}{{h}_{\ell v}} \right)$. Since Γ is a function of q''δ, eq. (3) does not provide an explicit relation for the interfacial heat flux. Assuming that ${{p}_{\ell }}$ and pv are the saturation pressures corresponding to ${{T}_{\ell }}$ and Tv, eq. (3) can be represented in the following form:

 ${{{q}''}_{\delta }}=\alpha {{h}_{\ell v}}\sqrt{\frac{{{M}_{v}}}{2\pi {{R}_{u}}}}\left[ \frac{\Gamma {{p}_{\text{sat}}}\left( {{T}_{v}} \right)}{\sqrt{{{T}_{v}}}}-\frac{{{p}_{\text{sat}}}\left( {{T}_{\ell }} \right)}{\sqrt{{{T}_{\ell }}}} \right]$ (8)

For evaporation and condensation of working fluids at moderate and high temperatures, a is usually very small according to its definition [see eq. (6)]. In such a case, eq. (2) can be approximated by

 $\Gamma =1+a\sqrt{\pi }$ (9)

An explicit relation for q''δ and ${{{\dot{m}}''}_{\delta }}$ was obtained by Silver and Simpson (1961) by substituting eq. (9) into eq. (3) and using ρv = pvMv / RuTv:

 ${{{\dot{m}}''}_{\delta }}=\frac{{{{{q}''}}_{\delta }}}{{{h}_{\ell v}}}=\left( \frac{2\alpha }{2-\alpha } \right)\sqrt{\frac{{{M}_{v}}}{2\pi {{R}_{u}}}}\left( \frac{{{p}_{v}}}{\sqrt{{{T}_{v}}}}-\frac{{{p}_{\ell }}}{\sqrt{{{T}_{\ell }}}} \right)$ (10)

which is referred to as the Kucherov-Rikenglaz equation (Kucherov and Rikenglaz, 1960) in the Soviet literature.

One can develop an alternative form of eq. (8) for small a by assuming that $\left( {{p}_{v}}-{{p}_{\ell }} \right)/{{p}_{v}}<<1$, $\left( {{T}_{v}}-{{T}_{\ell }} \right)/{{T}_{v}}<<1$, and by using the Clausius-Clapeyron relation.

 ${{{q}''}_{\delta }}=\left( \frac{2\alpha }{2-\alpha } \right)\left( \frac{h_{\ell v}^{2}}{{{T}_{v}}{{v}_{\ell v}}} \right)\sqrt{\frac{{{M}_{v}}}{2\pi {{R}_{u}}{{T}_{v}}}}\left( 1-\frac{{{p}_{v}}{{v}_{\ell v}}}{2{{h}_{\ell v}}} \right)\left( {{T}_{v}}-{{T}_{\ell }} \right)$ (11)

Using the above relation, the heat transfer coefficient at the interface hδ is obtained from the following equation:

 ${{h}_{\delta }}=\frac{{{{{q}''}}_{\delta }}}{\left( {{T}_{v}}-{{T}_{\ell }} \right)}=\left( \frac{2\alpha }{2-\alpha } \right)\left( \frac{h_{\ell v}^{2}}{{{T}_{v}}{{v}_{\ell v}}} \right)\sqrt{\frac{{{M}_{v}}}{2\pi {{R}_{u}}{{T}_{v}}}}\left( 1-\frac{{{p}_{v}}{{v}_{\ell v}}}{2{{h}_{\ell v}}} \right)$ (12)

If hδ is of the same order of magnitude as the other h values, the effects of interfacial resistances should be accounted for. It should also be noted that the above equations relating q''δ, hδ, and $\left( {{T}_{v}}-{{T}_{\ell }} \right)$ apply equally well to evaporation and to condensation, with the convention that q''δ is positive for condensation and negative for evaporation. It is clear that predicting the interfacial resistances using of any of the above equations depends on the value of the accommodation coefficient α, which varies widely in the literature. Paul (1962) compiled the accommodation coefficients for evaporation for a large number of working fluids. Mills (1965) recommended that α should be less than unity when the working fluids or the interface is contaminated.

## References

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

Kucherov, R.Y., and Rikenglaz, L.E., 1960, “The Problem of Measuring the Condensation Coefficient,” Doklady Akad. Nauk. SSSR, Vol. 133, pp. 1130-1131.

Paul, B., 1962, “Compilation of Evaporation Coefficients,” ARSJ, Vol. 32, pp. 1321-1328.

Mills, A.F. 1965, The Condensation of Steam at Low Pressures, Report No. NSF GP-2520, Series No. 6, Issue No. 39, Space Sciences Laboratory, University of California at Berkeley.

Schrage, R.W., 1953, A Thermal Study of Interface Mass Transfer, Columbia University Press, New York.