# Effects of Noncondensable Gas on Film Condensation

Figure 1 Mass transfer in the equivalent laminar film.

When non-condensable gas gas is present, heat and mass transfer in the vapor phase must be studied in addition to heat transfer in the liquid film (see Fig. 1). There exists a boundary layer in the vapor phase (δ < y < δ + Δ), in which the partial pressure of the condensable vapor, pv, decreases from a constant value ${{p}_{v,\infty }}$ at y = δ + Δ to the value pv,δ at y = δ, where the vapor is condensing to a liquid. The condensing vapor must diffuse through this vapor boundary layer to the liquid-vapor interface. The partial pressure of the noncondensable gas, pg, on the other hand, increases from its reservoir value, ${{p}_{g,\infty }}$, to the value pg,δ at the liquid-vapor interface. At any point in space and time the summation of the partial pressures of this binary system must equal the constant total pressure.

 pv + pg = p (1)

The partial pressure of the condensing gas decreases as it approaches the phase interface, and its corresponding saturation temperature Tsat(pv) also falls. Depending on the noncondensable gas content, the temperature at the interface can be much lower than if no such gas were present. The temperature difference across the interface would also be lower as a result, which would lead to a lower overall heat transfer coefficient. This clearly demonstrates the benefit of removing as much noncondensable gas from the system as possible. However, systematic purity cannot always be achieved, and the noncondensable gas content must be taken into account.

The mass transfer in the vapor boundary layer can be rigorously analyzed by solving the boundary-layer type of governing equations. Hewitt et al. (1994) presented an alternative model based on the concept of equivalent laminar film – a layer in the vapor phase in which the mass fraction of the condensable vapor, ωv, varies linearly from ωv,δ at the liquid-vapor interface to ${{\omega }_{v,\infty }}$ at y = δ + δv (see Fig. 2). Meanwhile, the partial pressure of the condensable vapor changes from pv to ${{p}_{v,\infty }}$ in the equivalent laminar film. Assuming that the noncondensable gas cannot be dissolved into the liquid condensate, the molar flux of the condensable vapor at any point within the equivalent laminar layer can be obtained by (see governing equation for condensation of binary mixture

 ${{\dot{{n}''}}_{v}}=-{{D}_{vg}}\frac{\partial {{c}_{v}}}{\partial y}+{{\dot{{n}''}}_{v}}{{x}_{v}}$ (2)

which can be rearranged to obtain

 ${{\dot{{n}''}}_{v}}=-\frac{{{D}_{vg}}}{1-{{c}_{v}}/{{c}_{T}}}\frac{\partial {{c}_{v}}}{\partial y}$ (3)

where cT is total molar concentration of vapor and noncondensable gas.

Integrating eq. (3) over the equivalent laminar layer and considering ${{\dot{{n}''}}_{v}}\text{ and }{{c}_{T}}$ are constants, one obtains

 ${{\dot{{n}''}}_{v}}\int_{\delta }^{\delta +{{\delta }_{v}}}{dy}=-{{D}_{vg}}{{c}_{T}}\int_{{{c}_{v,\delta }}}^{{{c}_{v,\infty }}}{\frac{1}{{{c}_{T}}-{{c}_{v}}}d{{c}_{v}}}$ (4)

Equation (4) can be rearranged to yield

 ${{\dot{{n}''}}_{v}}=\frac{{{c}_{T}}{{D}_{vg}}}{{{\delta }_{v}}}\ln \left( \frac{{{c}_{T}}-{{c}_{v,\delta }}}{{{c}_{T}}-{{c}_{v,\infty }}} \right)={{c}_{T}}{{h}_{m,G}}\ln \left( \frac{{{c}_{T}}-{{c}_{v,\delta }}}{{{c}_{T}}-{{c}_{v,\infty }}} \right)$ (5)

where

 ${{h}_{m,G}}=\frac{{{D}_{vg}}}{{{\delta }_{v}}}$ (6)

is mass transfer coefficient (m/s). Since the thickness of equivalent laminar film, δG, is still unknown at this point, the mass transfer coefficient can be approximately related to the heat transfer coefficient, hG, through the Lewis equation, i.e.,

 ${{h}_{m,}}_{G}=\frac{1}{{{\rho }_{vg}}}\frac{{{h}_{G}}}{{{c}_{p,vg}}}$ (7)

where cp,vg is the constant-pressure specific heat of the vapor-gas mixture.

If the condensable vapor and noncondensable gas can be treated as ideal gas, the molar flux in eq. (5) can also be expressed in term of mass flux, ${{{\dot{m}}''}_{v}}$, and partial pressure of the condensable vapor, pv, in the mixture

 ${{{\dot{m}}''}_{v}}={{\rho }_{vg}}{{h}_{m,G}}\ln \frac{p-{{p}_{v,\delta }}}{p-{{p}_{v,\infty }}}$ (8)
Figure 2 Energy balance at the liquid-vapor interface for film condensation on a vertical plate including the effects of non-condesable gases.

The energy balance across a differential control volume at the liquid-vapor interface, as shown in Fig. 2, is (Stephan, 1992)

 ${{h}_{\ell }}({{T}_{\delta }}-{{T}_{w}})={{{\dot{m}}''}_{v}}{{h}_{\ell v}}+{{h}_{G}}({{T}_{vg}}-{{T}_{\delta }})$ (9)

where hG is the heat transfer coefficient from the vapor-gas mixture to the liquid-vapor interface, and where Tδ is the temperature at the liquid-vapor interface.

Substituting eqs. (7) and (8) into the energy balance equation (9) and using hvg = ξhG, which corrects for the fact that vapor does not flow along the wall but stops at the liquid-vapor interface (where ξ is a correction factor), the following is obtained (Stephan, 1992):

 ${{T}_{\delta }}-{{T}_{w}}=\frac{{{h}_{G}}}{{{h}_{\ell }}}\left[ \frac{{{h}_{\ell v}}}{{{c}_{p,vg}}}\ln \frac{p-{{p}_{v,\delta }}}{p-{{p}_{v,res}}}+\xi \left( {{T}_{vg}}-{{T}_{\delta }} \right) \right]$ (10)

In the case of small inert gas content, the above equation will reduce to

 ${{T}_{\delta }}-{{T}_{w}}=\frac{{{h}_{G}}}{{{h}_{\ell }}}\left[ \frac{{{h}_{\ell v}}}{{{c}_{p,vg}}}\ln \frac{p-{{p}_{v,\delta }}}{p-{{p}_{v,\infty }}} \right]$ (11)

In the best-case scenario where there is no noncondensable gas in the vapor, the heat flux across the liquid film would be as follows:

 ${q}''=\frac{{{k}_{\ell }}}{\delta }\left( {{T}_{sat}}-{{T}_{w}} \right)$ (12)

However, if a noncondensable gas is present, the temperature drop across the film would be lower and the heat flux would be as follows:

 ${{{q}''}_{vg}}=\frac{{{k}_{\ell }}}{\delta }\left( {{T}_{\delta }}-{{T}_{w}} \right)$ (13)

where q''vg is the heat flux across the liquid film in the presence of a noncondensable gas. The ratio of the heat fluxes obtained by eqs. (13) and (12) is

 $\frac{{{{{q}''}}_{vg}}}{{{q}''}}=\frac{{{T}_{\delta }}-{{T}_{w}}}{{{T}_{sat}}-{{T}_{w}}}\le 1$ (14)

Substituting the expression for temperature difference across a liquid film in the presence of a noncondensable gas – eq. (11) – into the above ratio, the following is obtained:

 $\frac{{{{{q}''}}_{vg}}}{{{q}''}}=\frac{{{h}_{G}}{{h}_{\ell v}}}{\left( {{T}_{sat}}-{{T}_{w}} \right){{h}_{\ell }}{{c}_{vg,\infty }}}\ln \frac{p-{{p}_{v,\delta }}}{p-{{p}_{v,\infty }}}$ (15)

or, substituting ${{\dot{m}}_{cv}}$ from eq. (8):

 $\frac{{{{{q}''}}_{vg}}}{{{q}''}}=\frac{{{{{\dot{m}}''}}_{v}}{{h}_{\ell v}}}{({{T}_{sat}}-{{T}_{w}}){{h}_{\ell }}}$ (16)

It can be seen from the above expression that for large (TsatTw), the velocity or mass flow rate ${{\dot{m}}_{cv}}$, must be made sufficiently large to acquire a large heat transfer coefficient for the heat transfer from the vapor-gas mixture to the liquid-vapor interface. This must be done in order to make q''vg not too small and therefore remove the undesirable effects of the noncondensable gas as much as possible.

The effect of non-condensable gas on condensation in a stagnant vapor is very significant. For forced convective condensation, the effect of non-condensable gas is still significant, but is much weaker than its effect on the stagnant vapor condensation. Therefore, the effect of non-condensable gas on condensation can be minimized by allowing vapor to flow.

## References

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.

Hewitt, G.F., Bott, T.R., Shires, G.L., 1994, Process Heat Transfer, CRC Press, Boca Raton, FL

Stephan, K., 1992, Heat Transfer in Condensation and Boiling, Springer-Verlag, Berlin