# Filmwise condensation in a stagnant pure vapor reservoir

## Laminar Flow Regime

Figure 1 Overview of the control volume under consideration in the Nusselt analysis.

The classical analysis of laminar film condensation on a vertical or inclined wall was performed by Nusselt (1916). The physical conditions of laminar film condensation have been shown in Fig. 1. As is the case in any heat transfer analysis, the final goal is to obtain the heat transfer coefficient and the corresponding Nusselt number for the heat transfer device under consideration. Therefore, the objective of this section is to find the heat transfer coefficient and the Nusselt number for the laminar flow regime in film condensation on a vertical surface. The classical Nusselt analysis requires many assumptions in order to achieve a closed-form solution of the boundary layer type momentum equations. These assumptions include the following:

1.The flow is laminar.

2.Fluid properties are constant.

3.Subcooling of the liquid is negligible in the energy balance, i.e., all condensation occurs at the saturation temperature corresponding to the pressure in the liquid film near the wall.

4.Inertia and convection effects are negligible in the boundary layer momentum and energy equations, respectively.

5.The vapor is assumed stagnant, and therefore, shear stress is considered to be negligible at the liquid-vapor interface.

6.The liquid-vapor interface is smooth, i.e., condensate film is laminar and not in the wavy or turbulent stages.

Since the vapor phase is stationary, the governing equations for the vapor phase are no longer needed. The continuity and momentum equations for the liquid phase ard:

$\frac{\partial u_{l }}{\partial x}+\frac{\partial v_{l} }{\partial y}=0$
$u_{l}\frac{\partial u_{l }}{\partial x}+v_{l}\frac{\partial v_{l} }{\partial y}=v_{l}\frac{\partial^{2} u_{l}}{\partial y^{2}}+g-\frac{1}{\rho_{l}}\frac{dp}{dx}$

The pressure gradient in the momentum equation may be approximated in terms of conditions outside the liquid film, as required by boundary layer approximation. It follows that pressure in the liquid film satisfies

 $\frac{dp_{l}}{dx}=\frac{dp_{v}}{dx}=\rho_{v}g$ (1)

Substituting eq. (1) into the momentum equation, one obtains

 $\rho_{l}\left (u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y} \right ) = \mu_{l}\frac{\partial^{2}u}{\partial y^{2}}+g(\rho_{l}-\rho_{v})$ (2)

The left-hand side term represents the inertia effects in the slender film region, while the right-hand terms are the effects due to friction and the sinking effect. As stated by Assumption #4 of the above list, the inertia term is negligible in this analysis. Therefore, the above equation simplifies to

 $\frac{\partial^{2}u}{\partial y^{2}}=\frac{g}{\mu_{l}}(\rho_{v}-\rho_{l})$ (3)

Integrating this equation twice with respect to y and using the boundary conditions of nonslip at the wall (u=0 at y=0) and zero shear at the liquid-vapor interface (δu/δy=0 at y = δ, Assumption #5) yields the following:

 $u(x,y)=\frac{({{\rho }_{\ell }}-{{\rho }_{v}})g}{{{\mu }_{\ell }}}\left( y\delta -\frac{{{y}^{2}}}{2} \right)$ (4)

It can be seen from this equation that the downward velocity component is directly dependent on both the x and y coordinates due to the varying (increasingly thick in the x-direction) film thickness. The mass flow rate per unit width of surface, Γ, of this liquid film at any point can be found by integrating the above velocity profile across the liquid film thickness and multiplying by the liquid density, i.e.,

 $\Gamma = \rho_{l}\int_{0}^{\delta}udy=\frac{\rho_{l}(\rho_{l}-\rho_{v})g\delta^{3}}{3\mu_{l}}$ (5)

where it can now be seen that the mass flow rate is a function of the x coordinate.

Recalling that the heat transfer across the liquid film is by conduction only, and assuming that no subcooling exists at the liquid-vapor interface (Assumption #3), Fourier’s Law can be used to obtain the heat flux across the film thickness:

 $q^{''}=\frac{k_{l} \left (T_{sat}-T_{w} \right )}{\delta}$ (6)

where Tsat and Tw are the saturation temperature and wall temperature, respectively.

The heat transfer rate per unit width for the control volume shown in Fig. 1 is given as

 $dq^{'}=\frac{k_{l}\Delta T}{\delta}dx$ (7)

where ΔT = TsatTw. Since no subcooling in the liquid film exists, the latent heat effects of condensation dominate the process. Therefore, it can be said that

 dq' = hlvdΓ (8)

where dΓ is found by differentiating the expression for mass flow rate per unit surface, eq. (5). It is found to be

 $d\Gamma=\frac{\rho_{l}(\rho_{l}-p_{v})g\delta^{2}}{\mu_{l}}d\delta$ (9)

This expression and the expression for heat flow across the control volume are then substituted into the energy equation, eq. (8), and the following is found:

 $\delta^{3}\frac{d\delta}{dx}=\frac{k_{l}\mu_{l}\Delta T}{\rho_{l}(\rho_{l}\rho_{v})gh_{lv}}$ (10)

Finally, δ can be found by integrating the above and using the boundary condition that δ = 0 at x = 0:

 $\delta= \left [ \frac{4k_{l}\mu_{l}x\Delta T}{\rho_{l}(\rho_{l}-\rho_{v})gh_{lv}} \right ]^{1/4}$ (11)

The local heat transfer coefficient is found to be

 $h_{x}=\frac{q^{''}}{T_{sat}-T_{w}}=\frac{k_{l}}{\delta}=k_{l}^{3/4}\left [ \frac{\rho_{l}(\rho_{l}-\rho_{v})gh_{lv}}{4\mu_{l}x\Delta T} \right ]^{1/4}$ (12)

The local Nusselt number, Nux, of the laminar film condensation process on a vertical plate is found to be

 $Nu_{x}=\frac{h_{x}x}{k_{l}} = \left [ \frac{\rho_{l}(\rho_{l}-\rho_{v})gh_{lv}x^{3}}{4\mu_{l}x\Delta T} \right ]^{1/4}$ (13)

It is also desirable to obtain the average heat transfer coefficient and Nusselt number for a plate of length L. The mean heat transfer coefficient can be defined as

 $\overline{h}=\frac{1}{L}\int_{0}^{L}h_{x}(x)dx$ (14)

Substituting the expression for , eq. (12), into eq. (14) and integrating, the following expressions are obtained:

 $\overline{Nu}=\frac{\overline{h}L}{k_{l}}= 0.943 \left [ \frac{\rho_{l}(\rho_{l}-\rho_{v})gh_{lv}L^{3}}{\mu_{l}k_{l}\Delta T} \right ]^{1/4}$ (15)

The local and average heat transfer coefficient can also be nondimensionalized in term of Reynolds number Reδ, defined as

 $Re_{\delta}=\frac{4\Gamma}{\mu_{l}}$ (16)

Substituting eq. (5) into eq. (16), the Reynolds number for laminar film condensation becomes

 $Re_{\delta}=\frac{4\rho_{l}(\rho_{l}-\rho_{v})g\delta^{3}}{3\mu^{2}_{l}}$ (17)

Substituting eqs. (16) and (11) into eqs. (13) and (15), the following nondimensional correlations are obtained:

 $\frac{h_x}{k_l}\left[ \frac{\mu _l^2 }{\rho_l \left( \rho _l - \rho _v \right)g} \right]^{1/3} = 1.1Re_\delta^{-1/3}$ $\frac{\overline{h} }{{k_l }}\left[ {\frac{{\mu _l^2 }}{{\rho _l \left( {\rho _l - \rho _v } \right)g}}} \right]^{1/3} = 1.47Re_\delta ^{ - 1/3}$ (18)

where the bracket term on the left-hand side, together with its exponent (1/3), is the characteristic length.

The above Nusselt analysis assumes no subcooling in the liquid condensate. This assumption can be removed by making an energy balance at the interface that takes subcooling of the liquid into account, as follows:

 $\frac{dq^{'}}{dx}=\frac{k_{l}\Delta T}{\delta}=h_{lv}\frac{d\Gamma}{dx}+\frac{d}{dx}\int_{0}^{\delta}\rho_l c_{pl}u(T_{sat}-T)dy$ (19)

The final term of the right-hand side takes into account a temperature gradient across the liquid condensate film. Substituting the velocity profile in eq. (4) and using a linear temperature profile,

 $\frac{T_{sat}-T}{T_{sat}-T_{w}}=1-\frac{y}{\delta}$ (20)

to evaluate eq. (19), the energy balance can be written as

 $\frac{k_{l}\Delta T_l}{\delta} = h_{lv}^{'}\frac{d\Gamma}{dx}$ (21)

where

 ${h^{'}_v = h_{lv}\left \{1+\frac{3}{8}\left [ \frac{c_{pl}(T_{sat}-T_{w}}{h_{lv}} \right ] \right \} }$ (22)

Equation (22) is identical to the energy balance in eqs. (7) and (8) except that has been replaced by . Rohsenow (1956) improved this analysis even more by including the effects of convection in the liquid along with liquid subcooling, and thereby developing the following:

 $h^{'}_{lv} = h_{lv}\left \{1+0.68\left [ \frac{c_{pl}(T_{sat}-T_{w}}{h_{lv}} \right ] \right \}$ (23)

Finally, if it is desirable to include the effects of vapor superheat in the above analysis, the latent heat obtained by eqs. (22) or (23) can be further modified by adding cpv(TvTsat).

The above analysis can also be applied for condensation outside a vertical tube if δ/D = 1, where D is the diameter of the tube. For an inclined wall with an inclination angle of θ (the angle between the wall and the vertical direction), the component of the gravitational acceleration along the inclined wall is gcosθ. It follows from eqs. (12) and (15) that $h_x\propto (\cos \theta)^{1/4}$ and $\overline{h}\propto (\cos \theta )^{1/4}$ .

## Wavy Condensate Regime

Wavy flows of thin liquid films have higher heat transfer coefficients than smooth thin films, due to mixing action and an increase in the interfacial surface area. It has been shown by experiments that even during well defined laminar film flow, the film surface (liquid-vapor interface) can be wavy. These waves can occur on film flow over a rough wall or a polished wall. This wave formation can even occur when the vapor reservoir is stagnant; however, the waves can be more pronounced when the vapor reservoir has an average velocity, which leads to a higher shear stress at the liquid-vapor interface. These small disturbances can amplify under specific circumstances, i.e., when the film reaches a critical thickness and produces full waves that are regular with respect to time and therefore are not considered turbulent. These waves can lead to an improvement in the heat transfer coefficient by as much as 50%, compared to that of the above laminar Nusselt analysis. The mechanisms of heat transfer enhancement by wave include enlargement of liquid-vapor interfacial area and decrease in mean film thickness.

The flow regime of the liquid condensate is determined by the Reynolds number defined by eq. (17). While experimental observations of the condensate indicate that the laminar flow regime usually becomes wavy in the general vicinity of Reδ=30, Brauer (1956) suggested that the Reynolds number for onset of waves is related to the Archimedes number by

 $Re_{\delta} > 9.3Ar_{l}^{1/5}$ (24)

where Archimedes number is defined as

 $\frac{\rho_{l}\sigma^{3/2}}{\mu_{l}^{2}g^{1/2}(\rho_{l}-\rho_{v})^{3/2}}$ (25)

The wavy laminar film remains so until approximately Reδ=1800, where it becomes turbulent.

In heat transfer analysis of the wavy film regime, there is no one theory reliable for calculating the heat transfer across the wavy film. Many numerical analyses have been performed and different coefficients have been found for the heat transfer coefficient and corresponding Nusselt number. Kutateladze (1982) gave the following correlation for the mean Nusselt number of film condensation on a vertical plate where wave effects are present:

 $\overline{h}=\frac{k_l}{(v^{2}_l / g)^{1/3}}\frac{Re_{\delta}}{Re_{\delta}^{1.22}-5.2}, 30 \leq Re_{\delta} \leq 1800$ (26)

To use eq. (26) to determine the heat transfer coefficient, it is first necessary to know the Reynolds number, which depends on the mass flow rate of condensate per unit width, Γ, as indicated by eq. (17). The mass flow rate of the condensate can be determined by

 $\Gamma = \frac{q^{'}}{h_{lv}^{'}}=\frac{\overline{h}L(T_{sat}-T_w)}{h_{lv}^{'}}$ (27)

Substituting eq. (27) into eq. (16), one obtains

 $Re_{\delta}=\frac{4\overline{h}L(T_{sat}-T_w)}{h_{lv}^{'}}$ (28)

Rearranging eq. (28) yields

 $\overline{h}=\frac{Re_{\delta}h_{lv}^{'}v_{l}}{4L(T_{sat}-T_w)}$ (29)

Combining eq. (26) and (29) yields the Reynolds number for film condensation with waves as follows:

 $Re_{\delta} = \left [ 4.81+\frac{3.7Lk_l (T_{sat}-T_w )}{\mu_l h_{lv}^{'}} \left ( \frac{g}{v^{2}_{l}} \right )^{1/3} \right ]^{0.82}$ (30)

After the Reynolds number is determined by eq. (30), the heat transfer coefficient can be determined by eq. (26).

## Turbulent Film Regime

If one follows the film condensate farther down the vertical wall, the film reaches a critical thickness where the waves become irregular with respect to both time and space. This is where the turbulent film regime exists and can exist even with a stagnant vapor reservoir. It follows that this regime occurs when the Reynolds number, Reδ, becomes a critical value that is much higher than that of the wavy regime.

As stated in the previous section, the condensate film becomes turbulent at approximately Reδ=1800.The heat transfer rate in this turbulent regime is much larger than in laminar and wavy flow. The turbulent region in film condensation, as in any turbulent flow, is extremely difficult to model, and all accurate results come from empirical correlations or from detailed numerical modeling. For turbulent flow of condensation on a vertical plate, Labuntsov (1957) recommended the following empirical correlation:

 $\frac{h_x}{k_l} \left [ \frac{\mu_{l}^{2}}{\rho_l (\rho_l - \rho_v )g} \right ]^{1/3} = 0.023 Re_{\delta}^{0.25}Pr_{l}^{-0.5}, Pr_l \geq 10$ (31)

Chun and Seban (1971) obtained the following experimental correlation for the local turbulent heat transfer coefficient for evaporation of water from a vertical wall, which is applicable for local heat transfer coefficient for condensation:

 $\frac{h_x}{k_l} \left ( \frac{v_{l}^{2}}{g} \right )^{1/3} = 3.8x10^{-3}Re^{0.4}Pr_{l}^{-0.65}, Re>5800Pr_{l}^{-1.06}$ (32)

Butterworth (1983) obtained the average heat transfer coefficient for film condensation that covers laminar, wavy laminar, and turbulent flow by combining eqs. (19), (26), (31) and as follows

 $\overline{h}=\frac{k_l}{(v^{2}_l / g)^{1/3}} \frac{Re_{\delta}}{8750+58Pr_{l}^{-0.5}(Re_{\delta}^{0.75}-253)}, 1 (33)

The Reynolds number, Reδ, is needed in order to use eq. (33) to determine the heat transfer coefficient for turbulent film condensation. Equation (29) was obtained by energy balance and it is valid for all film condensation regimes. Combining eqs. (33) and (29), the Reynolds number for turbulent flow is obtained:

 $Re_{\delta}=\left [ \frac{0.069Lk_l Pr^{0.5}_l (T_{sat}-T_w)}{\mu_l h_{lv}^{'}} \left ( \frac{g}{v_{l}^{2}} \right )^{1/3} - 151Pr_{l}^{0.5}+254 \right ]^{4/3}$ (34)

which can be used together with eq. (33) to determine the heat transfer coefficient for turbulent film condensation.

## References

Brauer, H., 1956, “Stromung und Warmeubergang bei Reiselfilmen,” VDI Forschung, Vol. 22, pp. 1-40.

Butterworth, D., 1983, “Film Condensation of Pure Vapor,” Heat Exchanger Handbook, Chapter 2.6.2, Hemisphere, Washington, DC.

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.

Kutateladze, S.S., 1982, “Semi-empirical Theory of Film Condensation of Pure Vapors,” International Journal of Heat and Mass Transfer, Vol. 25, pp. 653-660.

Labuntsov, D.A., 1957, “Heat Transfer in Film Condensation of Pure Steam on Vertical Surfaces and Horizontal Tubes,” Teploenergetika, Vol. 4, pp. 72-80.

Nusselt, W., 1916, “Die Oberflächenkondensation des Wasserdampfes,” Z. Vereins deutscher Ininuere, Vol. 60, pp. 541-575.