Film condensation with vapor motion

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Laminar Condensate Flow

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

It was assumed in filmwise condensation in a stagnant pure vapor reservoir that the vapor reservoir was stagnant. This assumption was made in order to simplify the analysis of the heat transfer across a thin condensing film. In most real systems, however, the effect of vapor motion must be taken into account. This vapor motion can be due to free (natural effects) or forced (mechanical effects) convection processes. The analysis is conducted in the same way in both cases except that with vapor motion the shear stress at the liquid-vapor interface cannot be assumed to be zero.

In the analysis below, the vapor will be considered as having a downward motion. The analysis follows the same outline as the Nusselt analysis presented above for a case where the vapor reservoir is stagnant, producing zero interfacial shear. In other words, referring to Fig. 1, the shear stress at the $y = δ$ location of the interfacial control volume will now have a finite real value of shear stress. The boundary layer momentum equation for liquid is once again given as

${{\rho }_{\ell }}\left( u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y} \right)=-\frac{d{{p}_{\ell }}}{dx}+{{\mu }_{\ell }}\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}}+{{\rho }_{\ell }}g$	(1)

where $d p / d x$ has a value different from the Nusselt analysis, because another pressure gradient imposed by the motion of the adjacent vapor exists along with the hydrostatic pressure gradient in the liquid, i.e.,

$\frac{d{{p}_{\ell }}}{dx}={{\rho }_{v}}g+{{\left( \frac{dp}{dx} \right)}_{v}}$	(2)

where ρ_v is the vapor density and the subscript v denotes vapor motion. For the sake of convenience, this superimposed pressure gradient can be combined into a fictitious density, $\rho _{v}^{*}$ , denoted as

$\rho _{v}^{*}={{\rho }_{v}}+\frac{1}{g}{{\left( \frac{dp}{dx} \right)}_{v}}$	(3)

Substituting eq. (3) into eqs. (2) and (1) produces

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

The left-hand side of eq. (4) represents the inertia effects of the slender film region, while the right-hand side gives the effects of friction and the pressure gradient in the liquid. It is assumed once again that the inertia terms are negligible compared to the other terms in this analysis. Therefore, the above equation simplifies to

$\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}}=\frac{g}{{{\mu }_{\ell }}}(\rho _{v}^{*}-{{\rho }_{\ell }})$	(5)

Integrating eq. (5) twice with respect to y, using the nonslip boundary conditions at the wall (u = 0 at y = 0), and using the new boundary condition of a finite shear stress at the liquid-vapor interface ( $\partial u/\partial y$ = const at y = δ), the following is obtained:

$u(x,y)=\frac{({{\rho }_{\ell }}-\rho _{v}^{*})g}{{{\mu }_{\ell }}}\left( y\delta -\frac{{{y}^{2}}}{2} \right)+\frac{{{\tau }_{\delta }}y}{{{\mu }_{\ell }}}$	(6)

where $τ δ$ is the shear stress at the interface.

In this equation it can be seen once again that the downward velocity component depends directly on both the x and y coordinates due to the varying (increasingly thick in the x-direction) film thickness. In addition, u is now also a function of the interfacial shear stress.

The mass flow rate per unit width of surface, $Γ$ , of this liquid film at any point can now be found by integrating the velocity profile across the liquid film thickness and multiplying by the liquid density –

$\Gamma ={{\rho }_{\ell }}\int_{0}^{\delta }{udy=\frac{{{\rho }_{\ell }}({{\rho }_{\ell }}-\rho _{v}^{*})g{{\delta }^{3}}}{3{{\mu }_{\ell }}}+\frac{{{\tau }_{\delta }}{{\rho }_{\ell }}{{\delta }^{2}}}{2{{\mu }_{\ell }}}}$	(7)

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

The previously described procedure for Nusselt analysis is followed verbatim, using eq. (7) for the mass flux across the control volume. Since it is still assumed that heat transfer across the liquid film is by conduction only, and that no subcooling exists at the liquid-vapor interface, heat flux across the film thickness can be obtained using Fourier’s law:

${q}''=\frac{{{k}_{\ell }}\left( {{T}_{sat}}-{{T}_{w}} \right)}{\delta }$	(8)

where T_sat and T_w are the saturation temperature and wall temperature respectively (ΔT = T_sat – T_w). Rewriting eq. (8) in terms of heat transfer rate for the control volume shown in Fig. 1, the heat flow is given as

$d{q}'=\frac{{{k}_{\ell }}\Delta T}{\delta }dx$	(9)

Since no subcooling of the liquid film exists, the latent heat effects of condensation dominate the process. Thus

$d{q}'={{h}_{\ell v}}d\Gamma$	(10)

where $d Γ$ can be found by differentiating the above expression for mass flow rate per unit surface area, with the result

$d\Gamma =\left( \frac{{{\rho }_{\ell }}({{\rho }_{\ell }}-\rho _{v}^{*})g{{\delta }^{2}}}{{{\mu }_{\ell }}}+\frac{{{\tau }_{\delta }}{{\rho }_{\ell }}\delta }{{{\mu }_{\ell }}} \right)d\delta$	(11)

Substituting this expression and the expression for heat flow across the control volume, dq, into the conservation of mass and energy equation, the following is found:

$\frac{d\delta }{dx}=\frac{{{k}_{\ell }}{{\mu }_{\ell }}\Delta T}{{{\rho }_{\ell }}\left( {{\rho }_{\ell }}-\rho _{v}^{*} \right)g{{h}_{\ell v}}{{\delta }^{3}}+{{\tau }_{\delta }}{{\rho }_{\ell }}{{h}_{\ell v}}{{\delta }^{2}}}$	(12)

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

${{\delta }^{4}}+\frac{4{{\tau }_{\delta }}{{\delta }^{3}}}{3\left( {{\rho }_{\ell }}-\rho _{v}^{} \right)g}=\left[ \frac{4{{k}_{\ell }}{{\mu }_{\ell }}x\Delta T}{{{\rho }_{\ell }}\left( {{\rho }_{\ell }}-\rho _{v}^{} \right)g{{h}_{\ell v}}} \right]$	(13)

This equation can be nondimensionalized by using the following dimensionless numbers (Rohsenow et al., 1956):

${{\delta }^{*}}=\frac{\delta }{{{L}_{F}}}$	(14)

${{x}^{*}}=\left( \frac{x}{{{L}_{F}}} \right)\frac{4{{c}_{p\ell }}\Delta T}{{{\Pr }_{\ell }}{{h}_{\ell v}}}$	(15)

$\tau _{\delta }^{}=\frac{{{\tau }_{\delta }}}{{{L}_{F}}\left( {{\rho }_{\ell }}-\rho _{v}^{} \right)g}$	(16)

where

${{L}_{F}}={{\left[ \frac{\mu _{\ell }^{2}}{{{\rho }_{\ell }}\left( {{\rho }_{\ell }}-\rho _{v}^{*} \right)g} \right]}^{1/3}}$	(17)

is the characteristic length of the non-dimensional problem. Equation (13), which relates film thickness to the vertical location on the plate, can be rewritten as follows:

${{x}^{}}={{({{\delta }^{}})}^{4}}+\frac{4}{3}{{\left( {{\delta }^{}} \right)}^{3}}\tau _{\delta }^{}$	(18)

It directly follows that the mean Nusselt number and Reynolds number for laminar flow with finite vapor shear can be expressed as follows:

$\overline{Nu}=\frac{{{h}_{L}}}{{{k}_{\ell }}}{{\left[ \frac{\mu _{\ell }^{2}}{{{\rho }_{\ell }}({{\rho }_{\ell }}-\rho _{v}^{})g} \right]}^{1/3}}=\frac{4}{3}\frac{{{\left( {{\delta }^{}} \right)}^{3}}}{{{x}^{}}}+\frac{2{{\left( {{\delta }^{}} \right)}^{2}}\tau _{\delta }^{}}{{{x}^{}}}$	(19)

${{\operatorname{Re}}_{\delta }}=\frac{4\Gamma }{{{\mu }_{\ell }}}=\frac{4}{3}{{\left( {{\delta }^{}} \right)}^{3}}+2{{\left( {{\delta }^{}} \right)}^{2}}\tau _{\delta }^{*}$	(20)

From the above it can be seen that in order to solve for the average Nusselt number or Reynolds number, the dimensionless numbers x* and δ^* are needed. These numbers are directly dependent on each other; therefore, an iterative procedure is required to solve for the heat transfer parameters for each and every point. Additional complications follow from the fact that δ depends on position x and either the variation of heat removal rate across the film with respect to x or the varying temperature drop across the film thickness. In other words, unless the problem statement is oversimplified, the only easy way to solve this problem is numerically. For the case in which the gravitational force is negligible compared with the interfacial shear force imposed by the co-current vapor flow, Butterworth (1981) recommended the following correlation for local heat transfer coefficient:

${{\operatorname{Re}}_{\delta }}=\frac{4\Gamma }{{{\mu }_{\ell }}}=\frac{4}{3}{{\left( {{\delta }^{}} \right)}^{3}}+2{{\left( {{\delta }^{}} \right)}^{2}}\tau _{\delta }^{*}$	(21)

where the modified local Nusselt number is defined as

$Nu_{x}^{*}=\frac{{{h}_{x}}}{k}{{\left[ \frac{\mu _{\ell }^{2}}{{{\rho }_{\ell }}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g} \right]}^{1/3}}$	(22)

and the dimensionless interfacial shear stress is

$\tau _{\delta }^{+}=\frac{{{\rho }_{\ell }}{{\tau }_{\delta }}}{{{\left[ {{\rho }_{\ell }}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right){{\mu }_{\ell }}g \right]}^{2/3}}}$	(23)

Butterworth (1981) also recommended the following expression for the cases where both gravity and interfacial vapor shear are significant:

$h={{(h_{shear}^{2}+h_{grav}^{2})}^{1/2}}$	(24)

where h_grav is heat transfer coefficient for gravity-dominated film condensation – determined with eqs. (8.103) or (8.108) – and h _shear is heat transfer coefficient for shear-dominated film condensation, eq. (21). Lin and Faghri (1998) developed a model for predicting the condensation heat transfer coefficient for annular flow in rotating stepped-wall heat pipes. The theoretical result is compared with experimental data. The effect of vapor shear drag on the condensation heat transfer is discussed (see Problem 8.22).

Turbulent Condensate Flow

The velocity and pressure in turbulent flow experience large fluctuations and they can be expressed

$u=\bar{u}+{u}'$	(25)

$v=\bar{v}+{v}'$	(26)

$p=\bar{p}+{p}'$	(27)

where the bar notation denotes the mean value averaged over time and the prime notation denotes velocity fluctuations.

The boundary-layer equation for forced turbulent flow along a planar surface is

$\bar{u}\frac{\partial \bar{u}}{\partial x}+\bar{v}\frac{\partial \bar{u}}{\partial y}=-\frac{1}{\rho }\frac{\partial \bar{p}}{\partial x}+\frac{1}{\rho }\frac{\partial }{\partial y}\left( \mu \frac{\partial \bar{u}}{\partial y}-\overline{\rho {u}'{v}'} \right)$	(28)

where the bar notation denotes the mean value averaged over time and the prime notation denotes velocity fluctuations. Introducing the following notation:

$-\overline{\rho {u}'{v}'}=\rho \varepsilon \frac{\partial \bar{u}}{\partial y}$	(29)

which is known as the eddy shear stress. ε is an empirical function known as the momentum eddy diffusivity; it is a flow property, not a fluid property. A close look at eq. (28) shows that the shear stress expression has changed from the normal laminar flow form. The apparent shear stress expression for turbulent flow can be written as follows:

${{\tau }_{app}}=\mu \frac{\partial \bar{u}}{\partial y}-\overline{\rho {u}'{v}'}$	(30)

Substituting eq. (29) into eq. (30) gives

${{\tau }_{app}}=\mu \frac{\partial \bar{u}}{\partial y}+\rho \varepsilon \frac{\partial \bar{u}}{\partial y}=\rho \left( \nu +\varepsilon \right)\frac{\partial \bar{u}}{\partial y}$	(31)

which is the shear stress expression for turbulent flow consisting of both laminar and turbulent portions. If the following is defined

${{\varepsilon }^{+}}=1+\frac{\varepsilon }{\nu }$	(32)

eq. (31) can be simplified to

${{\tau }_{app}}=\rho {{\varepsilon }^{+}}\nu \frac{\partial \bar{u}}{\partial y}$	(33)

which shows that when ε⁺= 1, the problem simplifies to the laminar case. Further, it can be said that when $\varepsilon /\nu \gg$ 1, then ε⁺= $\varepsilon /\nu$ , and therefore eq. (33) becomes simplified to represent the fully turbulent case. The turbulent regime in film condensation with vapor motion is very difficult to model, so empirical correlations are often used to predict the heat transfer coefficient. Rohsenow et al. (1956) presented the variation of the average film condensation heat transfer coefficient with Reynolds number and the nondimensional number $\tau _{\delta }^{*}$ . The figure shows both the laminar and turbulent flow regimes, demonstrating that the above expressions for Reynolds number and average Nusselt number give a good solution up to approximately ${{\operatorname{Re}}_{\delta }}$ = 1100 at low nondimensional shear stress values. At that point the heat transfer rate rises sharply in response to the transition to turbulent film flow. It appears that wavy flow at these low shear stress values does not contribute to any change in the flow’s average heat transfer coefficient and Reynolds number. However, higher shear stress numbers allow for a more gradual change (shallow gradient) from laminar to fully turbulent flow, in turn allowing a wavy flow to exist. This shallow gradient shows neither the abrupt steep rise of a turbulent flow nor the consistent downward gradient characteristic of Nusselt flow. Instead, it is a combination of the two types of flow.

The Reynolds numbers at the transition points were presented by Rohsenow et al. (1956) as the following:

${{\operatorname{Re}}_{\delta ,tr}}=1800-246{{\left( 1-\frac{{{\rho }_{v}}}{{{\rho }_{\ell }}} \right)}^{1/3}}\tau _{\delta }^{}+0.667\left( 1+\frac{{{\rho }_{v}}}{{{\rho }_{\ell }}} \right){{\left( \tau _{\delta }^{} \right)}^{3}}$	(34)

Rohsenow et al. (1956) extended Seban’s (1954) falling film condensation analysis for turbulent flow from zero vapor shear to finite vapor shear cases to arrive at the following expression

$\bar{h}=0.065\Pr _{\ell }^{1/2}{{\left( \tau _{\delta }^{*} \right)}^{1/2}}{{k}_{\ell }}{{\left( \frac{g}{v_{\ell }^{2}} \right)}^{1/3}}$	(35)

which may be used to predict the average heat transfer coefficient beyond the transition point into turbulent flow. It is found to agree well with a relationship obtained by Carpenter and Colburn (1951).

References

Butterworth, D., 1981, “Simplified Methods for Condensation on a Vertical Surface with Vapor Shear,” UKAEA Rept. AERE-R9683.

Carpenter, F.S., and Colburn, A.P., 1951, “The Effect of Vapor Velocity on Condensation Inside Tubes,” Proceedings of General Discussion of Heat Transfer, Institute of Mechanical Engineers and American Society of Mechanical Engineers, pp. 20-26.

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.

Lin, L., and Faghri, A., 1998, “Condensation in a Rotating Stepped Wall Heat Pipe with Hysteretic Annular Flow,” AIAA Journal of Thermophysics and Heat Transfer, Vol. 12, No. 1, pp. 94-99.

Rohsenow, W.M., 1956, “Heat Transfer and Temperature Distribution in Laminar Film Condensation,” Transactions of ASME, Vol. 78, pp. 1645-1648.

Rohsenow, W.M., Webber, J.H., and Ling, T., 1956, “Effect of Vapor Velocity on Laminar and Turbulent Film Condensation,” Transactions of ASME, Vol. 78, pp. 1637-1643.

Seban, R., 1954, “Remarks on Film Condensation with Turbulent Flow,” Transaction of the ASME, Vol. 76, pp. 299-303.

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Film condensation with vapor motion

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Laminar Condensate Flow

Turbulent Condensate Flow

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