# Surface Waves on Liquid Film Flow

(Difference between revisions)
 Revision as of 18:08, 10 July 2010 (view source)← Older edit Revision as of 18:11, 10 July 2010 (view source)Newer edit → Line 182: Line 182: $\frac{\partial w}{\partial t}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}=\frac{\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g}{{{\rho }_{\ell }}}\left( \sin \theta -\cos \theta \frac{\partial \delta }{\partial z} \right)+\frac{\sigma }{{{\rho }_{\ell }}}\frac{{{\partial }^{3}}\delta }{\partial {{z}^{3}}}+{{v}_{\ell }}\frac{{{\partial }^{2}}w}{\partial {{y}^{2}}}$ $\frac{\partial w}{\partial t}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}=\frac{\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g}{{{\rho }_{\ell }}}\left( \sin \theta -\cos \theta \frac{\partial \delta }{\partial z} \right)+\frac{\sigma }{{{\rho }_{\ell }}}\frac{{{\partial }^{3}}\delta }{\partial {{z}^{3}}}+{{v}_{\ell }}\frac{{{\partial }^{2}}w}{\partial {{y}^{2}}}$ - (5.236) + (16) - Integrating eq. (5.236) and considering the continuity equation, two governing equations for $\delta (z,t)$ can be obtained: + Integrating eq. (16) and considering the continuity equation, two governing equations for $\delta (z,t)$ can be obtained: Line 191: Line 191: $\frac{3{{{\bar{w}}}^{2}}}{\delta }\frac{\partial \delta }{\partial z}=\frac{1}{{{\rho }_{\ell }}}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g\left( \cos \theta \frac{\partial \delta }{\partial z}-\sin \theta \right)-\frac{\sigma }{{{\rho }_{\ell }}}\frac{{{\partial }^{3}}\delta }{\partial {{z}^{3}}}={{v}_{\ell }}\frac{3\bar{w}}{{{\delta }^{2}}}$ $\frac{3{{{\bar{w}}}^{2}}}{\delta }\frac{\partial \delta }{\partial z}=\frac{1}{{{\rho }_{\ell }}}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g\left( \cos \theta \frac{\partial \delta }{\partial z}-\sin \theta \right)-\frac{\sigma }{{{\rho }_{\ell }}}\frac{{{\partial }^{3}}\delta }{\partial {{z}^{3}}}={{v}_{\ell }}\frac{3\bar{w}}{{{\delta }^{2}}}$ - |{{EquationRef|(16)}} + |{{EquationRef|(17)}} |} |} Line 201: Line 201: $\frac{\partial \delta }{\partial t}=3\bar{w}\frac{\partial \delta }{\partial z}$ $\frac{\partial \delta }{\partial t}=3\bar{w}\frac{\partial \delta }{\partial z}$ - |{{EquationRef|(17)}} + |{{EquationRef|(18)}} |} |} Line 212: Line 212: $\frac{1}{{{\rho }_{\ell }}}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g\sin \theta =\frac{3{{{\bar{w}}}_{0}}{{v}_{\ell }}}{\delta _{0}^{2}}$ $\frac{1}{{{\rho }_{\ell }}}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g\sin \theta =\frac{3{{{\bar{w}}}_{0}}{{v}_{\ell }}}{\delta _{0}^{2}}$ - |{{EquationRef|(18)}} + |{{EquationRef|(19)}} |} |} Line 223: Line 223: $\frac{3\bar{w}_{0}^{2}}{{{\delta }_{0}}}\frac{\partial {\delta }'}{\partial z}=\frac{1}{{{\rho }_{\ell }}}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g\cos \theta \frac{\partial {\delta }'}{\partial z}-\frac{\sigma }{{{\rho }_{\ell }}}\frac{{{\partial }^{3}}{\delta }'}{\partial {{z}^{3}}}$ $\frac{3\bar{w}_{0}^{2}}{{{\delta }_{0}}}\frac{\partial {\delta }'}{\partial z}=\frac{1}{{{\rho }_{\ell }}}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g\cos \theta \frac{\partial {\delta }'}{\partial z}-\frac{\sigma }{{{\rho }_{\ell }}}\frac{{{\partial }^{3}}{\delta }'}{\partial {{z}^{3}}}$ - |{{EquationRef|(19)}} + |{{EquationRef|(20)}} |} |} Line 234: Line 234: ${\delta }'={{\delta }^{*}}{{e}^{i\alpha \left( z-ct \right)}}$ ${\delta }'={{\delta }^{*}}{{e}^{i\alpha \left( z-ct \right)}}$ - |{{EquationRef|(20)}} + |{{EquationRef|(21)}} |} |} - where $\alpha$ is the wave number and c is the wave velocity.  For $\alpha$ > 0, which is the condition under which waves exist, it follows from eqs. (19) and (20) that + where $\alpha$ is the wave number and c is the wave velocity.  For $\alpha$ > 0, which is the condition under which waves exist, it follows from eqs. (20) and (21) that Line 295: Line 295: It can be seen from the above table that the wave becomes significant after ${{\operatorname{Re}}_{\delta }}$ is greater than 40. Waves increase the heat transfer coefficient during evaporation (also condensation) of wavy films as compared to a smooth surface due to an increase in interfacial surface area and mixing action ([[#References|Faghri and Seban, 1985]]). Heat transfer during condensation and vaporization over a wavy liquid film will be discussed in detail in Chapter 8 and 9, respectively. It can be seen from the above table that the wave becomes significant after ${{\operatorname{Re}}_{\delta }}$ is greater than 40. Waves increase the heat transfer coefficient during evaporation (also condensation) of wavy films as compared to a smooth surface due to an increase in interfacial surface area and mixing action ([[#References|Faghri and Seban, 1985]]). Heat transfer during condensation and vaporization over a wavy liquid film will be discussed in detail in Chapter 8 and 9, respectively. + + ==Additional Equations== + + + + ==References==

## Revision as of 18:11, 10 July 2010

As will become evident in later chapters, condensation or evaporation at the interface of a thin liquid film flowing over a solid surface is often encountered in engineering applications. This type of process is commonly found in chemical and mechanical engineering equipment such as condensers, long-tube evaporators, wetted wall columns, and cooling towers. The fact that current literature shows continued interest in this general area affirms its continued importance. The problem of the transport from a liquid layer running down a vertical wall has been considered in literature with particular attention to heating, evaporation, condensation, and gas absorption for various applications. The flow in such a liquid layer may be either (a) laminar with a constant thickness, or (b) laminar with variable thickness due to waves, or turbulent with waves moving down the interface. Kapitza (1964) predicted theoretically that the mean film thickness is reduced by approximately 8% due to ripples on the interface. Many experiments have focused on determining the values of the wave properties in both laminar and turbulent flows. Experimental data on the mean film thickness of wavy laminar flow falls between the predictions of Kapitza and Nusselt (smooth interface) theories. Although the wave effect does not have any significant effect on the mean film thickness in wavy laminar regions, it plays a major role in heat and mass transfer processes. The structure of the liquid film running down a plane usually shows a randomly distributed wave on the surface, except at low flow rate at the entry region, where a periodic motion exists. Figure 5.26, which is taken originally from Rogovan et al. (1969) shows such a variation as obtained at a Reynolds number of 54 for water.

Comparisons of the experimental and theoretical values of the transport in falling film should be supported by a specification of the nature of the waves, for it is through the interaction of the normal velocities produced by the waves and the distribution of the transported quantity that the additional transport arises. But this is complicated by the existence of an initial wave-free length, depending upon the way in which the film was formed by the liquid supply, and by a region of wave development that ostensibly culminates in a steady regime at a distance far enough down the height of the film. Brauer (1956), for example, made measurements on water, and mixtures of water and diethylene glycol, at a distance of 1.3 m from the point of film initiation, and found sinusoidal waves to begin at a Reynolds number, $\operatorname{Re}=4\Gamma /\mu =1.2/K{{a}^{1/10}},$ where Ka = ν4ρ3g / σ3 is the Kapitza number. The ratio of the crest height of the waves to the mean film thickness increased with the Reynolds number to about twice for this Reynolds number than for higher Reynolds number. This ratio remained the same up to a Reynolds number of 140 / Ka1 / 10. Between these Reynolds numbers the waves become distorted and no longer sinusoidal; above the higher Reynolds number there were secondary capillary waves on the surface. Throughout, up to 4Γ / μ = 1600, the average film thickness remains essentially that given by the Nusselt analysis. But in the range where the ratio of the crest height to the mean film thickness is almost constant, the wave frequency increases with the Reynolds number to indicate a basis for a proportional increase in heat and mass transfer. Koizumi et al. (2005) studied the behavior of a liquid film flowing down the inner surface of a pipe with countercurrent gas flow. In the experiment, the gas used was air, and silicone oils of 500, 1000, and 3000 cSt as well as water, were used for the liquid phase.

Three primary parameters were correlated: the film thickness, wave velocity, and wavelength. The film thickness was given three different values: minimum, maximum and mean film thickness. The film height was determined by taking photos of the film and measuring its thickness. The wave velocity was determined by seeing how far a wave peak traveled during the time interval between picture frames. The wavelength was determined in a similar manner; that is, photographs were taken, and the distance between peaks of the wave were measured.

Figure 5.27 presents experimental results for water and silicone oil with no airflow for dimensionless δ * versus the Reynolds number, where ${{\delta }^{*}}=\left( y/\nu \right)\sqrt{{{\tau }_{w}}/{{\rho }_{\ell }}}$, δ is the film thickness, τw is the wall shear stress and ${{\rho }_{\ell }}$ is the liquid density. When there is no airflow, the maximum film thickness of the 500 cSt and the 1000 cSt silicone and water films is much greater.

To correlate the wave properties using a non-dimensional analysis, the non-dimensional wave velocity, ${{N}_{w}}={{w}_{w}}/\nu _{g}^{1/3},$ and maximum film thickness, $\delta _{max}^{+}={{\delta }_{\max }}{{(g/{{\nu }^{2}})}^{1/3}}$, are found according to the Buckingham pi theorem. They are functions of the Reynolds number ( $\operatorname{Re}=4\Gamma /\mu$), the Morton number ( KF = ρ3ν4g / σ3) and the nondimensional wavelength (${{N}_{\lambda }}=\lambda {{(g/{{\nu }^{2}})}^{1/3}}$), where ww is the wave velocity and λ is the wavelength. Nosoko et al. (1996) developed the following correlations for wave velocity and maximum film thickness:

 ${{N}_{w}}=0.68K_{F}^{0.02}N_{\lambda }^{0.31}{{\operatorname{Re}}^{0.37}}$ (1)

 $\delta _{\max }^{+}=0.26K_{F}^{0.044}N_{\lambda }^{0.39}{{\operatorname{Re}}^{0.46}}$ (2)

Equations (1) and (2) require information about the wavelength; therefore, the correlations are not closed. The wave velocity and maximum film thickness should be evaluated only from the film flow rate and physical dimensions, if possible. The wavelength correlation was incorporated into the Nosoko correlations by Koizumi et al. (2005), and the constants and exponents were modified for better accuracy. The final forms are

 ${{N}_{\lambda }}=14.9\text{F}{{\text{r}}^{1.29}}{{\operatorname{Re}}^{-0.133}}$ (3)

 ${{N}_{w}}=0.88K_{F}^{0.008}\text{F}{{\text{r}}^{0.977}}{{\operatorname{Re}}^{0.214}}$ (4)

 $\delta _{\max }^{+}=1.09K_{F}^{0.021}\text{F}{{\text{r}}^{0.316}}{{\operatorname{Re}}^{0.424}}$ (5)

where $\text{Fr}={{w}_{w}}/\sqrt{g\bar{\delta }}$and $\bar{\delta }$ is the mean film thickness.

The thickness of these liquid films is small enough that boundary layer assumptions are valid. However, these films are not thin enough for disjoining pressure to play any significant role. Figure 5.28 shows a slow liquid flow on an inclined surface with negligible shear stress at the free surface. The gravitational and liquid viscous forces, as well as surface tension effects, are dominant. As the mass flow rate of the liquid increases, waves on the liquid film interface can be observed, as indicated by Fig. 5.28. The flow regimes of the liquid film include smooth laminar, wavy laminar, and turbulent, which correspond to different film Reynolds numbers. The Reynolds number is defined as

 ${{\operatorname{Re}}_{\delta }}=\frac{4\Gamma }{{{\mu }_{\ell }}}=\frac{4\bar{w}\delta }{{{\mu }_{\ell }}}$ (6)

where Γ is the mass flow rate per unit width and $\bar{w}$ is the mean velocity of the liquid film. The liquid film is laminar if the film Reynolds number is below 30. A wavy flow regime can be observed when the film Reynolds number is between 30 and 1600. The liquid flow becomes turbulent when the film Reynolds number is above 1600. For a fully-developed steady-state downward laminar flow, the momentum equation reads

 $\frac{\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g\sin \theta }{{{\rho }_{\ell }}}+{{v}_{\ell }}\frac{{{d}^{2}}w}{d{{y}^{2}}}=0$ (7)

Integrating eq. (7) twice and considering the boundary conditions w = 0 at y = 0 and dw / dy = 0 at y = δ, the velocity distribution in the liquid film is obtained.

 $w=\frac{({{\rho }_{\ell }}-{{\rho }_{v}})(2\delta y-{{y}^{2}})g\sin \theta }{2{{\mu }_{\ell }}}$ (8)

The mean velocity in the liquid film is then

 $\bar{w}=\frac{1}{\delta }\int_{0}^{\delta }{w\left( y \right)dy}=\frac{\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g\sin \theta {{\delta }^{2}}}{3{{\mu }_{\ell }}}$ (9)

which is obtained under assumption that the liquid flow is steady-state. Under these assumptions (Nusselt analysis) and assuming ${{\rho }_{\ell }}\gg {{\rho }_{v}}$, the dimensionless film thickness, and film velocity at the surface δ, wδ and the mean velocity $\bar{w}$ are given for $\theta ={{90}^{\circ }}$ below:

 ${{\delta }^{+}}=\delta {{\left( \frac{g}{{{\nu }^{2}}} \right)}^{1/3}}=0.908{{\operatorname{Re}}^{1/3}}$ (10)

 ${{w}_{\delta }}=\frac{g}{2\nu }{{\delta }^{2}}$ (11)

 $\bar{w}=\frac{2}{3}{{w}_{\delta }}$ (12)

When waves are present, the liquid film flow will be unsteady. However, it is assumed that eq. (9) is also valid for unsteady film flow. Consideration of hydrostatics and the Young-Laplace equation at the interface yields:

 $\frac{\partial {{p}_{\ell }}}{\partial z}=\frac{\partial {{p}_{v}}}{\partial z}+\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g\cos \theta \frac{\partial \delta }{\partial z}-\sigma \frac{{{\partial }^{3}}\delta }{\partial {{z}^{3}}}$ (13)

Since there is no motion in the vapor phase, the pressure gradient in the vapor phase satisfies

 $\frac{\partial {{p}_{v}}}{\partial z}={{\rho }_{v}}g\sin \theta$ (14)

The momentum equation for an unsteady-state flow in the liquid film is

 ${{\rho }_{\ell }}\left( \frac{\partial w}{\partial t}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z} \right)=-\frac{\partial {{p}_{\ell }}}{\partial z}+{{\rho }_{\ell }}g\sin \theta +{{\mu }_{\ell }}\frac{{{\partial }^{2}}w}{\partial {{y}^{2}}}$ (15)

Substituting eqs. (13) – (14) into eq. (15), the momentum equation becomes

$\frac{\partial w}{\partial t}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}=\frac{\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g}{{{\rho }_{\ell }}}\left( \sin \theta -\cos \theta \frac{\partial \delta }{\partial z} \right)+\frac{\sigma }{{{\rho }_{\ell }}}\frac{{{\partial }^{3}}\delta }{\partial {{z}^{3}}}+{{v}_{\ell }}\frac{{{\partial }^{2}}w}{\partial {{y}^{2}}}$

(16) Integrating eq. (16) and considering the continuity equation, two governing equations for δ(z,t) can be obtained:

 $\frac{3{{{\bar{w}}}^{2}}}{\delta }\frac{\partial \delta }{\partial z}=\frac{1}{{{\rho }_{\ell }}}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g\left( \cos \theta \frac{\partial \delta }{\partial z}-\sin \theta \right)-\frac{\sigma }{{{\rho }_{\ell }}}\frac{{{\partial }^{3}}\delta }{\partial {{z}^{3}}}={{v}_{\ell }}\frac{3\bar{w}}{{{\delta }^{2}}}$ (17)

 $\frac{\partial \delta }{\partial t}=3\bar{w}\frac{\partial \delta }{\partial z}$ (18)

These equations can be solved along with the energy equation. Our purpose, however, is to find the conditions of stability of the film. Assuming that δ = δ0 + δ' Z and $\bar{w}={{\bar{w}}_{0}}$, and noticing from eq. (9) that

 $\frac{1}{{{\rho }_{\ell }}}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g\sin \theta =\frac{3{{{\bar{w}}}_{0}}{{v}_{\ell }}}{\delta _{0}^{2}}$ (19)

the following is obtained:

 $\frac{3\bar{w}_{0}^{2}}{{{\delta }_{0}}}\frac{\partial {\delta }'}{\partial z}=\frac{1}{{{\rho }_{\ell }}}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g\cos \theta \frac{\partial {\delta }'}{\partial z}-\frac{\sigma }{{{\rho }_{\ell }}}\frac{{{\partial }^{3}}{\delta }'}{\partial {{z}^{3}}}$ (20)

The fluctuating component δ' is assumed to be a sinusoidal wave of the form

 ${\delta }'={{\delta }^{*}}{{e}^{i\alpha \left( z-ct \right)}}$ (21)

where α is the wave number and c is the wave velocity. For α > 0, which is the condition under which waves exist, it follows from eqs. (20) and (21) that

 $\bar{\delta }>{{\left[ \frac{3\mu _{\ell }^{2}\cos \theta }{\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right){{\rho }_{\ell }}g{{\sin }^{2}}\theta } \right]}^{1/3}}$ (21)

which can also be written in terms of film Reynolds number, ${{\operatorname{Re}}_{\delta }}=4{{\rho }_{\ell }}{{\bar{w}}_{0}}\bar{\delta }/{{\mu }_{\ell }}$, i.e.,

 ${{\operatorname{Re}}_{\delta }}>4\cot \theta$ (22)

Equation (22) indicates that the critical Reynolds number at which waves appear equals cotθ. For a vertical surface where

 $\theta ={{90}^{\circ }}$, (23)

become ${{\operatorname{Re}}_{\delta }}>0$, which means that waves can be present no matter how small the film Reynolds number is. This result contradicts experimental observation, because waves are present only when the film Reynolds number reaches a certain value. This contradiction can be solved by considering the amplification rate of wave amplitude. If amplification over the time interval required to travel 100 times the liquid film thickness is considered, the amplification factor is (Benjamin, 1957; Carey, 1992)

 $\frac{\left| \delta \right|}{{{\left| \delta \right|}_{t=0}}}=\exp \left\{ \left[ 0.31\nu _{\ell }^{4/3}{{g}^{1/3}}{{\left( \frac{\sigma }{{{\rho }_{\ell }}} \right)}^{-1}} \right]\operatorname{Re}_{\delta }^{8/3} \right\}$ (24)

For saturated water at one atmosphere, ${{\nu }_{\ell }}=2.91\times {{10}^{-7}}{{\text{m}}^{\text{2}}}\text{/s},$ σ = 0.0589N/m, and ${{\rho }_{\ell }}=958.77\text{kg/}{{\text{m}}^{\text{3}}}\text{,}$ and eq. (24) becomes

 $\frac{\left| \delta \right|}{{{\left| \delta \right|}_{t=0}}}=\exp \left( 2.08\times {{10}^{-5}}\operatorname{Re}_{\delta }^{8/3} \right)$ (25)

The dependence of disturbance amplification versus film Reynolds number is shown in the following table:

It can be seen from the above table that the wave becomes significant after ${{\operatorname{Re}}_{\delta }}$ is greater than 40. Waves increase the heat transfer coefficient during evaporation (also condensation) of wavy films as compared to a smooth surface due to an increase in interfacial surface area and mixing action (Faghri and Seban, 1985). Heat transfer during condensation and vaporization over a wavy liquid film will be discussed in detail in Chapter 8 and 9, respectively.