Evaporation from wavy laminar falling film

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<center><math>\operatorname{Re}_{L}^{1.22}=\operatorname{Re}_{0}^{1.22}-4\frac{{{k}_{\ell }}L\left( {{T}_{w}}-{{T}_{v}} \right)}{{{\mu }_{\ell }}{{h}_{\ell v}}}{{\left[ \frac{{{\rho }_{\ell }}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g}{\mu _{\ell }^{2}} \right]}^{{1}/{3}\;}}</math>
<center><math>\operatorname{Re}_{L}^{1.22}=\operatorname{Re}_{0}^{1.22}-4\frac{{{k}_{\ell }}L\left( {{T}_{w}}-{{T}_{v}} \right)}{{{\mu }_{\ell }}{{h}_{\ell v}}}{{\left[ \frac{{{\rho }_{\ell }}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g}{\mu _{\ell }^{2}} \right]}^{{1}/{3}\;}}</math>

Revision as of 15:52, 1 June 2010

Wavy flows of thin liquid films have higher heat transfer coefficients than smooth thin films. This effect is due to the former’s greater interfacial surface area and mixing action. Faghri and Seban (1985) analyzed a system in which a liquid film at an initial nondimensional temperature of zero flowed down a vertical wall and evaporated at the free liquid-vapor interface. The laminar liquid film thickness varied sinusoidally and the Reynolds number ranged between 35 and 472. The energy equation for a two-dimensional situation using the same vertical wall shown in Fig. 9.11 is as follows:

\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \left( \frac{{{\partial }^{2}}T}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}T}{\partial {{y}^{2}}} \right) (9.121)

The continuity equation for this system is

\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 (9.122)

Faghri and Seban (1985) assumed a quasi-parabolic velocity profile in the x-direction as follows:

u=3\bar{u}(x,t)\left( \eta -\frac{{{\eta }^{2}}}{2} \right) (9.123)

where the nondimensional coordinate η is defined as

\eta =\frac{y}{\delta } (9.124)

and \bar{u} is the local mean velocity. Equation (9.122) can be rearranged and integrated with respect to y to obtain the velocity component v:

v=-\int\limits_{0}^{y}{\frac{\partial u}{\partial x}dy=-3}\left[ \frac{\partial \bar{u}}{\partial x}\delta \left( \frac{{{\eta }^{2}}}{2}-\frac{{{\eta }^{3}}}{6} \right)-\bar{u}\frac{\partial \delta }{\partial x}\left( \frac{{{\eta }^{2}}}{2}-\frac{{{\eta }^{3}}}{3} \right) \right] (9.125)

It can be seen that eq. (9.123) has been substituted in order to find an expression for the v component of the velocity where the local mean velocity and film thickness δ are still unknown as functions of the wavy surface. To determine these two variables, the following mass balance can be written as a function of time and space:

\frac{\partial \delta }{\partial t}=-\frac{\partial }{\partial x}\int_{0}^{\delta }{udy=-\frac{\partial }{\partial x}\left( \bar{u}\delta  \right)} (9.126)

The film thickness can be written in terms of its average value, \overline{\delta } , and a local amplitude φ.

\delta =\bar{\delta }(1+\varphi ) (9.127)

Assuming that the wave formation has a periodic characteristic with a velocity c, the following relations can be written for the fluctuation of local mean velocity and film thickness:

\frac{\partial \bar{u}}{\partial t}=-c\frac{\partial \bar{u}}{\partial x} (9.128)
\frac{\partial \delta }{\partial t}=-c\frac{\partial \delta }{\partial x} (9.129)

Combining eqs. (9.126) – (9.129) and integrating, the following expression is obtained:

\left( c-\bar{u} \right)\left( 1+\varphi  \right)=\left[ c-{{{\bar{u}}}_{0}} \right] (9.130)

where {{\bar{u}}_{0}} is the average velocity for the average film thickness \bar{\delta }. For very small local amplitudes, φ < 1, \bar{u} may be approximated by expanding eq. (9.130) and neglecting the third-order terms.

\bar{u}={{\bar{u}}_{0}}+(c-{{\bar{u}}_{0}})\varphi -(c-{{\bar{u}}_{0}}){{\varphi }^{2}} (9.131)

Substituting eqs. (9.127) and (9.131) into eq. (9.125), one obtains the velocity profile in terms of the wavy surface parameters:

  & v=-3{{{\bar{u}}}_{0}}\bar{\delta }\frac{\partial \phi }{\partial x}\left\{ \left( \frac{c}{{{{\bar{u}}}_{0}}}-1 \right)(1-2\phi )\frac{\delta }{{\bar{\delta }}}\left( \frac{{{\eta }^{2}}}{2}-\frac{{{\eta }^{3}}}{6} \right) \right. \\ 
 & \left. \begin{matrix}
   {} & {}  \\
\end{matrix}-\left[ 1+\left( \frac{c}{{{{\bar{u}}}_{0}}}-1 \right)\phi -\left( \frac{c}{{{{\bar{u}}}_{0}}}-1 \right){{\phi }^{2}} \right]\left( \frac{{{\eta }^{2}}}{2}-\frac{{{\eta }^{3}}}{3} \right) \right\} \\ 
\end{align} (9.132)

Assuming the wave is sinusoidal,

\varphi =A\sin \left[ \left( \frac{2\pi }{\lambda } \right)\left( x-ct \right) \right] (9.133)

and introducing the nondimensional variable ξ defined as

\xi =\left( \frac{2\pi }{\lambda } \right)\left( x-ct \right) (9.134)

where A is magnitude of the wave, and λ is the wavelength, eq. (9.121) is transformed from the (t, x, y) to the (ξ, η) coordinate system,

{{C}_{1}}\frac{\partial T}{\partial \xi }+{{C}_{2}}\frac{\partial T}{\partial \eta }={{C}_{3}}\frac{{{\partial }^{2}}T}{\partial {{\eta }^{2}}}+{{C}_{4}}\frac{{{\partial }^{2}}T}{\partial \eta \partial \xi }+{{C}_{5}}\frac{{{\partial }^{2}}T}{\partial {{\xi }^{2}}} (9.135)

where the constants are as follows:

{{C}_{1}}=\frac{2\pi }{\lambda }(u-c) (9.136)
  & {{C}_{2}}=\frac{2\pi \eta }{\lambda }\frac{{\bar{\delta }}}{\delta }(c-u)A\cos \xi -{{\left( \frac{2\pi }{\lambda } \right)}^{2}}\frac{{\bar{\delta }}}{\delta }\alpha \eta A\sin \xi  \\ 
 & \begin{matrix}
   {} & {}  \\
\end{matrix}-2{{\left( \frac{2\pi }{\lambda } \right)}^{2}}\alpha \left( \frac{{\bar{\delta }}}{\delta } \right)\eta {{A}^{2}}{{\cos }^{2}}\xi +\frac{v}{\delta } \\ 
\end{align} (9.137)
{{C}_{3}}=\frac{\alpha }{{{\delta }^{2}}}+\alpha {{\left( \frac{2\pi }{\lambda } \right)}^{2}}{{\left( \frac{{\bar{\delta }}}{\delta } \right)}^{2}}{{\eta }^{2}}{{A}^{2}}{{\cos }^{2}}\xi (9.138)
{{C}_{4}}=-2\alpha {{\left( \frac{2\pi }{\lambda } \right)}^{2}}\eta \frac{{\bar{\delta }}}{\delta }A\cos \xi (9.139)
{{C}_{5}}=\alpha {{\left( \frac{2\pi }{\lambda } \right)}^{2}} (9.140)

These coefficients are evaluated with \bar{u} from eqs. (9.123) and (9.131), and v from eq. (9.132).

In the evaporating film with a fluid with a high latent heat of vaporization, the average film thickness over a wave length, \bar{\delta }, will not vary substantially with distance, x. Furthermore, the temporal average temperature profile will not vary with x far from where heating begins. Therefore, the boundary condition for eq. (9.135) that describes evaporation on a wavy film is approximated as below. The equality of temperatures at corresponding points in the period gives

T(0,η) = T(2π,η) (9.141)

Boundary conditions for dimensionless temperature at the heated vertical wall and wavy surface are, respectively,

T(ξ,0) = 1 (9.142)
T(ξ,1) = 0 (9.143)

Therefore, the energy equation with the above boundary conditions can be solved to obtain the wavy condensate film’s temperature profile. With this profile, the heat flux at the wall can be found from

{{{q}''}_{w}}=-\frac{k}{\delta }{{\left. \frac{\partial T}{\partial \eta } \right|}_{\eta =0}} (9.144)

and the heat flux normal to the wavy surface (η = 1) is as follows:

\frac{{{{\mathbf{{q}''}}}_{\delta }}}{k}=\left[ \frac{2\pi }{\lambda }\left( \frac{{\bar{\delta }}}{\delta }A\cos \xi  \right)\mathbf{i}-\frac{1}{\delta }\mathbf{j} \right]{{\left. \frac{\partial T}{\partial \eta } \right|}_{\eta =1}} (9.145)

where \mathbf{i} and \mathbf{j} are unit vectors in the x- and y-directions, respectively. The average heat flow over a period 0 < ξ < 2π is found by integrating eqs. (9.144) and (9.145). For a sinusoidal wavy layer, such as the one considered here, this average is identical to the average obtained for the actual surface length. The average heat flux at the wall (η = 0) is approximated by

\frac{{{{\bar{{q}''}}}_{w}}}{k}=\frac{1}{2\pi }\int_{0}^{2\pi }{\frac{1}{\delta }{{\left. \frac{\partial T}{\partial \eta } \right|}_{0}}d\xi } (9.146)

and the average normal heat flux at the wavy surface is as follows:

\frac{{{{\bar{{q}''}}}_{\delta }}}{k}=\frac{1}{2\pi }\int_{0}^{2\pi }{\frac{1}{\delta }{{\left. \frac{\partial T}{\partial \eta } \right|}_{\eta =1}}{{\left[ 1+{{\left( \frac{\partial \delta }{\partial x} \right)}^{2}} \right]}^{1/2}}d\xi } (9.147)

Table 9.1 Comparison of wavy film analysis with experimental data (Faghri and Seban, 1985).

References Rogovan et al. (1969) Kapitza and Kapitza (1975) Rogovan et al. (1969) Rogovan et al. (1969)
1 T(°C) 20 25 20 20
2 4Γ / μ 35 92 268 472
3 \bar{\delta } (10-4ft) 4.132 5.96 9.41 11.5
4 {{\bar{w}}_{0}} (ft/s) 0.279 0.383 0.836 1.164
5 2π / λ(ft-1) 191 241 159 136
6 c/{{\bar{u}}_{0}} 2.12 1.93 1.65 1.55
7 A 0.30 0.52 0.55 0.40 (0.55)
8 ν / α 7.2, 1.7 6.2, 1.7 7.2, 1.7 7.2, 1.7, 7.0, 1.7
9 {{\left. \bar{h}\bar{\delta }/k \right|}_{w}} 1.059, 1.057 1.309, 1.296 1.567, 1.543 71.393, 1.372, 1.724, 1.694
10 {{\left. \bar{h}\bar{\delta }/k \right|}_{\delta }} 1.058, 1.056 1.314, 1.300 1.611, 1.580 1.433, 1.405, 1.816, 1.772
11 1/\sqrt{1-{{A}^{2}}} 1.05 1.17 1.19 1.09, 1.19
12 {{\left. \bar{h}\bar{\delta }/k \right|}_{w}} 1.049, 1.048 1.171, 1.169 1.199, 1.199 1.092, 1.092, 1.199, 1.199
13 {{\left. \bar{h}\bar{\delta }/k \right|}_{\delta }} 1.049, 1.048 1.171, 1.169 1.197, 1.197 1.091, 1.091, 1.197, 1.197
  • Conduction. Reprinted with permission from Elsevier.

The calculations were made using 40 increments in both ξ and η for the value of 2π / λ, A, and c/{{\bar{u}}_{0}}. Table 9.1 lists these experimental determinations made by the cited authors. Rows 1 through 7 give the experimental conditions and measurements. Row 8 gives the Prandtl numbers for which the calculations were made. One, on the order of 7, corresponds to the experimental conditions, and the other, 1.7, was selected as a comparison to show the effect of Prandtl number. Row 9 gives the calculated average Nusselt number at the wall, and Row 10 gives the average Nusselt number at the outer edge of the layer. These values should be the same, and the difference indicates the failure of the calculation to satisfy the energy balance. This difference is small; it increases with the Reynolds number, reflecting some increase in truncation error. Row 11 gives the average Nusselt number as evaluated for unidimensional conduction. On this basis, the local Nusselt number is hδ / k = 1, and then \frac{{\bar{h}}}{k}=\frac{1}{2\pi }\int_{0}^{2\pi }{\frac{d\xi }{\delta }} (9.148) For the sinusoidal wave,

  & \frac{{\bar{h}}}{k}=\frac{1}{2\pi \bar{\delta }}\int_{0}^{2\pi }{\frac{d\xi }{1+A\sin \xi }} \\ 
 & \begin{matrix}
   {} & =\frac{1}{2\pi \bar{\delta }}\left[ \int_{0}^{\pi }{\frac{d\xi }{1+A\sin \xi }+\int_{0}^{\pi }{\frac{d\xi }{1-A\sin \xi }}} \right]=\frac{1}{\bar{\delta }\sqrt{1-{{A}^{2}}}}  \\
\end{matrix} \\ 
\end{align} (9.149)

The difference between the calculated average Nusselt numbers is due to the contribution of convection and to the two-dimensional nature of the conduction that exists because of the variation in layer thickness. The result for w = v = c = 0 corresponds to the two-dimensional conduction solution for the wave, and the Nusselt numbers for such a calculation are shown in Rows 12 and 13. These Rows should be identical, and the truncation error in the calculation creates a discrepancy between them, as it does in the cases in which there is fluid motion. These values are essentially the same as Row 11, for unidimensional conduction, and this correspondence shows that two-dimensional conduction effects are negligible. Therefore, the difference between the Nusselt numbers in Rows 9 and 11 reflect only the effect of convection.

Figure 9.12 shows the values of the Nusselt number (\bar{h}\bar{\delta }/k) as given by Row 9 of Table 9.1 for a Prandtl number of about 7. The value of 472 for (4Γ / μ) is also shown for an arbitrarily-higher value of A = 0.55, because the value of A probably should not decrease as the Reynolds number increases. Figure 9.11 also contains lines A – to show the specification of Zazuli given by Kutateladze (1963) – and B – to show that of Kutateladze (1982). Curve C is that of Hirschburg and Florschuetz (1982) for the intermediate wave solution designated by them as f+ = 0.65, which best fits the average of the data. Three data points from Chun and Seban (1971) for evaporation of water with a Prandtl number of about 5.6 are shown by plus (+) symbols, and four numerical results of Faghri and Seban (1985) are shown by circles (◦) in the same figure.

Figure 9.13 shows the variation of the local heat transfer coefficient as a function of ξ, normalized with respect to the average value over a period. The results are for the Prandtl number of 8.2 and the Reynolds numbers of 35 and 472 (the same portrayal for the Prandtl number of 1.7 is not much different.) The ratio {{h}_{w}}/\bar{h} for the wall varies considerably for the low Reynolds number, but not very much for the high Reynolds number. For the surface, the ratio {{h}_{\delta }}/\bar{h} varies substantially for both cases. The numerical results for the Nusselt numbers of Faghri and Seban (1985) are on the order of, but tend to be higher than, the correlation equations of Kutateladze (1963, 1982). Comparison of Rows 9 and 11 in Table 9.1 reveals the considerable effects of convection and two-dimensional conduction. For practical purposes, Chun and Seban (1971) suggested that the following correlation based on wavy laminar film condensation is valid for wavy laminar falling film evaporation:

{{h}_{x}}=0.876{{\left( \frac{\operatorname{Re}}{4} \right)}^{0.11}}{{h}_{Nusselt}} (9.150)

An empirical correlation for the local heat transfer coefficient for laminar-wavy flow is obtained by combining eq. (9.150) with eq. (9.108),

\frac{{{h}_{x}}}{{{k}_{\ell }}}{{\left[ \frac{\mu _{\ell }^{2}}{{{\rho }_{\ell }}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g} \right]}^{{1}/{3}\;}}=0.828{{\operatorname{Re}}^{-0.22}},\text{   }\operatorname{Re}>{{\operatorname{Re}}_{wavy}} (9.151)

Chun and Seban (1971) suggested the following empirical correlation to predict the onset of wavy laminar flow:

\operatorname{Re}\ge {{\operatorname{Re}}_{wavy}}=2.43K{{a}^{-1/11}} (9.152)

where Ka is Kapitza number, a ratio of surface tension to viscous force:

Ka=\frac{\mu _{\ell }^{4}g}{\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right){{\sigma }^{3}}} (9.153)

The average heat transfer coefficient is often relevant to the practical application and can be obtained by using eq. (9.112), which is also valid for laminar flow with waves. Substituting eq. (9.151) into eq. (9.112), one obtains an empirical correlation of the average heat transfer coefficient as follows:

\frac{\overset{\_}{\mathop{h}}\,}{{{k}_{\ell }}}{{\left[ \frac{\mu _{\ell }^{2}}{{{\rho }_{\ell }}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g} \right]}^{{1}/{3}\;}}=\frac{\left( {{\operatorname{Re}}_{o}}-{{\operatorname{Re}}_{L}} \right)}{\left( \operatorname{Re}_{o}^{1.22}-\operatorname{Re}_{L}^{1.22} \right)} (9.154)

Substituting eq. (9.109) into eq. (9.154), an equation correlating {{\operatorname{Re}}_{L}}, the Reynolds number at x = L, is obtained:

\operatorname{Re}_{L}^{1.22}=\operatorname{Re}_{0}^{1.22}-4\frac{{{k}_{\ell }}L\left( {{T}_{w}}-{{T}_{v}} \right)}{{{\mu }_{\ell }}{{h}_{\ell v}}}{{\left[ \frac{{{\rho }_{\ell }}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g}{\mu _{\ell }^{2}} \right]}^{{1}/{3}\;}} (9.155)