Evolution of metastable quasi-regular structures in heated wavy liquid films

Abstract

Formation and development of quasi-regular, metastable structures within laminar wavy falling films were studied using IR-thermography. These structures emerge within the residual layer between large waves. It is shown that the typical size of regular structures does not depend on the liquid flow rate and is in the order of magnitude of the critical length of the Rayleigh–Taylor instability. Also a model of thermal-capillary breakdown on the basis of a simplified force balance between the surface tension and the tangential stress as well as the energy balance in the residual layer is presented. Model predictions and experimental data are in good agreement.

Appendix

In this part different approaches for the determination of the critical dimensionless heat flux are presented.

Experimental data for laminar-wavy and turbulent films were described in [8] by the following empirical dependencies:

$$Ma_{q} = 0.522Re^{{0.4}} {\left({1 + 0.12{\left({\frac{{Re}}{{250}}} \right)}^{{4.5}}} \right)}^{{0.5}}\; \hbox{for}\; L\leq 1\,\hbox{m} .$$

(23)

For 100 < Re < 200 in [8] the scattering of data was up to 50 %, for Re < 100 no experimental data has been recorded. It can be seen in Fig. 9 that this dependence suggests lower values than the current experimental data. The difference can be explained by the fact that in [8] the experimental data was obtained only for a water film flow with a relatively long heated section. In this case evaporation effects and thus a shift in the thermophysical properties could have appeared.

Fig. 9

The dependence of dimensionless parameter Ma _q versus the Reynolds-number. Points experimental data

In [10] the empirical dependence of the critical Marangoni number on the Reynolds number for a short heat section (6.5 mm length along the flow) for laminar waveless falling films was obtained:

$$Ma_{q} = 8.14Re^{{0.98}}.$$

(24)

In this case the length of the heated section is in the same order of magnitude as the thermal entry length [10]. Therefore this curve indicates higher values than our experimental data.

It was shown in [9] that for the 2D case the modified critical Marangoni number is constant:

$$Ma_{\rm c} = \frac{{q_{\rm W} {\left| {\frac{{{\rm d}\sigma}}{{{\rm d}T}}} \right|}}}{{\lambda \rho U_{\rm s}^{2}}} = 0.23.$$

(25)

With the film surface velocity based on Nusselt’s film theory $U_{\rm s} = \frac{g}{{2\nu}}\delta^{2}_{m},$ expression (25) can be transformed into:

$$Ma_{q} = 0.248Re^{{4 \mathord{\left/ {\vphantom {4 3}} \right. \kern-\nulldelimiterspace} 3}} .$$

(26)

It can be seen, that only dependence (26) is in the same order of magnitude as our experimental data.

Other dimensionless parameters for generalisation of experimental data were used in [3] and [16]. In [3] the data for dimensionless breakdown heat flux is approximated in the form:

$$\frac{{q_{\rm W} {\left| {\frac{{{\rm d}\sigma}}{{{\rm d}T}}} \right|}}}{{\rho^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} c_{\rm p} \mu^{{5 \mathord{\left/ {\vphantom {5 3}} \right. \kern-\nulldelimiterspace} 3}} g^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}}}} = 4.78 \times 10^{5} Re^{{1.43}} .$$

(27)

With elementary transformations (27) can be transformed into:

$$\frac{{Ma_{q}}}{{Pr}} = 4.78 \times 10^{5} Re^{{1.43}} .$$

(28)

Figure 10 shows that (28) again leads to lower values than our experimental data. Here, as in case of Eq. 23, evaporation effects could have appeared, because this dependence was obtained for water and for a 30% glycerol–water solution at a 2.5 m long test section for Re > 959.

Fig. 10

The dependence of dimensionless parameter $\frac{{Ma_{q}}}{{Pr}}$ versus the Reynolds-number. Points experimental data

A generalisation for water and aqueous solution of alcohol is presented in [16]:

$$\frac{{Ma_{q}}}{{Pr}} = 0.155Re^{{0.65}} .$$

(29)

This correlation leads to results which exceed current data by more than one order of magnitude. It can be partially explained by the fact that dependence (29) was obtained for stable dry spots, whereas the new data was recorded for the formation of local instable dry spots.