Interdependency of Mass Transfer and Falling Film Thickness According to the continuity equation, if the mass flow

A w

m_f (7-28)

is kept constant, a theoretical comparison of the film thickness at these conditions can be performed

Here, it is assumed that the liquid covers the available vertical surface very well and thus, a thin film with a rectangular cross-section and a constant film width can be assumed. The ratio of the film thickness in Eq. 7-29 is estimated by means of the Nusselt and the wall models. If the tau model is applied

7.4 Interdependency of Mass Transfer and Falling Film Thickness

130

In Eq. 7-29 and 7-30 it can obviously be seen that the film thickness ratio depends, at the same mass flow rate, only on the material properties c , f , c and f .

This kind of film thickness comparison can be carried out for:

two different systems at the same operating conditions one same system at different operating conditions

one same system at the constant operating conditions but different degree of saturation.

The first alternative to compare the film thickness of two different systems provides a basis to predict the effectiveness of a hitherto unknown process regarding the mass transfer between the phases, for instance when an extraction column should be used for a process including a new material system. The available data of a system can be used to forecast the effectiveness of the unknown system by means of the film thickness ratio.

Varying the operating conditions means an alteration in the material properties as well. This is well known and is not astonishing if, due to the change in the material properties, the film thickness changes as well. This change of the film thickness can as well be predicted according to the given equations above.

When at the constant operating conditions the material properties change as the saturation process or the dissolution proceeds, it is interesting to discover, how the film thickness is affected. Upon employing Eq. 7-29 for the Nusselt and the wall model, and Eq. 7-30 for the tau model (1: pure, 2: saturated film phase) the film thickness ratio is calculated. The result for the system water-carbon dioxide at 313 K is given in Fig. 7-16. The mixture properties such as the viscosity f and the density f of the water saturated with carbon dioxide are taken from [131,185]. At 6 MPa the film thickness of the saturated water is 1% higher than the initial film.

However the value of the ratio decreases with increasing pressure so that at 30 MPa it becomes 0.91 (tau model) or 0.93 (Nusselt and wall model).

The fall of the film thickness ratio is even more radical when corn germ oil is brought in touch with pressurized carbon dioxide. The ratio of the film thickness at 333 K calculated by means of the Nusselt, the wall and the tau model can be seen in Fig. 7-17. In [185] the density of corn germ oil against pressurized nitrogen at 335 K is reported. The viscosity of pure corn germ oil at 338 K [92] is assumed as constant in the whole pressure range and applied for the calculation of the thickness ratio at 333 K. The density of the carbon dioxide saturated corn germ oil is reported also in [185] and the influence of the dissolved carbon dioxide on the viscosity of corn germ oil at 338 K is taken from [92].

131 0.9

0.92 0.94 0.96 0.98 1 1.02

0 3 6 9 12 15 18 21 24 27 30 33

p [MPa]

f, sat / f, pure

Nusselt + Wall model Tau model

Figure 7-16: Theoretically predicted film thickness ratio between the pure and with carbon dioxide saturated water film at 313 K.

0.4 0.5 0.6 0.7 0.8 0.9 1

0 3 6 9 12 15 18 21 24 27 30 33

p [MPa]

f, sat / f, pure

Nusselt + Wall model Tau model

Figure 7-17: Theoretically predicted film thickness ratio between the pure and with carbon dioxide saturated corn germ oil film at 333 K.

At 3 MPa and 333 K, the saturated film thickness becomes only 92.8% of the initial thickness where no carbon dioxide is mixed in the oil phase. The ratio falls consistently and at 30 MPa it reaches the value 0.45. At the end of the saturation process, that is when equilibrium is reached, the film becomes just one half of its initial thickness. Since the equilibrium process is time-dependent, the change in the film thickness does not happen suddenly too. As the amount of the

7.5 List of Equations

132

dissolved carbon dioxide in the oil phase increases, the film becomes thinner. This, in turn, gives the carbon dioxide a better chance to be transferred into the bulk phase of the film. Other molecules can thus be dissolved at the film surface which causes a further alteration of the material properties, the film becomes thinner and so on. This is a snow ball effect which enhances and thus from advantage for the mass transfer.

The difference in the thickness ratio calculated with help of the Nusselt and the wall model compared with the tau model shown in Fig. 7-16 and Fig. 7-17 is not appreciable. The solubility of carbon dioxide in the oil phase is much higher than that in the water phase. As a consequence, the material properties of the oil phase are affected more strongly too. That is why the fall in the ratio thickness can be seen more clearly in corn germ oil-carbon dioxide than in water-carbon dioxide system.

7.5 List of Equations

Table 7-2: Film phase mean velocity for 0 up to 90°.

Rectangle Circle Triangle

Nusselt ² equivalent radius R is given in Eq. 5-4 on page 79. The integration results ai as a function of the wetting angle from zero to 0

133 i=3:

" #

" #

" # " #

" #

1 13 3 3

cos sin sin cos cos

4 8 8 2

(7-33)

The relation between the angle 0 and 0 can be seen in Eq. 5-2 on page 79.

Table 7-3: Film phase mean velocity for 0 above 90°.

Tau Mean Velocity

Symmetric

204cos sin 90 cos sin 45

294 36 cos 312 cos

128sin 16sin cos 24sin cos

24 cos 30 24

24 2 sin 2

128sin 6sin cos 20sin cos

36 cos 6 45 24 cos

Wall Mean Velocity

" #

135