**4.2.2 Discussion**

634 Mass Transfer - Advanced Aspects

Fig. 10 shows that limiting values of Sh are approached for long distance, i.e. for low Graetz numbers. The dependence of these limiting values on the packing fraction φ, in the range

3.29 *Sh e* 1.93

3.3 *Sh e* 2.08

The two solutions for low Graetz numbers are reported in Fig. 11, together with the data for triangular array, after [Miyatake & Iwashita, 1990]. Apparently the equivalent annulus formulation gives practically the same results as the more elaborated model of triangular array for packing fraction φ less than 0.45; only for larger packing fraction the deviation

> Uniform wall flux Constant wall conc. Triangular array

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7

Fig. 11. Asymptotic Sherwood numbers (low Gz) for the equivalent annulus model under the conditions of uniform wall flux and constant wall concentration, compared with the data

As for the tube side, a simple expression for all the Graetz number can be obtained combining the two asymptotic solutions (entrance zone and long distance) by Eq. (32). By using n =3 in Eq. (32), Eq. (46) for long distance and the approximate expression (43) for

> <sup>10</sup> <sup>3</sup> <sup>3</sup> *Sh e* 7.2 1 2 *Gz* φ≈ ++ ⎡

( )

φ

1

<sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> (48)

The same qualitative behaviour was already observed for the solution of Eq. (40) under the conditions of uniform wall flux. In that case is possible to develop an analytical asymptotic solution also for large z values, i.e. low Gz, following a procedure similar to that reported in [Bird et al, 2007] for circular channel. The complete analytical solution was reported elsewhere [Gostoli & Gatta, 1980]. A more handy interpolation formula, for the range

φ

φ

<sup>∞</sup> = (46)

<sup>∞</sup> = (47)

φ

0.1<φ<0.6, is well described by the interpolation formula:

0.1<φ<0.65, is here reported:

becomes substantial.

0

of [Miyatake & Iwashita, 1990] for triangular array

the entrance zone the following, very handy expression is obtained:

5

10

15

**Sh**

20

The equivalent annulus model, presented in details in the previous section, gives a simple description of the mass transfer in the shell side, leading to an approximate explicit expression for the mass transfer coefficient, the Eq. (48), very handy for design and analysis of membrane contactors. As discussed above, for packing fraction up nearly 0.45, the model gives the same results of the more complicated models of regular arrangement in which the circumferential dependence of velocity and concentration are taken into account.

Of course its adequacy to describe mass transfer in actual apparatuses can be questioned.

The equivalent annulus model was compared to experimental results in [Asimakopoulou & Karabelas, 2006a, 2006b]. In their analysis only the entry length solution, Eq. (41), was considered. In the first paper the module used contained only 5 hollow fibres of outer diameter 0.3 mm with a very low packing fraction (φ = 0.093) and the agreement seemed satisfactory. In the second paper, aimed to investigate the effect of packing fraction, three further modules were used with packing fraction up to 0.4. The number of fibres contained in the modules was again quite limited, up to 68 and the modules were carefully hand made. The agreement between experiments and equivalent annulus model was again pretty good. The data showed a dependence on the packing fraction weaker than that predicted by Eq. (41), and the Authors suggested to use the simple expression Sh = 1.45 Gz0.33, to better represent the data in the range 0.093<φ<0.4. This equation corresponds to Eq. (41) or (42) for φ = 0.225, i.e. nearly the average value of the packing fractions tested.

Even if limited to the Leveque-type solution (the entrance zone), essentially the works of Asimakopoulou and Karabelas confirmed that the equivalent annulus adequately describes the mass transfer in small modules for packing fraction up to 0.4.

The situation in commercial modules, housing thousands of fibres, is somewhat more complicated. Many factors make the analysis complex and uncertain [Bao & Lipscomb, 2003]: 1) fibres are packed randomly in the module, 2) the fibres may not lie parallel to each other leading to a transverse flow component, 3) cross-flow regions are present near the entrance and exit ports, 4) fluid distribution into the bundle may not be uniform if the entrance and exit ports are not well designed. Finally the fibres are flexible and the arrangement may change with the flow.

Only the first factor, the effect of random fibre packing, received considerable attention. [Zheng et al, 2004] described the effect of randomness by a generalized free surface model, introducing a probabilistic distribution for the areas of the fictitious annuli circumventing each fibre. [Wu & Chen, 2000] used Voronoi tessellation to generate polygonal regions around each fibre. [Bao & Lipscomb G., 2002] modelled the fibre bundle as an infinite, spatially periodic medium. All the results indicate that channelling through randomly packed bundles reduces the average mass coefficient relative to regularly packed bundles, especially in the well developed limit.

A number of experimental correlations for shell side mass transfer appeared in literature, [Lipnizki & Field, 2001] provide an excellent summary and discussion. These correlations predict values that can be either much higher or much lower than the values calculated for regular and random fibres packings. This suggests that the other factors mentioned above can control the flow field. The presence of cross-flow regions due to shell port design or

Recovery of Biosynthetic Products Using Membrane Contactors 637

produced vanillin from the bioconversion broth removing only a minor extent of the substrates. Indeed the partition coefficient of vanillin is 21 (at pH 7) whereas for ferulic acid it is 0.2 and for vanillylamine 0.01. As shown in Tab 2, the partition coefficient of vanillin depends on pH, making possible to counter-extract vanillin from the vanillin rich solvent

pH 7 8 9 10 12 *m* 21 15.8 10.4 4.4 0.02 Table 2. Partition coefficient of Vanillin between water and n- butyl acetate at various pH

Accurel® polypropylene hollow fibres, supplied by Membrana GmbH, Germany, with inner/outer diameter 0.6/1 mm and porosity 60% were used to build the module used in the experiments. The module contained 30 hollow fibres with overall area 158 cm2 , the shell

The experimental set up is shown schematically in Fig. 6. The feed and the solvent were circulated counter-currently through the module and the respective reservoirs by two gear pumps with variable speed. The pressure of aqueous phase was kept a bit larger than that of the organic phase; to this purpose the feed reservoir was kept at a higher level with respect to the solvent reservoir. In addition a suitable overpressure (0.2-0.4 bar) can be created by a valve at the module exit. Both circuits were equipped with instruments for measuring the

Small samples of the organic and aqueous phases were taken at various times and analyzed with HPLC reverse phase system, equipped with a Beckman Ultrasfere 4.6 mm x 250 mm ODS C18 column (particle diameter = 5 μm) , at 35°C. The mobile phase was composed of 70% H20 added with 1% CH3COOH and 30% CH3OH added with 1% CH3COOH; column temperature was 35°C, injection volume was 20 μL. The isocratic elution was performed for

Four type of extraction experiments were performed: i) extraction from model solutions containing vanillin and ferulic acid or vanillylamine (the substrates of microbial and enzymatic conversion respectively), ii) extraction from the whole bioconversion broth, iii) extraction during bioconversion coupling bioconversion and separation, iv) counterextraction of vanillin from the solvent by NaOH-water solution. The feed flowed either through the lumen or in the shell side. In all the experiments the temperature was 30°C and

The bioconversion broth used in the experiments was obtained in a 3.5 L stirred tank bioreactor by *Pseudomonas fluorescens* BF13-1p at concentration 6 g/L with ferulic acid (2

Typical results obtained in the extraction experiments are shown in Figs 12 and 13, in the first case the aqueous solution contained vanillin and ferulic acid, in the second one vanillin

Apparently vanillin was rapidly removed and reached the concentrations corresponding to the partition coefficient measured in batch experiments, whereas the ferulic acid or vanillylamine concentrations in the aqueous phase remained almost constant. The membrane

the pH was 7 in the extraction experiments and 12 in the counter extraction.

diameter was 10 mm, the packing fraction was thus φ = 0.3.

temperature and the flow rate during the experiments.

g/L) as unique carbon source, at 30°C and pH 7.

phase by using alkaline water.

16 minutes.

**7. Results** 

and vanillylamine.

non-parallel fibres may lead to higher than expected mass transfer coefficients. Poor fluid distribution may lead to lower than expected values. A number of apparent errors in the literature experimental correlations were also pointed out [Liang & Long, 2005].

The comparison of the models and correlations is also complicated by the different parameters used to characterize the packing as well as the different definition of the characteristic diameter. In this paper the outer fibre diameter do has been used in the definition of Reynolds and Sherwood numbers; many authors instead used the hydraulic diameter dh, which depends on the packing fraction:

$$d\_h = d\_o \left(\frac{1}{\phi} - 1\right) \tag{49}$$

Both choices can be accepted, the outer diameter seems more rational to compare the performances of modules made with the same fibres, but with different packing density.
