Shorter Communications 403

k, gas film mass transfer coefficient, LT’ k, liquid tilm mass transfer coefficient, L-‘T

K, overall mass transfer coefficient, LT’ N, mass transfer flux, ML-*T-’ P pressure, ML-‘Tm2 R radius, L t time, T

V volume. L”

Greek symbols p dimensionless growth coefficient c defined in eqn (3) p density, ML-’ Cp driving force parameter

Dimensionless groups Ja Jakob number Re Reynolds number

SC Schmidt number Sh Sherwood number

Subscripts g gas phase i interface I liquid phase

[l] Bankoff S. G., Aduan. Chem. Engng 1966 6 l-60. [2] Epstein P. S. and Plesset M. S., J. Chem. Phys. 1950 18 1505. [3] Szekely J. and Martins J. P., Chem. Engng Sci. 197126 147. [4] Striven L. E., Chem Engng Sci. 1959 10 1. [5] Burman J. E., Ph.D. Thesis, University of London, 1974. [6] Whitman W. G., Chem Met. Engng 1923 29 147. [7] Co&on J. M. and Richardson J. F., Chemical Engineering,

Vol. II, p. 701, Pergamon Press, London 1966.

Chemical Engineming Science, 1976, Vol. 31, pp. 4OM04. Pergram PN.K Printed in Great kitah

Gas bubble formation from submerged ori6ces- “simultaneous bubbling” from two orifices

(Received 1 October 1975; accepted 26 Nooember 1975)

The formation of gas bubbles from submerged orifices has been the object of numerous investigations[l-111. However, none of such works deals with bubble growing from a multiple-orifices plate feeded from a single chamber. In such a case, for sufficiently large values of the gas flowrate, bubbles grow simultaneously from all the orifices. For smaller flowrate values the plate works discontinously and as the number of oritices increases “simultane- ous bubbling” becomes more ditficult.

In this work volumes of gas (nitrogen) bubbles grown in water from a two-orifices plate have been obtained from measured values of flowrate and frequency. The experimental apparatus is equal to the one described in[ll], but for the plate in which two

o’Q’

A A

0 00

A

A

0 0

f l

Q, ccisec 0.2L ,

0 5 10 15

Fig. 1. Bubble volume vs volumetric gas flowrate from single or&e. +, chamber volume 215cd, single orifice; A, chamber volume 415 cm3, two orifices; 0, chamber volume 415 cm’, single

orifice; A, chamber volume 785 cm>, two orifices.

conical or&ices were drilled, with the smaller section upwards. The minimum orifice diameter was 0.15 cm and the pitch was 0.5 cm in all cases but one for reasons discussed in the following. In order to verify the equivalence between the or&es they were alternatively occluded, and the volume of the outcoming bubbles resulted the same within a few percent.

Data taken when both oritices were working simultaneously (“simultaneous bubbling”) are reported in Figs. 1 and 2 together with others taken when one of the orifices was occluded. In Fig. 1 the bubble volume, V, is plotted vs the volumetric flowrate from single orifice, Q, obtained as the ratio between the volumetric flow- rate coming into the feeding chamber, G, and the number of orifices. The volume of bubbles outcoming from single or&es feeded by chambers of 215 and 415 cm3 is in Fig. 1 close to that of

_._

1 ’ ’ ‘I v, cc 0.7 -

0.6 -

0.6 -

0.4 -

0.3 -

a2 -

<: <

.

.

0 v

0

a

V

V

,

4’ 2

- 0.0

- 0.5

- 0.4

- 0.3

- 0.2

I s. cc I 0 200 400 000 800

Fig. 2. Bubble volume vs the ratio between the volume of the feeding chamber and the number of or&es. 7, single orilice, pitch 1 cm; V, single or&e, pitch 0.5 cm; I, two orifices, pitch 1 cm; Cl,

two orifices, pitch 0.5 cm.

401 Shorter Communications

“simultaneous bubbles” growing from chambers of 415 and 785 cm” respectively.

This result can be s urnmarized as follows: “The volume of bubbles outcoming from a single orifice approximates that of simultaneous bubbles growing from a chamber which has double capacitance and is fed by double flowrate”. With reference to the data reported in Fig. 1 this simple rule is not exactly verified especially for larger bubble volumes and it slightly underestimates the volume of “simultaneous bubbles”. This can be exulained in terms of interaction between the bubbles. Consequently:as shown in the following, the pitch atIects the volume of simultaneous bubbles and at larger pitch values the above rule is much better verified.

The volume of bubbles generally increases with the flowrate coming in the feeding chamber. For any value of the chamber capacitance, a value of G can be however found below which the bubble volume is practically constant [ll]. Data taken in this range of G have been plotted in Fig. 2 as bubble volume vs the ratio, S, between the volume of the feeding chamber and the number of orifices for two diierent pitch values: 0.5cm (as for the data shown in Fig. 1) and 1 cm. These data show that by increasing the pitch the volume of “simultaneous bubbles” decreases and the agreement to the equivalence rule between bubbles outcoming from a single orifice and “simultaneous bubbles” improves consid- erably. This result can be easily understood. When the pitch is large the two growing bubbles do not interact in the liquid side and they share the flowrate and the feeding chamber capacitance coherently to the above suggested rule. Decreasing the pitch below the bubble diameter stretches the bubble in the vertical direction. As a consequence of this change in shape the pressure inside the bubble keeps lower producing a larger gas Aowrate through the or&e and thus a larger detachment bubble volume.

In conclusion the volume of “simultaneous bubbles”, especially when the pitch is sufficiently large, can be brought back to either experimental or theoretical results of bubbles outcoming from a single orifice.

Istituto di Ingegnetia Chimica Universitci di Palermo Palermo Italy

G. TITOMANLIO G. RIZZO

D. ACIERNO

REFERENCES

[l] Davidson J. F. and Schiiller B. 0. G., Trans. Inst. Chem. Engrs 1960 38 335.

[21 Ramakrishnan S., Kurnar R. and Kuloor N. R., Chem. Engng Sci. 1969 24,731.

[3] Wraith A. E., Chem. Engng Sci. 1971 26 1659. [4] WraithA.E.andKukutaniT., Chem. EngngSci. 197429 1. [5] Satyanarayan A., Kumar R. and Kuloor N. R., Chem. Engng

bl

171

PI

191

PO1

[ill

_ - Sci 1%9 24 749. Lanauze R. D. and Harris I. J., Chem. Engng Sci. 1972 27 2102. McCann D. J. and Prince R. G. H., Chem. Engng Sci. 1%9 24 801. Khurana A. K. and Kumar R., Chem. Engng Sci. 1969 24 1711. McCann D. J. and Prince R. G. H., Chem. Engng Sci. 197126 1505. Lanauze R. D. and Harris I. J., Chem. Engng Sci. 1973 29 1663. Titornanlio G., Rizzo G. and Acierno D. Gas bubble forma- tion from submerged orifices. A simplified model for predict- ing the bubble volume, Quademi Ing. Chim. Ital., in press.

Che’mical Enginming Science, 1976, Vol. 31, pp. 404405. Pergamn hess. Printed in Great Britain

The prediction of the densities of n -alkanes

(Received 29July 1975;accepted 8January 1976)

A number of workers have recently applied the extended form of the Corresponding States Principle of Rowlinson and Watson[l] to predict the thermodynamic properties of pure fluids and their mixtures[2-71. Calculations of vapour-liquid equilibria in the argon t nitrogen t oxygen system by Watson and Rowlinson[2] showed that the biggest single source of error in the predictions was the failure of Pitzer’s acentric factor to reproduce the ex- perimental vapour pressures of the pure substances exactly. This discrepancy is small but would be larger if it were assumed that all substances had the same reduced vapour pressure curve (as implied by the two-parameter Corresponding States approxi- mation). In addition, the errors would increase as the pure substance and the reference became increasingly dissimilar--in going up the homologous series of the n-alkanes, for example.

Since the vapor-pressures of pure substances are generally known, Gunning and Rowlinson[3] in their calculations of vapour-liquid equilibria provided for the adjustment of acentric factors at each temperature to fit the known vapour pressures. This was done implicitly in their computer program and no attempt was made to correlate the acentric factor with the temperature.

Adjustment of the acentric factor to fit the vapour pressures, however, leads to large errors in the predictions of the densities of the pure fluids. Mollerup and Rowlinson[5] therefore chose o to give a good representation of both saturation pressure and density or saturation temperature and density. They did not, however,

attempt to correlate the acentric factor with the temperature except for carbon dioxide and ethylene, when two elliptic equa- tions were presented in a footnote to their Table 1.

We have used the Corresponding States method with a variable acentric factor to predict the densities of the n-akanes from propane to n-dodecane. In addition to yielding information on the variation of the acentric factor with temperature, our study also extends the method to the higher molecular weight n-alkanes (previous studies were limited to simple substances and the lower molecular weight hydrocarbons). We have retained methane as the reference, since an accurate equation of state is available for this substance. The choice of an equation of state is a matter of accuracy and convenience and we have used the equation of state of Bender [8] since it accurately represents the experimental data for methane throughout the temperature and pressure range where such data is available (including the two-phase region). Also, the analytical equation is relatively easy to manipulate mathemati- cally. The shape factor equations of Leach[9] have also been retained to describe the deviations of the pure substances from the reference in terms of the acentric factor.

We have varied the acentric factor until the predicted value of the density coincided with the experimental value at a given temperature and pressure. The acentric factor for the ten sub- stances studied tirst decreases with reduced temperatures up to a reduced temperature of approximately 0.5 and then increases up to a reduced temperature of approximately 0.9. The minimum