105
Ind. Eng. Chem. Fundam. 1981, 20, 105-107
Sir: We have read with interest the comments of Buffham and Kropholler on the article by Ritchie and Tobgy (1978) and feel that these emphasize more strongly our conclusions that in treating multi-input, multi-output systems careful attention must be given to the definition of the various volumetric flowrates and residence-time frequencies involved. The major difference in analyses arises from the treatment of specific inlet-outlet pairs. It is unfortunate that several misprints escaped attention during the preparation of our paper for publication and these may have contributed to a misunderstanding of our definition of the function f l j . These corrections have now been published (Ind. Eng. Chem. Fundam. 1980,19, 231). The communication of Buffiam and Kropholler is based on a linguistic misinterpretation of our definition of the residence time frequency function, f l j . Buffham and Kropholler’s function, is identical with our function, f l l , although the definitions are phrased differently. Any ambiguity in our definition off, could have been removed by stating the definition: flj(t)dt= the fraction of the ith inlet stream which leaves via the jth outlet stream, Ql1, with residence time between t and t + dt. We contend, therefore, that our eq 12 and 17 are consistent with the above definition and that, after correcting the misprints in the paper, the application of these equations yields the correct overall residence time frequency function, f , of a well-stirred vessel provided with two inlets and two outlets. The functions defined by Buffham and Kropholler as f I l l (or q j ) ,f l l l ” , and f l j I V are “weighting” or “response” functions and not normalized residence time frequency (density) functions. Consequently, none of the integrals
yields the mean residence time, T ~ of~ the , fluid both entering by the ith inlet stream and leaving by the jth outlet stream, Qi+This mean residence time can be calculated,
however, from our f i j function
However, the definitions of Buffham and Kropholler’s functions ti?,fijn, f i j m , and f i j N can be used to establish the relationship between these functions and our function, fi.. Since the flowrate of fluid both emanating from the it6 inlet and egressing via the j t h outlet with residence time between t and t + dt equals Q i f i J ( t ) dt, Q i j f i j n ( t ) dt, Q ; j f i j ( t ) dt, Q j f J T 1 ( t ) dt, and Qjfi?v(t) dt, it follows that
fit(t)=
(Qij/Qi)fijYt)
(2)
= fij(t)
(3)
(Qij/Qj)fij’(t)
(4)
(Qij/Qj)fijn(t)
(5)
fitl(t) fijTt)
=
fJv(t) =
Equations 2-5 and the relationship
should be the basis for checking the consistency and accuracy of the measurements of any of the above functions. We hope that this discussion will resolve any difficulties readers may find in interpreting the papers of Buffham and Kropoller (1970, 1973) and of Ritchie and Tobgy (1978). Literature Cited Buffham, B. A.; Kropholler. H. W. Math. Biosci. 1970, 6, 179. Buffham, 6 . A.; Kropholler. H. W. Chem. Eng. Sci. 1073, 28, 1081. Richie, 6. W.; Tobgy, A. H. Ind. Eng. Chsm. Fundam. 1970, 17, 207.
Department of Chemical Engineering University of Exeter Exeter, England EX4 4QF
B. W.Ritchie*
Department of Chemical Engineering University of Jordan Amman, Jordan
A. H.Tobgy
CORRESPONDENCE Comments on “Gas Holdup and Bubble Diameters in Pressurized Gas-Liquid Stirred Vessels” Sir: In a recent article (1980) T. Sridhar and 0. E. Potter of Monash University, Australia, presented results of a study on the effects of pressure on gas holdup, interfacial area, and bubble size in gas-liquid agitated vessels. Measurements of gas holdup H were made by a manometric method and interfacial area a by a light transmission technique. Mean bubble sizes D B M were calculated from the relationship. DBM = a/6H (1) Both gas holdup and interfacial area were found to increase and mean bubble size to decrease with increasing pressure. The attempt was made to explain the pressure effects as resulting from the increased kinetic energy PK dissipated in gas sparging 0 196-4313/8 1/ 1020-0105$01.00/0
Pk = 0.5Qgpp2 the increase being due to higher gas density p g at higher pressure. Total energy input from gas sparging is the summation of kinetic energy Pkand gas expansion energy Pq,where Pq = P&LQg (3) All nomenclature in this commentary is consistent with the Sridhar and Potter usage. A comparison was made of “extra area” vs. “extra energy” at higher pressure. Interfacial area measurements were more accurate than holdup, and these were therefore used in this comparative analysis. “Extra area” was taken to be the percentage increase in a at elevated pressure 0 1981 American Chemical Society
