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440 Shorter Communications spherical with an indented base and of volume less than the parent sphere When they are large the dhTerence in shape between two and three dimensinal bubbles is considerable Further treatment of al1the data referred to above shows that the bubble perimeter increase with size above that of a circle in a way that can be described by pe I rdB eO OzdB 2 which puts quantitatively the shape information sketched in Fig 1 Very smal1 two dimensional bubbles like al1 three dimensional ones have a wake that fills roughly one quarter of the full circle and a perimeter surface area little different from the truc circular Whatever the thickness of the containing vessel al1 fluidised bed bubbles and gas bubbles in liquids elongate during coalescence and when about to break surface Coalescing distortion has little effect on the average shape of threedimensional bubbles and al1 our observations are restricted to several centimeters below the free bed surface Elongation therefore is presumably principally a wal1effect This means that the results presented here should not be widely generalised for they must depend to some degree on the actual bed thickness used Chemical Engineering Science 1975 Vol 30 pp 440442 Pergamon Pres and this was not varied Other work has shown ref 3 that parameters describing the bubbles do vary with bed thickness and can be sensitive to its value when it is smal Department of Chemical Engineering Uniuersity College London Torrington Place London WClE IJE England J A GOLDSMITH P N ROWE NOTATION A area of a two dimensional bubble in the plane normal to the usual direction of view cm2 dB bubble diameter defined as its maximum width cm pB bubble perimeter cm Ir supert cial minimum fluidisation velocity cm s V volume of a three dimensional bubble cm REFBRBNCES l Goldsmith J A PLD Thesis University of London February 1974 2 RoweP N and Widmer A J Chem EngngSci 197328980 3 Rowe P N and Everett D J Trans L Chem E 1972 4049 Printed in Great Britain Existente of controls for a stirred tank a counterexample Received 5 August 1974 accepted 1 October 1974 Time optimal control for continuous flow stirred tank reactor has received particular attention for about a decade One of the first and most extensive studies in this area is that of Siebenthal and Aris l Other works 2 4 recently published refer to the theoretical developments achieved by them In their paper a proof for the existente of optimal control is established under very weak conditions The purpose of this communication is to draw attention to a defect of their proof Moreover a counterexample provided here shows that under the conditions mentioned in l control thus optimal control may not exist Siebenthal and Aris based their proof on the optimal control existente theorem by Lee and Markus see Appendix IV in ll When applying the theorem one has to show that the set of admissible controls steering the system from the



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