440 Shorter Communications spherical with an indented base and of volume less than the parent and this was not varied. Other work has shown (ref. [3]) that sphere. When they are large the dhTerence in shape between two parameters describing the bubbles do vary with bed thickness and and three-dimensinal bubbles is considerable. can be sensitive to its value when it is smal]. Further treatment of al1 the 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. Department of Chemical Engineering, Uniuersity College London, Torrington Place, London WClE IJE, England J. A. GOLDSMITH P. N. ROWE 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 three- dimensional bubbles and al1 our observations are restricted to several centimeters below the free bed surface. Elongation therefore is presumably principally a wal1 effect. 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 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,,,,,, (supertïcial) 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. Chemical Engineering Science, 1975, Vol. 30. pp. 440442. Pergamon Pres. 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 initial point to the selected steady state is not empty. Makhtg use of theorem V in Appendix of [IJ it was shown that unless the endpoint lies on a specified curve there exists a neighbourhood of the end point from which it is attainable by admissible control. Nevertheless it is not sure whether this neighbourhood contains the initial point. This was disregarded in [ll. To show that the existente theorem by Siebenthal and Aris cannot be proved in genera1 a counterexample is created. For simplicity we refer to the tûst example in [l] and the notation introduced there wil1 be used. Let US consider the irreversible, tirst order, exotherm reaction A, + A,. The dimensionless concentration c* and dimensionless temperature T* are governed by the differential equations (1) The manipulated variable h * is bounded, 0 s h * 5 2 h 3, where h ? is the dimensionless cooling rate at normal operation. In the following, since more then one steady states may exist at normal operation, suffix s refers to the one of highest c* value. Let Q. denote the heat removed at normal operation, namely Q. = ht(Tt- T:)>O (2) and u denote the differente between actual and normal cooling, that is u=h*(T*-T:)-Q.. (3) Introducing the linear transformation x =c* Yzc*-T* (4) Eq. (1) take the form X=-xt106(1-x)exp & ( > ,z\ Y=-ytQ,-Tytu. (2) Assume for the moment that the piecewise continuous control function IA can be chosen independently of state with restriction ]u] 5 8, 8 > 0. Note that this is not truc for system (1). Nevertheless the simpler control system (5) will be useful in the following. Fixing the value of u the steady state point(s) of system (5) lie on line y = QS - T’j t u. On the other hand equation 0=-xt106(1-x)exp & ( ) must be satisfied. Fortunately it is possible to be explicit about the (x, y) points satisfying (6). They lie on the graph of function y(x)= I 1 \ tx. (7) hl kö&JShorter Communications 441 Using classical calculus the following properties of y(x) can be stated: 1. It is continuous in the open interval 0 < x < (I, where 10’ a=l+lOó 2. $i10 y(x) = 0, liliO = -m. 3. Outside the interval 0 < x < q the points of its graph have no physical meaning (either negative concentration or negative temperature occurs). So we restrict the function to the interval mentioned above. 4. y(x) has an inflection point before which it is convex after which it is concave. 5. It is strictly monotone decreasing, then strictly monotone increasing and at last it is strictly monotone decreasing again. Properties 2 and 5 assure that, whatever value Q. has, one can always select a positive 7’: and a (small enough) S so, that system (5) has three steady states at any fixed u, (UI 5 6. This phenomenon is often referred to as ‘thermal instability’. From property 4 and 5 it can be verified that an instable steady state is intercepted by two stable ones. In Fig. (1) the steady states are denoted by A,B,C at u=O, and A’,B’,C’ at u=S, A-,B-,C- at u=-6. It can be seen that a trajectory starting from the domain between the lines y=Q,-TTtS and y=Q,-T$-8 can never get out of it, since y < 0 along the upper line and y > 0 along the lower