CP H hs 1 L m P S t T z = heat capacity at constant pressure, L2/t2T = generalized local transport coefficient, L3/T = inverse of surface resistance, L / t = thickness of stagnant film; average thickness of = thickness of liquid layer in wetted wall column, L = number of capacitances or capacitors in multiple = potential, pc, T in M/Lt2 or C in M/L3 = frequency or fractional rate of surface renewal, = time of process on the macroscopic scale, t = temperature, T = distance from interfacial plan, L fluid elements, L capacitances model, dimensionless t-' Greek Letters y = hS/K, L-' 6 = impulse function, t-1 0 = contact time or age, t K = generalized molecular diffusivity, Lz/t 1~ = 3.14159 . . ., dimensionless p = density of fluid or particles, M / L 3 7 = mean residence time, t 4 = contact time (or age) disMbution function, t-1 Jli = instantaneous local transport rate, ML2/t3 or M / t Jl = average local transfer rate, ML2/C? or M / t = average transfer rate for any time interval, ML2/t" or M / t Subscripts b 1 = quantity evaluated at bulk stream = infinite thickness of fluid elements 0 y = no surface resistance 00 = infinitely many capacitors LITERATURE CITED = quantity evaluated at interface 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.14. 15.Although much attention has been given recently to the development of methods for the determination of the optimal control of a batch reactor or the best operating conditions for a tubular reactor, a number of difficulties and uncertainties still remain, especially when the analysis involves an exothermic reversible reaction. Several investigators (1, 5, 8 ) have been concerned with the establishment of the optimum temperature profile along a tubular reactor, from which the optimal control (heat removal rate) must then be obtained. Others (2, 10 to 12) Daniel Y. C. KO is with Gulf Research and Development Conipany,have studied methods suitable for direct determination of the heat flux profile, some of which resulted in the possible appearance of singular control for a portion of the reactor length. The present authors have looked further into the occurrence of such singular problems during the applicatjop pf tbq theory of optimal control and have developed an improved approach to the determination of the optimal heat transfer coefficient distribution along a tubular reactor (6, 7 ) . This paper presents the details of an application of the method of solution presented in the companion paper (7)to the optimal design of a tubular reactor. It is shown that, in general, if the reactor is "sufficiently long," the optimal

For a variational problem in which the control variable is bounded and appears linearly in the system equation, the maximum principle (11) seems to indicate an optimal control (optimal operating condition) of the bang-bang type. However, a singular arc may occur when the Hamiltonian function H is not explicitly a function of the control over a finite time interval, and hence the maximum principle does not give adequate information for selecting the optimal control. In this situation, optimal control may actually consist of variable effort, with values intermediate between the upper and lower bounds of the control (called singular control on the singular arc). The possible appearance of singular arcs in a problem is usually accompanied by considerable analytic and computational difficulties.Siebenthal and Aris (13) and Paynter and Bankoff (10) showed the possible appearance of singular control in the optimal control of a continuous stirred tank reactor and the best design of a tubular reactor. Dyson and Horn ( 2 ) also indicated that a singular arc occurs in optimal distributed feed reactors. The problem of optimal catalyst distribution along a tubular reactor ( 4 , 5, 8 ) can also be shown to have singular control.Studies of singular control can be classified into three categories: the definition, characterization, and determination of the existence of singular control; the synthesis of an optimal singular solution; and the derivation of necessary conditions for optimality of the singular arc. Various ways of deriving the necessary conditions for optimality have been studied by several authors (3, 6, 7) and will not be discussed here. Construction of the singular surface (1, 6) has been very important and useful in characterization and determination of the existence of the singular control. However, each of these authors studied only the free time Bolza problem. For synthesis of the optimal singular solution, most authors (10, 12) report using the gradient method. However, the gradient method has the shortcomings of slow convergence, especially close to the optimum, and failure to give a precise optimal control function (12). In the present paper, efforts first will be directed to constructing the singular surface for various types of problems and to establishing the criteria for the existence of the singular surface which have not been discussed in the literature. Secondly, a computational algorithm, called the combined modes method, developed by the authors, will be presented. It is demonstrated in a companion paper (9) that when the new method is used as a terminal refinement scheme with the gradient method, the disadvantages of the latter are largely overcome.Consider the problem of finding a scalar control function u ( t ) which maximizes the scalar objective function of Bolza form H is linear with respect to u. Accordiiigly, the optimal control is given by a bang-bang type; that isIf, however, the switching function 4 vanishes identically over a finite time interval, then A u ( t ) = +[x(t), p ( t ) ] = 0, ...