Slow droplet motion on chemically heterogeneous substrates is considered analytically and numerically. We adopt the long-wave approximation which yields a single partial differential equation for the droplet height in time and space. A matched asymptotic analysis in the limit of nearly circular contact lines and vanishingly small slip lengths yields a reduced model consisting of a set of ordinary differential equations for the evolution of the Fourier harmonics of the contact line. The analytical predictions are found, within the domain of their validity, to be in good agreement with the solutions to the governing partial differential equation. The limitations of the reduced model when the contact line undergoes stronger deformations are partially lifted by proposing a hybrid scheme which couples the results of the asymptotic analysis with the boundary integral method. This approach improves the agreement with the governing partial differential equation, but at a computational cost which is significantly lower compared to that required for the full problem.

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that the efficiencies were known and that it was required to find the transient solution of the problem under consideration.It appeared that a suitable method for the determination of the transient plate efficiencies could be developed by replacing the steady state equations in the methods developed by Davis et al. and Taylor et al. by corresponding equations for the unsteady state models proposed by Waggoner et al. This approach, a logical extension of previous work, was successful, and the resulting methods are described in the first two sections that follow.A third procedure was developed for the determination of the transient plate efficiencies for the case where the compositions and plate temperatures are known functions of time. The equations needed in the application of this method are presented in a subsequent section.Also, a method was developed for separating the mixing effects of the liquid on each plate and in its downcomer which was not involved in mass transfer from the plate efficiencies of the liquid on each plate that was involved in mass transfer. This method was developed by replacing the steady state equations in the methods of Taylor et al. and Davis et al. by the unsteady state equations for the model described by Tetlow et al. (8). An example of this method is presented in which the plate efficiencies and mixing effects are determined simultaneously,In the method described in the next section, it is supposed that the product distributions (bi/di) and the tem- specifications such as the distillate rate, reflux rate, column pressure, type of condenser, number of plates, location of the feed plate, the complete definition of the feed, and the holdups are known. Information over and above this, such as any combination of product distributions (bi/di) and temperatures ( T j ) , is hereafter referred to as additional specifications. A D D I T I O N A L SPECIFICATIONS: ALL PRODUCT F U N C T I O N S OF T I M EOn the basis of these known values of the operating variables, the procedure that follows determines a set of plate efficiencies required to obtain agreement between results calculated by Waggoner's model (10) and the results of field tests.In the method developed, the transient operation period is subdivided into time increments. At the beginning of and the K values of component i (evaluated at the temperature and ressure at which the liquid leaves plate for Kjt, it is also evaluated on the basis of the temperature, pressure, and composition of the liquid leaving plate i.Since the sum of the yjls at the end of the time period (t, + A t ) under consideration has the value of unity, it follows that j ) by Kji. When t R e activity yj~ji is included as a multiplier 0 f j = 2 Eli Kji xji -1

Presented herein is the formulation of a more general model for the unsteady state operation of a distillation column than has been available previously. Suitable numerical methods for solving the equations required to describe the model are also presented. The reliability of the proposed numerical methods was established by solving a wide variety of problems ( 1 8 ) . To demonstrate some of the characteristics of the model as well as possible uses, such as the analysis of various control schemes, selected numerical results are presented in a subsequent section.In general, the models proposed for distillation columns in unsteady state operation may be divided into two groups: one group consists of those models used to obtain analytical and analog computer solutions; and the other group consists of those models used to obtain numerical solutions by use of digital computers. Analytical and analog models generally require many simplifying assumptions, such as linear equilibrium relationships, the lumping together of plates, and the treatment of multicomponent mixtures as binary mixtures. In the past these models have proved to be useful for the broad understanding of the operation of distillation columns ( 6 ) and are presently being used successfully as a basis for column control (9).Models of the second group used to obtain numerical solutions on digital computers need not contain many of the simplifying assumptions generally used for the simpler analytical and analog models. Models of the second group, which give an accurate description of column operation, Previous models have neglected the combination of mixing effects on the plates and in the downcomers, and in most models the mixing effects in the holdups of the reflux drum and transfer lines have been neglected also.The present model includes the effect of mixing in all of these holdups. Liquid passing through a column exhibits some degree of mixing in the direction of bulk flow. The mixing effects of a liquid in bulk flow, without dead volume, on a plate and its downcomer are bounded by those for the three limiting cases: perfect mixing, plug flow, and channeling. The third limiting case was introduced to acreasons for t K eir superiority over the approximate models