Determining the optimum number of trays in a distillation column has been all important process optimization problem for over 50 years, Which over the past decade has been addressed successfully, as a mixed-integer nonlinear-programming (MINLP) problem. But tools for solving MINLPs are not widespread, especially in connection with detailed simulation models. In a differentiable distribution function (DDF), all streams around a column, except top and bottom products, are directed to all of the column trays and the distributed flow rate of entry or exit streams is directed to a specific tray based oil the value of its DDF at that tray,. It allows the placement of feeds, sidestreams, and number of trays in the column to be continuous variables in the DDF and in the optimization problem it eliminates the need for integer variables. Instead, the tray optimization problem is formulated as an NLP using the original MESH model. It is critical to deal with trays that have no liquid or vapor flows. To describe this phenomenon properly, in the optimization, complementary, constraints are formulated and added to the NLP, by taking advantage of a smoothing algorithm developed by Gopal and Biegler for phase equilibrium. Also, since pressure in each tram, affects distillation calculations substantially, the pressure drops across each tray depend oil the optimal number of trays and a related smoothing function formulation removes the pressure drop on dry trays. pressure in the condenser o reboiler is then adjusted accordingly. The results of the methods applied to three distillation problems show significant cost reductions in distillation design.