The development of embedded and interlayer liquid cooling in integrated circuits (ICs) using silicon microchannels has gained interest in the recent years owing to the rise of on-chip heat uses that aggravate thermal reliability issues of the emerging 3D stacked ICs. Further development of such devices and their translation to commercial applications depend largely on the availability of tools and methodologies that can enable the "temperature-aware" design of liquid- cooled microprocessors and 2D/3D multiprocessor systems-on-chip (MPSoCs). Recently, two optimal design methods have been proposed for liquid-cooled microchannel ICs: one to minimize on-chip temperature gradients and the other, called GreenCool, to maximize energy eciency in the coolant pumping eort. Both these methods rely upon the concept of channel width modulation to modify the thermal behaviour of a microchannel liquid-cooled heat sink. At the heart of both these methods is a new semi-analytical mathematical model for heat transfer in liquid-cooled ICs. Such a mathematical model enables the application of gradient descent approaches, such as non-linear programming, in the search for the most optimally performing channel design in a huge multi-dimensional design space. In this paper, we thoroughly quantify the impact and efficiency of the semi-analytical model, combined with non-linear programming, when compared against several numerical optimization mechanisms. Our experimental evaluation shows that non-linear programming, alongside the semi-analytical model, is up to 23x faster than conventional randomized/heuristic design approaches such as genetic algorithms and simulated annealing using fully-numerical thermal models.