In this work models of a centrifugal compressor and radial inflow turbine are presented. The models serve as a way to perform the initial design of the turbomachines for application in small- to medium-scale power cycles. The models are based on the classical mean-line velocity triangle approach which is supplemented with loss coefficients. Restrictive assumptions are not placed on the fluid and therefore the models can be used with supercritical carbon dioxide as will be explored in this work. In order to gain insight into the relationship between the geometrical specifications, thermodynamic operating conditions and performance characteristics of the turbomachinery, mathematical optimisation is performed for specific objectives under given constraints. The benefit of this is that the inevitable trade-offs between the design variables can be studied a priori, before any physical prototypes are built. Results are presented as Pareto plots between pairs of design variables. The importance of choosing an appropriate objective function and upper and lower bounds on the variables is discussed as well.