This paper presents a transcritical / supercritical CO2 (sCO2) recompression Brayton cycle using a novel rotary liquid piston compressor (LPC). This new type of multi-phase compressor utilizes a pumped motive fluid that interfaces with sCO2 in a rotating ducted cylinder for efficient CO2 compression at lower hardware costs. The energy required to pump the motive fluid can be significantly lower than that required to compress CO2 in a traditional compressor. The compressor utilizes a low compressibility, low diffusivity, low solubility liquid as the motive fluid to pressurize process fluid (sCO2) stream. Its use as a replacement for the main compressor in a recompression sCO2 Brayton cycle is expected to reduce compression power by more than 10% while maintaining robust operation over a wide range of ambient temperatures and CO2 densities that are typical for dry-cooled sCO2 cycles in arid climates. The new rotary liquid piston compressor also eliminates the need for gas lubricated bearings & dry gas seals, thus providing added advantage of rotordynamic stability, mechanical robustness & life over traditional compressors. Thermodynamic cycle analyses and 1D compressible flow analysis of multi-phase compression inside the rotary LPC is presented. An advanced 3D multi-phase flow model is developed to study fundamental physics of multi-species transport, diffusion & mixing of species and liquid-supercritical interface compression & decompression. This 3D model is used to validate some of the assumptions made in the 1D model. Various performance curves are developed to study the effect of lead flow, rotational speed and compressor inlet temperature on CO2 exit mass flow rate, % mixing of the two species, compression power requirements and overall compression efficiency. Optimization study on above system variables is carried out and a set of guidelines for use of rotary LPC in sCO2 compression is established.