This thesis deals with Conjugate Heat Transfer analysis of an oil-free Roots blower which is a Positive Displacement Machine used for low pressure ratio applications. Prediction of temperature distribution is critical in the design of rotor-to-rotor and rotor-to-casing clearance gaps inside these compressors. The aim of the study is to numerically analyse the heat transfer from pressurized air to rotors and casing and provide the temperature distribution over structural elements. These problems are too complex to be solved analytically so this thesis will employ High Performance Computing to perform CFD simulations and find an approximate numerical solution of the heat transfer problem. Internal flow inside the blower is highly transient and to predict its characteristic we need to tackle one of the most challenging deforming mesh problems where working medium is displaced during simulation. To allow for mesh deformation, due to intermeshing of the rotors, the moving mesh technique will be employed in this thesis. A numerical framework for 3D simulation of complex flow field inside a Roots blower compressor is established and presented. The framework employs Finite Volume Method. Five cases are investigated, each for different operating condition. In order to evaluate the accuracy of the developed numerical model, extensive validation is performed for several operating conditions. The numerical solution is compared to the overall performance of an equivalent experimentally measured compressor. The experimental data were provided by City University London and they include measurements of surface and air temperature, mass flow rate, pressure ratio, and effective power measurements. The comparison between numerical results has shown good agreement with the experimental data. The outcome of this research contribute to the understanding of the effects of leakage flow phenomena and can help design more efficient compressors.