This research sets out to help computational modellers, to select the most cost effective Cloud service provider. This is when they opt to use Cloud computing in preference to using the in-house High Performance Computing (HPC) facilities. A novel Quality-aware computational Cloud Selection (QAComPS) service is proposed and evaluated. This selects the best (cheapest) Cloud provider‟s service. After selection it automatically sets-up and runs the selected service. QaComPS includes an integrated ontology that makes use of OWL 2 features. The ontology provides a standard specification and a common vocabulary for describing different Cloud provider‟s services. The semantic descriptions are processed by the QaComPS Information Management service. These provider descriptions are then used by a filter and the MatchMaker to automatically select the highest ranked service that meets the user‟s requirements. A SAWSDL interface is used to transfer semantic information to/from the QAComPS Information Management service and the non semantic selection and run services. QAComPS selection service has been quantitatively evaluated for accuracy and efficiency against Quality Matchmaking Process (QMP) and Analytical Hierarchy Process (AHP). The service was also evaluated qualitatively by a group of computational modellers. The results for the evaluation were very promising and demonstrated QaComPS‟s potential to make Cloud computing more accessible and cost effective for computational modellers.