The development of novel clinker production processes is typically a slow and incremental process. This work demonstrates that the pace of change can be greatly accelerated using computational tools. The single-stage production of C$A and C$A-Ternesite cements via a novel elemental-sulfur combustion process is examined here. This work collects a thermodynamic database and compares a simple equilibrium model with results from lab and pilot scale trials. The model appears to describe sufficiently accurately clinker phase stability up to the melting temperature for its use in clinker production process design and optimisation. In particular, the model accounts for the effects of atmospheric conditions within the kiln on clinker phase stability, which appears to be of primary significance in the production of calcium-sulfoaluminate cements. To accurately track process energetics and to allow for optimisation of the cement production process, a one-dimensional rotary kiln thermal model is also developed and found to agree well with both the existing and the novel pilot-scale trial kiln data collected here. This validates the model over a wide range of process conditions including temperatures, pressures, and energy efficiencies, confirming its broad applicability in modeling clinker production. Pilot plant trial products also conform closely to computational predictions and several commercially-viable C$A cements are produced at scale via sulfur combustion.