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The objective of this paper is to derive and test a set of algebraic equations describing the fluid flow based on the Euler equations. The derivation is based on the finite volume approach and the benefits of Roe's approach are preserved. A further objective is that the set of algebraic equations can be solved by incorporating an equation of state as complex as for example the EOS for CO 2 given by Span and Wager albeit this equation of state is not used during the derivation. The particular innovation stems from the assumption made during the derivation. Only small changes of fluid properties are assumed at first. This allows to use the equation of state of a thermally perfect gas (specific heat capacity constant) to derive Roe's matrix. Effectively this leads to the set of equations already derived by Roe. But the determination of any state variables and in particular the Roe-averaged speed of sound is done by means of the previously chosen EOS. Thereafter, the obtained set of equations is utilised to calculate flows with small to large gradients. It turns out that the derived scheme gives results in good agreement even for Sod's problem and flows in Laval nozzles with the compressibility factor changing from 1.5 to 0.5 across the shock. Subsequently, the solver is used for predicting condensing steam flows. Besides the calculation of thermodynamic properties also the radii of droplets are derived. The results obtained could be used for optimising the numerical stability of flow prediction in steam turbines. The integration of the EOS for CO 2 given by Span and Wager in the Roe-based solver could also be applied to the investigation of Joule cycles operated with Carbon Dioxide in a supercritical state, thus increasing the accuracy. Due to the novel way of integrating an arbitrary equation of state into a numerical scheme, it also represents a contribution to current developments in the field of computational fluid dynamics. • The developed GIRoe scheme is applicable to an arbitrary equation of state. • Both single- and two-phase flows can be described. • The scheme is characterised by a high numerical stability. [ABSTRACT FROM AUTHOR]