#### 2.1 Equations of State

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The equation of state can be written in different forms depending on the independent variables taken. Numerical algorithms should allow to calculate and optimize the axial turbine stages, both with an ideal and a real working fluid. It uses a single method of calculating the parameters of the state of the working fluid, in which as the independent variables are taken enthalpy i and pressure P: For a perfect gas equation of state with P and i variables are very simple: For the water steam approximation formula proposed in  is used, which established a procedure to calculate parameters of superheated and wet fluid. It is easy to verify that the knowledge of the value of the velocity coefficient allows to determine the value of losses at the expansion and obtain an expression that relates the enthalpies iT and i at the end of the isentropic and the actual process of expansion, as well as stagnated enthalpy in relative motion  The last expression in combination with isentropic process equation from point 1 with parameters P1,i1 and the value of the relative velocity allows to come, deleting from (2.3), for example iT, to the following process equation with unknowns P, : With the help of the equation (2.5) can be solved a number of problems related to the thermal calculations of stages, which statement depends on which parameter of the unknown is a given. If we assume a known specific enthalpy i at the end of expansion, we obtain the equation (2.5) relative to the pressure P. This problem arises, for example, based on a predetermined degree of reaction or determining the counter pressure by the theoretical enthalpy drop per stage.

Solution of equations of the form (2.5) with one unknown is carried out by means of minimizing the residual square using one-dimensional search of extreme.

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