# One flow and infinite number of stator and rotor

One Dimensional Flow in AxialCompressorsThe flow through axial compressor is essentially threedimensional as flow properties and velocities are function of r, ? and z.  in this module, the flow is assumed to be onedimensional to avoid the complicated flow analysis which require great effortfor analyzing.The flow properties and velocities are assumed to vary inthe z direction (i.e. meridional planes) in axial compressors.

A meridionalplane is any plane passing by the axis of rotation of the machine. Meridionalfor is a two-dimensional flow based on the assumption of asymmetry of flow andinfinite number of stator and rotor blades. The flow properties are assumed tobe constant in r direction. DR.

GALAL BOOK  A s we have discussed before the Stage of compressor consistof a work transfer component with negative work, a decelerating stator andsometime an accelerating stator may precede the rotor (inlet guide vanes).photo of 3 component of stages 0 1 2 3 Blade to Blade Flow PathThe velocity componentsof the working fluid canbe expressed in three velocity vectors, absolute, radial and relative velocity.Assuming the radial velocity equal zero. The air approaches the rotor with anabsolute velocity C1 at an angle ?1 from the axial direction. Combiningthe absolute velocity vectorially with the blade speed U gives the relative velocity W1 at an angle ?1.

After passing through therotor, which increases the absolute velocity of the air, the fluid leaves therotor with a relative velocity W2 at an angle ?2 determined by the blade outletangle. The fluid leaving the rotor is consequently the air entering the statorwhere a similar change in velocity will occur. Here the relative velocity W2will be diffused and leaving the stator with a velocity C3 atan angle ?3.The velocity vectors and associated velocity diagram for atypical stage are shown in Fig {number }Saravanamutto, HIH,Rogers, GFC och Cohen, H. Gas Turbine Theory Fifth Edition, Pearson Prentice Hall, 2001.

From Euler turbine equation  and Referring to thevelocity triangle in the figure it’s easy to get:         Combining the Euler equation with 1st lawof thermodynamics   We got:   The term I is therefore a constant for the rotor andis known as the relative total enthalpy. The flow is assumedadiabatic and there is no work transfer in the stator, then the total enthalpyis constant.           In the Figure 7.3 thethermodynamic states are displayed on the T-S diagram. The distances thatrepresent the absolute and relative kinetic energies are also shown. The relativestagnation enthalpy across the rotor remains constant. In the rotor rothalpyhas properties analogous to stagnation enthalpy in the stator.

KORPELLA   Stage Efficiency Is the ratio between isentropic work of compressor toactual work of compressor    Degree of reaction Is a measure of the enthalpy rise in the rotor to thetotal enthalpy rise in the stage. The degree of reaction R indicates theportion of energy transferred in the rotor blading. It may be defined based onthe actual enthalpy rise or the isentropic enthalpy rise.

or   Although that there are some differences in thementioned expressions of the degree of reaction, but all of them give the sameconcept. It’s an important parameter especially for axial machines which definesthe class of machine with particular characteristics, since it’s fixed, theshape of velocity diagrams and blade arrangement are also fixed. For axialcompressor the degree of reaction is usually 0.5 or higher as the flow isdecelerated through compressor hence high flow turning can’t be carried out. Toincrease energy transfer across the stage an increase in rotational speed ofcompressor is required.