Development of a software for axial turbomachinery preliminary design
The accurate design process of a turbomachine requires several steps. The strong three–dimensionality and the complexity of the flow leads to the use of sophisticated design tools and testing equipment. Such procedure requires a good theoretical solution to start from in order to save time and money. The best strategy is to design a starting geometry according to theory. Several parameters are involved in the turbomachinery modeling but simplifications can be made, in this phase, to reduce its complexity.
Nomenclature
ρ [Kg/m^{3}]  Density 
P [Pa]  Static pressure 
P_{D} [Pa]  Dynamic pressure 
P_{T} [Pa]  Total pressure 
R [m]  Radius 
C_{θ} [m/s]  Tangential velocity 
C_{r} [m/s]  Radial velocity 
C_{a} [m/s]  Axial velocity 
C_{L}  Lift coefficient 
C_{D}  Drag coefficient 
σ  Solidity of the airfoil cascade 
β_{1} [rad]  Relative inlet flow angle 
β_{2} [rad]  Relative outlet flow angle 
β_{∞} [rad]  Relative mean flow angle 
c [m]  Airfoil chord 
p [m]  pitch of the airfoil cascade 
η  Total efficiency of the fan 
ω [rad/s]  Rotational speed 
α  Forced vortex coefficient 
β  Free vortex coefficient 
Q [m^{3}/s]  Air flow rate 
Theoretical hypothesis
The computational procedure is divided in two main parts: the first one concerns the calculation of the velocity triangles, based on wellconsolidated theory of turbomachinery, the second one concerns the definition of the aerodynamic machine (cascade of airfoil) that provides the required performance.
Velocity triangles
The basic hypothesis are:

Radial Equilibrium: The pressure gradient is equal to the centrifugal components of velocity
;

Zero gradients: The gradients along tangential direction are zero
Starting from these hypotheses the velocity triangles are derived [1].
The Fan Inlet Total Pressure (FITP) is:
where the total pressure difference and the dynamic pressure averaged on the flow rate are function of the derived velocity triangles entities.
Cascade of airfoils
The turbomachinery entities obtained from the computed velocity triangles are connected with the aerodynamic performances of the cascade according to the following relation [2]:
where
; ; ; ;
The two dimensional airfoil characteristics are extracted from database [3]. The three dimensional polars are estimated by empirical correlations [4].
The FITP formulation is here written as:
where the static pressure rise averaged on the mass flow is related to the blade modeled polar.
The value of FITP expressed by the Cascade of airfoils and Velocity triangles formulation must agree.
Implementation
The program is written in Matlab. Four cycles are nested in the iterative process. The computation begins with a starting unity value of efficiency which is reduced during the progress of the computation. The other iterative cycles regard the sections lift coefficient, the Fan Inlet Total Pressure and the input air flow verification according to the flow chart reported in the following figure.
Fig. 1: Flow chart of the algorithm implementation.
Program description
The figure below shows the main window where to insert the input data concerning the required performance, the fan geometric configuration and the working parameters. The four graphs on the top detail the convergence histories of the iteration cycles. They are useful to control the progress of the computation. The graph on the right of the working parameter section details the imposed working distribution along the blade span, the sections position and the qualitative value of the Spanwise Lift Function.
Fig. 2: Main windows.
If the convergence is not reachable an information window explains the problem and gives a suggestion on how to modify the computational configuration. If a solution is obtained a new window displays the two families of results in several graphs.

When the “Turbo” radio button is selected (left windows of figure 3), the turbomachine entities are visualized. The figure displays the velocity triangles (that can be detailed in a separate window) and several graphs describing the spanwise distribution of several flow parameters (which can also be magnified in a bigger window). When the “Aero” radio button is selected (right windows of figure 3), the main figure displays the effective aerodynamic coefficients at the selected section compared with the aerodynamic characteristics of the isolated airfoil.

In this phase it is possible to modify the geometry of the blade using a windows that allow to control the sections dimension (left windows of figure 5). It is also possible to visualize the 3D blade geometry (right windows of figure 5). The new blade can be saved and the calculation restarted to obtain a new solution.

From the solution window it is possible to save a summary of the result in a text file or to visualize it in a separate window (figure 6). The file saved can be loaded and used as input for a new calculation. The geometry can be exported in a CAD format.
Fig. 6: Results summary window.
Conclusions
A software for axial turbomachinery preliminary design has been developed under the theoretical hypotheses of Radial Equilibrium and Zero Gradients along tangential direction. The purpose is to support the designer, in the preliminary phase, with a tool that permits to rapidly define the opportune starting geometry that fulfill the requirements. The software provides the solution, according to the well consolidated turbomachinery theory, that represent the best starting configuration from wich to proceed with more accurate design. A brief introduction on the theoretical hypotheses has been given.
A friendly Graphical User Interface helps to easily analyze the flow parameters and to review the estimated fan performance. The solution is provided in a matter of seconds. It is then possible to rapidly have the feeling of the consistencies of the targets and to iterate among several trial configuration to investigate the opportune design compromise.
Bibliography
[1] A.Kahane, “Investigation of Axial–Flow Fan and Compressor Rotors Designed for ThreeDimensional Flow” NACA Technical Note No. 1652, 1948.[2] R. J. Downie, M. C. Thompson, R. A. Wallis, “An Engineering Approach to Blade Designs for Low to Medium Pressure Rise RotorOnly Axial Fans” 6:376401 CSIRO (Au), 1993.
[3] S. J. Miley, “A Catalogue of Low Reynolds Number Airfoil Data For Wind Turbine Applications” United Technologies Library System.
[4] P. Bushnell, “Axial Flow Fan Design and Analysis Program”, Aeroacoustics & Vibration Research, Carrier Corporation, 1998.
[5] Wallis, E.Allen, “Axial Flow Fans and Ducts”, John Wiley & Sons, New York, 1983.
[6] B. Lakshminarayana, “Fluid Dynamics and Heat Transfer of Turbomachinery” John Wiley & Sons, New York, 1996.
Notes
The preliminary design tool here described was developed for Johnson Electric with the collaboration of Gabriele Milanese and Riccardo Ghio. The software is property of Johnson Electric.