Inﬂuence of input parameters in radial compressor design algorithm on the efﬁciency and its sensitivity analysis

Nowadays there are lots of methods using three-dimensional or quasi three-dimensional CFD analysis. Unfortunately, this approach is still very demanding, so that quick preliminary design algorithms have still its importance, even though simpliﬁed analytical model of radial compressor gives less accurate results. Obtained results can be used in later stages of the radial compressor (RC) design, such as deﬁnition of spatial impeller geometry and CFD computation. The article presents the inﬂuence of input parameters in the radial compressor design algorithm on the efﬁciency. The assembled mathematical model of RC is derived from the basic laws of continuum mechanics and can be used for a quick assessment of the preliminary design concept of the RC. A sensitivity analysis is performed on input parameters to select parameters that have the dominant effect on the monitored performance indicators. On the basis of the sensitivity analysis, a multicriteria optimization process was assembled to increase the performance parameters.


Introduction
Compressors are a key component in an aircraft engine and they can be divided into two groups based on the direction of the airflow leaving the rotor.Both types of compressors are commonly used in the aircraft industry.Centrifugal compressors are mostly used in smaller aircraft engines, especially in turbo shaft and subsidiary engines [8].Radial compressor is the most efficient and compact compression device for the flow range 0.3-95 m 3 s −1 [19].
Airflow entering the centrifugal compressor in axial direction is turned into radial direction along the impeller.Impeller delivers the kinetic energy into the airflow.Vane and vaneless diffusers convert kinetic energy of the air into pressure energy.Lastly, collector redirects the airflow back into axial direction and provides a connection following stages of aircraft engine.
Designing a state of the art centrifugal compressor requires a significant amount of engineering effort.Airflow inside the radial compressor is a complex 3D phenomenon.However, the importance of a preliminary compressor design cannot be omitted [11,20].Initial calculations regarding radial-flow compressor stage may be obtained via one-dimensional computation along the mean streamline.There is a large number of 1D design algorithms.
Ru ˚z ˇek and Kmoch in [14] applied fundamental laws of thermodynamics on radial compressor for aircraft applications.For the needs of 1D design, effort to quantify properties of a complex airflow in RC via several engineering constants is spent.This leads to a relatively simple and fast 1D design algorithm, which is able to calculate fundamental compressor parameters.
Vane ˇk and Matous ˇek in [17] presented an algorithm used for radial compressor design, namely impeller and diffuser.The need of one-dimensional design preceding the design of complex spatial geometry is emphasized.
Zurita-Ugalde in [21] presented design algorithm for stationary industrial radial compressor.Even though this paper does not deal with aerial turbomachinery, significant resemblance with [14] and [17] can be seen.Described approach is based on flow path mean-line design using circular arc presented by Smith in [16].
Xu in [19] and [20] described basic considerations which can be used for guiding industry centrifugal compressor design.Presented recommendations are based on author's experience.Preliminary design then can be done using herein presented diagrams.
Computation based on characteristics of known measured compressors is presented by Fo ¨zo ïn [8].Design approach is then based on similarity and dimensionless parameters.
Shiff presented a complex tool for centrifugal compressor stage in [15].In comparison with above mentioned algorithms, Shiff's method does require less input parameters.The computation is based on Aungier's calculation procedure described in [1] and [2].
It is possible to obtain the same performance of the centrifugal compressor with different computed geometries [20].Since the basic thermodynamic and geometric design is included in the initial 1D design, the spatial geometry of the impeller blades may be computed.If the design requirements are not satisfied, the design returns back to its former phase [11].
Nowadays there are lots of methods using three-dimensional or quasi three-dimensional CFD analysis [5,13].Unfortunately, this approach is still very demanding, so that quick preliminary design algorithms have still its importance, even though simplified analytical model of radial compressor gives less accurate results.Obtained results can be iteratively refined.Output of one-dimensional design is later used to determine spatial configuration of the radial compressor.Defined geometry can be subsequently tested and modified in chosen CFD software.
Presented one-dimensional design algorithm is based on [14] and [17].Verified analytical models are combined and extended by several features, such as optimization of inlet axial velocity.Applying pre-whirl is also possible.Mean-line, hub and shroud lines are created using Be ´zier polynomials.Similar approach is described in [3].Furthermore, sensitivity study is performed to evaluate the effect of input parameters on the performance of the radial compressor.Finally, results from parametric study are used in optimization tool.Optimization outlines the dependency of calculated results on design requirements.

Design algorithm
For the preliminary radial compressor design, one-dimensional methods are often used.Presented algorithm is mainly designed to compute both thermodynamic quantities at individual cross-sections and the basic geometry of the centrifugal compressor, that can be seen in Fig. 1.Obtained results may be used in more demanding three-dimensional methods.Two design approaches are possible.Radial compressor can be designed either at its strength limit (characterized by u 2 max1 ) or with respect to a specified total pressure ratio π ct and efficiency η ct .Even though the second approach is more common in the industry, the first design approach is preferred in this paper to examine the capabilities of the presented design algorithm.

Design procedure
Main design input parameters like total pressure ratio π ct , mass flow Q v , rotational speed n, impeller inlet hub diameter D 1i , maximal external diameter of radial compressor D 5e max and flight velocity c 0 depend on the general engine design.Static pressure p 0 and static temperature T 0 are computed from design altitude H via ISA model.Atmospheric total pressure p 0t and temperature T 0t are then computed using the equations of gas dynamics where M 0 is the Mach number at the compressor inlet and κ is the specific heat ratio.The last of main parameters is collector outlet airflow velocity c 5 that is determined according to the needs of the following engine component (100-120 m s −1 if combustion chamber is considered [14]).Impeller is the only component of radial compressor which delivers kinetic energy into the airflow.Both pressure and temperature increase in this section.At the impeller inlet there has to be defined absolute inlet velocity.It has two components -axial component c 1a (100-150 m s −1 [14]) and circumferential component c 1u (0-50 m s −1 [14]).Motion of an air particle is described by Euler's equation enhanced with friction loss [14] For rotational motion, equation ( 2) can be written as [14] By simplifying (3), the equations describing an impeller with infinite number of blades are derived.Due to a finite number of blades and inertial force influencing air particles, a recirculation of air in individual airflow channels is caused.As a result, circumferential component of absolute velocity at the impeller outlet c 2u is different for the impeller with finite and infinite number of blades (c 2u = c 2u∞ ).This phenomenon is quantified by slip coefficient μ defined as Work input into air flow can be expressed [14] as where w r represents aerodynamical and frictional losses along the impeller.Absolute circumferential velocity at the impeller outlet c 2u can be written as where c 2r is an absolute radial velocity at the impeller outlet and ϕ 2 stands for angle of blades at the impeller outlet.
The value of μ can be estimated by several semi-empirical equations.Each of these equations works for a specific group of radial compressors.For aircraft radial compressor with a number of impeller blades around 30, equation ( 7) derived by Eckert [6] is used where angle of blades ϕ 2 is chosen in span 45 • -90 • [4].The number of impeller blades z I is customizable.The empirical equation from [14] is used to estimate the upper and lower bound of z I in presented algorithm as , where The number of impeller blades is often more than twenty-two in aircraft engines [14].Other important parameters defining impeller geometry are average blade thickness at the impeller outlet t 2s and radial clearance δ m .The unfavorable effects caused by radial clearance are described in [4] and [20].Additional parameters describing the impeller geometry can be seen in Fig. 2.
Fig. 2. Impeller geometry, recommended spans are taken from [14] Presence of shock waves is undesirable since the transition to supersonic airflow causes additional aerodynamic losses.Impeller rotation induces circumferential velocity with specific magnitude proportional to the radial distance from axis of rotation.Relative velocity at the impeller tip w 1e affects the properties of airflow along the impeller.Thus, it is required that The magnitude of w 1e is obtained from the components of absolute inlet velocity c 1a and c 1u and circumferential velocity at the impeller inlet tip u 1e as From (10), it can be seen that there are two ways of reducing M w1e .Since u 1e is proportional to the rotational speed n which directly influences the pressure ratio π ct , the modification of c 1a and c 1u comes to mind.That is why the optimization of c 1a can be applied.If necessary, pre-whirl blading to increase c 1u may be designed, as well.
Density of air at the impeller outlet ρ 2 is necessary to determine the geometry of the impeller outlet, thus, an iterative calculation is used.The value of ρ 2 is unknown in the first iteration and the width of the airflow channel at impeller outlet b 2 is calculated.After that, impeller velocity field and aerodynamic losses are computed.Consequently, the temperature (and pressure) of air at the impeller outlet T 2 (p 2 ) are evaluated.Finally, new value of ρ 2 is determined.Iterative process is terminated when a convergence condition is reached.The impeller isotropic efficiency η Iis is calculated as When the dimensions of the impeller and the properties of air leaving the impeller are calculated, a diffuser can be designed.It transforms kinetic energy into pressure energy.Diffuser is divided into vane and vaneless part.Vane diffuser design assumes subsonic airflow, thus vaneless diffuser is placed after impeller outlet and following assumptions are taken into account where the superscript ' ' refers to the vaneless diffuser inlet.Vaneless diffuser ensures absence of shock waves even though airflow leaving impeller is supersonic.Shock waves appear unless radial component of velocity at impeller outlet c 2r is subsonic (M c2r < 1).Length of vaneless diffuser is calculated with respect to given Mach number at the vane diffuser inlet M 3 [14] as where and n 2 is determined from Since the vane diffuser has higher efficiency over the vaneless diffuser [14], the remaining velocity reduction is performed in the vane diffuser.Number of vanes in the vane diffuser z D is customizable as well as z I and according to [14] it is often chosen in the range 15 − 35.Other input parameters are Laval number at the vane diffuser outlet λ 4 and polytropic index n D .Diffuser geometric configuration of a single vane can be seen in Fig. 3. Coefficients describing losses take into account influences of boundary layer at the individual component i , pressure losses marked as σ i and friction losses ξ i .Ranges for coefficients i are derived from basic considerations about boundary layer.Pressure loss coefficients and friction loss coefficient at impeller ξ I are taken from [14].Range of friction loss coefficient at vaneless diffuser ξ I is based on measurement developed by Johnson and Dean in [10].Recommended spans of these loss coefficients are presented in Table 1.
The flowchart of the presented algorithm workflow is visualized in Fig. 4.

Comparison with another design algorithm
Compressor designs comparison with design algorithm described in [14] is presented in this section.Since the design algorithm presented in this paper is derived from methods used in [14] and [17], several input parameters used in [14] differ from those in the presented algorithm.That is the reason why particular results in [14] (rotational speed n, impeller inlet hub diameter D 1i , number of impeller blades n I , etc.) were used as input parameters in the presented algorithm.Methodology used in [14] does not include impeller and vane diffuser geometry 2 .Overview of the input parameters used for verification is ordered in Table 2. Parameters common in both algorithms are marked with a symbol '*'.
Value 12 15 500 124 900 0 120 101 288 Parameter Value 500 4.24 0.79 124 0 90 0.5 0.98 Computed results are compared in Table 3.The main design parameters π ct , η cis , D 5e correspond with the results in [14].Requirement on Mach number at the impeller tip, described by (9), was satisfied.Impeller dimensions D 1e , D 2 and request on its "strength" expressed with u 2 max are very similar.Higher difference in pressure at the cross-section 2-2 is caused by considering a different loss model than the author [14] used.On the other hand, the presented design algorithm assumes more losses in the diffusers and collector, so that thermodynamic quantities at the compressor outlet differ a little.
Example from [

Sensitivity study
There are 27 input coefficients entering the presented algorithm.Significant effort has been spent to determine the rate of change in the designed pressure ratio π ct and the isentropic efficiency η cis caused by the varying of individual input parameters.Due to a computational difficulty, a variation of one parameter was performed while others were fixed at their span centres.We gained an overview of the influence of individual parameters on the 1D radial-flow compressor design.The sensitivities of chosen parameters are plotted in Fig. 5.We can see that the design fundamentally depends on the choice of the u 2 max parameter, whereas the inlet casing pressure loss coefficent σ E affects the design to a very little extent.Influence of individual parameters is assessed as the difference between its minimum and maximum value in the efficiency and pressure ratio.The summary of the influences can be seen in Fig. 6.If the difference is negligible, then the parameter is marked with the symbol '×'.Several parameters were selected to perform a more detailed sensitivity study.These parameters can make the difference in efficiency more than 0.5 % or they influence the resulting pressure ratio (with respect to the chosen span of values).We gain a set of nine parametersu 2 max , ϕ 2 , c 1a , ξ i , n D , δ m , λ 4 , σ C and z r (descending order of influence).
From the computed results, we will focus on the parameters δ m and ϕ 2 .In Fig. 7, it can be seen that as the radial clearance δ m increases (in arrow direction), both efficiency and pressure ratio decrease.The increasing angle of impeller blades at the outlet ϕ 2 leads to higher efficiency and pressure ratio (Fig. 8).These results are in accordance with the knowledge in [4] and [7].In Figs. 9 and 10, it can be seen that there are two other parameters that affect the efficiency and pressure ratio, as well.The parameter σ C characterizes pressure loss in the collector.Its value depends on the design solution.Some authors state that this decrease is neglected for small compressors [18].Since we consider the total pressure p 5t estimation at the collector as p 5t = σ C p 4t , Fig. 9 shows that the difference in the isentropic efficiency can be up to 1 %.
The friction loss coefficient ξ I quantifies the friction loss along the impeller.These losses include the disc friction loss, recirculation loss, blade loading loss, skin friction loss and incidence loss (when prewhirl is applied [11,14]).These losses are thoroughly explained in [9].Fig. 10 justifies the importance of choosing the coefficient as realistically as possible.The difference in isentropic efficiency for various ξ I is up to 4 %.
In Fig. 11, there are results considering (and varying) all nine parameters that make the biggest difference in the compressor performance.Optimal set of inputs is indicated as the upper boundary curve constructed using the 4th order least squares method (LSM).Not all cases lead to satisfying designs.On the other hand, there can be seen that isentropic efficiency can be higher than it was computed in the example case (described in [14]).With the appropriate choice of inputs, especially u 2 max , the isentropic efficiency η cis could be higher than 80 %.Some notable results are summarized in Table 4. Furthermore, some cases lead to one-dimensional designs with 80 % in efficiency and value of pressure ratio up to 4.6.

Multi-criteria optimization
The optimization method for one-dimensional design of radial-flow compressor is presented in this section.Parametric study performed in Section 4 is taken into account.
One-dimensional radial-flow compressor design requires a significant amount of knowledge to determine the plausible values of various coefficients entering the design process.Many input parameters determine losses along the airflow channels.Including these loss coefficients into the optimization process clearly leads to their minimization.Parameters δ m , 1 , 2 , σ E , ξ I , ξ D , n D , 3 , 4 , σ D , σ C and 5 are not included into optimization process.Loss coefficients need to be specified and are considered constant during optimization.Radial clearance δ m between the impeller and shroud causes additional aerodynamic losses.Collector elbow radius coefficient K b4 needs to be as low as possible to minimize D 5e .These two parameters are excluded from parameter optimization as well.
Main design parameters (Q v , n, D 1i , D 5emax , c 0 , c 5 , H) are given from the general engine design.Parameters defining the geometry of the impeller airflow channel (see Fig. 2) do not affect one-dimensional design.The geometry of the vane diffuser directly affects the external diameter D 5e .On the other hand, the parametric study conducted in the previous section proves that input parameters defining vane diffuser geometry (Fig. 3) can be excluded from the optimization process.Mean blade thickness at impeller outlet t 2s is also excluded based on Fig. 6.Finally, Laval number at vane diffuser outlet λ 4 is not included into optimization process since it's value has been always maximized for wide range of weighting coefficient combinations.
Fixed parameters with corresponding values3 are ordered in Table 2. Optimization has been performed in Matlab using the fmincon function [12], which finds the minimum of the constrained nonlinear multi-variable function

Cost function
The goal of the optimization is to maximize both the pressure ratio π ct and compressor isentropic efficiency η kis .Furthermore, minimal dimensions, the especially outer diameter D 5e are desired.Finally, manufacturing costs are taken into account (u 2 , z I and z D ).Cost function CF is constructed as where a πct , a η kis , a D5e , a zI , a zD and a u2 , are the weighting coefficients.First of all, weighting coefficients a i were normed to ensure that ratios of individual terms in ( 16) change as much proportionally to their weighing coefficients4 as possible.This has been performed for vector p mean composed from mean values of individual parameters being optimized.
Fixed parameters were set to values in Table 2. Remaining parameters were set to: Results of the performed optimizations are in Table 5. Number of impeller blades z I and diffuser vanes z D are rounded at the end of optimization.Optimization has been performed for different sets of initial conditions.When a given parameter does not affect the cost function CF , the symbol '×' is used and the optimized parameter differ for individual initial conditions.Since the number of impeller blades and number of diffuser vanes cannot be precisely stated in certain cases, then some parameters vary, as well.Case \Unit Table 5 shows that the resulting pressure ratio π ct , compressor isotropic efficiency η cis and external diameter D 5e correspond with chosen weight coefficients.In the first case, when maximal pressure ratio is desired, we obtained the highest pressure ratio from all studied cases.However, efficiency was at it's minimum.When the highest possible efficiency was desired, resulting pressure ratio was significantly lowered.In the third case there were no requirements on neither π ct or η cis .The smallest external diameter was desired.Considering only spatial restriction leads to unacceptably low pressure ratio and compressor isotropic efficiency.When all three mentioned criteria are combined, we obtain compromise from former three cases.

Conclusion
The structure of the one-dimensional design algorithm for radial-flow compressor stage has been described.The presented algorithm was compared with the design method from [14].This comparison affirms that all kinds of one-dimensional design algorithms differ mainly in aerodynamic loss model.The parametric study unveils which parameters have the most significant effect on the performance indicators of the radial compressor stage.There was shown that eighteen parameters from twenty-seven considered influence compressor design very little.Optimization tool was assembled based on the results of the parametric study.It confirmed that during the design process there is a lot of contradictory requirements.For instance, demanding a minimal compressor outer diameter leads to unsatisfactory performance indicators.Furthermore, combining both the requirements on performance parameters and compressor dimensions leads to compromise.
Further work will concern the calculation of spatial impeller blade geometry.After that, one-dimensional CFD simulation through the radial compressor stage will be performed.Subsequently, three-dimensional analysis of airflow inside the centrifugal compressor should be carried out.Finally, fully parametric tool for a complete centrifugal compressor design combining initial one-dimensional computation with complex three-dimensional flow analysis accompanied with optimization processes should be developed.

Table 1 .
Loss coefficients summary

Table 2 .
Input parameters

Table 4 .
Results at the upper boundary curve