Impellers are the core dynamic components found in various turbo machines such as pumps, compressors and turbines. In case of a compressor, the function of the impeller is to create static pressure rise by diffusing the relative velocity component of the intake fluid by means of its rotating action. The key issue that needs to be addressed when designing impellers is to control the rate of relative velocity diffusion along the impeller passage length. This paper aims at rapidly designing an impeller for a centrifugal compressor to meet the operating design conditions by using the basic governing equations of fluid flow and CAD Modelling packages. The design optimisation was carried out in a qualitative manner. The design adhered to the physics of effectively diffusing compressible flow.
Nomenclature
b = tip width
B = Blockage due to blade thickness
c = absolute flow velocity
cp = specific heat at constant pressure
d = diameter
m = mass flow rate
M = Mach number
N = rotational speed
P = fluid pressure
r = radius
T = fluid temperature
U = peripheral velocity
W = relative velocity
α = flow angle, angle between axial and meridional velocity components.
β = relative flow angle
γ = ratio of specific heat
η = efficiency
θ = relative angular coordinate
φs = slip factor
ρ = fluid density
ω = angular speed of rotation
Subscripts
0 = stagnation conditions
1 = inlet condition
2 = exit condition
p = pressure
r = radial
θ = tangential
tip = with reference to inducer tip
hub = with reference to hub
abs = with reference to the absolute velocity co-ordinate system
rel = with reference to the relative velocity co-ordinate system
1. Introduction
Centrifugal compressors are widely used in aeronautical applications; mainly in small-scale gas turbine engines for helicopters due to their compactness and ability to produce high pressure ratios in a single compression stage. All centrifugal compressors employ an impeller; a rotating component that is used to accelerate the incoming flow causing static pressure rise in the impeller channel. Although the flow is accelerated in the absolute velocity co-ordinate system, the rise in static pressure is caused by diffusion in the relative velocity co-ordinate system. The remainder of the accelerated flow enters the divergent passages of the diffuser where the velocity is further reduced to produce pressure. In practice, it is usual to design the compressor so that about half the pressure rise occurs in the impeller channel and the other half in the diffuser.
The air mass flow through the compressor and the pressure rise depend on the rotational speed of the impeller; therefore, impellers are designed to operate at tip speeds of up to 500m/s (insert a reference for this!). By operating at such high tip speeds, the air velocity from the impeller is increases so that greater energy is available for conversion to pressure by the diffuser. The inlet air temperature is another factor influencing the pressure rise, for the lower the temperature of the air entering the impeller the greater the pressure rise for a given amount of work put into the air by the impeller. Also, in order to maintain the efficiency of the compressor, the clearance between the impeller and the casing are kept as small as possible to prevent excessive air leakage [1].
2. Literature Survey
2.1 Centrifugal compressors
Centrifugal compressors belong to the class of continuous flow, dynamic compressors that are used in a wide variety of applications mainly due to their smooth operation, large tolerance of process fluctuations and their higher reliability when compared to other types of compressors. Centrifugal compressors range in size from pressure ratios of 3:1 up to 12:1on experimental models. Proper selection of a compressor is a complex and important decision. To ensure the best selection and proper maintenance of a centrifugal compressor, the engineer must have adequate knowledge of numerous engineering disciplines [2].
Effects of various parameters on the compressor efficiency are well documented, for example Balje; O.E. [3] studied the effect of specific speed (shape factor) and specific diameter on the efficiency of centrifugal compressors as shown in Figure 1. It was concluded from his research that the most efficient region for centrifugal compressor operation region is in the specific speed range 60 < Ns < 1500. For specific speeds greater than 3000, axial flow compressors have to be used.
When compared to axial flow compressors, centrifugal compressors are more robust, easy to manufacture and cheaper to produce. Although centrifugal compressors employ at most up to two stages of compression, the pressure rise per stage is higher than that of axial flow compressors. Centrifugal compressors in general are used for higher pressure-ratios and lower flow-rates compared to lower-stage pressure ratios and higher flow-rates in axial compressors. While centrifugal compressors find their application in small scale gas turbines (helicopter engine) and auxiliary power units, axial flow compressors are used in commercial and military aircraft engines (Rolls Royce Pegasus 11-61 and EJ 200).
Figure 1: Centrifugal compressor map [3]
2.2 Components of a centrifugal compressor
The radial flow (centrifugal) compressor consists of four basic components or sections:
a. A stationary inlet, also known as inlet casing
b. A rotating impeller
c. A stationary; vaned or vane less diffuser
d. The collector or volute casing
The contribution of each component of the compressor in producing the pressure rise in the stage is illustrated in Figure 2.
Figure 2: Pressure rise across centrifugal compressor [4]
2.2.1 Inlet casing
The function of the inlet casing is to deliver the air to the impeller eye with minimum loss and to provide a uniform velocity profile at the eye. Design of the inlet casing of a centrifugal compressor with inboard mounted bearings is simple and presents no constraints. The inlet flange is axisymmetric and the inlet duct takes the form of a simple convergent nozzle. The stagnation enthalpy remains constant as there is no energy transfer in the inlet casing.
2.2.2 Impeller
Energy transfer occurs in the impeller of the compressor, hence there is an increase in stagnation enthalpy and pressure. To attain peak compressor efficiency, great care must be taken to achieve very efficient diffusion processes in the impeller and the diffuser. Since the diffusion processes are related to the flow Mach number, it is desirable to establish conditions which lead to a minimum relative Mach number at the impeller inlet and a minimum absolute Mach number at the impeller outlet.
2.2.3 Diffuser
The diffuser is a stationary component fitted directly around the impeller. The function of the diffuser is to convert the kinetic energy of the fluid leaving the impeller tip efficiently into static pressure. The influence of the diffuser on compressor efficiency is significant, since approximately half of the fluid energy at the impeller tip is kinetic energy. Centrifugal compressors are usually fitted with either a vane less or a vaned diffuser. Virtually all vaned diffusers also use a small vane less 'gap' to reduce fluid noise levels and decrease the Mach number at the entry to the vanes.
2.2.4 Collector or volute casing
The function of the collector or volute (scroll) is simply to collect the diffuser exit flow and to guide it as efficiently as possible to the exit pipe or manifold (sometimes via a conical diffuser to provide additional diffusion), without impeding the effectiveness of the diffuser. Only very mild diffusion can be achieved in the volute if circumferential pressure distortions at the diffuser exit are to be avoided. If friction in the volute is neglected, then the design may be based on the assumption that the angular momentum of the flow remains constant.
2.3 Impeller design
For a start, when designing high speed turbo machinery components such as impellers, the designer must have concrete background knowledge on the flow physics in order to accomplish a successful design of the impeller. From theory, we infer that the intake flow is rotated at high speed by the impeller. As a result, the relative velocity of the fluid in the impeller channels undergoes a rapid deceleration from the impeller eye through towards the impeller tip. In addition to the function of transferring energy to the air, the impeller should act as an efficient diffuser. A badly shaped channel will interfere with the diffusion processes, causing flow separation at the impeller walls, leading to high impeller losses. The key design point here is to design the impeller channel in such a way so as to delay the onset of flow separation, i.e. to obtain gradual deceleration of the flow in the channel by optimising the blade shape, hub and shroud profiles. The optimal design will allow for a smooth gradual change in the relative Mach number along the mean flow path.
Figure 3: The two major planes of an impeller [2]
Did you refer to the above figure in your text?
The impeller inducer plays an important role in determining the level of efficient diffusion which will be attained. The leading edge of the inducer requires very careful design consideration. The flow arrives at the leading edge with a relative Mach number in the high subsonic or, in the case of pressure ratios above 4:1 or 5:1, the supersonic range at the impeller tip. Further, the relative Mach number and incidence flow angle vary along the inducer leading edge from high values at the eye tip to much lower values at the eye hub. Consideration must be given to avoiding shock formation which can lead to boundary layer separation. Thus, approach Mach numbers exceeding 1.2 should be avoided at any section of the inducer. An attempt to visualize the flow was carried out by Boyce, M.P.,[5] which detailed the flow separation points in the impeller channel. To achieve maximum diffusion, one strategy in designing the inducer is to diffuse the flow to the separation limit before the impeller passage turns towards the radial direction. This leads to long axial inducers.
Figure 4: Flow in the meridional plane (hub-to-shroud plane) [5]
Figure 3: Flow in the meridional flow plane (hub-to-shroud plane)Another strategy that has been used in the past regarding lowering the relative Mach number at inlet is to fit the inlet casing with guide vanes; more commonly known as inlet guide vanes or IGV's. The purpose of fitting IGV's into a centrifugal compressor system is to allow certain control of the flow characteristics, one of them being to lower the Mach number. IGV's induce pre-whirl or rotation to the incoming fluid as a result of which the work done by the impeller to change the direction of the fluid from the axial to the radial direction is significantly reduced. Rodgers, C et al [6] studied the effect of pre-whirl on the efficiency of the centrifugal compressor as shown in Figure 5. It was concluded that for a given pressure ratio, increasing the vane angle increases the efficiency of the compressor. By lowering the inlet Mach number, shock formation is avoided thereby delaying the onset of boundary layer separation in the impeller channels.
Figure 5: Effect of inlet pre-whirl on compressor efficiency [6]
3. Design methodology
To sum up the design process, the designer begins by treating the flow to be 1-dimensional to determine the principle dimensions of the impeller using the basic governing equations of fluid flow. Based on the dimensions, numerous impeller designs are produced and analysed in terms of efficient diffusing capability with minimal losses. The optimum geometry is then produced using advanced CAD software packages which can then be later simulated for comprehensive 3-D flow and stress analysis. The geometry is also analysed for feasibility in manufacturing by developing accurate NC data to be fed into the 5-axis machining centre, which, however, is beyond the scope of this project study. The following is a flow diagram indicating the elements of the design process that fall well under the scope of this project:
Figure 6: Design flow process
3.1 Assumptions
In addition to the flow model assumptions, compressor operating conditions need to be assumed based on the operating range of centrifugal compressors, especially those found in helicopter engines namely the Turbomeca Arrius 2K1 and Boeing 502-6 Turbo shaft gas turbine.
Table 1: Operating conditions of helicopter engines
Operating conditions
Turbomeca Arrius 2K1
Boeing 502-6 Turbo shaft
Mass flow rate, á¹ (kg/s)
3.00
1.60
Pressure ratio, P02/P01
9:1
3.5:1
Rotational speed, N (rpm)
100,000
36,500
Efficiency, Æžc
90%
80%
Hence, the design will be carried out for a centrifugal compressor assumed to operate at the following conditions:
Mass flow-rate, á¹ = 1.00 kg/s
Pressure ratio, P02/P01 = 6:1
Rotational speed, N = 66,000 rpm
Efficiency, Æžc = 85%
3.2 Calculation of principle impeller dimensions
The impeller dimensions that are to be determined are illustrated in the Figure as shown below:
Figure 7: Side profile of the impeller
The velocity distribution at entry and exit to the impeller can be approximated to velocity triangles more commonly known as vector diagrams. The following figure illustrates the velocity vector diagrams at entry and at exit:
Figure 8: Velocity vector diagrams at inlet and outlet (shown along the radial axial profile of the impeller)
Euler's equation is used to describe the energy/work transfer process to the fluid by the rotor. Therefore, work transfer can be given as:
Since C1 is purely axial at entry,. Therefore, the equation now becomes;
For a radial vaned impeller, the whirl or tangential component of the absolute fluid velocity at exit; should be equal to the impeller tip speed, U2 (in ideal cases). However, in reality not all the air passes smoothly through the passage formed between the impeller blades and thus the tangential component of the absolute fluid velocity at exit tends to be less than the impeller tip speed. Therefore, a correction factor or more commonly known as the slip factor which is the ratio between the tangential component of the absolute air velocity at exit to the impeller tip speed has to be multiplied to the energy equation. For maximum work/energy transfer, implies that slip factor, = 1, but in reality the slip factor has a lesser value than that of one. Therefore, the slip factor can be calculated based on the number of blades by using the following empirical formula proposed by Stanitz for a radial vaned impeller which is:
Where, Z = number of blades
Solving the above equation iteratively yields the following results:
Z
0.80
10
0.85
13
0.90
20
0.95
40
Generally higher the number of blades yields better flow guidance leading to increased rate of work transfer by the impeller. However, it leads to loss in efficiency due to increase in frictional losses, blade blockage (area inaccessible to the flow) and tip losses. Also, a lesser number of blades might lower frictional and tip losses but there will be poor flow guidance in addition to a reduction in work transfer from the impeller. Consequently, the designer must be able to strike a balance in order to obtain the right number of blades in order to serve the proposed design criteria. In reality, it is good design practice to choose the number of blades in the range of 15-20. Therefore, the number of blades chosen for this instance is 18 which yields a slip factor, = 0.89.
The Euler's equation can also be expressed in terms of the compressor operating conditions as shown in the following:
The above equation can therefore be expressed in terms of a dimensionless performance parameter as shown below:
The expression on the left-hand-side of the equation is known as the speed parameter which can be used to predict the efficiency of the compressor. Re-arranging and substituting the assumed values for the operating conditions, yields the preliminary dimension of the impeller which is the impeller tip diameter,
d2 = 149mm (r2 = 74.5 mm)
The absolute Mach number, Maabs, 1 corresponding to the absolute velocity at inlet, C1 is usually in the range so in this instance, the absolute Mach number at inlet is assumed to be 0.4
Using isentropic flow equations for a compressible fluid,
So, Tabs, 1 = 288.76K and Pabs, 1 = 0.90 bar
Hence,
Also, from theory we infer that the relative Mach number at inlet is usually the highest at the inducer tip
In order to prevent the occurrence of shockwaves, the inducer must be designed for a subsonic relative velocity. Hence, the relative Mach number at inlet, Marel, 1 is assumed to be 0.8. Again, using the isentropic flow equations, Trel, 1 = 264.20K and Prel, 1 = 0.656 bar. This gives
an inlet relative velocity,
Recalling, the inlet velocity triangle, using values of C1 and W1; the mean inducer tip speed U1 and blade angle β1 can be calculated using trigonometric ratios
The mean inducer diameter is given by,
Therefore,
d1 is related to the tip and hub diameters by the following relation:
Re-arranging, gives us the following expression:
From the continuity equation, we have
Thereby, solving equations. Simultaneously, Inducer tip and hub diameters are obtained
dtip = 95mm (rtip = 47.5mm)
dhub = 18mm (rhub = 9mm)
Recalling the outlet velocity triangle, the following velocities and blade angles can be obtained.
Using the thermodynamic equations of state, the static pressure, temperature and density at the impeller exit can be found out. This when combined with the continuity equation yields the outlet area. The blockage factor must also be accounted for, which indicates the area that is inaccessible to the flow. Finally, substituting these values yields a tip width value of 6mm.
b2 = 6mm
3.3 Impeller blade optimization
From the dimensions obtained by using the flow analytical equations, four impeller designs were produced with varying axial lengths. The reason for this was that, the length of the impeller channel would determine the level of efficient diffusion that will take place in order to prevent pre-mature flow separation at entry to the inducer in turn minimizing losses. The following side-view profiles of the blade were generated for selection of the best suited design to meet the requirements:
3.4 CAD Modelling
3-D Modelling of the impeller is carried out after the design selection stage. The purpose of carrying out 3-D Modelling is to check for how well the impeller has been proportioned. Usually after the CAD Modelling stage, many designers try to optimize the design further by running FEA and CFD codes which are finally checked for validation. Although, these codes can improve upon existing designs, it can prove to be laborious, time-consuming and also carry certain limitations that cannot be completely eliminated. Manufacturing feasibility can also be carried out either by rapid prototyping or 5 axes machining both of which make use of the generated CAD geometry file.
4. Results
Impeller Dimensions
Impeller tip diameter, d2 = 149mm
Inducer tip diameter, dtip = 95mm
Hub diameter, dhub = 18mm
Tip width, b2 = 6mm
Impeller axial length = 37.25mm (50% of r2)
5. Discussions
For a successful blade selection, knowledge of the flow physics is extremely crucial. With reference to Figure 9, it can be noted that the impeller channel is extremely narrow. This does not prove to be a good design as a narrow channel indicates immediate flow separation upon entry to the inducer. Also, the rate of diffusion is extremely high increasing diffusion losses.
Figures 10 and 11 both fall under the optimal design category. Rate of diffusion is controlled indicating delayed separation of the incoming flow. This leads to lower losses maximizing work transferred by the impeller to the fluid.
With reference to Figure 12, the impeller blade channel has the highest axial length. Although long axial inducers control the growth/onset of flow separation, a very long axial inducer implies greater losses due to friction. Since frictional losses account for the highest percentage among all the other associated losses in the impeller, this design falls under the unacceptable region.
Therefore, in this instance, Figure 10 was chosen as the optimum blade design as it possesses the ability to delay the onset of flow separation, i.e. control the rate of diffusion as the flow turns radial from its primary axial direction of entry.
6. Conclusions
Principle impeller dimensions were obtained from the governing equations for a given set of operating conditions and assumptions (Mach numbers)
Four sample designs were analyzed and the best design was selected based on flow physics and flow nature.
The selected best design was then modeled in SolidEdge V20 as shown in the following figure. The impeller is well proportioned and can be deemed suitable for manufacture.
Figure 13: Front and isometric views of the chosen impeller design
