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Centrifugal compressors, sometimes termed radial compressors, are a subclass of dynamic axisymmetric workabsorbing turbomachinery.^{[1]}
The idealized compressive dynamic turbomachine achieves a pressure rise by adding kinetic energy/velocity to a continuous flow of fluid through the rotor or impeller. This kinetic energy is then converted to an increase in potential energy/static pressure by slowing the flow through a diffuser. The pressure rise in impeller is in most cases almost equal to the rise in the diffuser section.
Imagine a simple case where flow passes through a straight pipe to enter a centrifugal compressor. The simple flow is straight, uniform and has no vorticity. As illustrated below α1=0 deg. As the flow continues to pass into and through the centrifugal impeller, the impeller forces the flow to spin faster and faster. According to a form of Euler's fluid dynamics equation, known as "pump and turbine equation," the energy input to the fluid is proportional to the flow's local spinning velocity multiplied by the local impeller tangential velocity.
In many cases the flow leaving centrifugal impeller is near the speed of sound (340 metres/second). The flow then typically flows through a stationary compressor causing it to decelerate. These stationary compressors are actually static guide vanes where energy transformation takes place. As described in Bernoulli's principle, this reduction in velocity causes the pressure to rise leading to a compressed fluid.^{[1]}
Over the past 100 years, applied scientists including Stodola (1903, 1927–1945),^{[2]} Pfleiderer (1952),^{[3]} Hawthorne (1964),^{[4]} Shepard (1956),^{[1]} Lakshminarayana (1996),^{[5]} and Japikse (many texts including citations),^{[6]}^{[7]}^{[8]}^{[9]} have educated young engineers in the fundamentals of turbomachinery. These understandings apply to all dynamic, continuousflow, axisymmetric pumps, fans, blowers, and compressors in axial, mixedflow and radial/centrifugal configurations.
This relationship is the reason advances in turbines and axial compressors often find their way into other turbomachinery including centrifugal compressors. Figures 1.1 and 1.2^{[10]}^{[11]} illustrate the domain of turbomachinery with labels showing centrifugal compressors. Improvements in centrifugal compressors have not been achieved through large discoveries. Rather, improvements have been achieved through understanding and applying incremental pieces of knowledge discovered by many individuals.
Figure 1.1 represents the aerothermo domain of turbomachinery. The horizontal axis represents the energy equation derivable from The First Law of Thermodynamics.^{[1]}^{[11]} The vertical axis, which can be characterized by Mach Number, represents the range of fluid compressibility (or elasticity).^{[1]}^{[11]} The Zaxis, which can be characterized by Reynolds Number, represents the range of fluid viscosities (or stickiness).^{[1]}^{[11]} Mathematicians and Physicists who established the foundations of this aerothermo domain include:^{[12]}^{[13]} Sir Isaac Newton, Daniel Bernoulli, Leonhard Euler, ClaudeLouis Navier, Sir George Gabriel Stokes, Ernst Mach, Nikolay Yegorovich Zhukovsky, Martin Wilhelm Kutta, Ludwig Prandtl, Theodore von Kármán, Paul Richard Heinrich Blasius, and Henri Coandă.
Figure 1.2 represents the physical or mechanical domain of turbomachinery. Again, the horizontal axis represents the energy equation with turbines generating power to the left and compressors absorbing power to the right.^{[1]}^{[11]} Within the physical domain the vertical axis differentiates between high speeds and low speeds depending upon the turbomachinery application.^{[1]}^{[11]} The Zaxis differentiates between axialflow geometry and radialflow geometry within the physical domain of turbomachinery.^{[1]}^{[11]} It is implied that mixedflow turbomachinery lie between axial and radial.^{[1]}^{[11]} Key contributors of technical achievements that pushed the practical application of turbomachinery forward include:^{[12]}^{[13]} Denis Papin,^{[14]} Kernelien Le Demour, Daniel Gabriel Fahrenheit, John Smeaton, Dr. A. C. E. Rateau,^{[15]} John Barber, Alexander Sablukov, Sir Charles Algernon Parsons, Ægidius Elling, Sanford Alexander Moss, Willis Carrier, Adolf Busemann, Hermann Schlichting, Frank Whittle and Hans von Ohain.
<1689  Early turbomachines  Pumps, blowers, fans 
1689  Denis Papin  Origin of the centrifugal compressor 
1754  Leonhard Euler  Euler's "Pump & Turbine" equation 
1791  John Barber  First gas turbine patent 
1899  Dr. A. C. E. Rateau  First practical centrifugal compressor 
1927  Aurel Boleslav Stodola  Formalized "slip factor" 
1928  Adolf Busemann  Derived "slip factor" 
1937  Frank Whittle and Hans von Ohain, independently  First gas turbine using centrifugal compressor 
>1970  Modern turbomachines  3DCFD, rocket turbopumps, heart assist pumps, turbocharged fuel cells 
Many types of dynamic continuous flow *turbomachinery are treated in Wikipedia. As stated in the main turbomachinery article, what is notable about turbomachinery is that the fundamentals apply almost universally. Certainly there are significant differences between these machines and between the types of analysis that are typically applied to specific cases. This does not negate that they are unified by the same underlying physics of fluid dynamics, gas dynamics, aerodynamics, hydrodynamics, and thermodynamics.
A few of these machines have physical characteristics related to the centrifugal compressors, such as the following;
Similarities to axial compressor Centrifugal compressors are similar to axial compressors in that they are rotating airfoil based compressors as shown in the adjacent figure. ^{[5]}^{[7]} It should not be surprising that the first part of the centrifugal impeller looks very similar to an axial compressor. This first part of the centrifugal impeller is also termed an inducer. Centrifugal compressors differ from axials as they use a greater change in radius from inlet to exit of the rotor/impeller. 
Similarities to centrifugal fan Centrifugal compressors are also similar to centrifugal fans of the style shown in neighboring figure as they both increase the flows energy through increasing radius.^{[1]} In contrast to centrifugal fans, compressors operate at higher speeds to generate greater pressure rises. In many cases the engineering methods used to design a centrifugal fan is the same as those to design a centrifugal compressor. As a result they can at times look very similar. This relationship is less true in comparison to a squirrelcage fans as shown in figure farthest right. For purposes of generalization and definition, it can be said that centrifugal compressors often have density increases greater than 5 percent. Also, they often experience relative fluid velocities above Mach number 0.3^{[5]}^{[16]} when the working fluid is air or nitrogen. In contrast, fans or blowers are often considered to have density increases of less than five percent and peak relative fluid velocities below Mach 0.3. 
Similarities to centrifugal pump Centrifugal compressors are also similar to centrifugal pumps^{[1]} of the style shown in the adjacent figures. The key difference between such compressors and pumps is that the compressor working fluid is a gas (compressible) and the pump working fluid is liquid (incompressible). Again, the engineering methods used to design a centrifugal pump are the same as those to design a centrifugal compressor. Yet, there is one important difference: the need to deal with cavitation in pumps. 
Similarities to radial turbine Centrifugal compressors also look very similar to their turbomachinery counterpart the radialinflow turbine as shown in the figure. While a compressor transfers energy into a flow to raise its pressure, a turbine operates in reverse, by extracting energy from a flow, thus reducing its pressure.^{[7]} In other words, power is input to compressors and output from turbines. 
A partial list of turbomachinery that may use one or more centrifugal compressors within the machine are listed here.
A simple centrifugal compressor has four components: inlet, impeller/rotor, diffuser, and collector.^{[1]} Figure 3.1 shows each of the components of the flow path, with the flow (working gas) entering the centrifugal impeller axially from right to left. As a result of the impeller rotating clockwise when looking downstream into the compressor, the flow will pass through the volute's discharge cone moving away from the figure's viewer.
The inlet to a centrifugal compressor is typically a simple pipe. It may include features such as a valve, stationary vanes/airfoils (used to help swirl the flow) and both pressure and temperature instrumentation. All of these additional devices have important uses in the control of the centrifugal compressor.
The key component that makes a compressor centrifugal is the centrifugal impeller, Figure 01. It is the impeller's rotating set of vanes (or blades) that gradually raises the energy of the working gas. This is identical to an axial compressor with the exception that the gases can reach higher velocities and energy levels through the impeller's increasing radius. In many modern highefficiency centrifugal compressors the gas exiting the impeller is traveling near the speed of sound.
Impellers are designed in many configurations including "open" (visible blades), "covered or shrouded", "with splitters" (every other inducer removed) and "w/o splitters" (all full blades). Both Figures 0.1 and 3.1 show open impellers with splitters. Most modern high efficiency impellers use "backsweep" in the blade shape.^{[6]}^{[17]}^{[18]}
Euler’s pump and turbine equation plays an important role in understanding impeller performance.
The next key component to the simple centrifugal compressor is the diffuser.^{[7]}^{[18]}^{[8]} Downstream of the impeller in the flow path, it is the diffuser's responsibility to convert the kinetic energy (high velocity) of the gas into pressure by gradually slowing (diffusing) the gas velocity. Diffusers can be vaneless, vaned or an alternating combination. High efficiency vaned diffusers are also designed over a wide range of solidities from less than 1 to over 4. Hybrid versions of vaned diffusers include: wedge, channel, and pipe diffusers. There are turbocharger applications that benefit by incorporating no diffuser.
Bernoulli's fluid dynamic principle plays an important role in understanding diffuser performance.
The collector of a centrifugal compressor can take many shapes and forms.^{[7]}^{[18]} When the diffuser discharges into a large empty chamber, the collector may be termed a Plenum. When the diffuser discharges into a device that looks somewhat like a snail shell, bull's horn or a French horn, the collector is likely to be termed a volute or scroll. As the name implies, a collector’s purpose is to gather the flow from the diffuser discharge annulus and deliver this flow to a downstream pipe. Either the collector or the pipe may also contain valves and instrumentation to control the compressor.
Below, is a partial list of centrifugal compressor applications each with a brief description of some of the general characteristics possessed by those compressors. To start this list two of the most wellknown centrifugal compressor applications are listed; gas turbines and turbochargers.^{[5]}
While illustrating a gas turbine's Brayton cycle,^{[12]} Figure 5.1 includes example plots of pressurespecific volume and temperatureentropy. These types of plots are fundamental to understanding centrifugal compressor performance at one operating point. Studying these two plots further we see that the pressure rises between the compressor inlet (station 1) and compressor exit (station 2). At the same time, it is easy to see that the specific volume decreases or similarly the density increases. Studying the temperatureentropy plot we see the temperature increase with increasing entropy (loss). If we assume dry air, and ideal gas equation of state and an isentropic process, we have enough information to define the pressure ratio and efficiency for this one point. Unfortunately, we are missing several other key pieces of information if we wish to apply the centrifugal compressor to another application.
Figure 5.2, a centrifugal compressor performance map (either test or estimated), shows flow, pressure ratio for each of 4 speedlines (total of 23 data points). Also included are constant efficiency contours. Centrifugal compressor performance presented in this form provides enough information to match the hardware represented by the map to a simple set of enduser requirements.
Compared to estimating performance which is very cost effective (thus useful in design), testing, while costly, is still the most precise method.^{[9]} Further, testing centrifugal compressor performance is very complex. Professional societies such as ASME (i.e. PTC–10, Fluid Meters Handbook, PTC19.x),^{[25]} ASHRAE (ASHRAE Handbook) and API (ANSI/API 6172002, 6722007)^{[22]}^{[24]} have established standards for detailed experimental methods and analysis of test results. Despite this complexity, a few basic concepts in performance can be presented by examining an example test performance map.
Pressure ratio and flow are the main parameters^{[12]}^{[22]}^{[24]}^{[25]} needed to match the Figure 5.2 performance map to a simple compressor application. In this case, it can be assumed that the inlet temperature is sealevel standard. Making this assumption in a real case would be a significant error. A detailed inspection of Figure 5.2 shows:
As is standard practice, Figure 5.2 has a horizontal axis labeled with a flow parameter. While flow measurements use a wide variety unit specifications, all fit one of 2 categories:
Also as is standard practice, Figure 5.2 has a vertical axis labeled with a pressure parameter. The variety of pressure measurement units is also vast. In this case, they all fit one of three categories:
Other features common to performance maps are:
Choke  occurs under one of 2 conditions. Typically for high speed equipment, as flow increases the velocity of the flow can approach sonic speed somewhere within the compressor stage. This location may occur at the impeller inlet "throat" or at the vaned diffuser inlet "throat". In contrast, for lower speed equipment, as flows increase, losses increase such that the pressure ratio eventually drops to 1:1. In this case, the occurrence of choke is unlikely.
To weigh the advantages between centrifugal compressors it is important to compare 8 parameters classic to turbomachinery. Specifically, pressure rise (p), flow (Q), angular speed (N), power (P), density (ρ), diameter (D), viscosity (mu) and elasticity (e). This creates a practical problem when trying to experimentally determine the effect of any one parameter. This is because it is nearly impossible to change one of these parameters independently.
The method of procedure known as the Buckingham π theorem can help solve this problem by generating 5 dimensionless forms of these parameters.^{[1]}^{[7]}^{[13]} These Pi parameters provide the foundation for "similitude" and the "affinitylaws" in turbomachinery. They provide for the creation of additional relationships (being dimensionless) found valuable in the characterization of performance.
For the examples below Head will be substituted for pressure and sonic velocity will be substituted for elasticity.
The three independent dimensions used in this procedure for turbomachinery are:
According to the theorem each of the eight main parameters are equated to its independent dimensions as follows:
Flow:  ex. = m^3/s  
Head:  ex. = kg*m/s^2  
Speed:  ex. = m/s  
Power:  ex. = kg*m^2/s^3  
Density:  ex. = kg/m^3  
Viscosity:  ex. = kg/(m*s)  
Diameter:  ex. = m  
Speed of sound:  ex. = m/s 
Completing the task of following the formal procedure results in generating this classic set of five dimensionless parameters for turbomachinery. Full similitude is achieved when each of the 5 Piparameters are equivalent. This of course would mean the two turbomachines being compared are geometrically similar and running at the same operating point.
Flow coefficient:  
Head coefficient:  
Speed coefficient:  
Power coefficient:  
Reynolds coefficient: 
Turbomachinery analysts gain tremendous insight into performance by comparisons of these 5 parameters with efficiencies and loss coefficients which are also dimensionless. In general application, the flow coefficient and head coefficient are considered of primary importance. Generally, for centrifugal compressors, the velocity coefficient is of secondary importance while the Reynolds coefficient is of tertiary importance. In contrast, as expected for pumps, the Reynolds number becomes of secondary importance and the velocity coefficient almost irrelevant. It may be found interesting that the speed coefficient may be chosen to define the yaxis of Figure 1.1, while at the same time the Reynolds coefficient may be chosen to define the zaxis.
Demonstrated in the table below is another value of dimensional analysis. Any number of new dimensionless parameters can be calculated through exponents and multiplication. For example, a variation of the first parameter shown below is popularly used in aircraft engine system analysis. The third parameter is a simplified dimensional variation of the first and second. This third definition is applicable with strict limitations. The fourth parameter, specific speed, is very well known and useful in that it removes diameter. The fifth parameter, specific diameter, is a less often discussed dimensionless parameter found useful by Balje.^{[27]}
Corrected mass flow coefficient:  
Alternate#1 equivalent Mach form:  
Alternate#2 simplified dimensional form:  
Specific speed coefficient:  
Specific diameter coefficient: 
It may be found interesting that the Specific speed coefficient may be used in place of Speed to define the yaxis of Figure 1.2, while at the same time the Specific diameter coefficient may be in place of Diameter to define the zaxis.
The following "affinity laws" are derived from the five Piparameters shown above. They provide a simple basis for scaling turbomachinery from one application to the next.
From flow coefficient:  
From head coefficient:  
From power coefficient:  
From flow coefficient:  
From head coefficient:  
From power coefficient: 
The following equations outline a fully threedimensional mathematical problem that is very difficult to solve even with simplifying assumptions.^{[5]}^{[28]} Until recently, limitations in computational power, forced these equations to be simplified to an Inviscid twodimensional problem with pseudo losses. Before the advent of computers, these equations were almost always simplified to a onedimensional problem.
Solving this onedimensional problem is still valuable today and is often termed meanline analysis. Even with all of this simplification it still requires large textbooks to outline and large computer programs to solve practically.
Modern turbomachinery design is a compromise between fluid and thermodynamics, structural mechanics and manufacturability.^{[6]} Making the correct compromises requires appropriate experimental data and design experience. As precise as the fundamental equations above are, resultant design methods are many and varied.
Focusing on the thermofluid element, two logically opposed approaches are used to solve the fundamental equations. The first is to analyze the existing performance of existing hardware: this analysis is a postexperiment activity used to explain what has happened. The second is to design new hardware: here design is a predictive activity similar to forecasting the weather.
Currently, popular turbomachinery design methods alternately solve two problems: first the design problem, to define the air foil, and then the analysis problem, to detail the aerodynamic performance of the design. For "end users/customers" requiring the highest performance, designers incorporate numerical optimization to direct these iterations, while sometimes pursuing a parallel experimental program.
In practice, design and analysis uses simplified 1D and 2D equation sets before performing the final 3D solutions. After this expansive aerodynamic problem is solved, an equally expansive set of design problems remain involving the structural design and the manufacture of the centrifugal compressor.
Ideally, centrifugal compressor impellers have infinitely thin airfoil blades that are infinitely strong, each mounted on an infinitely light rotor. This material would be infinitely easy to machine or cast and infinitely inexpensive. Additionally, it would generate no operating noise, have an infinite life while operating in any environment.
From the very start of the aerothermodynamic design process, the centrifugal impeller’s material and manufacturing method must be accounted for within the design. Whether it be plastic for a vacuum cleaner blower to aluminum alloy for a turbocharger, steel alloy for an air compressor or titanium alloy for a gas turbine. It is a combination of the centrifugal compressor impeller shape, its operating environment, its material and its manufacturing method that determines the impeller’s structural integrity.
