Centrifugal compressor

Cutaway showing an axi-centrifugal compressor gas turbine

Centrifugal compressors are similar to axial compressors in that they are rotating airfoil-based compressors. Both are shown in the adjacent photograph of an engine with 5 stages of axial compressor and one stage of centrifugal compressor.[5][7] 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 significant change in radius from inlet to exit of the impeller to produce a much greater pressure rise in a single stage (eg 8[16] in the Pratt & Whitney Canada PW200 series of helicopter engines) than does an axial stage. The 1940s-era German Heinkel HeS 011 experimental engine was the first aviation turbojet to have a compressor stage with radial flow-turning part-way between none for an axial and 90 degrees for a centrifugal. It is known as a mixed/diagonal-flow compressor. A diagonal stage is used in the Pratt & Whitney Canada PW600 series of small turbofans.

A low speed, low pressure centrifugal compressor or centrifugal fan, with upward discharging cone used to diffuse the air velocity

A squirrel-cage fan, without a discharge diffuser

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 are the same as those to design a centrifugal compressor, so they can look very similar.

This relationship is less true in comparison to the squirrel-cage fan shown in the accompanying figure.

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][17] 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.

A 3D-solids model of a type of centrifugal pump

Cut-away of a 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.

Centrifugal compressors also look very similar to their turbomachinery counterpart the radial 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.

• Auxiliary power unit
• Centrifugal-type supercharger
• Gas turbine
• Turbocharger
• Turboprop
• Turboshaft
• Turbofan

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 0.1, which contains a 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 high-efficiency 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][18][19]

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][8][19] 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. Some turbochargers have 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][19] 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.

Pressure ratio and flow are the main parameters[12][23][25][26] 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 sea-level standard. This assumption is not acceptable in practice as inlet temperature variations cause significant variations in compressor performance. Figure 5.2 shows:

• Corrected mass flow: 0.04 – 0.34 kg/s
• Total pressure ratio, inlet to discharge (PR_t-t = P_t,discharge/P_t,inlet): 1.0 – 2.6

As is standard practice, Figure 5.2 has a horizontal axis labelled with a flow parameter. While flow measurements use a variety of units, all fit one of 2 categories:

Mass flow units, such as kg/s, are the easiest to use in practice as there is little room for confusion. Questions remaining would involve inlet or outlet (which might involve leakage from the compressor or moisture condensation). For atmospheric air, the mass flow may be wet or dry (including or excluding humidity). Often, the mass flow specification will be presented on an equivalent Mach number basis, ${\displaystyle m{\sqrt {\theta }}/{\delta }}$.[27] It is standard in these cases that the equivalent temperature, equivalent pressure and gas is specified explicitly or implied at a standard condition.

In contrast, all volume flow specifications require the additional specification of density. Bernoulli's fluid dynamic principal is of great value in understanding this problem. Confusion arises through either inaccuracies or misuse of pressure, temperature and gas constants.

Also as is standard practice, Figure 5.2 has a vertical axis labelled with a pressure parameter. There is a variety of pressure measurement units. They all fit one of two categories:

• A delta pressure, ie increase from inlet to exit (measured with a manometer)
• A discharge pressure

The pressure rise may alternatively be specified as a ratio which has no units:

• A pressure ratio (exit/inlet)

Other features common to performance maps are:

The two most common methods for producing a map for a centrifugal compressor are at constant shaft speed or with a constant throttle setting. If the speed is held constant, test points are taken along a constant speed line by changing throttle positions. In contrast, if a throttle valve is held constant, test points are established by changing speed and repeated with different throttle positions (common gas turbine practice). The map shown in Figure 5.2 illustrate the most common method; lines of constant speed. In this case we see data points connected via straight lines at speeds of 50%, 71%, 87%, and 100% RPM. The first three speed lines have 6 points each while the highest speed line has five.

The next feature to be discussed is the oval shaped curves representing islands of constant efficiency. In this figure we see 11 contours ranging from 56% efficiency (decimal 0.56) to 76% efficiency (decimal 0.76). General standard practice is to interpret these efficiencies as isentropic rather than polytropic. The inclusion of efficiency islands effectively generates a 3-dimensional topology to this 2-dimensional map. With inlet density specified, it provides a further ability to calculate aerodynamic power. Lines of constant power could just as easily be substituted.

Regarding gas turbine operation and performance, there may be a series of guaranteed points established for the gas turbine’s centrifugal compressor. These requirements are of secondary importance to the overall gas turbine performance as a whole. For this reason it is only necessary to summarize that in the ideal case, the lowest specific fuel consumption would occur when the centrifugal compressors peak efficiency curve coincides with the gas turbine's required operation line.

In contrast to gas turbines, most other applications (including industrial) need to meet a less stringent set of performance requirements. Historically, centrifugal compressors applied to industrial applications were needed to achieve performance at a specific flow and pressure. Modern industrial compressors are often needed to achieve specific performance goals across a range of flows and pressures; thus taking a significant step toward the sophistication seen in gas turbine applications.

If the compressor represented by Figure 5.2 is used in a simple application, any point (pressure and flow) within the 76% efficiency would provide very acceptable performance. An "End User" would be very happy with the performance requirements of 2.0 pressure ratio at 0.21 kg/s.

Surge - is a flow phenomenon at low mass flow rate operation for which the impeller cannot add enough energy to overcome the system resistance or backpressure.[28] At low mass flow rate operation, the pressure ratio over the impeller is high. The high back pressure, downstream of the impeller, pushes flow back over the tips of the rotor blades towards the impeller eye (inlet).[29] This rapid flow reversal (i.e., surge) exhibits a strong rotational component, which affects the flow angles at the leading edge of the blades. The deterioration of the flow angles causes the impeller to be inefficient and less flow is delivered downstream. (Therefore, surge is sometimes referred to as axi-symmetric stall.) Thereby, the plenum downstream of the impeller is emptied and the (back) pressure drops. As a result, less flow reverses over the rotor tips and the impeller gains becomes again efficient. This cyclic events cause large vibrations, increase temperature and change rapidly the axial thrust. These occurrences can damage the rotor seals, rotor bearings, the compressor driver and cycle operation. Most turbomachines are designed to easily withstand occasional surging. However, if the machine is forced to surge repeatedly for a long period of time, or if it is poorly designed, repeated surges can result in a catastrophic failure. Of particular interest, is that while turbomachines may be very durable, the cycles/processes that they are used within can be far less robust.

The surge-line shown in Figure 5.2 is the curve that passes through the lowest flow points of each of the four speed lines. As a test map, these points would be the lowest flow points possible to record a stable reading within the test facility/rig. In many industrial applications it may be necessary to increase the stall line due to the system backpressure. For example, at 100% RPM stalling flow might increase from approximately 0.170 kg/s to 0.215 kg/s because of the positive slope of the pressure ratio curve.

As stated earlier, the reason for this is that the high-speed line in Figure 5.2 exhibits a stalling characteristic or positive slope within that range of flows. When placed in a different system those lower flows might not be achievable because of interaction with that system. System resistance or adverse pressure is proven mathematically to be the critical contributor to compressor surge.

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.

The speed lines of gas turbine centrifugal compressors typically exhibit choke. This is a situation where the pressure ratio of a speed line drops rapidly (vertically) with little or no change in flow. In most cases the reason for this is that close to Mach 1 velocities have been reached somewhere within the impeller and/or diffuser generating a rapid increase in losses. Higher pressure ratio turbocharger centrifugal compressors exhibit this same phenomenon. Real choke phenomena is a function of compressibility as measured by the local Mach number within an area restriction within the centrifugal pressure stage.

The maximum flow line, shown in Figure 5.2, is the curve that passes through the highest flow points of each speed line. Upon inspection it may be noticed that each of these points has been taken near 56% efficiency. Selecting a low efficiency (<60%) is the most common practice used to terminate compressor performance maps at high flows. Another factor that is used to establish the maximum flow line is a pressure ratio near or equal to 1. The 50% speed line may be considered an example of this.

The shape of Figure 5.2's speed lines provides a good example of why it is inappropriate to use the term choke in association with a maximum flow of all centrifugal compressor speed lines. In summary; most industrial and commercial centrifugal compressors are selected or designed to operate at or near their highest efficiencies and to avoid operation at low efficiencies. For this reason there is seldom a reason to illustrate centrifugal compressor performance below 60% efficiency.

Many industrial and commercial multistage compressor performance maps exhibits this same vertical characteristic for a different reason related to what is known as stage stacking.

Minimum operating speed

The minimum speed for acceptable operation, below this value the compressor may be controlled to stop or go into an "Idle" condition.

Maximum allowable speed

The maximum operating speed for the compressor. Beyond this value stresses may rise above prescribed limits and rotor vibrations may increase rapidly. At speeds above this level the equipment will likely become very dangerous and be controlled to lower speeds.

• 
  Inlet velocity triangles for centrifugal compressor impeller
• 
  Exit velocity triangles for centrifugal compressor impeller

The three independent dimensions used in this procedure for turbomachinery are:

• ${\displaystyle M}$ mass (force is an alternative)
• ${\displaystyle L}$ length
• ${\displaystyle T}$ time

According to the theorem each of the eight main parameters are equated to its independent dimensions as follows:

Flow	${\displaystyle Q=}$	${\displaystyle {\frac {L^{3}}{T}}}$	ex. = m3/s
Head	${\displaystyle H=}$	${\displaystyle {\frac {ML}{T^{2}}}}$	ex. = kg·m/s2
Speed	${\displaystyle U=}$	${\displaystyle {\frac {L}{T}}}$	ex. = m/s
Power	${\displaystyle P=}$	${\displaystyle {\frac {ML^{2}}{T^{3}}}}$	ex. = kg·m2/s3
Density	${\displaystyle \rho =}$	${\displaystyle {\frac {M}{L^{3}}}}$	ex. = kg/m3
Viscosity	${\displaystyle \mu =}$	${\displaystyle {\frac {M}{LT}}}$	ex. = kg/m·s
Diameter	${\displaystyle D=}$	${\displaystyle L}$	ex. = m
Speed of sound	${\displaystyle a=}$	${\displaystyle {\frac {L}{T}}}$	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 Pi-parameters are equivalent. This of course would mean the two turbomachines being compared are geometrically similar and running at the same operating point.

Flow coefficient	${\displaystyle \Pi _{1}=}$	${\displaystyle {\frac {Q}{ND^{3}}}}$
Head coefficient	${\displaystyle \Pi _{2}=}$	${\displaystyle {\frac {gH}{N^{2}D^{2}}}}$
Speed coefficient	${\displaystyle \Pi _{4}=}$	${\displaystyle {\frac {ND}{a}}}$
Power coefficient	${\displaystyle \Pi _{3}=}$	${\displaystyle {\frac {P}{\rho N^{3}D^{5}}}}$
Reynolds coefficient	${\displaystyle \Pi _{5}=}$	${\displaystyle {\frac {\rho ND^{2}}{\mu }}}$

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 y-axis of Figure 1.1, while at the same time the Reynolds coefficient may be chosen to define the z-axis.

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.[30]

1	Corrected mass flow coefficient	${\displaystyle \Pi _{1}\Pi _{4}=}$	${\displaystyle {\frac {m}{pD^{2}}}\left({\frac {Rt}{k}}\right)^{0.5}}$
2	Alternate#1 equivalent Mach form	${\displaystyle {\frac {m_{1}}{\rho _{1}a_{1}{D_{1}}^{2}\cos(\alpha _{1})}}=}$	${\displaystyle {\frac {m_{2}}{\rho _{2}a_{2}{D_{2}}^{2}\cos(\alpha _{2})}}}$
3	Alternate#2 simplified dimensional form	${\displaystyle {\frac {m_{1}{t_{1}}^{0.5}}{p_{1}}}=}$	${\displaystyle {\frac {m_{2}{t_{2}}^{0.5}}{p_{2}}}}$
4	Specific speed coefficient	${\displaystyle {\frac {{\Pi _{1}}^{0.5}}{{\Pi _{2}}^{0.75}}}=}$	${\displaystyle {\frac {NQ^{0.5}}{(gH)^{0.75}}}}$
5	Specific diameter coefficient	${\displaystyle {\frac {{\Pi _{2}}^{0.25}}{{\Pi _{1}}^{0.5}}}=}$	${\displaystyle {\frac {D(gH)^{0.25}}{Q^{0.5}}}}$

It may be found interesting that the specific speed coefficient may be used in place of speed to define the y-axis of Figure 1.2, while at the same time, the specific diameter coefficient may be in place of diameter to define the z-axis.

The following affinity laws are derived from the five Π-parameters shown above. They provide a simple basis for scaling turbomachinery from one application to the next.

From flow coefficient	${\displaystyle {\frac {Q_{1}}{Q_{2}}}=}$	${\displaystyle {\frac {N_{1}}{N_{2}}}=}$	${\displaystyle \left({\frac {D_{1}}{D_{2}}}\right)^{3}}$
From head coefficient	${\displaystyle {\frac {H_{1}}{H_{2}}}=}$	${\displaystyle \left({\frac {N_{1}}{N_{2}}}\right)^{2}=}$	${\displaystyle \left({\frac {D_{1}}{D_{2}}}\right)^{2}}$
From power coefficient	${\displaystyle {\frac {P_{1}}{P_{2}}}=}$	${\displaystyle \left({\frac {N_{1}}{N_{2}}}\right)^{3}=}$	${\displaystyle \left({\frac {D_{1}}{D_{2}}}\right)^{5}}$

Also termed continuity, this fundamental equation written in general form is as follows:

${\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho \mathbf {v} )=0}$

Also termed the Navier–Stokes equations, this fundamental is derivable from Newton's second law when applied to fluid motion. Written in compressible form for a Newtonian fluid, this equation may be written as follows:

${\displaystyle \rho \left({\frac {\partial \mathbf {v} }{\partial t}}+\mathbf {v} \cdot \nabla \mathbf {v} \right)=-\nabla p+\mu \nabla ^{2}\mathbf {v} +\left({\frac {1}{3}}\mu +\mu ^{v}\right)\nabla \left(\nabla \cdot \mathbf {v} \right)+\mathbf {f} }$

The first law of thermodynamics is the statement of the conservation of energy. Under specific conditions, the operation of a Centrifugal compressor is considered a reversible process. For a reversible process, the total amount of heat added to a system can be expressed as ${\displaystyle \delta Q=TdS}$ where ${\displaystyle T}$ is temperature and ${\displaystyle S}$ is entropy. Therefore, for a reversible process:

${\displaystyle dU=TdS-pdV.\,}$

Since U, S and V are thermodynamic functions of state, the above relation holds also for non-reversible changes. The above equation is known as the fundamental thermodynamic relation.

The classical ideal gas law may be written:

${\displaystyle {\ pV=nRT}.}$

The ideal gas law may also be expressed as follows

${\displaystyle {\ p=\rho (\gamma -1)U}}$

where ${\displaystyle \rho }$ is the density, ${\displaystyle \gamma =C_{p}/C_{v}}$ is the adiabatic index (ratio of specific heats), ${\displaystyle U=C_{v}T}$ is the internal energy per unit mass (the "specific internal energy"), ${\displaystyle C_{v}}$ is the specific heat at constant volume, and ${\displaystyle C_{p}}$ is the specific heat at constant pressure.

With regard to the equation of state, it is important to remember that while air and nitrogen properties (near standard atmospheric conditions) are easily and accurately estimated by this simple relationship, there are many centrifugal compressor applications where the ideal relationship is inadequate. For example, centrifugal compressors used for large air conditioning systems (water chillers) use a refrigerant as a working gas that cannot be modeled as an ideal gas. Another example are centrifugal compressors design and built for the petroleum industry. Most of the hydrocarbon gases such as methane and ethylene are best modeled as a real gas equation of state rather than ideal gases. The Wikipedia entry for equations of state is very thorough.