centifugal compressor

10/26/13 CH. 9 THE CENTRIFUGAL COMPRESSOR

www.mie.utoronto.ca/labs/mfl/propulsion/lectures/lecture7/ 1/23

THE CENTRIFUGAL COMPRESSOR

Notes from: Hill, Philip and Carl Peterson.

Mechanics and Thermodynamics of Propulsion, Second Edition.

Reading: Addison-Wesley Publishing Company, 1992.

I. INTRODUCTION

USES

- used in the first 2 independently developed jet engines

- used in small gas turbines

ex: small turbojet engines, to drive propellers, to drive helicopter rotors

*in these smaller sizes, the centrifugal compressor is as efficient as an axial compressor

*for smaller horsepower ranges (<10,000 hp) and compression ratios (15:1 to 25:1),the blade height at the back end of an axial compressor is very small

Þ this causes tip leakage and therefore reduces efficiency

- not used in larger engines, where the axial compressor is better

WHY? * the axial compressor has a smaller

cross-sectional area : airflow rate ratio

Þ this means a smaller machine and therefore less drag

* axial is better for multistaging to obtain large pressure ratios

* axial compressors are typically more efficient than centrifugal ones

CAPABILITIES

- a single stage centrifugal compressor can have pressure ratios of 5:1 (h =.85) and even 10:1 (h>.80)

- the compressor can have rotor tip speeds as high as 650m/s (giving a supersonic rotor exitvelocity)

EXAMPLE OF CENTRIFUGAL COMPRESSOR: PW 100



*2 centrifugal stages, 2000hp, 15:1 pressure ratio

- higher efficiency than a multistage

- shorter axial length axial compressor with

- same weight the same flow rate

II. STAGE DYNAMICS

- air leaving the impeller has no axial velocity component

3 MAIN PARTS OF THE CENTRIFUGAL COMPRESSOR

1. Impeller

2. diffuser

3. scroll

HOW IT WORKS

1. FLUID ENTERS THE IMPELLER THROUGH THE INDUCER

2. THE INDUCER TURNS THE FLOW THAT ENTERS IT

3. AIR LEAVES THE IMPELLER FLOWING IN THE RADIAL DIRECTION



the flow enters the inducer part of the impeller axially the flow leaves radially after the inducer turns it

4. FLOW PASSES THROUGH A RADIAL DIFFUSER WITH VANED PASSAGES

-these passages decrease the flow velocity, turning the momentum into higher pressure

5. THE SCROLL/COLLECTOR COLLECTS THE FLOW LEAVING THE DIFFUSER ANDREDIRECTS IT TOWARD THE OUTLET

CENTRIFUGAL COMPRESSOR STAGES / STATES

1 – impeller entrance

2 – impeller exit / diffuser entrance

3 – diffuser exit

SOME EQUATIONS…..

For a steam line through a rotor:

The sum of the torques, acting = the change in the angular

on a control volume surrounding momentum, rcq , of the flow

a compressor rotor through the rotor

We can approximate S t as the torque applied to the rotor from the shaft, t s

Then, the work per unit mass, work, done by the rotor on the flow is:

with U = W r.

Next, look at the control volume energy equation:



where is the heat transfer rate

is the work transfer rate

Fbody = body forces

The flow through the rotor is steady (d/dt = 0) and adiabatic ( ) with negligible gravitationaland body forces (Fbody = 0, gz = 0). Applying these simplifications to the energy equation yields:

But:

So:

This shows that the work done on the fluid is equal to the change in stagnation enthalpy:

Or, substituting the work from the equation above:

Now, if we fix the reference frame to the rotor, the absolute and relative velocities, c and w, arerelated by:



Substituting these relations into the enthalpy equation above, and rearranging, yields:

For an incremental change along a streamline:

since U2 = W 2r2.

Solving the entropy equation,

for dh, substituting into the dh equation above and assuming isentropic flow (ds=0) yields:

* The presence of the d(W 2r2/2) term indicates that the rotor pressure can rise

even if there is no change in relative velocity, w. (In an axial compressor, this term is equalto zero).

* This type of pressure increase can cause boundary layer separation on the

blade

WHY? The pressure increases but the relative boundary layer and freestreamvelocities don’t change

WHAT DO YOU DO? Design the rotor so that the freestreamvelocity decelerates. Boundary layer separation may stilloccur, but this does prevent some separation.

We can assume that all streamlines entering the compressor have the same angular momentum



(aka swirl free inlet flow), but many compressor have swirl at the rotor entrance. This can be from aprevious compressor stage or inlet guide vanes. If we assume that the uniform swirl upstream anddownstream, the work equation becomes:

Dividing both sides by cpT01, rearranging, and using the following relations:

with a01 = speed of sound at T01, yields:

On a typical machine, the velocity has only an axial component at the inlet.

So:

Then the above equation becomes:

3 TYPES OF IMPELLER BLADES

1. Forward leaning2. Straight radial3. Backward leaning



b 2 = angle of blade with respect to radial direction

wr = cr because U is tangential

More Equation Stuff….

From the velocity triangles of all 3 impellers:

Inserting into the stagnation temperature difference equation changes it to:

Now look at continuity at the impeller exit:

density, r , is just the impeller exit density, r 2 the velocity, V, is only in the radial direction, so V = wr2

the area, A, is the area perpendicular to the flow direction. It is the surface area of a cylinder(minus the top and bottom bases) with a radius equal to the impeller outer radius, r2, and a

height equal to the blade height, b. So: A = 2p r2b

Substituting into the continuity equation gives:

Solving for wr2 gives:



This equation can be used to find the value of wr2 used in the stagnation temperature equation.

COMPARISON OF 3 IMPELLER BLADES

- For fixed inlet conditions

fixed tip speed, U2

zero mass flow rate

Þ All 3 impellers give the samestagnation temperature rise,

- As you increase the mass flow rate

Þ the stagnation temperature rise in thebackward leaning impeller decreases

Þ the stagnation temperature rise in theforward leaning impeller increases

Þ the stagnation temperature rise in theradial bladed impeller doesn’t change

- If there is no swirl at the entrance, the relative stage temperature rise is a linear function of wr2/U2.

Then, wr2/U2 is directly proportional to the mass flow rate and inversely proportional to the shaft

speed.

- Backward leaning vs. Radial

* for a given tip speed the radial impeller provides a greater pressure rise than thebackward leaning impeller

BUT

it has a higher Mach number flow entering the diffuser

at pressure ratios above 4:1, the air leaving the impeller and flowing into the diffuserbecomes supersonic

the resulting shocks cause flow irreversibilities that decrease the efficiency of the diffuser



therefore, as the pressure ratio increases, the diffuser efficiency decreases

* Modern centrifugal compressors therefore use backward leaning impellers toprevent the diffuser efficiency loss

the bending stresses characteristic of backward impellers are designed against throughmaterial choice and the design itself

the designs can have b 2 values as high as 30°

* Some designs also obtain a backsweep effect in impellers with radial lineelements at the outlet.

in the tangent plane, the blade is angled away from the axial direction and against thedirection of rotation

this imparts an extra 3-D effect on the flow field that creates the effect of backsweep acrossthe impeller outlet

STAGE STAGNATION PRESSURE RATIO &

IMPELLER EXIT MACH NUMBER

Pressure Ratio

(equation is the same as for an axial compressor)

But, the diffuser has adiabatic flow, so, T03 = T02. Also, from before:

Therefore:

Impeller Exit Mach Number

absolute Mach number at the impeller exit:



also,

c22 = cq 2

2 + cr22

or

(speed of sound at T2)2

with

and the isentropic relation

stagnation temperature relation from before

Substituting the c22, a2

2, and T02/T01 relations into the M2 equation gives:

where



* This equation can be used to generate plots of M2 and P03/P01 versus U2/a01 for different

impeller exit blade angles (b 2’s) and flow coefficients (wr2/U2’s). These plots can be used to

determine the necessary conditions for a given pressure ratio or the pressure ratio attainable froma given set of conditions.

* The plots also show that pressure ratios above a certain value can lead to supersonic diffuserflow and thereby decrease the diffuser efficiency.

h c = 0.85



h c = 0.85 b 2 = 15°

III. THE INDUCER AND IMPELLER

INDUCER: impeller entrance section

changes the tangential fluid motion

helps the flow enter the impeller smoothly WHY IS IT IMPORTANT?

* Without an inducer, there is flow separation and violent mixing nearthe leading edge of the impeller vane. The separation and mixing canbe very noisy.

* By directing the flow smoothly into the impeller, the inducer preventsthe separation and mixing, making the compressor much quieter.

INDUCER MAXIMUM TURNING ANGLE

Consider the flow entering the centrifugal impeller.



-At the impeller tip, the angle between the flow and the relative velocity vector can be very large.

For the flow to enter the inducer without boundary layer separation at the leading edge, theinlet van angle must be set at a large angle with respect to the flow.

This limits the amount the inducer may turn the flow

- If we assume that the inducer can turn the relative velocity without changing the radius…..

There is no inducer radial velocity component

The axial velocity component therefore remains constant

For the tip, the relative velocity at the inducer outlet is:

* if w’ < w1, there is diffusion in the inducer

- To find the maximum turning angle, or the largest angle the flow may be turned before boundarylayer separation:

1. Find Cpmax, the pressure coefficient at separation, where:

2. Find the maximum turning angle from;

- Turbulent boundary layer behavior is affected by surface curvature, though. It may thereforeseparate earlier on more curved surfaces than on less curved surfaces.

COMPRESSIBILITY EFFECTS

- High Mach numbers can cause shocks near the impeller tips



- The shock irreversibilities can then cause the compressor performance to deteriorate

- For example, shocks in the diffuser (from supersonic flow leaving the impeller) decrease itsefficiency

EFFECT OF INLET TIP-RADIUS BLADE ANGLE, b 1t,

AND RELATIVE MACH NUMBER M1rel

The inducer inlet Mach number and blade angle are related by:

If we know the inlet stagnation temperature and pressure we can find the inlet temperature anddensity from their isentropic relations and the perfect gas law:

r 01 = P01 / RT01

We can then get the inlet absolute velocity from the equation for the Mach number as c1/a1:

Next, we can find the blade-tip velocity, U1t, and inlet tip diameter, Dt1, from:

The shaft rpm, N, is then related to the tip velocity and diameter by:

- Plotting the shaft speed, N, versus the M1 for several M1rel’s and b 1t’s then shows the



necessary shaft speeds for certain inlet Mach number and tip-radius blade angle combinations.Please see Figure 9.11 in Hill and Peterson’s Mechanics and Thermodynamics of Propulsion foran example of this kind of plot.

- These quantities then limit the compressor design in two ways:

1. aerodynamically: the inlet relative Mach number limits the rotational speed, N2. stress-related: the impeller stresses limit the tip speed at the impeller exit, thus

affecting the pressure ratio

- These two sets of limits then combine to limit the machine size

IMPELLER PASSAGE SHAPE

- Boundary layer separation occurs at a certain overall velocity ratio, w2/w1

- You want the compressor design to decrease the possibility of boundary layer separation

Þ try to keep the average flow velocity constant in the radial part of the impeller

HOW?

decrease the axial width of the impeller as you increase its radius

Then, instead of obtaining a pressure rise by decreasing the velocity, centrifugal forcesincrease the pressure rise in the flow direction

THE CORIOLIS FORCES

- The pressure change from the centrifugal forces does not cause boundary layer separation

- Consider a fluid particle moving radially outward with:

w = particle velocity relative to the impeller

W = impeller rotational velocity



**The solid lines represent conditions at time t. The dotted lines represent conditions at time t+dt.

* w experiences no change in magnitude, but does experience a change in the q -direction

in dt the particle moves dq = W dt and dr = wdt then dcq = W dr + wdq = 2W wdt

divide both sides by dt to get the Coriolis acceleration, aq

* This acceleration requires a tangential Coriolis pressure gradient of

- from before:

- canceling d(W 2r2 /2) since the acceleration is in the tangential direction and dividing both sidesby rdq gives:

- combining this equation with

gives:



Since this equation contains a dw/dq term, it shows that there is a tangential variation in w. Thisvariation is shown below.

THE CORIOLIS PRESSURE GRADIENT

- There is a pressure difference across the blades. It is indicated by the + and – signs in twoprevious figures

- The static pressure in the tangential direction at the impeller exit is constant

- The Coriolis pressure gradient causes 3 effects near the tip:

1. As the pressure gradient near the outlet disappears, the fluid particles in themiddle of the channel can’t continue their radial movement and slip back againstthe direction of rotation. Then the tangential component of the absolute velocity,

cq 2, will be much less than U2. To prevent this, the blades must be spaced very

closely together.2. As the boundary layer on the (+) pressure side approaches the tip diameter, the

fluid accelerates.3. On the (–) pressure, or suction, side, the pressure gradient tends to cause the

boundary layer to separate. The separation may increase as the secondary flowdriven by the Coriolis pressure gradient moves the slower particles to the suctionsurface.

SLIP FACTOR, s s

For radial blades:



- depends on the momentum averaged tangential velocity component, cq 2, or on the number of

blades, Nb

For nonradial blades:

- depends on the momentum averaged tangential velocity component, the number of blades andthe impeller exit blade angle, B2

OUTLET BLADE HEIGHT

- The centrifugal compressor has a very small blade height at the impeller outlet

WHY?

- large density increase across the wheel

- the expansion of the diameter between the impeller inlet and outlet

- the limitation of relative velocity reduction

- The outlet blade height is related to the stage efficiency since the efficiency depends on the rotorand on the casing adjacent to the blade edges. The stage efficiency, h c, is related to these

quantities by:

where: D 2 is the tip clearance

b2 is the blade height

Ã is equal to 0.3 ± 0.05 for the clearance range: 0< D 2/b2 <1

SIMULATION OF INTERNAL FLUID DYNAMICS

- Since impellers are used in small gas turbines, they should be highly efficient, but weigh as littleas possible



- Simulating the internal fluid dynamics helps in the design

- To simulate the internal flow:

1. Divide the passage into a large number of volume elements2. Use an iterative numerical procedure to satisfy the continuity, momentum, and energy

equations for each element

- The pressure gradients calculated at the wall then indicate any danger of flow separation

IV. THE DIFFUSER

Velocities flowing out of the impeller, relative the impeller are quite low, but with a highimpeller tip speed (35,000 RPM +), the absolute velocity leaving the impeller could reach valuesabove Mach 1. The diffuser is used to reduce this velocity, while at the same time increase thepressure. In this section, our interest lies in flow through a vaneless diffuser. The vaneless diffuseris a simple annular channel in which the radial velocity component is reduced by area increase andthe tangential velocity component by the requirement of constant angular momentum.

We begin our analysis for an incompressible flow in a vaneless region of constant width h. Inthis region, both continuity and angular momentum of the air must be conserved.

Along a streamline, continuity becomes

which may be reduced to the general form

The general equation for angular momentum is

If we combine the two equations above, we see that

or



where a is the angle between the radial direction and the velocity c, as shown in the figure below.

***Angles in a diffuser are always with respect to the radial direction.

Thus the streamline makes a constant angle with the radial direction inside the vaneless section ofthe diffuser and since a is constant, the velocity is inversely proportional to the radius

One can also analyze the flow through a vaneless section of a diffuser for compressible flow. Herewe adopt the relation between fluid properties at any point in a flow with their values at a point werethe Mach number is 1 (denoted by the *).

The continuity and angular momentum equations used now are

Noting that

The continuity and angular momentum equations may now be written as

Once again we combine these equations (solve rcsin a for rc and plug into

r rccos a ) to yield



If we assume one dimensional, adiabatic, reversible flow, r */r may be written as

Therefore

And a is now a function of M. We can then use rcsin a in the form

of which, upon using the isentropic relations for T/T*, we obtain

With all said, what have we actually achieved. To explain this best, let’s go into an example.Lets say we only know the velocity and the angle of the air flowing out of the impeller, M1 = 1.3 and

a 1 = 70° . (Tan a */Tan a ) can be used to find a * = 75° , which is constant throughout the

vaneless section. From here we then use (r*sin a */rsin a ) and solve for (r*/r)1 = 0.8 at the

beginning of the section. Now if we wish to diffuse the air to M2 = 0.4, we can get the angle a 2from

(Tan a */Tan a ), knowing a * = 75° and use (r*sin a */rsin a )to solve for (r*/r)2 = 2.3 at the end of

the vaneless section. Again, knowing the conditions and parameters of the impeller, we can nowfind the ratio

(equaling 2.9) and the total width of the vaneless section needed to diffuse air flowing out of animpeller at M1 = 1.2 to M2 = 0.4. So as you can see, the equations above are very useful to quickly

approximate how big your compressor needs to be.



The advantage of using a vaneless diffuser is that it can provide deceleration of supersonicflows to subsonic flow. However there is a price to pay and that is the large radius ratio across thevaneless section. Where one must avoid large engine diameters, a vaneless diffuser isundesirable and a vaned diffuser is used. In the vaned diffuser, the vanes are used to guide theflow so that the overall rate of diffusion is higher then that obtainable in a vaneless diffuser, allowingfor smaller compressor sizes. Great care is taken for the angles of these vanes. Without properangles, stall could occur, creating losses in the flow and decreasing the efficiency of thecompressor.

IMPELLER SHAFT SPEED

In the earlier section we were able to calculate the impeller shaft speed by considering onlythe impeller inlet conditions:

Mass flow rate

Stagnation Temperature T01

Stagnation Pressure P01

Inducer angle b 1t

Hub Diameter Dh

Mach number M1 rel

By changing one of these variables, we needed to make numerous calculations to get the shaftspeed N or W . This may become a little tedious and it was found that another equation could beused to make one calculation when changing any variable. The equation to be used is

Where z is

IMPELLER DIAMETER RATIO



One can calculate the diameter ratio for the impeller from

Where

IMPELLER EXIT BLADE WIDTH RATIO

If one would like, they may use this equation to approximate the blade width ratio for acentrifugal compressor by

Where h c is

And s s is the slip factor given by

centifugal compressor

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