Turbomachinery Lecture Notes 1 2007-10-04

Radial Compressors

Damian Vogt Course MJ2429

Nomenclature

Symbol Denotation Unit c Absolute velocity m/s h Enthalpy J/kg m& Mass flow rate kg/s r Radius m u Tangential velocity m/s w Relative velocity m/s α Absolute flow angle deg β Relative flow angle deg ω Rotational speed rad/s

Subscripts

0 Total 1 Inlet rotor 2 Outlet rotor (inlet stator) 3 Outlet stator n Normal r Radial component x Axial component θ Tangential component

KTH/EKV/DV

Turbomachinery Lecture Notes 2 2007-10-04

Introduction and Velocity Triangles Due to the working principle radial compressors are also referred to as “centrifugal compressors”. These denotations indicate the following:

• “Radial” denotes the outflow from the stage. Other than axial compressors, which feature axial in- and outflow, the flow direction in radial compressors is turned from axial to radial

• “Centrifugal” denotes the working principle, i.e. whereas the total enthalpy increase in axial compressors is almost exclusively due to a change in tangential velocity component, in radial compressors a major contribution arises from the centrifugal effect

Figure 1 depicts a radial compressor stage in side view (cut through rotor) and front view (seen in flow direction).

stator (diffuser)

rotor (impeller)

stator (diffuser)

rotor (impeller)

α2

θ pos

α, β pos

θ neg

α, β neg

Circumferential speed

Absolute velocity

u2

wθ1

cθ2

w2

c2

Absolute circumferential velocity component

w1 c3 c1

u1

Relative velocity

β1

Figure 1. Sketch of radial compressor stage and velocity triangles

KTH/EKV/DV

Turbomachinery Lecture Notes 3 2007-10-04

Note that the following synonyms are used:

• “Impeller” for “rotor” it is the rotor that gives impetus to the flow • “Diffuser” for “stator” it is the stator that diffuses (or decelerates) the flow to

increase static pressure The contribution of centrifugal effects to the overall rise in stagnation enthalpy gets apparent in the Euler equation as follows

22110201 θθ cucuhh −=− Eq. 1 In case of the depicted stage the absolute circumferential velocity component at rotor inlet is equal to zero as there is no device upstream of the rotor that creates swirl. The Euler equation therefore simplifies to

220201 θcuhh −=− Eq. 2 wherefrom it is recognized that the tangential speed of the rotor at rotor outlet plays a major contribution to the total enthalpy increase. From the sketch of the stage and the velocity triangles it is also recognized that the flow in the rotor undergoes deflection in two directions as follows:

• The first deflection is in the meridional plane, i.e. the blade-to-blade plane and leads to a change on relative flow angle β (note: this deflection is similar to the one occurring in axial compressors)

• The second deflection is in the axial-radial plane, i.e. apparent in the side view and leads to a change in flow direction from axial to radial

On a first view radial compressor impeller blades seem to have a complicated geometry that seems to be hard to understand. However once these two types of deflections are understood the geometry can be broken down to an understandable level. At impeller inlet the blade direction is such that the relative inflow is met at minimum incidence angle, i.e. the blade direction should be close to β1. At rotor outlet the blades are often purely radial (to minimize mechanical stresses) or as outlined further below backswept (to minimize aerodynamic losses). In-between the blade is such that it leads to a smooth deflection of the flow in the aforementioned directions. Note that the fore-part of the impeller is called “inducer”. The Euler equation and the impeller geometry can explain two major characteristics of radial compressors:

• Radial compressors feature superior pressure increase (up to pressure ratio 5 or 6) thanks to the centrifugal effect

• Due to the double deflection of the flow radial compressors generally feature lower efficiency than axial counterparts

KTH/EKV/DV

Turbomachinery Lecture Notes 4 2007-10-04

Diffuser The diffuser (i.e. the stator) downstream of the rotor is used to decelerate the flow and transform the kinetic energy at impeller exit to pressure recovery. The diffuser does not necessarily need any blades as the circular geometry itself leads to an area increase (for parallel-walled diffusers) as follows

122

2

3

2

3 >⋅⋅⋅⋅

=rbrb

AA

ππ

Eq. 3

Assuming a vaneless diffuser (i.e. a diffuser without blades) a fluid particle released at impeller exit would trace out a logarithmic spiral from inlet to outlet. Depending on the absolute flow angle at diffuser inlet this flow path can be considerably long (note: large angle α2 lead to longer flow path). Diffuser blades are therefore used to guide the flow and thus to reduce the flow path from diffuser inlet to outlet. At diffuser outlet the flow is collected in a volute that contributes to further pressure recovery. If no high pressure increase in the stage is aimed at (e.g. blower) the diffuser might be omitted. The impeller exit flow is then directly discharging into the volute as depicted in figure 2.

Figure 2. Radial blower without diffuser Note also that the depicted impeller does not feature an inducer, i.e. the blades are shortened at the leading edge. Furthermore it can be recognized that the blades are swept back at impeller exit, i.e. swept against direction of rotation.

KTH/EKV/DV

Turbomachinery Lecture Notes 5 2007-10-04

Slip At impeller exit the relative flow leaves the blade approximately at the blade metal angle. In reality a phenomenon called slip leads to the relative outflow being deflected against the direction of rotation. The underlying physical phenomenon is the superposition of a relative eddy onto the relative main flow direction. This relative eddy establishes in a blade passage due to the rotation of the impeller as sketched in figure 3. For a given rotation a counter-rotating eddy will establish in a blade passage (figure 3a). This eddy affects the outflow from the blade passage such that the relative outflow is deflected against the direction of rotation (figure 3b).

Figure 3. Concept of relative eddy (adapted from Dixon) A practical consequence of the slip is that the relative circumferential component at impeller exit is reduced against the direction of rotation. This in turn implies that the absolute circumferential component is reduced as well, which leads to a reduction in total enthalpy increase. The effect of slip on the velocity triangle is graphically expressed in figure 4.

cΘ' (ideal)

c2’

w2’ c2 w2

cΘ (real)

Figure 4. Effect of slip on velocity triangle at impeller exit

KTH/EKV/DV

Turbomachinery Lecture Notes 6 2007-10-04

The slip is expressed by a slip factor as follows

'2

2

θ

θσcc

= Eq. 4

Ideal outflow is thus represented by σ=1 whereas real outflow features slip factors of σ<1. Several researchers have presented theoretical expressions for determining the slip factor. The common idea behind this theoretical consideration is that an infinite number of blades would lead to a slip factor of unity while decreasing number of blades tends the slip factor towards zero. Stodola (1927) has presented one of the earliest and simplest expressions for the slip factor as follows

'tan1'cos

122

2βφ

βπσ−

⋅−=n

Eq. 5

The expression contains the effect of blade count and blade metal angle at impeller exit. Stanitz (1952) recognized that the influence of the blade angle was not of major importance and thus simplified Stodola’s expression to

'tan163.01

22 βφπσ

−⋅−=

n Eq. 6

For impellers with radial blades β2’ gets zero and thus the expression simplifies further to

n63.01 ⋅

−=πσ Eq. 7

For preliminary estimates of the slip factor it is therefore often accurate enough to apply

n21−=σ . By employing the slip factor the Euler equation simplifies considerably for radial bladed impellers with axial inflow. Note that for such impellers the theoretical relative circumferential speed equals to zero. Consequently the theoretical absolute circumferential speed equals the tangential speed, which would give . Taking into account the slip the Euler equation is then expressed by

220201 uhh −=−

2

20201 uhh ⋅−=− σ Eq. 8

KTH/EKV/DV

Turbomachinery Lecture Notes 7 2007-10-04

Benefit of Backsweep By applying backsweep, i.e. bowing the impeller blades against the direction at impeller outlet, it is apparent from the Euler equation that the total enthalpy increase is reduced. This is due to the relative impeller outflow velocity now featuring negative circumferential component (note: for purely radial blades the circumferential component would ideally be zero). Nevertheless backsweep is applied to reduce the losses associated to the absolute outflow velocity from the impeller. As radial compressors wheels usually are rotating at high speed (circumferential Mach number close to unity) the absolute impeller outflow velocity gets supersonic. By reducing this velocity the impeller discharge losses due to supersonic flow phenomena can be reduced greatly. To compensate for the reduction in total enthalpy increase due to backsweep the rotational speed of the wheel is often increased to some degree. The amount of backsweep and increase in rotational speed are consequently chosen such that the overall performance of the radial compressor stage is increased.

KTH/EKV/DV