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In this article we will discus about:- 1. Construction and Principle of Operation of Centrifugal Compressor 2. Velocity Diagrams of a Centrifugal Compressor 3. Work Requirement (Euler’s Work) 4. Slip Factor 5. Pressure Ratio of Compression 6. Influence of Impeller Blade Geometry 7. Influence of Compressor Geometry on the Performance 8. Pre-Whir 9. Losses.

**Contents:**

- Construction and Principle of Operation of Centrifugal Compressor
- Velocity Diagrams of a Centrifugal Compressor
- Work Requirement (Euler’s Work) for a Centrifugal Compressor
- Slip Factor of Centrifugal Compressors
- Pressure Ratio of Compression in a Centrifugal Compressor
- Influence of Impeller Blade Geometry on Centrifugal Compressor
- Influence of Compressor Geometry on the Performance of Centrifugal Compressor
- Pre-Whir in a Centrifugal Compressor
- Losses in a Centrifugal Compressor

#### 1. Construction and Principle of Operation of Centrifugal Compressor:

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This is a dynamic compressor. The compression and the pressure rise of air is achieved by dynamic action. The compressor is often directly coupled to a prime mover and is driven at high speed.

Figure 16.9 shows a schematic diagram of a centrifugal compressor. It consists of inlet (suction) pipe (1), an impeller (rotor) with blades or vanes (2), a vaned or vanless diffuser (3), a volute or scroll casing (4), an outlet or delivery pipe (5), a shaft (6) and shaft seal (7). The impeller is keyed to the shaft and the rotating assembly is dynamically balanced.

The thermodynamic process of compression and pressure-rise over the rotor and in the diffuser is shown on h-S diagram in Fig. 16.10.

Due to the rotation of impeller, vacuum is created at the eye of the impeller. Thus, air is sucked in from the surrounding ambient, at initial stagnation pressure p_{Io} and stagnation temperature T_{Io}, represented by point on h-S diagram.

The static ambient condition is represented by point o. The air is somewhat accelerated in the inlet pipe. The inlet pipe admits or directs air for smooth and shockless entry into the eye of the impeller. In the inlet pipe, the direction and value of absolute velocity of air changes.

Due to the acceleration of flow in the inlet pipe, velocity increases from V_{0} to V_{1} thereby decreasing pressure and temperature of air. The acceleration process is non-isentropic due to friction and is shown by process 0 – 1 on h-S diagram. Therefore, at inlet to the impeller, the pressure and temperature are p_{1} and T_{1} respectively.

The vanes or blades of the impeller may have radial, backward or forward curvature.

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Air leaves the impeller outlet at very high absolute velocity V_{2} which is further converted into pressure in a vaneless or vaned diffuser and volute (scroll) casing. In the vaneless diffuser the absolute velocity of air decreases from V_{2} to V_{3}, process 2-3 on h-S diagram. It also stabilises the flow leaving the vaned impeller tips for shockless entry into the vaned diffuser. The velocity of air further decreases from V_{3} to V_{4} in the vaned diffuser to increase pressure further. This process is represented by line 3-4 on h-S diagram.

The vaned diffuser discharges air into the volute (scroll) casing and finally air leaves from the outlet pipe. Process 4 – e on h-S diagram represents the process in the volute casing and outlet pipe. Centrifugal compressors used in gas turbine plants have a 90° bend in place of volute casing, to direct the air towards the combustion chamber as shown in Fig. 16.11. Finally, air has static and stagnation pressure of p_{e} and p_{eo} at the exit of the compressor. If there would be no frictional losses whatsoever, the final pressure would have been together, as represented by p_{eo(wl)}.

** 2. Velocity Diagrams of a Centrifugal Compressor****: **

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Figure 16.13 shows velocity diagrams at the inlet and outlet of impeller and at inlet and outlet of diffusers.

**Nomenclature:**

The following notations shall be adopted hereafter for energy analysis of the centrifugal compressor. Suffix ‘1’ denotes parameters at inlet and ‘2’ at outlet.

β_{1} = Angle of the rotor blade at inlet

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β_{2 }= Angle of the rotor blade at outlet

α_{1} = Angle made by entering air or exit angle of guide blade

α_{2} = Angle made by the outgoing air from rotor blade

V_{1} and V_{2 }= Absolute velocity of air at inlet and outlet of rotor, m/s

V_{r1} and V_{r2} = Relative velocity of air at inlet and outlet of rotor, m/s

V_{f1} and V_{f2} = Velocity of flow at inlet and outlet of rotor, m/s

V_{w1} and V_{w2} = Velocity of whirl at inlet and outlet of rotor, m/s

u_{1} and u_{2 }= Mean peripheral velocity of blade tip at inlet and outlet, m/s

r_{1} and r_{2} = Inner and outer radii of rotor, m

m = Mass flow rate of air, kg/s

α_{3} = Vaned diffuser inlet angle or vaneless diffuser outlet angle

At the inlet to rotor, air enters with absolute velocity V_{1} making an angle α_{1} to the direction of motion of blade (usually α_{1} = 90°), without any shock and its whirl component V_{w1 }= 0.

Inlet triangle of velocities is now drawn to scale setting V_{1} in the radial direction, u_{1} in the tangential direction to inlet periphery. The vector joining end points of V_{1} and u_{1} represents relative velocity V_{r1} at inlet. The blade tip at inlet has angle β_{1}, i.e., its curvature at inlet lies in the direction of V_{r1}.

At the outlet from rotor, air leaves with a relative velocity V_{r2} at an angle β_{2}, with the direction of motion. Now, as u_{2} is known, V_{2} can be found by vectorial addition.

In a vaneless diffuser, the flow is assumed to be logarithmic spiral and free-vortex. Air enters the vaned diffuser with velocity V_{3} at an angle α_{3} and leaves the diffuser with a velocity.

** 3. Work Requirement (Euler’s Work) for a Centrifugal Compressor****: **

The work required/kg of air in a stage of a centrifugal compressor can be found by applying the moment of momentum theorem.

As per the Newtonian equation, force is given by rate of change of momentum.

Similarly, rate of change of moment of momentum about the centre of rotation gives torque.

Consider 1 kg of air flowing through the impeller (rotor). Theoretical torque supplied by impeller,

**Dimensionless Parameters**** of Centrifugal Compressors:**

For the design and performance analysis of centrifugal compressor, several dimensionless parameters are very useful.

**These are defined and explained in what follows:**

** 4. Slip Factor ****of Centrifugal Compressors: **

Under ideal conditions, fluid particles follow exactly the same path of blade profile such that relative velocity at impeller outlet tip is inclined with the tangential direction at blade tip angle β_{2} only, irrespective of mass flow rate, speed etc.

Such an ideal flow is possible when impeller has infinite number of blades of no thickness. Figure 16.14 shows velocity triangle for radial blades. Dotted lined diagram represents ideal conditions while full lined diagram represents actual conditions.

In actual practice, when impeller has finite number of blades, fluid is trapped between the impeller vanes due to its inertia and the fluid is reluctant to flow over the impeller.

This causes a pressure difference across the blades. There is a high pressure at the leading face and low pressure at the trailing face. This pressure difference generates a relative velocity gradient and formation of eddies.

The fluid is thus discharged at a certain average angle β’_{2} which is less than β_{2}.

Therefore fluid is said to have slipped with respect to impeller during its flow across it.

Slip factor depends upon number of blades and is usually 0.9. It does not reduces the efficiency but only reduces the head developed.

** 5. Pressure Ratio of Compression in a Centrifugal Compressor****: **

Slip factor is also defined as the ratio of work required under actual conditions to the work required under ideal conditions.

The slip factor µ and the power input factor Ѱ are different and independent from each other. Whereas the power input factor Ѱ signifies the increase in the work input required which is lost in overcoming friction. This energy lost gets converted into heat and causes temperature rise.

The temperature rise of compressed air is often not a loss as in gas turbine plants the hot compressed air elevates the temperature in the combustion chamber without increase of fuel supply. Yet the power input factor should be low and compression should ideally be isentropic.

But the slip factor limits the work capacity of the compressor even with isentropic working. The slip factor should be as high as possible. High slip factor increases V_{w2}. When V_{w2} = u_{2}, work input is maximum which can be gainfully utilised.

Slip factor increases with greater number of-vanes which in turn decrease area of flow passage for air. A compromise is often necessary. The present practice is to adopt about 20 vanes to obtain a slip factor of 0.9. An empirical formula proposed by Stanitz is –

Where, η = number of blades

Higher tip speed increases the pressure ratio.

Maximum tip speed is restricted to 460 m/s for the common materials.

Also reducing the inlet temperature raises the pressure ratio for a given work input.

** 6. Influence of Impeller Blade Geometry** on Centrifugal Compressor:

The blades of the impeller have various shapes which can be described based on the blade angle at outlet β_{2}. Depending on the value of the blades have different curvatures compared with the forward direction of rotation of the impeller.

**The various shapes of blades are: **

(i) Backward curved blades β_{2} < 90°

(ii) Radial blades; β_{2} = 90°

(iii) Forward curved blades; β_{2} > 90°

Figure 16.15 shows the velocity triangle for each case of the above.

In all cases, blade tip velocity at exit u_{2} is kept same.

It is seen from Fig. 16.15 that for the backward curved blades, the tangential component V_{w2} is least compared with those in other cases. Therefore, for a given speed work input required for the impeller is low, as given by Euler’s work (Eq. 6).

In the case of forward curved blades, V_{w2} is maximum, thus the impeller needs maximum work input. Too, the value of V_{2} is highest.

And in case of radial blades, V_{w2} lies in between, that is, backward and forward curved blades. Therefore, backward curved blades are best giving high efficiency.

The influence of blade geometry can also be explained by dimensionless parameters as follows. The head coefficient is one dimensionless parameter which is used to express the work required per stage of centrifugal compressor.

For a working fluid treated as ideal gas, isentropic efficiency η_{isent} = 1. As such head coefficient λ and pressure coefficient Ѱ are same, the work input to the fluid is then given by-

** 7. Influence of Compressor Geometry on the Performance of Centrifugal Compressor****: **

Figure 16.20 (a) and (b) illustrates variation of reaction of the stage Ω with pressure coefficient Ѱ.

** 8. Pre-Whir in a Centrifugal Compressor****: **

The gas turbines and jet engines use centrifugal compressors running at high speed. There is often a possibility of development of a shock wave in the flow passage. This can be avoided if the Mach number at any point in the flow passage is restricted below unity.

The maximum value of Mach number referred to relative velocity is found at inlet.

For a given flow rate, if V_{r1} and u_{1} is high for shockless operation of the compressor, the fluid is given an initial pre-rotation or pre-whirl with the help of fixed guide blades attached to the casing at inlet.

The entering air passing through the fixed guide blades is given pre-whirl. Thus V_{r1} is reduced without affecting V_{f1} or mass flow rate.

** 9. Losses in a Centrifugal Compressor****: **

The frictional losses in a centrifugal compressor occur mainly at two places viz. over the impeller and in the diffuser.

**The fluid friction losses over the impeller are given by-**

The losses in the diffuser (vaned or vanless) are due to irreversible inversion of friction force in boundary layer on the surface of diffuser.

All these losses tend to reduce the pressure ratio and increase the work input required thereby decreasing the efficiency of the compressor.