VELOCITY DIAGRAM OF CENTRIFUGAL COMPRESSOR

We were discussing the basic principle of operations, functions of various parts and performance characteristics of different types of fluid machines such as hydraulic turbines, centrifugal pumps and reciprocating pumps in our previous posts. We have seen there that working fluid for above mentioned fluid machines was liquid such as water.

Now it’s time to discuss few other types of fluid machines such as centrifugal compressors, axial flow compressors, fans and blowers where working fluid will be steam, air or gases.

We have already seen the working principle of centrifugal compressor in our last post, where we have also seen the parts and their functions of a centrifugal compressor. Today we will be interested here to discuss the velocity diagram of centrifugal compressor. We will also find out here the work done on air or energy transferred to the fluid in the centrifugal compressor.

Velocity diagram of centrifugal compressor

Following figure shown here indicates the velocity diagram of a centrifugal compressor. Inlet velocity triangle and outlet velocity triangle are drawn here in following figure.

Let us first understand here the various nomenclatures that we are using here.

u₁= Mean blade velocity at inlet = πD₁N/60

u₂= Mean blade velocity at outlet = πD₂N/60

V₁ = Absolute velocity of air at inlet to rotor or impeller

V₂ = Absolute velocity of air at outlet to rotor or impeller

Vr₁ = Relative velocity of air at inlet

Vr₂ = Relative velocity of air at outlet

Vω₁ = Velocity of whirl at inlet or tangential component of absolute velocity of air at inlet

Vω₂ = Velocity of whirl at outlet or tangential component of absolute velocity of air at outlet

V_f1 = Velocity of flow at inlet

V_f2 = Velocity of flow at outlet

α₁=Absolute angle at inlet or outlet angle

α₂=Inlet angle to the diffuser

β₁=Inlet angle to the rotor blade

β₂= Outlet angle to the rotor blade

m = Mass flow rate of air in Kg/sec

Centrifugal compressor parts are designed in such way that air will enter and leave the compressor without any shock. Let us assume that condition is ideal and there is no slip and no pre whirl.

We will see here two cases and velocity diagram will be different for these two cases. So let us understand here the first case.

Energy transferred to the fluid

Case -1: Air enters the impeller eye in an axial direction (α₁= 90⁰) and leaves the impeller radially

Following figure indicates the velocity triangle at inlet and outlet of the impeller blade.

Air enters at the inlet axially and hence α₁i.e. absolute angle at inlet will be 90⁰.

α₁= 90⁰,

Vω₁ = 0, V_f1 =V₁

As compressed air will leave the rotor radially and hence we can write following equation as mentioned here.

β₂= 90⁰,

Vr₂ =V_f2

Vω₂ =u₂

Now we will determine the work done by the impeller on air or energy transferred to the fluid and it could be written as mentioned here

E = m [Vω₂u₂- Vω₁u₁]

E = m [u₂x u₂]

Energy transferred per unit mass of air could be written by following equation

E/m = u₂²

Case -2: Air enters the impeller eye in an axial direction (α₁= 90⁰) but not leaving the impeller radially i.e. β₂< 90⁰

This is a case, where we are going to consider the phenomenon of slip in centrifugal compressor. We will see the phenomenon of slip in centrifugal compressor in our next post, but we must understand here that due to this phenomenon of slip in centrifugal compressor, velocity triangle at the outlet of the impeller blade will be changed.

Work done by the impeller on air or energy transferred to the fluid will be determined by following equation as mentioned here.

E = m [Vω₂u₂- Vω₁u₁]

E = m Vω₂u₂

Energy transferred to the fluid per unit mas, E/m = Vω₂u₂

Let us see here the outlet velocity triangle and we can write the following equation

Tan β₂=V_f2/ (u₂- Vω₂)

Cotβ₂= (u₂- Vω₂) / V_f2

Vω₂= u₂-V_f2Cot β₂

Therefore, we can say here that with the consideration of phenomenon of slip in centrifugal compressor, velocity of whirl at outlet i.e. Vω₂will be less than the Mean blade velocity at outlet i.e. u₂.

Vω₂< u₂

Here, we will consider one important factor i.e. slip factor in deriving the expression for energy transferred to the fluid. Slip factor will be discussed in detail in our next post.

Slip factor, σ = Velocity of whirl at outlet i.e. Vω₂/ Mean blade velocity at outlet i.e. u₂

σ = Vω₂/ u₂

Vω₂ = σ u₂

Now we will use this value of Vω₂ in above mentioned energy equation and we will have following equation for energy transferred to the fluid per unit mas.

Energy transferred to the fluid per unit mas, E/m = σ u₂²

Energy transferred to the fluid per unit mas, E/m = σ u₂²

We must note it here that the value of slip factor i.e. σ will be less than 1 and hence we can easily conclude that less amount of energy will be transferred to the fluid with consideration of slip factor as compared to the amount of energy transferred to the fluid in no slip.

Here, if we considered one more factor i.e. power input factor (ψ), we will have following equation of energy transferred to the fluid per unit mass as mentioned below.

Energy transferred to the fluid per unit mas, E/m = ψ σ u₂²

So, we have seen here the velocity triangles at inlet and outlet of impeller blades. We have also secured here the expression for the energy transferred to the fluid per unit mas or work done on air in a centrifugal compressor.

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Further we will find out, in our next post, slip phenomenon and slip factor for centrifugal compressor.