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

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Therefore, we can say here that with the consideration ofÂ phenomenon of slip in centrifugal compressor, velocity of whirl at outlet i.e.Â VÏ‰

Now we will use this value of VÏ‰

Energy transferred to the fluid per unit mas,Â E/m =Â ÏƒÂ u

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Now itâ€™s time
to discuss few other types of fluidÂ machinesÂ such as centrifugal
compressors, axial flow compressors, fans and blowers where working fluid will
be steam, air or gases.Â

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.Â

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**Velocity diagram of centrifugal compressorÂ **

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.Â

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

u

_{1 }= Mean blade velocity at inlet = Ï€D_{1}N/60
u

_{2 }= Mean blade velocity at outlet = Ï€D_{2}N/60
V

_{1}= Absolute velocity of air at inlet to rotor or impeller
V

_{2}= Absolute velocity of air at outlet to rotor or impeller
Vr

_{1}= Relative velocity of air at inlet
Vr

_{2}= Relative velocity of air at outlet
VÏ‰

_{1}= Velocity of whirl at inlet or tangential component of absolute velocity of air at inlet
VÏ‰

_{2}= 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
Î±

_{1}=Absolute angle at inlet or outlet angle
Î±

_{2}=Inlet angle to the diffuser
Î²

_{1}=Inlet angle to the rotor blade
Î²

_{2}= Outlet angle to the rotor blade
m = Mass flow
rate of air in Kg/secÂ

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.Â

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

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**Energy transferred to the fluid**

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**Case -1: Air enters the impeller eye in an axial
direction (****Î±**_{1 }= 90^{0}**) and leaves the impeller radiallyÂ **

_{1 }= 90

^{0}

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

Air enters at
the inlet axially and hence Î±

_{1}i.e. absolute angle at inlet will be 90^{0}.
Î±

_{1}= 90^{0},
VÏ‰

_{1}= 0, V_{f1}=V_{1Â }
As compressed
air will leave the rotor radially and hence we can write following equation as
mentioned here.

Î²

_{2}= 90^{0},
Vr

_{2}=V_{f2}
VÏ‰

_{2}=u_{2Â }
Now we will
determine the work done by the impeller on air orÂ energy transferred to the fluid and it could be written as mentioned hereÂ

E = m [VÏ‰

_{2}u_{2}- VÏ‰_{1}u_{1}]Â
E = m [VÏ‰

_{2}u_{2}- VÏ‰_{1}u_{1}]Â
E = m [u

_{2}x u_{2}]Â
Energy transferred per unit mass of air could be written by following equationÂ

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**E/m = u**_{2 }^{2Â }

_{2 }

^{2Â }

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**Case -2: Air enters the impeller eye in an axial
direction (****Î±**_{1 }= 90^{0}**) but not leaving the impeller radially i.e. ****Î²**_{2 }< 90^{0}

_{1 }= 90

^{0}

_{2 }< 90

^{0}

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

Â

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

Â

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

E = m [VÏ‰

_{2}u_{2}- VÏ‰_{1}u_{1}]
E = m VÏ‰

_{2}u_{2Â }
Energy transferred to the fluid per unit mas, E/m = VÏ‰

_{2}u_{2Â }
Let us see
here the outlet velocity triangle and we can write the following equation

Tan Î²

_{2}=V_{f2}/ (u_{2 }- VÏ‰_{2})
CotÎ²

_{2}= (u_{2 }- VÏ‰_{2}) / V_{f2}
VÏ‰

_{2}= u_{2 }-V_{f2}Cot Î²_{2Â }Therefore, we can say here that with the consideration ofÂ phenomenon of slip in centrifugal compressor, velocity of whirl at outlet i.e.Â VÏ‰

_{2Â }will be less than theÂ Mean blade velocity at outletÂ i.e.Â u

_{2}.Â

VÏ‰

_{2Â }<Â u_{2}
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.Â

Slip factor,Â Ïƒ = Velocity of whirl at outlet i.e.Â VÏ‰

Slip factor,Â Ïƒ = Velocity of whirl at outlet i.e.Â VÏ‰

_{2Â }/Â Mean blade velocity at outletÂ i.e.Â u_{2}
Ïƒ = VÏ‰

_{2Â }/Â u_{2}
VÏ‰

_{2}Â =Â ÏƒÂ u_{2}Now we will use this value of VÏ‰

_{2}Â in above mentioned energy equation and we will have following equation for energy transferred to the fluid per unit mas.Â

Energy transferred to the fluid per unit mas,Â E/m =Â ÏƒÂ u

_{2}

^{2}

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**Energy transferred to the fluid per unit mas,Â E/m =Â ÏƒÂ u**_{2}^{2}

_{2}

^{2}

We must note it here that the value of slip factor i.e.Â Ïƒ 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 theÂ amount of energy transferred to the fluid in no slip.Â

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

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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 orÂ work done on air in a centrifugal
compressor.Â

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

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**Energy transferred to the fluid per unit mas,Â E/m =Â Ïˆ ÏƒÂ u**_{2}^{2}

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 orÂ work done on air in a centrifugal
compressor.Â _{2}

^{2}

Do you have
any suggestions? Please write in comment box and also drop your email id in the
given mail box which is given at right hand side of page for further and
continuous update fromÂ www.hkdivedi.com.Â Â

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

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**Reference:****Â Â **

Fluid
mechanics, By R. K. BansalÂ

Image
courtesy: GoogleÂ Â Â

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