VELOCITY DIAGRAMS

The velocity diagrams at inlet and outlet of the impeller for the centrifugal compressor are shown in Fig. 14.6(a).

Figure 14.6 (a) Actual velocity diagrams

Let

u₁ = mean blade velocity at entrance

u₂ = mean blade velocity at exit

v_a1 = absolute velocity at inlet to rotor

v_a2 = absolute velocity at outlet to rotor

v_r1 = relative velocity at inlet to rotor

v_r2 = relative velocity at outlet to rotor

v_w1 = velocity of whirl at inlet

v_w2 = velocity of whirl at outlet

v_f1 = velocity of flow at inlet

v_f 2 = velocity of flow at inlet

D₁, D₂ = mean diameter of rotor at inlet and outlet, respectively

α₁ = exit angle from stator or guide vanes at entrance

β₁ = inlet angle to rotor in impeller blade angle at inlet

α₂ = inlet angle to the diffuser or the stator

β₂ = outlet angle from rotor or impeller blade angle at outlet.

If it is assumed that the entry of the air to the rotor is axial, then whirl component v_w1 = 0 and v_a1 = v_f1.

1 Theory of Operation

1. Ideal Velocity DiagramsBy Newton’s second law of motion, the rate of change of angular momentum of air is equal to the torque applied to the body causing that change. Considering 1 kg/s of mass flow rate of air,where For ideal case, let us assume that the impeller is radial vaned, and there is no frictional loss, no heat transfer and the air leaves the impeller with a tangential velocity v_w2 = u₂. Also for axial entry, v_w1 = 0Work done by impeller for 1 kg/s air flow rate,Figure 14.6(b) Ideal velocity diagramswhere ω = angular speed of rotor, rad/sNow, r₂ω = u₂Since air cannot leave the impeller at a velocity greater than the impeller tip velocity, Eq. (14.11) gives the maximum work capacity of the impeller.Also, by steady flow energy equation, we haveIn most practical problems,v_a1 = v_a2= h₂ − h₁where  = stagnation pressure ratiowhere,  = static pressure ratioComparing Eq. (14.11) and (14.14), we haveIf v_a1 = v_a2, thenTotal temperature increase across an impeller,where ṁ = mass flow rate of air, kg/s, and u₂ is in m/s.Therefore, for a centrifugal compressor working under ideal condition, the power input depends on the following:
   1. The mass flow rate of airThe total pressure ratio of the compressor which, in turn, depends on the square of the impeller velocityThe total inlet temperatureThe total inlet temperature rise between the inlet and outletThere is a maximum work capacity of an impeller depending on the tip velocity
   The maximum pressure produced depends on the following:
   1. Inlet temperature
   2. Square of the impeller velocity
   3. It is independent of the impeller diameter
2. Actual Velocity DiagramsThe actual velocity diagrams are shown in Fig. 14.7.Theoretical torque, T = v_w2r₂ – v_w1r₁Work done on 1 kg/s of air.w = (v_w2r₂ – v_w1r₁)wwhere, ω = angular speed of rotor, rad/s Figure 14.7 Actual velocity diagrams for centrifugal compressor  = ΔKE + Δp due to diffusion action + Δp due to centrifugal actionTherefore, the fraction of kinetic energy imparted to air and converted into pressure energy in impeller is given by,where ρ = density of air.If v_a1 = diffuser outlet velocity, then

2 Width of Blades of Impeller and Diffuser

Air mass flow rate per second,

where, v_s = specific volume of air

b₁, b₂ = width or height of impeller blades at inlet and outlet, respectively

k_b = blade factor

n = number of blades

t = thickness of blades

The width or height of the diffuser blades at the outlet is given by,

where suffix ‘d’ represents the quantities at the diffuser outlet. The width or height of the diffuser blades at the inlet is the same as that of the impeller blades at outlet.