Page 192 - Fluid Mechanics and Thermodynamics of Turbomachinery

Three-dimensional Flows in Axial Turbomachines 173
For a normal stage (˛ 1 D ˛ 3 ) with c x constant across the stage, the reaction was
shown to be
c x
R D .tan ˇ 1 C tan ˇ 2 /. (5.11)
2U
Substituting values of tan ˇ 1 and tan ˇ 2 into eqn. (5.11), the reaction becomes

k
R D 1 , (6.9)
r 2
where
k D .K 1 C K 2 //.2/.

It will be clear that as k is positive, the reaction increases from root to tip. Likewise,
2
from eqn. (6.1) we observe that as c /r is always positive (excepting c D 0), so

static pressure increases from root to tip. For the free-vortex flow rc D K, the
2
static pressure variation is obviously p/ D constant K/.2r / upon integrating
eqn. (6.1).
EXAMPLE 6.1. An axial flow compressor stage is designed to give free-vortex
tangential velocity distributions for all radii before and after the rotor blade row.
The tip diameter is constant and 1.0 m; the hub diameter is 0.9 m and constant for
the stage. At the rotor tip the flow angles are as follows
D 30 deg.
Absolute inlet angle, ˛ 1
D 60 deg.
Relative inlet angle, ˇ 1
Absolute outlet angle, ˛ 2 D 60 deg.
Relative outlet angle, ˇ 2 D 30 deg.

Determine,
(i) the axial velocity;
(ii) the mass flow rate;
(iii) the power absorbed by the stage;
(iv) the flow angles at the hub;
(v) the reaction ratio of the stage at the hub;

given that the rotational speed of the rotor is 6000 rev/min and the gas density is
3
1.5 kg/m which can be assumed constant for the stage. It can be further assumed
that stagnation enthalpy and entropy are constant before and after the rotor row for
the purpose of simplifying the calculations.

Solution. (i) The rotational speed,  D 2 N/60 D 628.4 rad/s.
Therefore blade tip speed, U t D r t D 314.2 m/s and blade speed at hub, U h D
r h D 282.5 m/s.
From the velocity diagram for the stage (e.g. Figure 5.2), the blade tip speed is
p p
°
°
U t D c x .tan 60 C tan 30 / D c x . 3 C 1/ 3/.
Therefore c x D 136 m/s, constant at all radii by eqn. (6.6a).