Page 25 - Fluid Mechanics and Thermodynamics of Turbomachinery

6 Fluid Mechanics, Thermodynamics of Turbomachinery
variables on the performance must now be included. The size of machine is char-
acterised by the impeller diameter D, and the shape can be expressed by a number
of length ratios, l 1 /D, l 2 /D, etc.

Incompressible fluid analysis

The performance of a turbomachine can now be expressed in terms of the control
variables, geometric variables and fluid properties. For the hydraulic pump it is
convenient to regard the net energy transfer gH, the efficiency , and power supplied
P, as dependent variables and to write the three functional relationships as

l 1 l 2
gH D f 1 Q, N, D, , , , ,... , .1.1a/
D D

l 1 l 2
D f 2 Q, N, D, , , , ,... , .1.1b/
D D

l 1 l 2
P D f 3 Q, N, D, , , , ,... , .1.1c/
D D
By the procedure of dimensional analysis using the three primary dimensions, mass,
length and time, or alternatively, using three of the independent variables we can
form the dimensionless groups. The latter, more direct procedure, requires that the
variables selected, , N, D, do not of themselves form a dimensionless group. The
selection of , N, D as common factors avoids the appearance of special fluid terms
(e.g. , Q) in more than one group and allows gH, and P to be made explicit.
Hence the three relationships reduce to the following easily verified forms.
Energy transfer coefficient, sometimes called head coefficient
2
gH Q ND l 1 l 2
D D f 4 , , , ,... , .1.2a/
.ND/ 2 ND 3 D D
2
Q ND l 1 l 2
D f 5 3 , , , ,... . .1.2b/
ND D D
Power coefficient
2
P Q ND l 1 l 2
O P D D f 6 , , , ,... . (1.2c)
3
N D 5 ND 3 D D
3
The non-dimensional group Q/.ND / is a volumetric flow coefficient and
2
ND / is a form of Reynolds number, Re. In axial flow turbomachines, an
3
alternative to Q/.ND / which is frequently used is the velocity (or flow) coefficient
D c x /U where U is blade tip speed and c x the average axial velocity. Since
Q D c x ð flow area / c x D 2
and U / ND.
then
Q c x
/ .
ND 3 U