Yamaguchi H. Engineering Fluid Mechanics (Fluid Mechanics and Its Applications)
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192 4 Perfect Flow
The theoretical pressure head rise
th
H in the case of a pump (or pres-
sure head drop in case of turbine) across the cascade low can then be cal-
culated based on the forces acting upon a representative airfoil in the iso-
lated cascade as illustrated in a lower diagram in Fig. 4.38(c). Setting
LF
A
(lift force) and DF
w
(drag force) on the airfoil for the average at-
tack angle
D
with reference to Fig. 4.38(c), we can write, following the
definitions of Eqs. (25) and (28) in Exercise 4.1.7
cLA
lwCF
2
2
1
U
(4.2.22)
cDw
lwCF
2
2
1
U
(4.2.23)
where
L
C and
D
C are the lift and drag coefficient per unit length of the
airfoil respectively and
c
l
is the cord length of the airfoil. Denoting
F
as
the resultant force by
w
F and
A
F , and
F
is decomposed in the peripheri-
cal direction and the axial direction written as
u
F and
w
F respectively, as
indicated in Fig. 4.38(c) with a representative angle
J
. So that, we have
JE
JE
S
r
¿
¾
½
¯
®
r
sin
2
cos
F
FF
u
(4.2.24)
and furthermore
EJE
EE
cossin
cossin
r
A
wAu
F
FFF
(4.2.25)
where
J
is a turbine,
J
is a pump (or fan) and together are given as
JJ
ลtan
L
D
A
w
C
C
F
F
(4.2.26)
It is now desired to derive an expression for the theoretical pressure
head
th
H
to characterize the turbomachine. This can be achieved directly
by considering the work transfer in energy balance. Taking an annulus
element (vane element) between
r and drr , as indicated in Fig. 4.39,
the work transfer
p
Wd between the fluid and the annulus element can be
written, from Eq. (4.2.25), as
4.2 General Theories of Turbomachinery 193
Fig. 4.39 Vane element
druFs
udrFsWds
Ac
ucpc
EOE
cossin r
(4.2.27)
where
c
s is the number of vanes. The flow rate through the element dQ is
written as
mcc
drctsdQ
(4.2.28)
where
c
t is the pitch of the vanes in the cascade. For
pc
Wds in Eq.
(4.2.27), we have the following relationship
thpc
HdQWds g
U
(4.2.29)
so that Eq. (4.2.27) is rewritten as
druFsdrctHs
A
cmcthc
EJEU
cossin r g
(4.2.30)
For a two dimensional assumption, Eq. (4.2.30) is integrated with respect
to
r
for a hub-tip length, where we have
mc
A
mc
th
ct
uF
ct
H
g
g
U
EJE
U
JE
cossin r
r
(4.2.31)
Equation (4.2.31) can be further expanded to give a design parameter for
the two dimensional cascade, by applying Eq. (4.2.22) with the expression
of
J
cos
A
FF
as follows
Fusin
194 4 Perfect Flow
J
JE
cos
sin
2
2
r
g
w
c
u
t
l
CH
mc
c
Lth
(4.2.32)
and
JE
J
r
sin
cos2
2
uw
cH
t
l
C
mth
c
c
L
g
(4.2.33)
With reference to the velocity diagram in Fig. 4.38(c),
w and
E
will be
given in the following expression
2
21
2
2
¸
¹
·
¨
©
§
uu
m
cc
ucw
(4.2.34)
and
¿
¾
½
¯
®
2
tan
21
1
uu
m
ccu
c
E
(4.2.35)
It is noted here that with the lower hub-tip radius ratios, the vanes of a
turbomachine will have an appreciable amount of twist along their length
so that, in many cases, a three dimensional approach in axial turbomachi-
degree of accuracy, a two dimensional approach will give a close approxi-
mation for axial flow machines with a high hub-tip ratio, such as given in
Eqs. (4.2.32) and (4.2.33).
4.2.3 Efficiency and Similarity Rules of Turbomachinery
Definitions of efficiency are always confusing and they, in fact, differ, case
by case. Nevertheless, in this section the most commonly used and ac-
cepted definitions in the literature of turbomachines are introduced. The to-
tal (or overall) efficiency
K
is defined in such a way that
fluidofenergymaximumpossible
shaftoutputofcouplingatavailableenergymechanical
K
(4.2.36)
The total efficiency
K
is further decomposed by three independent effi-
ciencies, such as the hydraulic efficiency
h
K
(or adiabatic efficiency
t
K
for
nes, the so-called vortex design, is required. However, with a reasonable
4.2 General Theories of Turbomachinery 195
gas turbine), the volumetric efficiency
v
K
and the mechanical efficiency
m
K
, which are respectively defined as follows
fluidofenergymaximumpossible
rotorthetosuppliedenergymechanical
h
K
(4.2.37)
H
H
H
hH
thl
h
K
(hydraulic turbine)
(4.2.38)
or
thth
lth
h
H
H
H
hH
K
(pump or fan)
(4.2.39)
In both cases
l
th
hHH r ( turbinehydraulic ; + pump or fan)
(4.2.40)
where
H
is the actual head,
l
h the loss head between the inlet and outlet
of turbomachines, and
th
H is the theoretical pressure head.
Q
QQ
l
v
K
(hydraulic turbine)
(4.2.41)
or
l
v
QQ
Q
K
(pump or fan)
(4.2.42)
where Q is the volumetric flow rate and
l
Q is the volumetric rate of leak-
age from turbomachines.
powerrotor
powershaft
m
K
(4.2.43)
Altogether, with the definitions of Eqs. (4.2.37–4.2.43), we have the total
efficiency
K
to write
mvh
K
K
K
K
(4.2.44)
In the design and development of turbomachinery, the widest compre-
hension of the general behavior is obtained from similitude and dimen-
sional analysis. In engineering practice, for design purposes, we have to
depend on test results obtained from experiment with “models”, which are
196 4 Perfect Flow
often smaller in size than the “prototype”. The “prototype” is the full-size
device at a preliminary stage of a commercial product. Dimensional analy-
sis based on this similitude can then be applied to predict a prototype’s per-
formance from tests conducted on a scale model, and also to determine the
most suitable type of machine for its maximum efficiency, for a specific
range of head, speed and flow rate. In a similar manner the dimensional
analysis enables data taken from a test machine to reduce into a smaller
number of dimensionless groups and given experimental relations between
variables to be found with the greatest economy of effect.
We will apply the idea of similitude and dimensional analysis devel-
oped in Chapter 6 and in Appendix C to turbomachines in this section.
First, we will define our system of a turbomachine, introducing a control
volume as shown in Fig. 4.40. The significant parameters for a turbo-
machine are; discharge Q , power (work transfer) W , revolutional (rota-
tional) speed
n , representative diameter of rotor
D
, head Hg ( g is con-
ventionally inclusive or pressure difference
P
), efficiency
K
, density of
fluid
U
, viscosity
0
K
and some geometric representative scales
1
l and
2
l .
The system then can be described, in an arbitrary function, as
0,
21
0
¸
¹
·
¨
©
§
WH
D
l
D
l
DnQf
,,,,,,,,
KKU
g
(4.2.45)
With an application of dimensional analysis, say Buckingham
S
-
theorem
Fig. 4.40 Control volume of turbomachine
with reference to Appendix C-1, we are able to reduce a group of non-
dimensional parameters by a functional relationship, using three of the in-
dependent variables
DN , ,
U
as common factors (control variables) when
considering the three primary (basic) dimensions, i.e. mass, length and
time, as follows
'
4.2 General Theories of Turbomachinery 197
¸
¸
¹
·
¨
¨
©
§
D
l
D
lnD
nD
Q
f
Dn
H
21
0
2
3
1
2
,
,
,
K
U
M
g
(4.2.46)
¸
¸
¹
·
¨
¨
©
§
D
l
D
lnD
nD
Q
f
21
0
2
3
2
,
,
,
K
U
K
(4.2.47)
¸
¸
¹
·
¨
¨
©
§
D
l
D
lnD
nD
Q
f
Dn
W
C
W
21
0
2
3
3
53
,
,
,
K
U
U
(4.2.48)
In Eqs. (4.2.46–4.2.48), we have the nondimensional parameters
2
nD
H
g
M
(or alternatively,
22
D
P
ZU
M
with angular velocity
Z
)
{ Head coefficient (or pressure coefficient)
(4.2.49)
53
Dn
W
C
W
U
(or alternatively,
53
D
W
C
W
ZU
)
{ Power coefficient
(4.2.50)
3
nD
Q
C
Q
(or alternatively,
3
D
Q
C
Q
Z
)
{ Volumetric flow rate coefficient (4.2.51)
0
2
K
U
nD
Re
(or alternatively,
0
2
K
ZU
D
Re
)
{ Reynolds number
(4.2.52)
The performance parameters in Eqs. (4.2.46–4.2.48) are thus correlated
with a group of nondimensional parameters defined in Eqs. (4.2.49–4.2.52).
The volumetric flow rate coefficient
Q
C in Eq. (4.2.51) is alternatively de-
fined by using the peripherical speed u and the representative fluid speed
x
c
at a datum point (such as the mean axial speed in a rotor)
nDu v
(4.2.53)
2
DcQ
x
v
(4.2.54)
'
198 4 Perfect Flow
Fig. 4.41 Dimensionless radial flow pump performance curve (Courtesy
of Teral Kyokuto Inc.)
so that
u
c
nD
Q
C
x
Q
3
(4.2.55)
For engineering purposes, the kinematic viscosity
U
K
/
0
v is very
small if compared to the inertia
2
Dn
U
that results in a Reynolds number
high. Correspondingly, the flow regime in a turbo machine is usually very
turbulent (
4
10!!Re ), and experiments confirm that the performance
characteristics of a turbomachine is almost independent from the effects of
a Reynolds number and can be ignored in a first approximation. It is also a
usual practice not to make a correlation between the roughness of the flow
channel and the pump losses, assuming that the hydraulic efficiency
h
K
(or
adiabatic efficiency
t
K
) is constant. With geometric similarity, the model
and the prototype are identical in shape, but differ in size. The model ratios
Dl
/
1
and Dl /
2
are kept constant and may be eliminated forthwith. Thus,
considering the facts above mentioned, the functional relationship for
geometrically similar hydraulic turbomachines are then expressed as
Q
Cf
1
M
(4.2.56)
Q
Cf
2
K
(4.2.57)
4.2 General Theories of Turbomachinery 199
Fig. 4.42 Dimensionless axial flow fan performance curve (Courtesy of
Teral Kyokuto Inc
.)
QW
CfC
3
(4.2.58)
It is noted that the actual form of the functions
31
ff ~ in Eqs. (4.2.56–
4.2.58) must be determined from results of experimental measurements (or
computer simulations). In Figs. 4.41 and 4.42, typical curves of
31
ff ~
are displayed for a radial flow pump and an axial flow fan respectively. It
should be mentioned that curves for
31
ff ~ may vary for types of turbo-
machines (Refs. Schetz and Fuhs, 1996 and Potter and Wiggent, 1997).
Designers, who wish to obtain a best match with changing flow condi-
tions, can consider off-design operation by considering an additional vari-
able
E
into Eqs. (4.2.56–4.2.58). For example, in an axial flow pump (or
turbine), considering
E
as the average vane angle
E
in Fig. 4.38, which
can be set at various values, and we can write as
E
M
,
Q
Cf
1
(4.2.59)
E
K
,
Q
Cf
2
(4.2.60)
Alternatively, we can write
),(,
K
M
E
QQ
CfCf
54
(4.2.61)
200 4 Perfect Flow
Therefore, from Eq. (4.2.61), the performance parameter, say
K
, can be
written as
M
K
,
Q
Cf
6
(4.2.62)
where
K
is a function of both
Q
C and
M
, as illustrated in Fig. 4.43 (de-
rived from Fig. 4.42 for varying
M
).
Fig. 4.43 Off-design operation (variable
M
)
From the relationship in Eqs. (4.2.53–4.2.55), similar relationships be-
tween any two machines from the same geometric family can be straight-
forwardly derived. Any two machines, identifying
M
and
P
as suffix, are
PM
Dn
H
Dn
H
»
¼
º
«
¬
ª
»
¼
º
«
¬
ª
2222
const.;
gg
M
(4.2.63)
PM
Q
nD
Q
nD
Q
C
»
¼
º
«
¬
ª
»
¼
º
«
¬
ª
33
;const.
(4.2.64)
PM
W
Dn
W
Dn
W
C
»
¼
º
«
¬
ª
»
¼
º
«
¬
ª
5353
;const
UU
.
(4.2.65a)
4.2 General Theories of Turbomachinery 201
PM
HD
W
HD
W
»
»
»
¼
º
«
«
«
¬
ª
»
»
»
¼
º
«
«
«
¬
ª
2
3
2
2
3
2
or
UU
(4.2.65b)
Furthermore, for any two identical machines with a different operating
condition, for example with a variety of revolutional speeds, can set
PM
DD to Eqs. (4.2.63–4.2.65), identifying the operating conditions by,
where we have
3
3
2
2
n
n
W
W
n
n
Q
Q
n
n
H
H
c
cc
cc
c
,,
(4.2.66)
Eliminating
H
and n
c
from above relationship, we can write
2
3
2
3
22
H
W
H
W
H
Q
H
Q
c
c
c
c
,
(4.2.67)
There is often the case that the turbomachinery designer faces the basic
problem of deciding what type of turbomachine will be the best choice.
The problem can be solved by correlating a turbomachine of a given fam-
ily to a nondimensional number that characterizes its operation of optimum
conditions. The nondimensional number is termed the specific speed
s
n ,
which is calculated by a preliminary design data and is usually provided to
the designer at initial stage of development, with such as
H
, Q and n for
a pump, or
W
,
H
and
n
for a turbine. For any hydraulic turbomachine
with fixed geometry, i.e. keeping
D
as constant, there is a unique relation-
ship between the efficiency and the volumetric flow rate, represented by
dimensionless number
M
and
Q
C , as seen in Fig. 4.41 and in Fig. 4.42. If
the optimum efficiency
max
K
(with reference to Fig. 4.43) is given by a
unique value of
0Q
C
and
0W
C
, we can write the similarity relationships
in Eqs. (4.2.63–4.2.65) for
max
K
,
0Q
C and
0W
C . Thus, eliminating
D
from the relationships, combining Eqs. (4.2.63) and (4.2.64) for pump, and
Eqs. (4.2.63) and (4.2.65) for turbine, we have
4
3
max
2
1
0
4
3
2
1
4
3
2
1
K
Q
PM
s
C
H
Q
n
H
Q
nn
»
»
»
¼
º
«
«
«
¬
ª
»
»
»
¼
º
«
«
«
¬
ª
gg
(4.2.68)