Yamaguchi H. Engineering Fluid Mechanics (Fluid Mechanics and Its Applications)
Подождите немного. Документ загружается.
142 4 Perfect Flow
(iv) The thrust force exerted by a rotating blade is a major concern for
driving marine vessels (screw), pumping air (fan), flying planes (propeller),
wind-driven power generators (wind turbine) and so on. The theory of the
thrust force by a rotating blade is originally due to W.J.R. Rankine, and
developed by R. E. Froude, and it is known as the actuator-disk theory. The
theory applies to a rotating blade to produce the thrust force by a given
power, bearing in mind that it can be replaced by a stationary disk which
provides a pressure rise across, as depicted in Fig. 4.12(d)(1) and (2). Tak-
ing appropriate control volume for the system, find an expression for the
thrust force
F
and associated power to a propeller and estimate the effi-
ciency. Assume that the propeller is a thin actuator-disk so that both side of
the propeller
2
A and
3
A are the same in area. Assume further that the in-
coming and exiting flows are uniform with velocities
1
u and
4
u respec-
tively, as depicted in Fig. 4.12(d)(2). The control volume is assumed to be
formed between side boundaries, called the slipstream, as shown in Fig.
4.12(d)(1) and (2). The fluid density is
U
.
Ans.
(i) The pressure loss can be determined with the mass continuity, the inte-
grated momentum and Bernoulli equation for the control volume in Fig.
4.12(a). The set of equations are respectively:
2211
AuAu
U
U
(1)
22112
AppuuQ
U
(2)
l
hzpuzpu gg
UUUU
2
2
21
2
1
2
1
2
1
(3)
2211
AuAuQ
the volume flow rate and z are levels of the reference
point
٤
1
and ٤
2
, which are the same. From Eqs. (1)–(3), solving for the
head loss
l
h , we have
2
2
1
2
1
1
2
¸
¹
·
¨
©
§
A
Au
h
l
g
(4)
Further, by definition of the correction factor
[
, Eq. (4) can be written by
g
U
Exercise 143
2
2
1
2
1
1
2
¸
¹
·
¨
©
§
A
Au
h
l
[
g
(5)
Alternately, to the head loss
l
h by defining the pressure loss coefficient
]
¸
¸
¹
·
¨
¨
©
§
gg 22
2
1
2
21
uuu
h
l
]
(6)
In Eqs. (5) and (6) we used a relationship between the area ratio
2
1
AA
and
]
as
2
2
1
1
¸
¸
¹
·
¨
¨
©
§
A
A
[]
(7)
Note that
[
is defined as the correction factor which is normally near unity.
Fig. 4.13 Conical diffuser
(ii) In contracting flow, a flow separation at the entrance of the outlet pipe
occurs, where the flow is accelerated to the point of vena contracta (0), and
decelerated from (0) to
٤
2
in which the flow is fully developed. In the
same manner as question (i), we can arrive at the head loss
٤
1
–٤
2
as fol-
lows
gg 2
1
2
2
2
2
1
2
2
2
u
A
Au
h
l
¸
¹
·
¨
©
§
¸
¸
¹
·
¨
¨
©
§
[]
(8)
[
144 4 Perfect Flow
In general, major head loss occurs in decelerating flow, while in accelerat-
ing flow the head loss is rather small. Thus, taking the control volume be-
tween (0)–
٤
2
, we may be able to write the head loss
s
h as follows
gg 2
1
1
2
2
2
2
2
20
u
C
uu
h
c
s
¸
¹
·
¨
©
§
(9)
c
C is the contraction coefficient defined by
20
AAC
c
.
Equation (8) covers overall head loss between
٤
1
and ٤
2
, in which Eq.
(9) dominates the head loss, thus consequently taking the following rela-
tion
sl
hh | and
2
1
1
¸
¸
¹
·
¨
¨
©
§
|
c
C
]
(10)
So that we have for
٤
1
–٤
2
g2
1
1
2
2
2
u
C
h
c
l
¸
¹
·
¨
©
§
(11)
In calculating Eq. (11), we have the following well-known expression of
c
C (ref. D.N. Roy, 1988)
2
1
12
170701
1
AA
C
c
.
(12)
c
C is the same as the contraction coefficient of the orifice plate. For flows
from a large reservoir through a pipe (i.e.
0
12
|AA ),
]
may be 50.|
]
.
The loss can be minimized by rounding the corner of the entrance pipe,
shaping the Bell-mouth, which yields 050
.|
]
in practical engineering
problems.
With the similar treatment as above, we can extend our problem to the
cases of effuser or diffuser. The head loss in a diffuser is greater than an ef-
fuser in the same contraction ratio
21
AA . In a conical diffuser for a diver-
gence angle
D
as indicated in Fig. 4.13, the correction factor
[
given by
Eq. (12) in question (i) takes the values 1.1 to 1.2 for
D
being between
q60 and q70 , and drops until it reaches approximately 1.0–1.05 at
q 180
D
. In the diffuser, the diffuser efficiency
d
K
of the pressure recov-
ery is defined as
Exercise 145
g2
1
recovery pressure lTheoretica
recovery pressure Actual
2
2
2
1
12
12
uu
h
pp
pp
l
th
d
K
(13)
Additionally, using
[
defined by Eq. (14) in question (i), we have
¸
¸
¹
·
¨
¨
©
§
21
21
1
1
1
AA
AA
d
[K
(14)
(iii) Imagine that the pressure (the static wall pressure) distribution along
the wall is known, either by experimental measurement or numerical result.
We can then plot the pressure data for locations of pressure taps as shown
٤
1
–٤
12
in Fig. 4.12(c)(1) (or for wall points in numerical results). In the
case of a long channel, there would be a strong pressure gradient in up-
stream and downstream regions due to viscous (frictional) losses as indi-
cated in Fig. 4.12(c)(2). As seen in Fig. 4.12(c)(2), at locations
٤
1
–٤
4
and
locations
٤8 –٤
12
, the pressure gradients in the fully developed regions are
almost the same, which is unaffected by the minor loss due to the bend in
locations
٤
4
–٤8 . The minor loss is the pressure drop caused by the flow
configuration in the bend, usually by flow separation and local circulation
at local points
٤
4
–٤8 . The minor loss
loss
p
is the difference between the
two parallel lines of fully developed flow regions, i.e. between lines of lo-
cations
٤
1
–٤
4
and locations٤8 –٤
12
.
On the other hand, in the case of relatively short channel, the viscous
losses at upstream and downstream regions, before and after the bend, are
small, yielding the trend that the pressure gradients of the regions are al-
most null. Namely, the lines of pressure distribution at locations
٤
1
–٤
4
and
locations
٤7 –٤
12
are almost horizontal, keeping their values almost constant
in the two regions. The pressure loss
loss
p due to the bend in a short
channel (as typically seen in such channels as shown in Fig. 4.12 (c)(3)) is
representatively displayed in Fig. 4.12(c)(4). As demonstrated, the pressure
loss
loss
p can be simply determined as a pressure difference between any
two arbitrary points between the upstream and downstream (fully devel-
oped) flow regions. Figure 4.12(c)(3) is a typical flow configuration in a
$
90 bend in a short channel.
The pressure loss coefficient
]
at this bend is often defined as
'
'
'
146 4 Perfect Flow
2
2
1
u
p
loss
U
]
'
(15)
where
u
is the average flow velocity.
]
is usually a function of the Rey-
nolds number and geometric parameters.
(iv) The thrust force
F
given by the pressure is the net force acting on
the control volume in Fig. 4.12(d)(1). Thus, the integrated momentum
equation applied to the large control volume gives
14
uuQF
U
(16)
Q is the volumetric flow rate propelled by the propeller when a control
volume as indicated in Fig. 4.12(d)(2) is drown in the vicinity of the pro-
peller such that
32
uu . Thus, the momentum equation would give
F
by
the following form
23
ppAF
(17)
When we assume no energy loss or gain due to viscous effects between
points
٤
1
and ٤
4
, and there is no area change between ٤
2
and ٤
3
, we can
write the conditions as follows
32
AAA
(18)
32
uuu
(19)
The Bernoulli equations of the points between
٤
1
and ٤
2
and ٤
3
–٤
4
are
written
2
22
2
11
2
1
2
1
upup
UU
(20)
2
44
2
33
2
1
2
1
upup
UU
(21)
The potential head is the same at all points. Using the fact that the pressure
is the same all around the control volume, we can recognize that
atm41
ppp
(22)
atm
p
is the surrounding pressure. From Eqs. (20), (21) and (22), we have
Exercise 147
32
2
4
2
1
2
1
ppuu
U
(23)
With Eqs. (16), (17) and (23), we have the relation
4
1
2
4
2
1
2
1
uuAvuuA
UU
(24)
Thus, consequently we can obtain v as
41
2
1
uuv
(25)
v is the flow velocity through the propeller. Equation (25) shows that the
flow velocity through the propeller is the average of the upstream and
downstream velocity.
The power
fluid
W required to propel the fluid between points ٤
1
and ٤
4
,
i.e. to produce the fluid motion, not considering losses, will be
2
1
2
4
2
1
uuQW
fluid
U
(26)
A moving propeller attached to a vessel (ship or aircraft moving with ve-
locity v ) requires the power
prop
W given by
1
FuW
prop
141
uuQu
U
(27)
The theoretical propeller efficiency
p
K
is then given by the following ex-
pression, from Eqs. (26) and (27)
v
u
uu
u
W
W
fluid
prop
p
1
14
1
2
K
(28)
The maximum efficiency of a propeller can be obtained by designating the
condition that
prop
W becomes maximum. So that, eliminating
4
u using Eq.
(25) from Eq. (27) and differentiating Eq. (27) with respect to v , while
keeping
1
u constant, we have
148 4 Perfect Flow
1
2
1
uv
(29)
Therefore, the maximum efficiency of a propeller is ideally obtained by
%50
4
1
2
1
2
1
2
1
Qu
Qu
W
W
fluid
prop
P
U
U
K
max
(30)
Fig. 4.14 Wind turbine
Similarly the maximum efficiency, that a wind turbine as depicted in
Fig. 4.14 can attain, can be found, using the same control volume as
treated in Fig. 4.12(d), while reversing the wind direction from
٤4 to ٤1 . In
the same manner as the propeller,
fluid
W is calculated by differentiating
with respect to v , while keeping
4
u constant, so that we have
4
3
4
uv
(31)
Exercise 149
The maximum efficiency of a wind turbine is ideally estimated by
%359
2
1
27
8
2
4
2
4
.
max
Qu
Qu
W
W
turb
fluid
T
U
U
K
(32)
In actual cases for both propeller and wind turbine, the efficiency is
less than the ideal situation as obtained in Eqs. (30) and (32), due to vis-
cous loss and the complex flow configuration through the rotor. There
would be a strong swirl component of velocity. However, the expressions
of Eqs. (30) and (32) would give the maximum attainable efficiency, which
may provide a target design.
Exercise 4.1.5 The Rayleigh-Plesset Equation
There is a spherical bubble in a perfect, incompressible liquid of infinite
extent. The bubble growth is due to a pressure variation at a distance from
the bubble. Referring to Fig. 4.15, the radius of the bubble at any time t >
0 is
tRR , and
r
is the radius to any point in the liquid, where the ori-
gin o of coordinates is at the bubble center, which is at rest in the inertial
reference frame. Derive an equation of motion for a spherical bubble in a
liquid for given external pressure
tp , which varies with time, with a con-
dition that the pressure at the bubble surface is
Rp .
Fig. 4.15 Growing bubble
150 4 Perfect Flow
Ans.
Let us assume that the flow is irrotational, so that the velocity field is
written in terms of the velocity potential
I
given by Eq. (4.1.2). With
spherical symmetry the surface velocity of the growing bubble is the radial
velocity u of the fluid flow, and it is a function of
r
, so that the velocity
potential )(r
I
I
satisfies the Laplace equation
0
2
r
I
(1)
For the spherical coordinates system with spherical symmetry, we have
0
1
2
2
¸
¹
·
¨
©
§
dr
d
r
dr
d
r
I
(2)
The integration of Eq. (2) will give the general solution as
c
r
m
r
I
(3)
where m and c are constants. With Eq. (3), the radical velocity u is thus
given as follows
2
r
m
dr
d
u
I
I
r
(4)
With the boundary condition at the bubble surface, we may write
R
dt
dR
u
at
R
r
(5)
Thus, that we can write Eq. (4) as
2
R
m
R
and RRm
2
(6)
Using m of Eq. (6) into Eq. (3) and setting c in Eq. (3) to zero, we can
obtain an expression of
I
as
r
RR
r
2
I
(7)
Equation (7) satisfies the problem condition that is
0 u for
f
r
(8)
and
Exercise 151
f u for
0
r
(9)
Knowing the velocity field as the velocity potential, we can apply the
pressure equation if Eq. (4.1.35), ignoring the gravitational potential, to
write
UU
I
I
tprp
dt
d
2
2
1
(10)
rp is the static pressure at
r
. With Eqs. (4) and (6) we can write the
time variation of
I
as
4
24
2
r
RR
I
(11)
and
RRRR
r
t
22
2
1
w
w
I
(12)
Eqs. (11) and (12) are to be applied to the bubble surface, i.e.
R
r
, and
with that the following expressions are obtained by
2
2
R
I
(13)
further
RRR
dt
d
2
2
I
(14)
Substituting Eqs. (13) and (14) into Eq. (10), at the bubble surface, i.e.
R
r
, we can finally obtain the equation of motion for the bubble radius
to write
RRR
tpRp
2
2
3
U
(15)
which can alternatively be expressed
23
2
2
1
RR
dt
d
RR
tpRp
U
(16)
Equations (15) and (16) are referred to as the Rayleigh-Plesset equation,
which can be used to estimate the growth and collapse of a vapor bubble
for known pressure change
tp
. In the Rayleigh-Plesset equation,
Rp
is
often assumed from the surface tension effect on the bubble surface as
follows