Duan C.G., Karelin V.Y. Abrasive Erosion and Corrosion of Hydraulic machinery
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204
Abrasive Erosion and Corrosion of Hydraulic Machinery
where w is the weighted settling velocity.
The efficiency ration E
R
was suggested by K. Holzenberger in equation
(4.21)
E
R
=l-265.3(>;
246
-(l-265.3(>; )-3.7«
(4.21)
where n
q
is the specific speed
n„ —n-
Q
1/2
H
3/4
where Q is capacity in m
3
/s, H is head in m and n rotational speed in rpm. The
simplified relations were given by Frazier and Kervrgessbe as follows:
E
R
=H
R
=(l-C
w
) or
1-C,
\-c
v
+s
s
c
v
(4.22)
The hydraulic efficiency of slurry pumps can be estimated in the
following statistic formula:
1 03
T
lh=
l
—^- =
1-0.0108231
n"
0.35
D
1.05
(4.23)
where D
2
is the outlet diameter of impeller. For the volumetric efficiency:
5.79
7v=l-
n
1.4
(4.24)
The transformation of efficiency from model (subscript: m) to prototype
pump (p) can be performed by using the following efficiency difference:
Ar
]
=
K(l-r
1m
)
1-
K
em
\
R
>PJ
(4.25)
Design of Hydraulic Machinery Working in Sand Laden Water
205
where K is reference value, which may take the value of 0.4 to 0.6. And
Reynolds number is:
Re
D,
JlgHr
where H
r
is the head of pumps at the design condition.
The task of hydraulic design of slurry pumps is at first to selection of the
geometry of their impellers, based on using the statistical formulas [4.18] as
follows, or based on concerning the influence of existence of particles to the
pumps [4.19,4.20].
Diameters of Impeller
When designing a centrifugal slurry pump, the slurry head H
sr
and capacity Q
at the design condition must be given. At first, the suction diameter of the
pump, d
0
should be chosen, then selection of the rotating speed, n, is need
according to Table 4.8.
Table 4.8 Selection of rotating speed of slurry pump
Suction diameter
d
0
(mm)
<125 125-200
250-300
300-500 500-800 >800
Rotating speed
n (rpm)
1470 1470-980 980-740 740-590 590-375 <375
The inlet diameter of the impeller D
0
can be chosen according to the
following statistic formula:
D,=K
m
l\ or D
n
Q
3.6 + 0.280
(/») (4.26)
where K
D0
= 3.5-4.5.
The inlet diameter of impeller blade D\ can be take in the same value as
D
0
or can be calculated as
Di = (0.187 In n
q
-0.24) D
2
(4.27)
206
Abrasive Erosion and Corrosion of Hydraulic Machinery
At the beginning of the design the outlet diameter of impeller can be
estimated according to equation (4.28)
D,J-™LJH
n
(4.28 a)
where
if «^ = 48 ~ 54.8, Z = 3and/?
2
= 20°-35°, then K
2
= 4.12 ~ 4.30
if nq
=
43.8 -71.2, Z= 5 and/?
2
= 20° -35°, then K
2
= 3.90 ~ 4.96
D
2
=4.18
r n^
-0.585
v
100
y
— for 2 = 5
D
2
= 5.00
/ -v -0.474
v
100
y
for 2=4 (4.28 b)
D
2
= 4.86
r
n
\
-0.525
v
100
y
for 2 = 3
The final determination of the diameter D
2
must be calculated based on
the head required and other geometrical parameters to be chosen. The ration
of
D
2
to D\ should be chosen in the range of
2.25
to 2.8, That is,
D
2
/D
x
= 2.25 -2.8
If it is larger than 2.8, the hydraulic efficiency of the pump may be higher,
but the pump is too big to be repaired.
Vane Number (Z)
Three to five vanes should be chosen for slurry centrifugal pump impellers in
order to transport solids. The vane number, Z, does not effect the
performances of the pumps definitely in the above range, and it may be
calculated according to the following formula:
Design of Hydraulic Machinery Working in Sand Laden Water
(\-y/
2
)7rD
2
207
Z<INT
'u2
S
u2m
,
m
=
16
-
43
+
1022
"
O
uZmim q
S
u2m
^=5.12n
q
-20.2
(4.29)
where y/
2
is the vane coefficient at the impeller outlet, which is defined as:
¥
= \
zs
nD •
sin p
A
where Sis the thickness of the vane and y/
2
can be estimated as follows:
^
2
= 0.954-0.774
n„
100
Vane
Angles (P
AX
and f3
A2
)
The inlet vane angle of impeller, ft
M
can be estimated from the inlet flow
angle, /?, adding a angle of attack, a , that is
PM
=A + «
where
tgfti
=
'
ml
U,
6.080.
D^b,\i/
X
rj
v
a = 5°-15°
y/
x
=1-0.04Z
or
(4.30)
y
x
= 0.479«°'
208 Abrasive Erosion and Corrosion of Hydraulic Machinery
where V
mi
is the meridional component of absolute velocity at the inlet and bj
is the width of impeller at the inlet, which may be estimated by the following:
b, = (0.06 ~ 0.20)£>
2
.
The outlet vane angle P
A2
can be taken in the value of the range of
\5°<P
A2
<50°.
For the transported solid particles with large diameters, fi
A2
is 30° to 35°,
for small particles, about 10°. And the angle
J3
A2
can also be estimated by
using the following equation:
/^^OZ"
0772
(4.31)
Outlet Width (b
2
)
The outlet width of slurry pumps is much larger than that of conventional
pumps for transporting solids. It can be estimated in the following equation:
b
2
=K
b2
D
2
(4.32 a)
where K
b2
= 0.12 ~ 0.18. If the specific speed is small or the particle
concentration is dilute, it takes small value. It can also be calculated by using
another equation:
b
2
= (o.084681n«
9
-0.12096)Z)
2
(4.32 b)
The inlet width of impeller can be estimated by the equation (4.33)
b,=(0.06~0.20)D
2
(4.33)
4.3.3 Vane Pattern
The pattern of vanes of slurry centrifugal pumps, which possess the low
specific speed, is basically two-dimensional. The logarithmoid is used to form
Design of Hydraulic Machinery Working in Sand Laden Water
209
the vane pattern, because it was verified by Hebich's experiment that the
impeller with the logarithmoid has higher efficiency than that with the circle-
shaped vane. In the collar coordinate system (r, 0), the logarithmoid is
described by equation (4.34)
r = r,e
kd
(4.34)
where r\ is the radius at the leading edge of the logarithmoid, i.e., at the inlet
of impeller vane, k is the coefficient and
k= ctga = tg/3
(4.35)
where a is the angle of
helix,
and /?the vane angle. Equation (4.34) will be a
equal-angle logarithmoid if k is a constant. The differential equation of the
vane pattern is as follows
d$_
dr
1
r-tgfi
(4.36)
where variation of vane angle /? from the impeller inlet to its outlet has three
types of function, that is, the linear function, the convex and the concave as
shown in Figure 4.29.
P
P
P
AX)
-9
Figure 4.29 Three types of
/?
distribution curves
210
Abrasive Erosion and Corrosion of Hydraulic Machinery
The Pattern of Vane by Using Equation P = f\6, sin 0)
The variation of angle
J3 is
assumed to follow equation (4.37)
p = fi
Ai
+ L*l P^O + A
■
sin
n
J
e
(4.37)
where
J3A\
and
PAI
are the inlet and outlet angles of
vane,
(j>
is the central angle
of the vane, and A is the coefficient of the oscillation of sin function. The
following two cases are used to form the vane pattern.
* The case of A - 0
If the coefficient^ is zero, introducing equation (4.37) into the differential
equation (4.36), then one integrates the equation to obtain the vane pattern
equation (4.38 a):
r
=
r,
cos/?
A\
COS
A..+
PA2 PA\ Q
(4.38 a)
where the central angle of
vane
is
ln
fc
/r
') (R -R )
1 ( O I O \W
A2
H
A\
)
In (cos fi
M
lcosP
A2
)
and the curvature radius R of the vane pattern is calculated as follows:
R
=
r-
Design of Hydraulic Machinery Working in Sand Laden Water
211
* The case of A * 0
When the coefficient A is a constant, then the similar equation can be
obtained by integrating equation (4.36)
2 3
n
a+
bO + cO
+ d9
+ e sin
— 9
+
/
r
=
r -e
K
1
- cos — (
where
In n
+
g
sin
— 9
+ h
■ 9
■
cos
— 6
(4.38 b)
a =
tg0M
A
2
sinytf
c =
2 cos
3
1
cos
2
p
M
sin £,,
cos
3
P
A
sin £,,
P*
(PA2-PA
2<j>
)
(£.2-fl«)i
*
2^(/?,
2
•
3
-A
)
/ =
g
=
h =
cos
3
/?^,
;r"
A<f>
7TCOS
2
^,
cos
3
/?^,
4;r
sin/7,,
2A(fi
A2
-fi
M
)
cos
3
/?.,
;r
and
212 Abrasive Erosion and Corrosion of Hydraulic Machinery
R = r,\\
+
2 .
e
„-K a , /
a
+
b
+
c6
+
dO + — sin—6 +
e
<t> e
1-cos—
6
—sin—0
+
hcos—d
6 <j> (j)
The Pattern of Vane by Using Equation p
=
f(&
b
)
The variation of
angle j5 is
assumed to follow equation (4.39)
p = arctg(aO
h
+ c)
(4.39)
where
<P
then the function of the vane pattern is as follows:
tgpA2-tg0A\
l+b
y _ j, Q L
+tgP
AX
(4.40)
where
In
tgP
A2
-tgP
A
,
-tgp
M
-l
\
r
u
+ =
lnl
r
*'
tgP
M
-tgf>«
+tg
p
M
l+b
4.3.4 Hydraulic Design of Centrifugal Slurry Pumps
Theoretical Head of Slurry Developed by Centrifugal Pumps
Design of Hydraulic Machinery Working
in
Sand Laden Water
213
Based
on the
governing equation
of
particle-liquid two-phase flow,
the
theoretical head
of
slurry,
H
st
h,
developed
by a
centrifugal pump impeller
can
be deduced
as
follows:
H
sth
= ^{U
2
V
u2p
-U
l
V
u]p
)+
l
-^^-{u
2
V
u2f
-liy
uU
) (4.41)
where subscripts
/ and p
indicate
the
fluid phase
and the
particle phase
respectively,
C
w
is the
weight fraction
of
solid particles,
U\ and U
2
are the
peripheral velocities
of
impeller
at
the inlet
and
the outlet respectively,
V
u
, and
V
u2
are the
circumferential components
of
absolute velocity
at the
inlet
and
outlet respectively,
g is the
gravity.
If the
inlet flow
is
meridional,
the
circumferential component of absolute velocity
is
zero, then
H
sth
=^ [C
w
V
u2p
+(\-CJV
u2f
] (4.42)
At
the
pure water operation condition,
the
theoretical head
of
the pump,
H
wt
h
can
be
got
from the equation (4.42)
as
follows:
wth
r
u2p
H
wth
=~^V
u2v
(4.43)
According
to
equation (4.16),
the
head ratio
of
the
slurry theoretical head
to one of water
is
H
Rth
=7^-= 1-C„(1
-£**-) (4.44)
From
the
experimental results
V
u2p
is
less than
V
u2
f,
so
that E
Rth
is
less
than one.
The Active Slurry Head Developed by Centrifugal Pumps