Duan C.G., Karelin V.Y. Abrasive Erosion and Corrosion of Hydraulic machinery

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224

Abrasive Erosion and Corrosion of Hydraulic Machinery

conditions. That means, the velocity distribution of the clear water flow is

regarded as the reference quantity and the solid particles are as boundary

conditions, then the two-phase flow field can be calculated according to the

Ref

[4.20].

Because of different inertial forces between solid and liquid through a

pump, solid phase and liquid phase are moving with different velocities from

its inlet to the outlet. While the velocity of solid particles is slower than that of

water flow, particles effects are like a function of 'relative reduction' to the

pump passage and a function of 'relative clog' to the water flow. Then in the

liquid-solid flow there exists a velocity field distortion of the water flow

compared with its original velocity field without particles. This type of the

distortion makes the water velocity to be going up. Otherwise, if the velocity

of particles is higher than that of water flow, there is a 'relative draw' effect

of the particle to the water flow and the 'particle's existence is like a function

of 'relative expansion' to the pump passage and makes the water velocity to

be decreased.

The Suction Chamber of Pump

The distortion velocity of two phase flow in the suction chamber of a pump

can be derived as follows. At first, the original energy head of clear water

flow is H

, that is,

H=z

+^-

+ -S- (4.60)

YL 2g

where the z is the height, p is the pressure and V

is absolute velocity along the

streamline x, y is the gravity rate. The subscripts c are refereed to the clear

water, and L and s to liquid and solid at the two phase condition respectively.

Due to the influence of solid particles, the energy head of distortion of

velocity and pressure fields H\ is in the following form:

v V

{

-^ (4.61)

YL 2g

Design of Hydraulic Machinery Working in Sand Laden Water

225

If neglecting the difference between two potential, then p

. And the

input energy of the pump for both cases are the same, that is, H,

. So far,

the distortion values of both velocity and pressure are as follows:

PC-PL _Vu- V

a (4 62)

Then the distortion equation of the liquid flow velocity between the two-

phase flow and the pure water flow can be obtained

[4.20]:

Vl-Vl=K

(4.63)

where K

is the velocity distortion coefficient

\Agd{y

-Y

)i^me+c

cos0)

'"{

And

the drag coefficient along the flow direction,

the drag

coefficient

the direction perpendicular

to the

direction,

6 is the

angle

between the flow direction and the horizontal direction,

is the diameter of

particles.

According

to the

spherical particle motion equation, that

is, the

Bssinesque-Basset-Oseen equation, the following equation,

called

the

transmission equation

two-phase flow,

can be got at the

steady flow

condition:

V^-V

(4.64)

which shows that

if V^

> K

, the particle can be transmitted by the liquid

flow. If the velocity field at the clear water flow condition, V

, is known and

the parameters of

particle,

such as, the diameter, the gravity rate and the drag

coefficient, are given, the distortion velocity field,

VLX,

the

water phase and

the particle-phase velocity, V

at the liquid-solid flow, can be got by solving

Equation. (4.63) and (4.64).

226

Abrasive Erosion and Corrosion of Hydraulic Machinery

In engineering practice, particles should be considered as the spheroids

with the equivalent diameter d

that is,

(4.65)

where N

is the discharge percentage of the particle, B

the transformation

coefficient of particle and D

the equivalent diameter of suction chamber of

pump.

When the equivalent diameter is used in the calculation, the gravity rate of

particles must be replaced by wet gravity rate,

If the liquid flow velocity

field is distorted at the two-phase flow condition, the 'relative block' diameter

of particles d\ will be given in the following equation:

(v X

f-[t)

\ \

SxJ

Figure 4.35 shows the velocity triangle at the blade leading edge at the

two-phase flow condition.

Figure 4.35 Calculation of velocity field at blade inlet

In Figure 4.35, V

is the clear water flow velocity at the inlet. And the

blade inlet jostle coefficient s[ = 1.1 ~ 1.3. At the two-phase flow condition,

the distortion velocity of liquid flow V

is larger than V

. The velocity of

Design of Hydraulic Machinery Working in Sand Laden Water 227

particles

zero because of their staying

the stationary point on the blade

surface. The stationary particles make the flow passage

'absolute block',

then the blade jostle coefficient

and the blade thickness

will be getting

large. The attack angle

the inlet,

, is

going up

the value

/?,

The

inlet circulation is not zero any more and get

positive value, which results

the decrease of the head of pump.

Pump Impeller

The same consideration as in the last section can be used in

pump impeller.

But the two-phase flow velocity distributions are complex through the pump

impeller. Because the absolute velocity V is composed

relative velocity

and

the

rotational velocity

U =

R (where co

is the

rotational angular

velocity of the impeller,

radius). Water and particles in the impeller are

acted by the inertial, centrifugal and Coriolis forces in the coordinate system

rotating with the impeller. As the results of the actions, particles move faster

than the water flow gradually through the impeller, especially near its outlet.

That means

the outlet the 'relative drawing'

the particles

the water

flow appears. The difference between the of the two phases are as follows:

=co

R(m

-m

)

(4.66)

Then the relative drawing force

particles

meridional plane

expressed as:

C^£

L.V

mL)

(4.67)

where

the

subscripts

refers

to the

meridional plane,

is the

particle

diameter and

Cr is the

drag coefficient

this plane. Then

the

following

equation is derived according to the mechanics basic equations:

mL ~

r (4.68)

where

228

Abrasive Erosion and Corrosion of Hydraulic Machinery

K =

\4a>

-y

)

r/L

The diameter d

will be replaced by the mean diameter d

when

considering infinite number of particles. The similar distortion equation of

liquid flow on the meridional plane of the impeller is obtained as follows:

-V

(4.69)

If the liquid velocity at the two-phase flow condition is larger than zero, it

is possible for the particles to transport through the impeller. In order to make

the transporting effective, it is necessary to make F^ larger than K

that is,

> K

For getting the higher efficiency, the values of velocities at the

impeller outlet should not be selected to be large.

The Coriolis force also make particles accumulate near the outlet, which

results in change of the particle equivalent d

, such as,

-ft^

<470)

where D

and b

are the outlet diameter and the width of impeller respectively,

Z is the number of blades, s

is their displacement at the outlet, B

the

incorporate coefficient of particles there. The same 'absolute clog' also occurs

at the outlet as at the inlet. The effects of blade thickness S

and the equivalent

diameter of particles make the direction of water flow change, which

gradually results in the transition of the flow to the state of 'the relative

suction'. Then the distortion of particle diameter becomes:

Ad=J\-

I V

(4.71)

And the relative suction diameter of particle on the meridional section at

the outlet is:

Design of Hydraulic Machinery Working in Sand Laden Water

229

(4.72)

Therefore, regarding the liquid flow velocity on the meridional section

the outlet and at the clear water flow condition as the reference value, and the

particles as the moving boundary layers, the velocity triangle at the outlet may

be solved.

The

outlet circulation will decrease

at the

two-phase flow

condition, which causes the pump head getting lower than that

the clear

water condition.

mentioned before

the

increase

the inlet circulation

make the head decrease too.

Volute Chamber of Pumps

At the clear water condition,

the

streamline equation

the pump volute

casing is such as:

tmS =

—

const (4.73)

where V

and V

are the radial and the circumferential velocity components. At

the two-phase flow condition, the tangential velocity component V

the

impeller outlet will decrease. So will the tangential velocity V

in the volute.

And the particle velocity V

on the meridional section

larger than that of

water flow, so that the discharge on

unit section of the volute,

will also

increase, the spiral angle

the volute should

large, then the width

volute should be greater than that of conventional pumps. In order to increase

the hydraulic efficiency of slurry pumps, the two-phase flow design should be

used

calculate the volute.

first, the distortional velocity

the volute

chamber V

should be taken in the following values:

v3y

[2tfl-AV

(4.74)

where K

the velocity coefficient

the two-phase flow condition, AV

the value of decrease of

at the outlet at the condition. The area at the eighth

section in the volute

for the water flow at the two-phase flow condition

should be indicated in the following equation:

230

Abrasive Erosion and Corrosion

of Hydraulic

Machinery

K„=^ (4-75)

For the solid flow, the similar area Fy\

also can be got:

Ku=^-

(4-76)

where

V =V -K

3L ^3

Hff*»(7w -

YL)(«"

cosfl)

i 3C

=fi

Nl'

And D

is the discharge diameter of

pump.

The largest section area of the

volute is F

„ that is,

^vm

^vrn ^ ^vm (4.77)

Then all section areas of the volute can be calculated, as that:

Fvu

7/8 (Fyiu)

F„ = 6/8(F

ra/

)

=5/%(F

)

Fj =1/8(F«

)

4.4.3 Examples of the Design

The foregoing design method has been used to design some types of slurry

pumps. Table 4.9 and Table 4.10 show the performances of LXL-H-36 and

Design of Hydraulic Machinery Working in Sand Laden Water 231

150 ND types of the slurry pump designed by using the method respectively.

From the tables it is seen that at the two-phase

flow

condition the efficiency of

the pump designed by the foregoing method could be higher than that at clear

water condition. These types of pumps also get good cavitation performances.

Table 4.9 Performances of the slurry pump of LXL-H-36 type

Discharge

QO/s)

65.72

60.03

55.31

63.29

52.05

44.07

64.82

62.04

58.57

Head

H(m)

39.19

40.10

40.74

39.20

39.60

39.00

26.10

35.80

36.30

Shaft

power

N(kw)

38.16

36.55

35.59

39.08

36.14

33.66

47.48

46.72

45.98

Efficiency

Tl (%)

66.10

64.50

62.70

73.40

66.00

59.00

68.20

65.70

63.90

Rotating

speed

n (r/min)

1451

1449

1451

1450

1452

1447

1448

Solid

weight

content

S (kg/m

)

S = 0.0

S = 235

S = 538

Gravity

rate

1.0

1.18

1.41

Table 4.10 Performances of the slurry pump of 150 ND type

Discharge

Q(l/s)

260.0

320.0

382.0

261.9

321.3

382.4

272.08

314.1

372.4

Head

H(m)

41.0

40.0

39.5

39.0

37.4

36.5

37.7

36.8

34.7

Shaft

power

N(kw)

44.7

51.3

58.4

67.6

76.6

86.4

60.8

78.1

84.6

Efficiency

Tl (%)

65.0

68.0

70.5

72.0

75.0

77.0

72.0

74.2

76.5

Rotating

speed

n (r/min)

980

Gravity

rate

1.0

1.75

1.83

232

Abrasive Erosion and Corrosion of Hydraulic Machinery

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