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

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144

Abrasive Erosion and Corrosion of Hydraulic Machinery

PAV

i=\

where n is the number of grid stations in a section of the domain, N

is the

particle number at the station, V

is velocity and A

the area at the station.

And N

is defined as the particle number in the unit volume, that is,

#,=% (3-81)

where C

is the volumetric concentration and d

the particle diameter.

* Conditions for gas phase flow.

For the gas-liquid two-phase flow, at this dilute condition the flow pattern

is the bubble flow in the liquid. The diameter of gas bubble is assumed to be

1.0 mm at the inlet of impeller, which will change according to the equi-

thermal process equation.

Calculated Results of Centrifugal Pump Impellers

The calculation was conducted through a centrifugal pump impeller. The

geometrical and flow parameters of the impeller are as follows:

Inlet diameter:

Outlet diameter:

Inlet angle:

Outlet angle:

Particle density:

Volume concentration:

Particle diameter:

Gas density:

Volume concentration:

Gas bubble diameter:

Liquid density:

Inlet velocity of liquid and;

Rotating speed of impeller:

Figure 3.2 shows the velocity

impeller, at different flow conditions

gas:

fields

, such

Dl = 130 mm

D2 = 330 mm

= 45°

jS2=18°

1.295 kg/m

= 5%

1.295 kg/m

= 5%

dp =

= 1.Ox

kg/m

= 6 m/s

n = 1450 rpm.

continuum phase through the

as in Figure 3.2 (a) at the particle-

Analysis and Numerical Simulation of Liquid-Solid Two-Phase Flow

145

liquid flow condition,

(b) at

the gas-liquid flow condition

and (c) at

the single

phase flow condition. From these figures

it is

clear that

the

gas-liquid flow

has

little influence

on the

liquid phase flow,

and

that

on the

contrary,

the

particle -liquid flow has

definite influence on the liquid flow.

105

0 0.05 0.1 0.15 0.2 -0.05 0 0.05 0.1 0.15 0.2

[The continuous phase (liquid) velocity profile (S-L)] [The continuous phase (liquid) velocity profile (G-L)]

(a)

(b)

(a)

particle-liquid two phase flow

(b) At gas-liquid two-phase flow

(c)

single liquid phase flow

Figure 3.2 Velocity field

liquid phase

146

Abrasive Erosion and Corrosion of Hydraulic Machinery

Figure 3.3 is the slip velocity distributions between the liquid phase and

the particle phase (in Figure 3.3 (a)), and between the liquid phase and the gas

phase (in Figure 3.3 (b)). The slip velocity Au between two phases can be

defined as

-u

(3.82)

where u

is the velocity of dispersed phase, and u

that of continuous phase.

-0.15

'■•'■'

-0.05 0 0.05 0.1 0.15 02

[The velocity difference of two phases (S-L)]

(a) At particle-liquid condition

[The velocity difference of two phases (G-L)]

(b) At gas-liquid condition

Figure 3.3 Slip velocity between two phases

Analysis and Numerical Simulation of Liquid-Solid Two-Phase Flow 147

The velocities between gas- phases and liquid-phases are not too much

difference. But the velocity difference between particle and liquid is large

because of large response time of particles, which results in the increase of

particle velocities at the outlet of centrifugal impeller.

Figure 3.4 is the pressure distribution through the impeller, which is not

much difference with that of pure liquid flow. The same pressure distribution

for liquid particle flow can be found in references

[3.21,

3.27]. Figure 3.5 is

distributions of the particle number (a) and the gas bubble number (b), which

is the relative number to the number at inlet. The particle number at the inlet

is N

- 9.55 xlO

at the condition C

= 5% and d

= 1.0 mm. From these

figures, it is shown the particle number is denser than that at the inlet, and

also denser near blade surfaces than that at the middle part of the flow

passage. From the results it can be obtained that the calculation model and the

computer program can be used to predict the main flow characters of gas-

liquid flow through the impeller.

200

150

100

-50

-100

-150

-5© 0 50 100 150 200

Figure 3.4 Pressure distribution at gas-liquid condition

148

Abrasive Erosion and Corrosion of Hydraulic Machinery

200

150

100

-SO

-100

-ISO

-50 0 50 100 150 200

(a) Particle phases

200

150

100

-50

-100

-150

-50 0 50 100 150 200

(b) Gas bubble

Figure 3.5 Number distribution

Three-Dimensional Particle-Liquid Two-Phase Turbulent Flow Calculation

through a Hydraulic Turbine Runner

The three-dimensional turbulent particulate-liquid two-phase flow have been

simulated through a hydraulic Francis turbine runner by using the k- s -A

turbulence model and the SIMPLEC algorithm on the bases of the theory of

two-fluid model for multiphase flow.

Analysis and Numerical Simulation of Liquid-Solid Two-Phase Flow

149

Parameters of the turbine runner and boundary conditions

Parameters of the turbine runner are as follows

Diameter of

runner:

D=lm

Head: H=

Rotating speed: n = 82 rpm

Discharge: Q

0.98 m

Blade number: Z= 14

Particle diameter: d

1.0 mm

Particle volume concentration: C

The boundary conditions at the inlet of runner, for example, the inlet

velocities of the continuous phase and dispersed phase, the particle number,

the turbulent kinetic energy and the turbulent energy dissipation rate, are

specified according to the known conditions. The outlet conditions are given

as the Neumann condition. The velocity of continuous phase near walls and so

on is determined according to the wall function. For the pressure and the

particle number on other boundaries the Neumann condition is also specified

except one station in the domain, where the value of pressure is fixed in order

to get a definite solution.

Calculated Results of Turbine Runners

In the grid system of the runner, the flow passage through the runner is

divided several parts from the blade pressure side to the suction side. The

surface K

2 is the geometrical surface close to the pressure side. And from

the crown (hub) to the band (shroud) of the runner, the passage also is

divided, for example, J

2 indicates the surface close to the crown. Figure

3.6 shows the relative velocity distribution on the surfaces of J

6 and J =

14.

The velocity increases from the pressure side to the suction side, except

near the trailing edge of the blade. And so does it from the inlet to the outlet

(L/LO

from 0 to 1) of the runner.

As in Figure 3.6, the same relative velocity distribution of dispersed phase

is shown in Figure 3.7. This velocity is much smaller than the continuous

phase velocity, because the velocity increases from the inlet to the outlet, the

friction drag and other forces between two phase resist the increase of the

continuous phase velocity. Figure 3.8 shows the pressure distributions

through the runner on different surfaces, i.e., J- 2, 6, 14 and 18. It increases

from suction side to pressure side, and decreases from inlet to outlet.

150

Abrasive Erosion and Corrosion of Hydraulic Machinery

2.8

2.6

* 2

1.8

1.6

1.4

/ /

■1.

. .

^-"-"/

6/^

T ^

' *\

/- L2

02 0.4 0.6 0.8

L/LO

5.5

4.5

S 3J

2.5

: /// /

s -"

/ /"■■

' / •

' /

02 0.4 0.6

L/LO

(a) J=6

(b) J=14

K = 2,

K = 8,

K = 4, K = 6,

K=10,

K=12,

Figure 3.6 Velocity distribution of continuous phase

1.4

0.9

0.8

0.7

>-■ --^s

'.'" ~

%£L

/,.

' </—

=====

/ \

^mz—

iii

02 0.4 0.6 0.8

L/L0

(a) J=6

2.8

2.6

2.4

t 2

^ 1.8

1.6

1.4

1.2

'■■/'

///

'/'

' /

/".-

V '''

■ /

\ /

" X

\\ N

\ \^

02 0.4 0.6 0.8

L/L0

(b) J=14

Figure 3.7 Velocity distribution of particulate phase

Analysis

and

Numerical Simulation

Liquid-Solid Two-Phase Flow

151

63000

6200C

61000

60000

59000

58O0O

57000

56000

55000

\ \

V\ s

0.4 0.6

L/LO

63000

62090

61000

60000

5SOO0

58000

57000

56000

55000

54000

■

\\x

; V>

V \^ -

"^s^

^~i

(a) J=2

L/LO

(b) J=6

64000

62000

60000

58000

56000

■?

540D0

52000

50000

48000

46000

44000

42000

fer-^-

$\-

^C<-

NN-

N ^

\ ^

\ \

\ \\

^ - *

0.2

0.4 0.6

L/LO

(c)J=14

0.8

65000

60000

55000

50000

45000

40000

35000

;____

\. V

\ x

\ \

0 0.2 0.4 0.6 0.8

L/LO

(d)J=18

Figure

3.8

Pressure distribution

152 Abrasive Erosion and Corrosion of Hydraulic Machinery

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