FEATURE •

A solution for profile generation in twin-screw multiphase pumps Capable of handling continuous gas volumes of over 98% as well as 100% slugging, twin- screw multiphase pumps are ideal for multiphase use. The rotor geometry of these pumps play a crucial role in their performance, but, say Cao Feng, Xing Ziwen and Shu Pengcheng from Xi'an Jiaotong University, it has not been clarified yet. A numerical solution based on divergence theorem for screw rotor profile generation is presented for further validation.

M ' u l t iphase fluid transfer technology has been

developed mainly for the oil industry. It is capable ~of moving crude oil containing large amounts of

gas, water and sand, and offers the potential for significant equipment and operational savings. The twin-screw multiphase pump is the most suitable model for multiphase duties. It facilitates the handling of multiphase products up to the highest void fractions, including even 100% gas for short periods. Other pump types are generally limited to GVFs of less than 50%.

The twin-screw multiphase pump is a self-priming, double- ended positive displacement pump with external timing gears and bearings. It is designed with intermeshing screws on parallel shafts operating inside close fitting bores (see Figure 1). Flow entering the pump splits into two. The split flow is introduced to the outermost ends of shafts where the rotors begin their meshing action. With the turning of the rotors, flow is pumped to the centre of the shafts, then discharged later. This approach results in hydro-dynamic balance in the axial direction, which greatly reduces the bearing load. The suction pipe centre is above the centre of

Figure 1. Twin-screw multiphase pump rotors

a 2 a 1

C 2 C 1

f f

Figure 2. Rotor pro f i le .

pump body, which keeps a small amount of liquid inside the pump casing to allow dry running for short periods.

To a large extent, twin-screw multiphase pump performance depends on the definition of rotor geometry. Longer sealing line lengths and larger blowhole areas mean significant slippage from the high-pressure stage to the lower one, which leads to poor performance.

A numerical solution based on the divergence theorem for screw rotor profile generation is presented here. A new rotor profile for multiphase duties is offered for validation. Also discussed are geometrical characteristics including sealing line length and blowhole area.

Profile generation Although the rotor profile plays a dominant role in developing the twin-screw multiphase pump, it has not been clarified in published literature yet. The special profile different from that of liquid pump and gas compressor is required for multiphase products pumping.

Rotor profiles for multiphase duties, as indicated by number 1 and 2, respectively, are shown in Figure 2. Two rotors roll on their pitch circles about their centres 01 and 02 by angle and (01 and q~2 = ¢Pl//• The coordinates of rotor profile 1 are defined as functions of an angular parameter.

x U = x t (t D )

y,j =y~(t,,) (1)

Referring to gear tooth engagement theory, the rotor profiles can be expressed with each other as follows:

XIj =--X2j COS k(Di j -Y2j sin k(p~j + A cos(itpl ) ) Ylj =-x2j sin k%j + Y2j cos krpu + .4 sin(i~l j ) (2)

X2j =--Xlj coskqo]j-YD sin ktPU +Ac°stPu Y2j =-Xl)sin ktPU +Yly c°skfPlj +Asintpu (3)

From equation (2) and (3), the conjugacy condition,

ax, ay, ax, ayt = 0 , can be derived as follows: at ~% 0(p~ at

Xly COSOIj + Ylj sinOu q01j = arccos(- -)+01j

RIt

0 Lj = arctg(YlJ (t)) x~j (t)

(4)

341. WORLD P U M P S September 2000 0262 176210015 - see front matter © 2000 Elsevier Science Ltd. All rights reserved

FEATURE •

Where YtJ (t), x!j (t) can be calculated by the explicit numerical solution according to Richardson Extrapolation (Ward Cheney, 1991) with the given discrete point values of profile 1, so xd., Yd' can be defined from equation (3) and (4). An example of numerical solution for segment a i b 1

and its conjugacy point b 2 characterized in Figure 2 is listed

in Table 1.

Geometry calculation In order to understand and simulate the thermodynamic performance of twin screw multiphase pump, the volume

curve, the contact line length and the blowhole area must

be defined first.

Cavity volume The twin-screw multiphase pump works on the principle of enclosing the multiphase mixture in feed chambers and continuously increasing the pressure on the fluid as it travels through the pump. It is helpful to visualize the working process as equivalent to a piston sliding down a cylinder (see Figure 3).

Discharge Inlet

. . . . . . . ii

Figure 3. Working process model

In the suction phase, the multiphase flow is drawn in until the trapped volume isolated from the suction area. As

rotation continues, the volume of the trapped multiphase flow remains constant while the flow moves from the inlet

to the outlet end. When the trapped volume is opened to the outlet end, the back-flow from the discharge area enters the pump until the internal pressure reaches this discharge

level, and then the flow is discharged as normal. The volume curve is shown in Figure 4 for an sample case. It is evident that the trapped volume remains constant during

the closed volume phase. This type of design enables the pump to handle a slug of liquid.

The difference between the real and theoretical flow rate is determined by back-flow between the first chamber and the

suction area. The thermodynamic behaviour of the twin

screw multiphase pump is influenced more strongly by the presence of the back-flow within the machine. Slippage of

fluid back through finite clearances of adjacent stages causes the pressure increase with the compressible fluid, and most of the pressure rise is in the final stages of the

pump.

Back-flow rate is dependent on a number of factors, including different pressure, inlet pump pressure, inlet gas

void fraction and clearances. As GVF rises, slippage rate decreases until the inlet-volume flow rate is equal to pump

displacement. Internal back-flow occurs at three gaps:

namely back-flow through contact lines, back-flow through

blowholes and back-flow across lobe tip clearances.

Contact line length The contact line for the sample rotors characterized in Figure 4 is shown in Figure 5. The interlobe back-flow path

area is obtained as the product of the contact line length

and the clearance. The contact line length varies with the rotational angle. It can be determined as

L.(~] ) = Z li(~l )

I,(~,) = j" 4X~ +Y,2 +Z?S, (5)

where the coordinates of the contact points are defined as

follows:

XIj : X l / COS(~I --YIj sinOt Yb = xjj sin¢~ +Yu cos01 Z,, =T, x¢, /2~ (6)

3x~ a~, L 3x t 3y~ =o at a~L a% at

The conjugacy condition can also be calculated with the

method mentioned above. If we express the wrap angle as rlz = 2k,r + a (a < 2re, k = 0,1,2....), then the total length

of the contact line can be written as follows:

When mt + a < 2 ~

Lr(q~l)=(k+l)xLp(2Ir)+Lp(a+(Pl-2Ir) Lp (q~l)

0.00035

~ . 0.00030

g o 0.00025

0.00020

0.00015

"6 0.00010

0.00005

0.00000

X R = 0.075m T= 0.060m

I ~,. I 0 5 li0 li5 20 25

Rotational angle (rad)

Figure 4. Volume curve

www.wor ldpumps.com WORLD P U M P S September 2000 . . ~ . . . .

:EATURE '

When ~ + ce > 27r

L r (~ol) = k x Lp (2a') + Lp (at + ~o,) - Lp (cpl) (7)

Blowhole area The presence of the leakage triangle (blowhole) in the twin-screw multiphase pump is an inevitable consequence of the rotor profile geometry. It is formed between the housing cusp and the meshing rotors. One side of this triangle is the housing cusp between leading and trailing crests of the two rotors. The other two sides are paths from the crest-cusp intersection, along the respective rotor surfaces to a common point where the two rotors make contact (see Figure 6). Presence of the blowhole results in increasing slippage rate along Z-axis direction. The blowhole area can be calculated as follows:

2 2 2 2 . XB • XC ] R2 xTxl3AB ( ~ - ~ ) - X c ( a r c s i n - - - a r c s , n ~ ) S (XB--Xc)Xn R2 R2 (8)

2 A2 2 R, 2 + A 2 _ R 2 rt/+ -R 2 where x~ , x c

2xd 2xA

H'

P A

Figure 6. Schematic diagram of the blowhole

Using a few computer programs driven from the analytical representation above, we can calculate the back-flow channel areas. We find that back-flow across the rotor tips prevails among all the channels.

Conclusions The rotor profile affects performance and torque radial loads to a great extent. A method of screw rotor profile generation has been demonstrated here for twin-screw multiphase pump profiles. Using a few computer programs driven by the analytical representation in this paper, volume curve, the contact line length and the blowhole area can be calculated. The calculation of the geometrical characteristics provides the basis of thermodynamic performance simulation of the twin-screw multiphase pump.

Figure 5. Contact line between two rotors

A c k n o w l e d g e m e n t

The work described in this paper is funded by Doctorate Foundation of Xi'an Jiaotong University.

References Allan J Prang, Selecting Multiphase Pumps, Chemical Engineering, New York, Feb 1997, P 74-79.

Buqing S u e t al, Practical Differential Geometry Theory, Science Press, 1998, P 117-127. (In Chinese)

D F Dal Porto, L A Larson, Multiphase Pump Field Trials Demonstrate Practical Applications for the Technology, 1996 SPE Annual Technical Conference and Exhibition, Denver, Colorado, USA, 1996, P 181-192.

K Egashira, S Shode, T Tochikawa, A Furukawa, Backflow in Twin.Screw.type Multiphase Pump, 1996 SPE Annual Technical Conference and Exhibition, Denver, Colorado, USA, 1996, P 221-230.

P J Dolan, R A Goodridge, J S Leggate, Development of a Twin-screw Pump for Multiphase Duties, 1996 SPE European Petroleum Conference, London, UK, 1986, P 293-298.

Ward Cheney, David Kincaid, Numerical Mathematics and Calculation, Fudan University Press, 1991. (In Chinese)

CONTACT Cao Feng, Xing Ziwen, Shu Pengcheng, School of Energy and Power Engineering, ,Xi'an Jiaotong University, Xi'an, 710049, P.R.China. Tel: +86 29 266 8216; Fax: +86 29 323 7910; E-mail: zwxing@xjtu.edu.cn

WORLD P U M P S September 2000 www.worldpumps.com