Chemical and Petroleum Engineering, Vol. 31, Nos. 7-8, 1995

U S I N G AN E N E R G Y C R I T E R I O N T O O P T I M I Z E T H E

P A R A M E T R I C R E L A T I O N S H I P S O F C E N T R I F U G A L P U M P

R O T O R S

N. S. Yalovoi and A. M. Kats UDC 621,67.001.5

One method of investigating centrifugal pump rotors is to look for the optimum parametric relationships of a statistical

mathematical model obtained from the results of a multifactor experiment [1]. In order to do this for the rotors of low-speed

(40-60) centrifugal pumps, a series of experimental rotors were designed, manufactured, and balance tested. The test results

were used to find the hydraulic efficiency of the rotors as a function of their operating conditions [2].

When a rotor is operating under optimum conditions, the maximum hydraulic efficiency r/h max uniquely determines

the minimum energy losses, which occur in the channel between the blades. The optimization problems are solved by fmding

those dynamic, kinematic, and geometric criteria which optimize %max for any previously specified reference parameter.

Separation on the leading edges of rotor blades depends on the flow conditions around them. For data reduction,

separation is well approximated by the overall parameter ql = vlmp/{ulp tan/31bp} which was used in analyzing cavitation

phenomena [3]. Here t)lm p and Ulp are the meridional and circumferential components of the velocity at the peripheral point

where the leading edge of the blade intersects the cover plate and 131b p is the blade angle coming into that point. The reference

parameter -- the rotor inlet coefficient K 0 -- had a substantial effect on the cavitation margin of the pump. The inlet coefficient,

which mainly determines the kinematic parameters at the rotor inlet, affects both cavitation and the energy criteria of the rotor.

Separation behind the rotor, after the flow leaves the blades, is determined by the interaction of the rotor and the outlet

channel. Well designed outlets have minimal energy losses under design conditions, while separation flows, which increase

the losses, are very weak under conditions close to the design condition. In this case, the similitude of the flows at the rotor

inlet and outlet can be expressed by the overall parameter q2 = V2m/{U2 tan ~2b}, which considers the characteristics of the

flow and the rotor only at its outlet cross section. Here v2r n and u 2 are the meridional and circumferential components of the

velocity, and ~2b is the blade angle at the rotor outlet.

Separation flows in the channel between the rotor blades depends on the interactions of the principle forces in this

channel and on the geometric characteristics of the channel. Research [4] indicates that the Rossby criterion Ro is the dynamic

criterion that substantially affects liquid flow (mainly the boundary layer characteristics) in the channel between the rotor

blades. For stationary flow, Ro is usually determined in terms of kinematic parameters: Ro = u/w, where u is the

circumferential velocity of flow rotating along with the rotor and w is the relative velocity of the flow in the rotor.

As a rule, the rotor blades of centrifugal pumps are highly curved. Research shows that the main parameter which

affects the boundary-layer characteristics in curved channels is the relative curvature of the channel (blade) L/Rav, where L is

the blade length and Ray is the average radius of curvature of the blade [4].

Also, losses in the blade channel can be characterized by the parameter ~, which is the diffuser coefficient of the blade

channel, and a parameter which characterizes the profile losses of fluid flow in the rotor. Here [~ = ~ -- x ~ /x / -~T c, where

F 1 and F 2 are the inlet and outlet areas of the channel and 1 c is the average channel length.

Balance tests of the test rotors that their maximum hydraulic efficiency varied up to 6% for different operating

conditions. Thus, one can say that ~/h max depends substantially on rotor operating conditions.

The results of the balance tests of the experimental rotors were used to obtain an adequate multifactor model in the

following form

422

Translated from Khimicheskoe i Neftyanoe Mashinostroenle, No. 8, pp. 8-10, August, 1995.

0009-2355/95/0708-0422512.50 �9 Plenum Publishing Corporation

t~r -O, ZJ

o, 0 - tI ~9

o~5 -tft5

o, .~ . o/I s,5

J f

j j j J

!

4/r 5 ~5 6 ~'o a

~2

a,r,~

a,,t

r

el

b

'-// \ O,965 \

3,5 ~ ~,s 5 5,5 6 /r C

Fig. 1. Determining the optimum relationships of centrifugal pump parameters: a) RO*s2 and K*Qout = f(Ko);

b) q*2 and (L/Rav) = f(K0); c) ~max = f(Ko).

)l~a x = 1,390 - 0,2531Row2 - 0,3281KQout +

L + 0,2858q2 - 0,01973K 0 - 0 , 0 9 7 4 0 ~ .

The model parameters were calculated as follows: The Rossby criterion is

~Dh2 R o ~ = - - ,

~t~ 2

where ~o is the angular velocity of the rotor; Dh2 = 2[tzb2/(tz+b2)]; t2 is the blade step at the rotor outlet; and b 2 is the width

of the rotor at the outlet. The pumping coefficient at the outlet is

Q~

where Qr is the pumping rate of the rotor, and D 2 is outer diameter of the rotor. The rotor inlet coefficient is

where D O = ~/DI 2 - d2hub is the reduced rotor inlet diameter, D 1 is the diameter of the rotor inlet, dhu b is the hub diameter

at the inlet, and Q is the pumping rate.

The average radius of curvature Rav of a blade was determined at 2/3 its length from the outer diameter of the rotor.

The model very adequately characterizes the experimental data: the multiple correlation coefficient was 0.98, and the

average error of the model was 0.19%. When independent variables were introduced into the model, all the coefficients were

statistically significant according to the Fisher criterion. Other parameters considered did not have a significant effect on the

total function and therefore were not introduced into the model.

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This mathematical model was used to conduct a search for the optimum relationships of the parameters as a function

of K 0, which was chosen as the reference parameter because the rotor inlet parameters depend on the customer' s requirements to control the pump cavitation properties, which uniquely determine the required value of K o.

The results of the optimization search on the parameters [ROw2, K*Q~ x, q'a, (L/Rav)*] = f(K0) and the optimization criterion rib max = f(K o) are shown in Fig. 1. As can be seen, the optimum K o for maximizing r/hmax lies in the range K o = 4.3-5. Rotor losses are minimal (about 2%) at these values of K 0 and for the corresponding values of the other p~.rameters. This corresponds roughly to the minimum possible hydraulic losses in the rotor (from data in [5]) within the limits of

experimental error.

REFERENCES

,

2.

3.

.

5.

N. S. Yalovoi, Optimizing the Designs and Quality Indices of Machines [in Russian], Izdatel'stvo Standartov, Moscow

(1988). N. S. Yalovoi and A. M. Kats, "Test results of low-speed centrifugal pumps," Khim. Neft. Mashinostr., No. 4, 1-3

(1995). A. M. Kats, "Multifactor cavitation experiment on rotors," in the information collection: Advanced Production and Scientific Experience [in Russian], Central Institute of Scientific and Technical Information on Petrochemical Machinery (TslNTIkhimneftemash) (1992), No. 2, pp. 14-16. K. P. Seleznev and S. N. Shkarbul', "Some criteria for determining flow in elements of the flow channels of turbines," l~nergomashinostroenie, No. 9, 19-22 (1972).

A. I. Stepanov, Centrifugal and Axial Pumps [in Russian], Mashgiz, Moscow (1960).

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