Introduction to fluid machines Contents:

Dynamical similaritude Same machine at different rotations

System curves Operation point Operation optimization of turbomachine and

system Maximum efficiency conditions

Exercise

Application of theorem to hydraulic turbomachines (constant ) As shown in previous class, for geometrically

similar machines :

2

352,

ND

ND

QF

DN

L

Torque coefficientflow coefficient

Reynolds number

352 ND

QF

DN

L

Neglecting Re effects (fully turbulent flow):

For any other independent variable (P, H, ,…):

322 ND

QF

DN

gHH

353 ND

QF

DN

PP

3ND

QF

Torque L was arbitrarily chosen

ect.

XfY There is only one idependent non-dimensional coefficient for fully rough flows (large Re)

Application of theorem to hydraulic turbomachines (constant )

For the same family of gemetrically similar machines, the non-dimensional performance curves overlap:

1000 rpm, D=25 cm1200 rpm, D=20 cm1350 rpm, D=15 cm1500 rpm, D=15 cm

22DN

gH

3ND

Q

H

Q

Application of theorem to hydraulic turbomachines (constant )

Dynamical similaritude

If 1000 rpm1200 rpm1350 rpm1500 rpm

22DN

gH

3ND

Q

23

13

ND

Q

ND

Q

222

122

DN

gH

DN

gHand 1 = 2

Points 1 and 2 are said to be dynamically similar points (same adimensional groups, same proportions of dynamic and kinematic quantities)

12

Dynamically similar points for the same machine at different rotations N Same machine: D1=D2

1000 rpm1200 rpm1350 rpm1500 rpm

22DN

gH

3ND

Q

32

23

1

1

DN

Q

DN

Q

222

222

1

1

DN

gH

DN

gH

122

1

21

2

1

2

1

N

N

H

H

Q

Q

Dynamically similar points for the same machine at different rotations N - D1=D2 Same machine

Points on the same parable in the H,Q diagram are dynamically similar points obtained at different rotations for the same machine.

2

1

21

2

1

2

1

N

N

H

H

Q

Q

H

Q

N1 = 1000 rpm

N2 = 1200 rpm

Parables H=kQ2

P1

P2

Q1

H1

Q2

H2

221

1 QQ

HH

k

Exercise 1

Take the Francis turbines of Cabora Bassa plant (Mozambique): H=113,5m; N=107,1rpm, P=415MW, D=6,56m.

It is inteded to test a 1/20 scale model in the laboratory with a head of 22m.

What is the rotational speed, shaft power and volume flow rate in the model for nominal conditions? Neglect the influence of Re; assume an efficiency of 95%.

Answer: N’ = 943 rpm, P’ = 88 kW, Q’ = 0,43 m3/s.

System curve

zB-zA Q

pA

pB

Applying Bernoulli equation between the 2 free surfaces of the system in the figure:

222

1Q

gAd

lfzz

g

ppH

eqAB

AB

Mechanical energy accumulated in the form of pressure and potential energy

Required net mechanical energy supplied to the fluid in the pump

Mechanical energy dissipated in the system

H=F(Q) is the system curve

System curve Curve that gives the mechanical energy per unit weight

H requied to be provided to the fluid if a given flow rate Q is expected to flow in the system.

Q

HH=F(Q)

System curve

Energy dissipated in the system

Mechanical energy accumulated by the fluid.

222

1Q

gAd

lfzz

g

ppH

eqAB

AB

k

If the flow is fully turbulent in the pipe f f(Re)

System curves

Ventilation systems

Closed circuit piping systems have similar curves (no energy storage).

222

1Q

gAd

lfzz

g

ppH

eqAB

AB

=0Q

H

H=F(Q)

System curve

Energy dissipation in the system

System curves

Hydroelectric plants

222

1Q

gAd

lfzzH

eqBA

Q

H

H=F(Q)

System curve

Energy dissipation

Operation point

Flow and head to which the provided energy by the pump balances the system energy requirements:

Q

HSystem curve

Pump performance curve at rotation N

Q1

H11

Maximum efficienty conditions For which rotation is maximum efficiency achieved?

222

2 QQ

HH

Q

HSystem curve

Pump curve at rotation N

Q1

H11

Q2

H2

Maximum efficiency points for different N and same pump:

Q3

2

3

2

3´Q

QNN

Point 2: Maximum efficiency point at original rotation

Point 3: maximum efficiency point (at a different rotation speed) and also a point in the system curve

Series association of machines What is the flow provided by the two pumps in series?

Same flow, added heads

Q

HResultant curve of the series association

Pump B curve at rotation NB

H=HA+HB

Pump A curve at rotation NA

System curve

BA

A+B

BA

BB

Q

Parallel association of machines What is the flow provided by the two pumps in parallel?

Same head, added flows

Q

H

Resultant curve of the parallel association

Pump B curve at rotation NB

H=HA=HB

Pump A curve at rotation NA

System curve

Q=QA+QB

BA BB

Q

B

AA+B

Series and parallel association in hydraullic powered machines Series association:

Parallel association:B

B

A

A

BA

B

B

A

A

BA

r

s

HHHH

HHgQ

HHgQ

P

P

)(

)(

B

B

A

A

BA

B

B

A

A

BA

r

s

QQQQ

QQgH

QQgH

P

P

)(

)(

Exercise: 1st Test 2010-11

es

10,5 m

A radial hydraullic pump, pumps water ( = 1000 kg/m3; = 10-6 m2/s) from a river to a reservoir at atmospheric pressure, as shown in the figure. The equations of the the pump curves at 3000 rpm are :

and

with H in meters and Q in m3/s. The flow in the pipes can be taken as fully turbulent, with a overal friction coefficient (suction and discharge pipes) of 5000 m/(m3/s)2.

21200045 QH 2780004670 QQ

a) Compute the flow rate:

25 l/s 31 l/s 40 l/s 45 l/s 52 l/s 60 l/s

b) And the dissipated energy in the pipe?

1,3 kW 2,1 kW 4,5 kW

6,1 kW 7,0 kW 8,5 kW

Exercise: 1st Test 2010-11

es

10,5 m

c) Compute the rotational speed for pump maximum efficiecy?

1525 rpm 1685 rpm 1784 rpm

1936 rpm 2352 rpm 2842 rpm

A radial hydraullic pump, pumps water ( = 1000 kg/m3; = 10-6 m2/s) from a river to a reservoir at atmospheric pressure, as shown in the figure. The equations of the the pump curves at 3000 rpm are :

and

with H in meters and Q in m3/s. The flow in the pipes can be taken as fully turbulent, with a overal friction coefficient (suction and discharge pipes) of 5000 m/(m3/s)2.

21200045 QH 2780004670 QQ

Bibliography

Chapters 2 and 3

Turbomáquinas, A. F. O. Falcão, Folhas AEIST, 2004.