Baker R.C. Flow Measurement Handbook: Industrial Designs, Operating Principles, Performance, and Applications

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10.A.1 DERIVATION OF TURBINE FLOWMETER TORQUE EQUATIONS

247

/ and

will be very small, we may rewrite Equation (10.A.I) as

= pN

rV\s [tan(£

tan(£

i)]dr (10.A.2)

rV\s

tmPS

»±L±)

(10.A.3)

yi + 5tan£ 1 + i tarijSy

v ;

Ignoring terms in is, etc. (1° = 0.0175

= pN

rVJs^f^^dr

(10.A.4)

+ (z+5)tan£

or if we ignore (i +

compared with unity,

= pN

s(i

S)(l +

tan

p)dr (10.A.5)

The alternative airfoil theory approach is to use lift and drag coefficients. These

are given per unit length of the blade by

L =

-pcKcA--^-\

(10.A.6)

where K is the factor that allows for the change in lift coefficient between an isolated

airfoil and a cascade. The torque on the rotor is now given by

= 1-pN

' '

c(KC

tan p

)dr

(10.A.8)

COS^

This is essentially the term given by Blows (1981) for the aerodynamic lift torque

minus the aerodynamic drag

torque.

Next, Equations (10.A.2) and (10.A.8) are shown

to be equivalent.

Equating the axial force to the force due to change in static pressure:

(pi-

+•

(Poi

Poz)s

Equating the tangential force to the rate of change of angular momentum,

s[tan(p -8)- tan(j8

/)]

Substituting for V\ and

we obtain

-4p^

5[tan

tan

8)]

+ s(p

poz)

= -pV

5[tan(^

tan(j8

8)]

tan^

+ s(p

P02)

We may write the lift in terms of Y and Z as

L = Ycosfi

248 TURBINE AND RELATED FLOWMETERS

and, substituting for Y and Z,

-8)

L = pV?s[tan(j8 - 5) -

and for the drag

- /)] cos£

s[t<m(/3 - i) - tantf -

COSp

- /)]

sec

+ s(p

i -

and substituting for Y and Z

D = p^

s[tan(£ -

- tan(fi -

i)]

sin^

4- s(p

i -

P02)

cos^

^ = 5(^01 -p02)COS£

Substituting into Equations (10.A.6) and (10.A.7), we obtain

= 2^[tan(£ - 8) - tan(p - /)] sec f5

cos

= — [tan(p -8)- tan()3 - /)] cos p

+ C

tan ^

^ sin^

cos

(10.A.9)

where

tan An = —

sin p

cos

which shows that Equations (10.A.2) and (10.A.8) are equivalent.

The theoretical expression given by various authors for the lift coefficient is

2JT

sin a

or in fuller form by Batchelor (1967)

= 27rsina(l +O.77£

/c)

Lift curves are reproduced in Figure 10.9, and drag curves can be found in Wallis

(1961).

The angle a is that between the far field and the line of zero lift of the blade,

which for a flat plate is the angle with the plate. It appears to be conventional to

take the direction of the far flow field for a cascade as the mean direction between

inlet and outlet and a for a cascade as the angle between this mean direction and

the blade angle (cf. Thompson and Grey 1970). Thus for a cascade of closely spaced

flat plates, the exit angle relative to the blades will be zero, whereas the inlet angle

10.A.1

DERIVATION OF TURBINE FLOWMETER TORQUE EQUATIONS

249

will

equal

to the far

field. Thus

the

value

of a

will

half

the

incidence angle

So following convention,

= 2nsini/2

(10.

10)

for

the

closely spaced flat plates,

and, in

this limit, according

Weinig (1932):

= -—-—

(10.A.11)

cos p

Thus

s4_sin^5^i

and Equations (10.A.5)

and

(10.A.8)

are

identical provided

= 0 and

= 0.

For values

sic

not

tending

zero,

shall find

useful

use

the

values

obtained

Weinig (1932)

and

given

Wislicenus (1947),

and it

will

necessary

to write

the

lift coefficient

terms

of the

true incidence angle

From Figure 10.3,

= tan

so that

but

P-<*

i +

and

= lit (^jr) = n(i +

(10.A.13)

and hence

rV\

COSp

If we make

the

assumption that both

and

may be neglected

in the

case

of a

well-designed turbine wheel,

have

cK(r)i(r)

_,__

rtA

Finally,

it is

useful

consider

the

actual values

and

We can

first

of all

obtain from Equations (10.A.5), (10.A.8),

and

(10.A. 10)

c „ . 5 .

2 cos

cos

2S l

COS

250 TURBINE AND RELATED FLOWMETERS

Then applying this to

and

and making use of Equations (10.A.5) and (10.A.

14),

1 cKn8 1 c „ s8

2 COS £

2 COS /3

COS

Hence,

c sin £

From Wallis (1961), as i tends to zero, C

tends to 0.017. For p = 45° and sic ~ 1,

~ 0.017-4^25

so that

8 ~ 0.003

~ (1/6)°

So if

neglect the drag terms (Equation 10.A.

14),

v * r * *

_L*

4s 8

j\7To

— L>n

tan p

_ z o

—

—8

ccos^

sin £

we are actually neglecting

-2-——

~-0.0085

ccosfi

We can compare this with the size of the drive torque

C COS

Thus the neglected term is

8/i

of the drive term.

Drag Terms

Tsukamoto and Hutton (1985) identified four drag torques:

bearing drag;

hub

disk friction drag;

tip

clearance drag;

hub

fluid drag.

the

first three,

the

torque

obtained from

equation

the form

Shear stress

Area

Radius

Reynolds number

x Function

Reynolds number

For instance,

for

the bearing drag,

obtain

10.A.2 TRANSIENT ANALYSIS OF GAS TURBINE FLOWMETER

251

where

— for Re

1,000

f(Re)=

i i i i i *i

for Re

1,000

where

Re is the

Reynolds number based

on the

radial

gap and

circumferential

speed. Blows (1981) obtained the same expression

Re <

1,000.

From the work of

Tsukamoto and Hutton (1985), the dependence on

ranges from

<x co

At low speeds, the dependence

is in

the range

whereas

high velocities

the dependence tends to

The hub fluid drag is

where

is the

relative velocity

the hub

and C

is a

constant

for a

particular

design.

will therefore behave like

adjustment

the main aerodynamic drag

term coefficient.

Flowmeter Equation

We may now equate

Tft

+ T

and we may note that, with increasing speed, the terms

, and

will become less

important compared with

and

7h.

If we combine the

drag with the aerodynamic

drag, we obtain from Equation (10.A.8)

[

, r, „,

/ —-— (K C

Cp tan

)dr

COS

where C

has incorporated T

. If we then substitute for Ci =

and approximate C^

as constant,

t r

COS

COSp

iteration, this equation enables the wheel speed to be obtained for a given flow

profile

the annulus (cf. Newcombe

al. 1972 for an expression for gas meters).

10.A.2 TRANSIENT ANALYSIS

GAS TURBINE FLOWMETER

Lee and Evans (1970) used

simple analysis

derive the behavior

of a

meter

observing the spin down as

indicator

changed bearing friction. The transient

equation (Lee

al. 1975,

cf.

Bonner 1977)

^-7,

(10.A.17)

252 TURBINE AND RELATED FLOWMETERS

where the driving torque T

may be obtained from Equation (10.A.I) for one radius

and unit blade length as

= pNrV

s(tan#> -tanfa) (10.A.18)

or, since V

tan^i =

rco

and putting

, (^tariff-ra>)

= pNrsv

—

where

allows for the blade deviation factor. Also putting T

= pAr

Qco

/2 + T

where the first term is fluid drag and the second term 7^ is nonfluid drag assumed

constant, Lee et al. (1975) then obtained

A-i—

+ VW +

= V

-D

(10.A.19)

where

pAV

rtanppA

and V =

,W=

rco/

tan

and r = t/T, the fundamental period of the pulsat-

ing flow. This equation is soluble on a computer. For certain cases, an exact solution

is possible. Lee and Evans (1970) give the speed decay curve for no flow (V = 0) as

Dl/

' "*—' "

--tan-' " ) (10.A.20)

This curve is shown in Figure 10.16. We can obtain from this curve, as shown by

Lee and Evans (1970), that the rotor coast time to standstill is

and just before this point

= -D

(10.A.22)

so that, from the slope 5, we have

5 = -^- (10.A.23)

CHAPTER

Vortex-Shedding, Swirl,

and Fluidic Flowmeters

11.1 INTRODUCTION

This chapter deals with measurement techniques based on instabilities in fluid be-

havior. In preparing this chapter, I benefited from an unpublished note in 1988

by Dr. M. V. Morris, of Cranfield University, who drew my attention to some early

references on this type of meter.

11.2 VORTEX SHEDDING

The detailed behavior as fluid flows past a cylindrical bluff body is given in Ap-

pendix

.A.

in relation to the size of the Reynolds number based on the diameter

of the cylinder. Figure 11.1 shows the development of the flow past such a bluff body

until, at a Reynolds number of about 40, the familiar shedding phenomenon starts.

However, small changes of detail in the pattern of the shedding are illustrated by the

diagram for certain Reynolds numbers. Frequency changes occur at these Reynolds

numbers. These changes could result in nonlinearity and poor repeatability in a

meter.

Figure 11.2 is a simple diagram to suggest how, as a vortex rolls up downstream

of a noncircular bluff body, it first forms, fills with rotating fluid, and then draws in

fluid with opposite rotation, which eventually cuts off the vortex and allows it to

move downstream. In Appendix 11.A.2 a simple argument is used to suggest that

the frequency of shedding f is about

f=V/2nw

(11.1)

where V is the velocity past the body of width w. An important parameter, the

Strouhal number, is given by

S= fw/V (11.2)

This relationship has been applied to long slender cylinders. Our application

is to the vortex-shedding flowmeter where the flow is in a pipe and the width of

the shedder (bluff body) is between a quarter and a third of the pipe diameter, and

thus is not a long slender cylinder. However, as we shall see, this length/width ratio

is advantageous in terms of the strength of vortex shedding and leads to coherent

shedding, whereas longer and more slender shedding bodies result in weaker and

less coherent shedding.

253

254

VORTEX-SHEDDING, SWIRL, AND FLUIDIC FLOWMETERS

40 < Re <

150

(e)

300<Re<3x10

(f) 3x10

<Re<10

Figure

11.1.

Flow patterns around a cylindrical

body:

(a) For

< 4; (b) Initiation of separation

for 4 < Re and up to about Re = 30; (c) Lack of separation symmetry behind cylinder for about

30 < Re < 40; (d) Stable and viscous shedding for 40 < Re < 150; (e) Stable shedding with

turbulent shear layer for 300 < Re < 3 x 10

; (f) Stable shedding with turbulent boundary and

shear layers for 3 x 10

< Re < 10

11.3 INDUSTRIAL DEVELOPMENTS OF

VORTEX-SHEDDING FLOWMETERS

The use of this instability for velocity measurement was recognized at an early stage

(cf. Roshko 1954) and has led to its use in flowmeters and probes, (cf. Medlock 1976).

Most modern meters use bluff bodies that are not circular in cross-section and

that have a sharp shedding edge to remove the shifts resulting from changes in the

boundary layer. Coherent shedding appears to take place along the length of the

bluff body if the length (in this case the diameter of the pipe) is in the range of

about 2.5 to 4 bluff body widths (Zanker and Cousins 1975). By

coherence

we mean

that the vortex is shed from end to end of the bluff body on the same side at the

same time.

(a)

Figure

11.2.

Diagram of vortex formation behind a bluff body: (a) Start of forma-

tion of new vortex; (b) Opposite vorticity starting to be entrained; (c) Opposite

vorticity breaks and separates vortex.

11.3 INDUSTRIAL DEVELOPMENTS OF VORTEX-SHEDDING FLOWMETERS

255

For a cylindrical bluff body, it is found that the strength of the force caused by

the vortex shedding is greatest when D/w is about 2.8 because the shedding becomes

more coherent and enhanced. By this means, the shedding becomes approximately

two-dimensional. For a flowmeter where the shedder spans the pipe, it is advan-

tageous if its width is about a quarter of the pipe diameter. The wall of the tube

then acts as end plates for the shedder and encourages coherent and strong vortex

shedding. Coherence and stability of the shedding range has been achieved down

to about Re = 10

, where the Reynolds number is based on bluff body width (cf.

Takamoto and Komiya 1981 who described a positive feedback system to extend the

range below Re = 2,000).

In flow measurement, strong pressure pulsations caused by the shedding will

enhance the signal, resulting in easier detection of shedding.

Three-dimensionality in the flow stream (e.g., turbulence, velocity gradients,

and swirl) has detrimental effects on the quality of the vortex shedding (Zanker and

Cousins 1975, cf. El Wahed and Sproston 1991 and Robinson and Saffman 1982).

Except in carefully controlled conditions, a slender cylinder in the flow will create

vortex shedding, which varies in strength and phase along its length (cf. Saito et al.

1993).

11.3.1 EXPERIMENTAL EVIDENCE OF PERFORMANCE

The vortex flowmeter, therefore, consists of

bluff body, usually of noncircular cross-

section, spanning the pipe and having a width of a quarter to a third of the pipe

diameter (Figure 11.3). Zanker and Cousins (1975) provided a useful review of early

work on vortex flowmeters (cf. Tsuchiya et al. 1970 and Cousins et al. 1973) and the

various effects that take place.

As the body becomes thicker (w increases) and the velocity past the body, there-

fore,

increases for a fixed volumetric flow rate, Equation (11.1) suggests that the

SENSOR CONNECTION

FLOW TUBE

FLOW

BLUFF

BODY

Figure 11.3. Diagram of vortex flowmeter.

SHED

VORTICES

256 VORTEX-SHEDDING, SWIRL, AND FLUIDIC FLOWMETERS

co 1000

800

600

<400

(a)

PRACTICAL

u u

-W02

—iQI

du-0 1

OUJ

too

/°

iii

i i i

01 02 03 04 05 06

w/D

01 02 03 04 05 06

w/D

(b)

Figure

11.4.

Pulses per unit volume and lift force for cylindrical bodies (from Zanker and Cousins

1975;

reproduced with permission from

NEL):

(a) Pulses per unit volume against blockage ratio;

(b) Lift coefficient against blockage ratio.

frequency and hence the pulses per unit volume of fluid may change. We shall refer

to this ratio as the K factor (see Section 1.4). Figure 11.4(a) for a cylindrical bluff

body shows this factor reducing to a minimum at a w/D of about 0.35. When w is

of order 0.1D, the strength of the lift coefficient is low [Figure 11.4(b)]

;

but when it

has the value 0.35D, it is at a peak. Zanker and Cousins suggested that this strength

was due to two effects that fortunately occur together:

• coherent shedding due to the shorter length of the bluff body between the end

plates of the wall of the tube;

• the accelerating flow through the smaller space between the body and the tube

wall, leading to a more uniform profile and less dependence on the upstream

flow profile.

An important reason for the choice of the minimum is that small variations in

w/D have a minimal effect on the frequency.

The constancy of the meter factor is shown in Figure 11.5 for a Reynolds number

range of nearly

1,000:1

with air and water where the scatter was usually within

±0.5%

(Zanker and Cousins 1975). White et al. (1974) gave a calibration curve

E 2430r

AUDIBLE

CAVITATI0N

8 10

2 A 6 8

PIPE REYNOLDS NUMBER

Figure 11.5. Typical calibration curve for a 3-in. (75-mm) vortex meter (from Zanker and Cousins

1975;

reproduced with permission from NEL).