THE VENTURI AND NOZZLE.pdf

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hydrostatic. Thus, the pressure difference measured at the DP cell is the

same as that at the elevation of the probe, because the static head between

point 1 and the pressure device is the same as that between point 2 and the

pressure device, so that P P2 P1.We usually want to determine the total flow rate (Q) through the

conduit rather than the velocity at a point. This can be done by using

Eq. (10-1) or Eq. (10-2) if the local velocity is measured at a sufficient

number of radial points across the conduit to enable accurate evaluation

of the integral. For example, the integral in Eq. (10-2) could be evaluated

by plotting the measured vr values as vr vs. r2, o r a s rvr vs. r

[in accordance with either the first or second form of Eq. (10-2), respec-

tively], and the area under the curve from r 0 to r R could be deter-

mined numerically.The pitot tube is a relatively complex device and requires considerable

effort and time to obtain an adequate number of velocity data points and to

integrate these over the cross section to determine the total flow rate. On the

other hand the probe offers minimal resistance to the flow and hence is very

efficient from the standpoint that it results in negligible friction loss in the

conduit. It is also the only practical means for determining the flow rate in

very large conduits such as smokestacks. There are standardized methods

for applying this method to determine the total amount of material emitted

through a stack, for example.

III. THE VENTURI AND NOZZLE

There are other devices, however, that can be used to determine the flow

rate from a single measurement. These are sometimes referred to as

obstruction meters, because the basic principle involves introducing an

obstruction (e.g., a constriction) into the flow channel and then mea-

suring the pressure drop across the obstruction that is related to the flow

rate. Two such devices are the venturi meter and the nozzle, illustrated in

Figs. 10-2 and 10-3 respectively. In both cases the fluid flows through a

reduced area, which results in an increase in the velocity at that point.

The corresponding change in pressure between point 1 upstream of the

constriction and point 2 at the position of the minimum area (maximum

velocity) is measured and is then related to the flow rate through

the energy balance. The velocities are related by the continuity equation,

and the Bernoulli equation relates the velocity change to the pressure

change:

1V1A1 2V2A2 10-5

Flow Measurement and Control 295


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For constant density,

V1 V2A2

A1 10-6

and the Bernoulli equation is

P2 P1

1

2 V22 V

21 ef 0 10-7

where plug flow has been assumed. Using Eq. (10-6) to eliminate V1 and

neglecting the friction loss, Eq. (10-7) can be solved for V2:

V2 2P

1 4

1=2

10-8

where P P2 P1 and d2=D1 (where d2 is the minimum diameter

at the throat of the venturi or nozzle). To account for the inaccuracies

introduced by assuming plug flow and neglecting friction, Eq. (10-8) is

written

V2 Cd2P

1 4

1=2

10-9

296 Chapter 10

FIGURE10-2 Venturi meter.

FIGURE10-3 Nozzle.


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where Cd is the discharge or venturi (or nozzle) coefficient and is deter-

mined by calibration as a function of the Reynolds number in the conduit.

Typical values are shown in Fig. 10-4, where

NReD D1V1

and NRed

d2V2

NReD=:

Because the discharge coefficient accounts for the non-idealities in the

system (such as the friction loss), one would expect it to decrease with increas-

ing Reynolds number, which is contrary to the trend in Fig. 10-4. However, the

coefficient also accounts for deviation from plug flow, which is greater at lower

Reynolds numbers. In any event, the coefficient is not greatly different from

1.0, having a value of about 0.985 for (pipe) Reynolds numbers above about

2 105, which indicates that these non-idealities are small.

According to Miller (1983), for NReD >4000 the discharge coefficient

for the venturi, as well as for the nozzle and orifice, can be described as a

function ofNReD and by the general equation

Cd C1 b

NnReD10-10

where the parameters C1, b, and n are given in Table 10-1 as a function of.

The range over which Eq. (10-10) applies and its approximate accuracy are

given in Table 10-2 (Miller, 1983). Because of the gradual expansion


FIGURE 10-4 Venturi and nozzle discharge coefficient versus Reynolds number.

(From White, 1994.)


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298 Chapter 10

T

ABLE

10-1

Va

lues

for

Disch

arge

Coe

ffic

ien

tParame

ters

a

inEq

.(10

-10)

Reyno

ldsnum

ber

term

Coe

ffic

ien

t

Exponen

t

P

rimary

dev

ice

Disc

hargecoe

ffic

ien

tC1

atinfin

ite

Reyno

ldsnum

ber

b

n

V

en

turi

Mac

hine

dinlet

0.9

95

0

0

Roug

hcas

tinlet

0.9

84

0

0

Roug

hwe

lde

ds

hee

t-iron

in

let

0.9

85

0

0

U

niversa

lven

turi

tubeb

0.9

797

0

0

L

o-L

oss

tubec

1:

05

0:

471b

0:

564b

2

0:

514b

3

0

0

N

ozz

le

ASME

longra

dius

0.9

975

6:

53b

0:

5

0.5

ISA

0.9

900

0

.2262b

4:

1

1708

8936b

1.1

5

19;

779b

4:

7

Ven

turinozz

le(ISA

inlet)

0.9

858

0

.195b

4:

5

0

0


5/8


O

rifice

Corner

taps

0:

5959

0:

0312b

2:

1

0:

184b

8

91

.71b

2:

5

0.7

5

Flange

taps

(D

inin

.)

D

2:

3

0:

59590:

0312b

2:

1

0:

184b

80:

09

b4

D1

b4

;

0:

0337

b3

D

91:

71b

2:

5

0.7

5

2D

2:

3d

0:

59590:

0312b

2:

1

0:

184b

80:

039

b4

1

b4

0:0337

b3

D

91:

71b

2:

5

0.7

5

Flange

tapsD*inmm

)

D*58:

4

0:

59590:

0312b

2:

1

0:184b

82:

286

b4

D*1

b4

0:

856

b3

D*

91:

71b

2:

5

0.7

5

50:

8D*58:

4

0:

59590:

0312b

2:

1

0:

184b

80:

039

b4

1

b4

0:

856

b3

D*

91:

71b

2:

5

0.7

5

D

an

dD=2taps

0:

59590:

0312b

2:

1

0:

184b

80:

039

b4

1

b4

0:

0158b

3

91:

71b

2:

5

0.7

5

2

1 2D

an

d8D

tapsd

0:

59590:

461b

2:

10:

48b

80:

039

b4

1

b4

91:

71b

2:

5

0.7

5

a

DetailedReynoldsnumber,line

size

,betaratio

,andotherlimita

tionsaregiveninTable10-2.

b

From

BIFCALC-4

40/441;them

anufacturershouldbeconsulted

forexactcoefficientinformation

.

c

Derivedfrom

theBadgermeter

,Inc

.Lo-Losstubecoefficientcu

rve;themanufacturershouldbe

consultedforexactcoefficientin

formation

.

d

From

Stolz(1978).

Source:Miller(1983).


6/8

300 Chapter 10


7/8



8/8

designed into the venturi meter, the pressure recovery is relatively large, so

the net friction loss across the entire meter is a relatively small fraction of the

measured (maximum) pressure drop, as indicated in Fig. 10-5. However,

because the flow area changes abruptly downstream of the orifice and noz-zle, the expansion is uncontrolled, and considerable eddying occurs down-

302 Chapter 10

FIGURE 10-5 Unrecovered (friction) loss in various meters as a percentage of

measured pressure drop. (From Cheremisinoff and Cheremisinoff, 1987.)

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THE VENTURI AND NOZZLE.pdf