3. FLOW RATE MEASUREMENT

WITH ORIFICE METERS

3.1 INTRODUCTION

The orifice meters are devices used to measure the flow rate of fluids

through closed ducts. The principle underlying the operation of the orifice meters

is based on increasing in the velocity of flow, which leads to a pressure drop of

the fluid through orifice. Thus, the quantity of fluid can be expressed in terms of

the pressure variation across the orifice meter.

Fig. 3.1 – Working principle of the orifice meter

Figure 1 shows the principle of this method of flow metering, with an

orifice meter having a sharp-edged circular orifice and fitted into a duct between

two flanges. Upstream the orifice, in the cross section , the fluid has the (mean)

velocity and the pressure . Downstream the orifice a vena contracta occurs

and the cross sectional area decreases to a minimum value (Vena

contracta is the section in a fluid stream where the diameter of the stream is the

least, and consequently the fluid velocity is maximum); the parameters of the

2

fluid in this section are and . Further downstream, the fluid stream gradually

expands to the normal flow.

Continuity equation of incompressible fluids gives the (volumetric) flow

rate ( ) as

(3.1) The (maximum) velocity may be expressed from Bernoulli’s

equation between section (1) and section (2)

(3.2)

where , are Coriolis' coefficients of the velocity correction in both sections

(or the energy-head coefficients)

is the coefficient of head loss (energy loss) due to the change in

velocity (for details see Head losses in forced pipes). Denoting with

(3.3)

the coefficient of contraction, or expansion factor (the ratio between the area of

the stream at the vena contracta to the area of the orifice), the continuity

equation (3.1) gives

(3.4)

where is the diameter of the orifice,

is the inner diameter of the duct ( ). Replacing the speed according to the equation (3.4) into equation

(3.2), the fluid velocity in the minimum section of flow may be expressed as

(3.5)

Hence, the flow rate in Eq. (3.1) may be written as

(3.6)

3

The ratio what follows the contraction coefficient in previous equation

represents the flow rate coefficient of the orifice meter. Usually, it is denoted by

α, and for any orifice type it is a function of the Reynolds number ( ) and

diameters ratio .

According with standards (STAS 7374-83), the volumetric flow rate ( )

and mass flow rate ( ) must be computed with the following equations

(3.7)

(3.8)

because in engineering practice the pressure drop across the orifice

is measured.

3.2 ORIFICE METER CALIBRATION WITH PITÔT-PRANDTL TUBES

This paragraph presents two of traditional and efficient methods for

local velocity measurement with the Pitôt-Prandtl tubes (probes) and

computation of the mean velocity in a stream section, and

flow rate measurement with the orifice meters. There is also revealed a method for the calibration of the orifice meters.

3.2.1 Laboratory principle The goal of this practical work is to measure the flow rate of the axial fan

7 through the duct 6 as function of the pressure drop across the orifice meter (1).

All of these are performed for several air flow regimes, which are set with the

flap-type flow meter 5, also used to control the flow rate.

Later on, for each case, there will be computed the flow rate coefficient of

the orifice meter and the variation must be established.

In order to compute the average speed of the air stream, the multiple-

point velocity method is employed in this experiment. Thus, a series of velocity

measurement at points of a stream section is made with a couple of Pitôt-Prandtl

2, the first tube placed in the vertical symmetry plane of the duct, and the second

one in the horizontal symmetry plane. Both tubes have the possibility of sliding

along the corresponding diameter.

The experimental setup is shown in Figure 3.2.

4

5

Air velocity in any point of radial coordinate (measured with a ruler 3)

may be done according to the equation

(3.9)

where the following are density of the piezometric liquid,

air density (working fluid in this experiment),

vertical deflection of the piezometric liquid shown by

manometers 4 connected to the Pitôt-Prandtl tubes

(3.10) manometer factor, which depends on the angle of tube and the

piezometric liquid

length of the piezometric liquid in the manometer tube. In order to compute the average velocity ( ), the cross section of the

duct must be divided into annular sectors of equal areas, as shown in Figure

3.3.

Fig. 3.3 – Principiul de stabilire al punctelor pentru

determinarea vitezelor locale

This experiment will include 20 points of velocity measurement in the

duct cross section ( ) using the Pitôt-Prandtl tubes placed in the middle of

each sector. Thus, the mean velocity can be computed by arithmetic integration

of the equation

6

(3.11)

Thus, for current experiment, the volumetric flow rate ( ), Eq. (3.1), may

be computed with

(3.12)

and the mass flow rate ( )

(3.13)

In the upstream and the downstream from the orifice meter, there is

connected a second manometer 4, which gives a vertical deflection of the

piezometric liquid due to the pressure drop

(3.14) where

(3.15) manometer factor,

length of the piezometric liquid in the manometer tube connected

to the orifice meter. Replacing (3.14) in the expression of the mass flow, Eq. (3.8), yields

(3.16)

Since the elocities are relati ely small we can consider ε = 1. Equating

the two formulations of the mass flow, Eqs. (3.13) and (3.16), the coefficient of

the orifice meter can be expresed as

(3.17)

The coefficient ( ) must be computed for each flow case, and finally the

variation . The Reynolds number (Re) is a dimensionless quantity that

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expresses the ratio of the inertial forces to the viscous forces for a given flow, and

it is also used to characterize different flow regimes

(3.18)

where , are the dynamic viscosity and the kinematic viscosity.

(3.19)

The Sutherland's formula can be used to compute the dynamic viscosity

as function of their temperature, hence

(3.20)

where terms with index ”0” are the reference parameters.

Their density can be computed with the Equation of State, applied for two

conditions, one of them being known (as reference):

(3.21)

3.2.2 Experimental procedure

check the horizontal planes of manometers (perform the adjustments, if

necessary);

check the atmospheric pressure ( ) and temperature ( ) at time of the experiment, then compute the air density ( ) with Eq. (3.21), and

viscosity ( ), Eqs. (3.20), (3.19); set a flow case with the flap-type flow mwter and start the fan;

for each position of the Pitôt-Prandtl tubes (see coordinate in Table)

read the corresponding values of the lengths of the piezometric liquid in

the manometer tube;

read the value of the length of the piezometric liquid in the manometer

tube connected to the orifice meter;

compute vertical deflections of the piezometric liquids , Eq. (3.10) and

, Eq. (3.15), local velocities ( ), Eq. (3.9), mean velocity ( ), Eq.

(3.11), mass flow rate ( ), (3.13), flow rate coefficient of the orifice

meter ( ), Eq. (3.17), Reynolds number ( ), Eq. (3.18) and fill the Table;

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repeat the previous operations for another minimum two cases of flow;

draw the calibration curve of the orifice meter ) and the

variation of ; similarly graphs are shown in Figure 3.4;

Fig. 3.4 – Graphical layout of ) and

Phisical factors and constant used

air density for standard conditions of

temperature and pressure: și ,

dynamic viscosity for și conditions

, sau ,

]

Sutherland's constant of air,

density of the piezometric (water) liquid

density of the piezometric (alcohol) liquid

inner diameter of the duct,

diameter of the orifice meter.

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TABLE

FLOW CASE 1 2 3

Ver

tica

l Pit

ôt-

Pra

nd

tl t

ub

e

[mm] [mm] [m] [m/s] [mm] [m] [m/s] [mm] [m] [m/s]

7.20

22.6

40.3

60.3

94.3

181.5

213.5

235.5

253.5

268.5

Ho

rizo

nta

l Pit

ôt-

Pra

nd

tl t

ub

e

[mm] [mm] [m] [m/s] [mm] [m] [m/s] [mm] [m] [m/s]

7.20

22.6

40.3

60.3

94.3

181.5

213.5

235.5

253.5

268.5

10