Introduction

The Bernoulli Equation

In a flow metering device based on the Bernoulli Equation the downstream pressure after an obstruction will be lower than the upstream pressure before. To understand orifice, nozzle and venturi meters it's therefore necessary to explore the Bernoulli Equation.

Assuming a horizontal flow (neglecting minor elevation differences between measuring points) the Bernoulli Equation can be modified to:

p1 + 1/2 ρ v12 = p2 + 1/2 ρ v2

2         (1)

where

p = pressure

ρ = density

v = flow velocity

The equation can be adapted to vertical flow by adding elevation heights h1 and h2. Assuming uniform velocity profiles in the upstream and downstream flow - the Continuity Equation can be expressed as

q = v1 A1 = v2 A2         (2)

where

q = flow rate

A = flow area

Combining (1) and (2), assuming A2 < A1, gives the "ideal" equation:

q = A2 [ 2(p1 - p2) / ρ(1 - (A2 / A1)2) ]1/2         (3)

For a given geometry (A), the flow rate can be determined by measuring the pressure difference p1 - p2. The theoretical flow rate q will in practice be smaller (2 - 40%) due to geometrical conditions. The ideal equation (3) can be modified with a discharge coefficient:

q = cd A2 [ 2(p1 - p2) / ρ(1 - (A2 / A1)2) ]1/2         (3b)

where

cd = discharge coefficient

The discharge coefficient cd is a function of the jet size - or orifice opening – the area ratio = Avc / A2

where

Avc = area in "vena contracta"

"Vena Contracta" is the minimum jet area that appears just downstream of the restriction. The viscous effect is usually expressed in terms of the no dimensional parameter Reynolds Number - Re.

Due to the Bernoulli and Continuity Equation the velocity of the fluid will be at its highest and the pressure at the lowest in "Vena Contracta". After the metering device the velocity will decrease to the same level as before the obstruction. The pressure recovers to a pressure level lower than the pressure before the obstruction and adds a head loss to the flow.

Equation (3) can be modified with diameters to:

q = cd π/4 D22 [ 2(p1 - p2) / ρ(1 - d4) ]1/2         (4)

where

D2 = orifice, venturi or nozzle inside diameter

D1 = upstream and downstream pipe diameter

d = D2 / D1 diameter ratio

π = 3.14

Equation (4) can be modified to mass flow for fluids by simply multiplying with the density:

m = cd π/4 D22 ρ [ 2(p1 - p2) / ρ(1 - d4) ]1/2         (5)

When measuring the mass flow in gases, its necessary to considerate the pressure reduction and change in density of the fluid. The formula above can be used with limitations for applications with relatively small changes in pressure and density.

The Orifice Plate

The orifice meter consists of a flat orifice plate with a circular hole drilled in it. There is a

pressure tap upstream from the orifice plate and another just downstream. There are in general

three methods of placing the taps. The coefficient of the meter depends upon the position of taps.

Flange location - Tap location 1 inch upstream and 1 inch downstream from face of orifice

"Vena Contracta" location - Tap location 1 pipe diameter (actual inside) upstream and 0.3 to 0.8 pipe diameter downstream from face of orifice

Pipe location - Tap location 2.5 times nominal pipe diameter upstream and 8 times nominal pipe diameter downstream from face of orifice

The discharge coefficient - cd - varies considerably with changes in area ratio and the Reynolds number. A discharge coefficient cd = 0.60 may be taken as standard, but the value varies noticeably at low values of the Reynolds number.

Vena Contracta

When a tank fitted with an orifice, the particles, in order to Flow out through the orifice, moves towards the orifice from all directions. A few of the particles first moves downward, then take a turn to enter into the orifice, and then finally flow through it. While taking turn to enter into to the orifice, the liquid particles lose some energy. Due to which, it is observed that the Jet after leaving the orifice, gets contracted.

The maximum contraction takes place at a section slightly on the downstream side of a orifice, where the Jet of Water is more or less horizontal. Such a section is known as "Vena Contracta" now we will see the four important hydraulic coefficients. These coefficients are also known as Orifice Coefficients.

Coefficient of Contraction

The ratio of area of the jet at vena contracta, to the area of the orifice, is known as "coefficient of contraction”. Its value will varies slightly with the available head of the liquid, size and shape of the orifice. An Average Value of Coefficient of Contraction is 0.64

Coefficient of Velocity

The ratio of actual velocity of jet at vena contracta, to the theoretical velocity is known as "coefficient of velocity”. The difference occurs between the velocities due to friction of the orifice. For the sharp edge orifice, the value of coefficient of velocity increases with the head of water.

Coefficient of Discharge

The ratio of an actual discharge through an orifice to the discharge, is known as "coefficient of discharge”. The value of coefficient of discharge varies with the value of coefficient of contraction and coefficient of velocity. An average is about 0.62

Apparatus

· Cussons Hydraulics Bench P6100

Cussons Elementary Orifice Set P6223 with 3, 5 ,and 8mm diameter square edged orifices.

Cussons Constant Head Inlet Tank P6103

Hook Gauge and horizontal rails

Stopwatch; Steel Rule and Internal Calliper

The Hydraulics Bench: This consists of a steel frame which supports a fibreglass worktop with

integral weir channel and volumetric measuring tank, a sump tank, variable speed centrifugal

water pump(s) with associated pipe work and valves. The measuring tank is provided with a

sight glass and scale and is stepped, with a 10 litre lower portion and a 35 litre upper portion to

allow accurate measurements of both low and high flow rates. For low flow rates only the lower

section of the measuring tank should be used. For high flow rates use the 15 litre mark as the

starting point, to ensure that the hold up of water as it flows across the tank does not cause an

error. A V-notch weir at the end of the weir channel has a scale calibrated in litres/min, and this

provides a crude indication of high flow rates. The measuring tank discharges into a fibreglass

sump tank of about 120 litres capacity, via a quick acting ball valve located in the connecting

pipe work. An overflow pipe prevents the sump from running dry. A centrifugal pump delivers

water to the supply hose at the bench top. The flow rate of water is controlled both by a chrome-

plated valve, and by a variable pump-speed control. Various accessories can be mounted on the

bench to perform different experiments in fluid mechanics.

The Constant Head Inlet Tank: This provides a constant head of water at a maximum of either

250 mm or 500 mm depending on the overflow pipe arrangement that is selected. The tank is

fitted with two screwed connection points, one in the side and one in the base for the attachment

of experimental components. Water enters the base of the tank from the bench supply hose, and

is then distributed within the tank by a vertical perforated sparge pipe. This arrangement avoids

excessive turbulence and enables a steady level surface to be maintained. The “Zero” of the scale

coincides with the centre of the side outlet position.

Horizontal Rails & Hook Gauge: The rails are PVC rectangular flat strips drilled to fit precisely

over pegs at the far right on either side of the weir channel. The rails have locating 31 holes at

50mm spacing to receive the locating spigots on the Hook Gauge. The Hook Gauge is a vertical

frame with shoes that ride the horizontal rails. A vernier scale on the vertical leg is adjusted

along the main scale by means of a knurled nut. Attached to the vernier piece is a “JHook”

holder. This is so arranged that when the J-Hook is inserted, the top of the “J” is flush with the

top of the holder when the “cross-wire” at the “J” is at the “Zero” mark of the Constant Head

Inlet Tank measurement scale.

Procedure

Two plastic rails were placed on the worktop of the bench, engaged onto the locating

pegs. The rail with the distance calibrations were to the front of the bench. the Constant

Head Inlet Tank was positioned onto the worktop of the bench, at the left end of the weir

channel, such that the right hand support feet engaged with the locating pegs.

One of the three orifices was selected for use in the experiment. It was fitted into the

screwed hole in the side of the tank. The hole in the base was blanked off with the

blanking plug. The supply hose was connected from the bench to the connection at the

rear of the inlet tank base. The overflow hose was fitted to the front of the inlet tank base,

and the end placed into the overflow pipe of the volumetric measuring tank.

It was ensured that the water level in the sump tank was near the top. The inlet and outlet

valves for pump #1 were fully opened and the bench regulating valve (chrome plated)

was closed. The knob on the speed control unit was set to the mid-range position. The

pump was then switched on and the bench regulating valve and speed control adjusted

until the level in the tank was stable at 500 mm and there was a small overflow.

The flow rate was measured through the orifices by means of the volumetric measuring

tank and stopwatch. The measuring tank outlet valve was closed. The stop watch was

started when the water level in the measuring tank was at zero/another convenient mark.

The collection of a suitable quantity of water was timed. this was considered a "Low

Flow Rate" experiment. However, for very low discharge, a two minute collection time

was considered adequate. When the measurement was made, the measuring tank outlet

valve was opened fully. The Procedure was repeated with sequentially reduced values of

head down to about 250 mm, inclusive so that at least 5 sets of readings were obtained.

The e head was set back at 500 mm, The hook gauge was placed in position on top of the

rails to trace the trajectory of the jet. The locating holes in the rails determined the x

(horizontal)values of the trajectory. The hook gauge scale was reading zero (y or vertical

value) when the cross wired was leveled with the center of the orifice.The hook gauge

was moved along the rails, to record the values and x and y for the trajectory of the jet. at

each position , it was ensured that the cross wire was in the center of the jet. The hook

gauge was then removed.

The bench regulating valve was closed with the head adjusted to a constant value of 500

mm. The time taken for the water level in the inlet tank to fall from 450 mm to 300 mm

was measured. This measurement was repeated and the average of the two times taken.

The procedure was repeated timing the fall from 450 mm to 200 mm and then fom 450

mm to 100 mm. 32 Times for the fall from 300 to 200 mm, from 300 to 100 mm, and

from 200 to 100 mm was obtained by subtraction. The internal diameter of the inlet tank

was measured with a steel rule. The bench regulating valve was closed, ensuring that the

measuring tank outlet valve was opened. The power supply was switched off, the pump

inlet valve closed, and the pump speed control turned to the minimum speed position.

Results table

Table 1.

Volume Time

Discharge Head ReadingQ H0 √Ho

(mL) (s) 10-6 m/s (mm) (√m)12 120.5 0.10 500 22.36111 120.25 0.09 450 21.21310 120.22 0.08 400 20.000

9.5120.07

5 0.08 350 18.7089.8 120.21 0.08 300 17.3219 120.5 0.07 250 15.811

Table 2.

Trajectory Jet @ Ho = 500mm

X value Y value √Y(mm) (m) (mm) (m) (√m)

10 0.01 6 0.006 0.07715 0.02 13 0.013 0.11420 0.02 22 0.022 0.14825 0.03 36 0.036 0.19030 0.03 50 0.05 0.22435 0.04 68 0.068 0.26140 0.04 86 0.086 0.29345 0.05 110 0.11 0.332

Table 3.

Head Reading Time

H1 H2 H1 H2 √H1 √H2 √H1 - √H2 T1 T2 Taverage

(mm) (mm) (m) (m) (√m) (√m) (√m) (s) (s) (s)450 300 0.450 0.300 0.671 0.548 0.123 13.48 13.48 13.48450 300 0.450 0.300 0.671 0.548 0.123 13.66 13.88 13.77450 200 0.450 0.200 0.671 0.447 0.224 24.62 24.62 24.62450 200 0.450 0.200 0.671 0.447 0.224 24.56 24.49 24.53450 100 0.450 0.100 0.671 0.316 0.355 38.89 38.48 38.69450 100 0.450 0.100 0.671 0.316 0.355 28.74 38.6 33.67

Graphs

Graph 1. graph of flow rate Q vs Head reading√Ho

0.450 0.500 0.550 0.600 0.650 0.700 0.7500

0.00000002

0.00000004

0.00000006

0.00000008

0.0000001

0.00000012

f(x) = 1.07827253625882E-07 x + 1.93389080279337E-08

graph of flow rate Q vs Head reading√Ho

head reading √Ho

flow

rate

Q (m

3/s)

Graph 2. showing graph of √y vs. x for the trajectory of the jet

0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.0500.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350f(x) = 7.25433885626226 x + 0.00535977867178886R² = 0.999607677306934

graph of √y vs. x for the trajectory of the jet

x (m)

√y (m

)

Graph 3. Showing Graph of Discharge Time (T) vs. Head Reading √H1 - √H2

0.100 0.150 0.200 0.250 0.300 0.350 0.40005

1015202530354045

f(x) = 108.161370504415 x + 0.342242382894636R² = 0.999920704132636

Graph of Discharge Time (T) vs. Head Read-ing √H1 - √H2

√H1 - √H2 (m)

Tim

e (s

)

Discussion

The results were tabulated after the lab was completed to be analyzed it was observed that there

was a gap between the experimental values and the theoretical values. When the graph of Q vs.

√Ho was plotted the coefficient of discharge was found to be 0.18 compared to the theoretical

value of 0.62. The calculated value of the coefficient of discharge had a 71% error. For the

trajectory of the jet √y was plotted against x the coefficient of velocity was found to be 1.37 with

a percentage error of 40% and the coefficient of contraction was found to be 131 with a

percentage error of 79%. From observing the graph of √y was plotted against x, the intercept was

found to be -0.0021m from the origin. The coefficient of discharge of the orifice was to be 0.12

compared to the theoretical value 0.62 which had percentage error of 80%. The value of the

experimental and the theoretical value had a large gap. However, this did not prove that derived

equation was incorrect but that there were errors encountered in the lab.

Source of error and uncertainty

Air could have been trapped in the tubes which would result in incorrect pressure head

readings. It was ensured that air was removed from the tubes by tapping them.

The reaction time of the experimenter could have been too slow or too fast resulting in

incorrect time readings which would result in an incorrect volume flow rate reading. This

was reduced by using two experimenters and taking the averages of both the times

obtained.

Incorrect manometer readings could have been taken. It was ensured that the manometer

tapings were read at eye level.

References

o Douglas, J.F. Gasiorek, J.M and Swaffield, J.A. (2001)Fluid Mechanics 4th ed. London:

Prentice Hall.Chapter 6 & 17.

o Munson, B.R. Young, D.F and Okiishi, T.H. (2002) Fundamentals of Fluid Mechanics.

4th ed. New York: John Wiley & Sons, Inc. Chapter 8.

o Discharge Coefficient, McGraw-Hill Dictionary of Scientific and Technical Terms. McGraw- High Companies, Inc., 2003

o http://www.engineeringtoolbox.com/orifice-nozzle-venturi-d_590.html

Sample calculations

From second row in Table 1.1 Ho = 455mm

Converting from mL to m3

1 litre = 10-3 m3

12 * 10-3 litre = 12 x10-6 m3

Discharge, Q:

Q = Volume / Time = ( 12x10-6 m3) / (120.5 s) = 0.1x0-6 m3/s

Head Reading, Ho1/2:

Ho1/2 = 0.675 = 0.077 m1/2

Using the first row of values from Table 2, at X = 10cm

Y1/2 = (.006m) = 0.077 m1/2

Using the first row of values from Table 3

Discharge time, Tavg.:

Tavg = ½ ( T1 + T2 ) = ½ ( 13.48 + 13.48) = 13.48s

Head Reading (difference), H11/2 , H2

1/2:

H11/2 - H2

1/2 = (0.671 – 0.548) = 0.123 m1/2

Analysis and Interpretation:

Defining Parameters:

Q – Flow rate

Cd – coefficient of discharge

d – diameter

Ao - Area

g – Acceleration due to gravity at UTECH

Ho – Head Reading

Section A:

The relationship between Q and √Ho is given by:

Q = (Cd Ao√2g) √Ho

It can therefore be deduced that the slope m is given by:

m = (Cd Ao√2g) therefore

Cd = m / Ao√2g

Ao = (π/4)(d2) = (π/4)(3 * 10-3m)2 = 7.069 * 10-6 m2

g = 9.784 m/s2 (at UTECH)

m = 22.26 * 10-6 m5/2 / s (gradient from plot of Q vs. H01/2)

Cd = (22.26 * 10-6 m5/2 / s) / (7.069 * 10-6 m2 * √2 * 9.784 m1/2)

Cd = 0.712

Section B:

The relationship between Cv and X is given by:

Cv = ((X2)/(4YHo))1/2

Rearranging the above equation produces:

√Y = ((1/2 Cv √Ho) X

From relationship written in the previous line then the slope m, is given by

m = 1/2 Cv √Ho where m = 0.7162 m-1/2 (From plot of Y1/2 vs. x)

Cv = 0.6981 / √Ho , where Ho = 0.500m

Cv = 0.6981 / 0.500 = 0.9873

Cv = 0.9873

Since Cd = Cc * Cv therefore,

Cc = Cd / Cv = 0.712 / 0.9873 = 0.721

Coefficient of Contraction for Orifice

Cc = 0.721

Part C:

The relationship between T and √H1 -√H2 is given by:

T = (2At / Ao Cd√2g) √H1 -√H2

It can therefore be deduced that the slope m is given by:

m = (2At / Ao Cd √2g) therefore

Cd = (2At / Ao m √2g)

Ao = (π/4)(d2) = (π/4)(3 * 10-3m)2 = 7.069 * 10-6 m2

At = (π/4)(d2) = (π/4)(104 * 10-3m)2 = 0.0085m2

g = 9.784 m/s2 (at UTECH)

m = 86.547 s /m1/2 (gradient from plot of T vs. √H1 -√H2 )

Cd = (2 *0.0085m2 ) / (7.069 * 10-6 m2 *86.547 s /m1/2* √2 * 9.784 m1/2 / s )

Cd = 6.28

Coefficient of Discharge Cd , for orifice Cd = 6.28