Part 4. P6223 & P6224 Flow through orifices

TABLE OF CONTENT

1. INTRODUCTION

2. DESCRIPTION

2.1 General Description

2.2 P6223 Elementary Orifices

2.3 P6224 Advanced Orifices

3. THEORY

3.1 Introduction

3.1.1 List Of Symbols

3.2 Flow Through A Small Orifice

3.3 Trajectory Of Horizontal Jet

3.4 Time To Empty A Vessel

3.5 Bell Mouthed Orifice

3.6 Borda Mouthpiece

3.7 Borda Mouthpiece Running Full

3.8 Square And Triangular Orifices

3.9 Form Of The Jet

3.10 Orifice With External Tube

4. EXPERIMENTS

4.1 List Of Experiments

4.2 Setting Up The Orifice Experiment

4.3 Operation

 

1. INTRODUCTION

The analysis of the quantity of water which can be discharged through an orifice is arrived at in a simple, straightforward manner by the application of Bernoulli's equation. However, experimental tests typically produce a result which is only some 65% of the solution indicated by the simple analysis. The study of water flow through an orifice is therefore a classic topic to illustrate the need for a semi-empirical approach which is so often required in Mechanics of Fluids.

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2. DESCRIPTION

2.1 General Description

The Cussons Inlet Head Tank P6103 can be used for the investigation of the flow of water through a horizontal or a vertical orifice. This tank is detailed in Part 1 of the manual. Water is supplied to the tank via a hose connection to the base inlet, 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. Two threaded holes are cut into the tank in which to mount the orifice being studied, one in the tank base for 'vertical' discharge, and the other in the tank side for 'horizontal' discharge. An orifice can be screwed into either of the threaded holes and the unused aperture sealed with the blanking plug provided. The union adaptor piece supplied with the Inlet Head Tank is not required for the orifice experiments and should be removed before insertion of the orifice under test. A scale is mounted on the side of the tank to enable the height of the water above either orifice position to be determined. The zero of the scale coincides with the centre of the side outlet position, but note that the face of the bottom outlet position is 38mm below the centre line of the side outlet. When an orifice is fitted in the horizontal discharge position a Trajectory Profile Hook Gauge P6107 can be used to determine the jet profile. The details of the gauge are given in Part 1 of the manual. Details of the orifices are given below and are illustrated in figure 1.

2.2 P6223 Elementary Orifices

A set of three circular orifices are supplied in a plastic case. Each orifice is mounted in a 1" BSP threaded orifice holder, secured between an 'O' ring and a circlip. The orifice details are :-

a) 3mm diameter orifice, square edged 061mm thick
b) 5mm diameter orifice, square edged 122mm thick
c) 8mm diameter orifice, square edged 122mm thick

2.3 P6224 Advanced Orifices

A set of four orifices each mounted in a threaded orifice holder, a pair of calipers and an orientation tool are supplied in a plastic case. The orifice details are :

a) Borda mouthpiece consisting of an orifice with extended upstream inlet tube which projects into the inlet head tank thus preventing flow of water across the face of the orifice. The leading edge of the inlet tube is reduced to a knife edge.

Orifice diameter 8mm
Length of inlet tube 7mm
Internal Diameter of inlet tube 8mm

b) Bell mouthed orifice having a bell shaped entry section.

Entry radius 2mm
Bell mouth semi angle 23
Orifice diameter 8mm

c) Triangular shaped orifice of side 10mm (equilateral), square edged 122mm thick.

d) Square shaped orifice of side 7mm square edged 122mm thick.

 

P6223 Elementary Orifice Set

P6224 Advanced Orifice Set

Figure 1 Orifice Details

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3. THEORY

3.1 Introduction

There are two methods of analysing the discharge of a vessel through an orifice. The first of these applies when the orifice is small in comparison with the head above the orifice, this is known as small orifice analysis. In this case variations in velocity with height within the jet of water can be ignored and the velocity is assumed to be constant.

The alternative analysis for large orifices takes into account the variation of velocity with height within the jet of water issuing from the orifice.

3.1.1 List Of Symbols

a

cross sectional area of orifice

m2

ac

cross sectional area at vena contracta

m2

A

cross sectional area of orifice tank

m2

Cc

coefficient of contraction

 

Cd

coefficient of discharge

 

Cv

coefficient of velocity

 

d

length of side of square

m

F

force

N

g

acceleration due to gravity 9807

m/s2

h

difference in height

m

H

head

metres

k

constant

 

l

length of side of equilateral triangle

m

m

area ratio

 

M

mass flow rate

kg/s

n

constant

 

P

pressure

bars

volume flow rate

m3/sec

t

time

secs

V

velocity

m/s

W

mass

gm

x

horizontal distance

m

y

vertical distance

m

Z

height

m

3.2 Flow Through A Small Orifice

Consider a small orifice in either the base or side of a vessel with the head of water above the orifice kept constant.

Figure 2 Discharge Through an Orifice

Applying Bernoulli's theorem between the surface of the water 1 and the orifice o yields

However P1 = Po = atmospheric pressure

hence substituting these into Bernoulli's equation gives

 

In other words, the theoretical velocity of the water passing through the orifice is given by

and hence the quantity of water being discharged through the orifice is given by

=

However in practice the discharge is always less than this theoretical amount due to the viscosity of the fluid, to surface tension and due to resistance of the air. The disparity between the theoretical discharge velocity and the actual discharge velocity is allowed for by introducing a factor Cv known as the Coefficient of Velocity so that

If the discharge from a sharp edged orifice is examined closely it will be observed that the minimum diameter of the jet of water discharging from the orifice is smaller than the orifice diameter. The plane at which this occurs is known as the Vena Contracta, which is the plane where stream lines first become parallel. Applying the discharge equation at the vena contracta

which can be written as

where

or more simply as

where

Typical values of Cd range from 06 to 065, i.e. the actual flow through a sharp edged orifice is approximately 60% of the theoretical value. The value of the Coefficient of Discharge may be determined by measuring the quantity of water discharged over a period of time whilst the head is maintained at a constant level.

3.3 Trajectory Of Horizontal Jet

Consider the trajectory of a jet formed by the discharge of water through an orifice mounted in the side of a tank. The jet will be subjected to a downward acceleration of g due to gravity.

Figure 3 Trajectory of Horizontal Jet

Taking the origin of co-ordinates at the vena-contracta and applying the laws of motion in the horizontal and vertical planes then ignoring any effect of air resistance on the jet.

In the horizontal direction

In the vertical direction

solving simultaneously by eliminating t

 

but

therefore

It may be difficult to accurately locate the position of the vena contractor in which case measurements may be taken from any convenient datum for two points in the trajectory and in which case it can be shown that

3.4 Time To Empty A Vessel

Consider a vessel being emptied through an orifice in the base (or side) of the vessel.

Figure 4 Emptying a Vessel

Whilst the head falls from height H1 to height H2 in time T seconds, consider the situation in some small time interval d t when the head h decreases by a small amount d h.

If the cross sectional area of the vessel is A and that of the orifice is a then let the volume of liquid leaving the tank during this time interval be d q which must also be equal to the flow through the orifice.

separating the variables

and integrating between H1 at T = 0 and H2 at time t = T then

\

3.5 Bell Mouthed Orifice

For a bell mouthed orifice which has a curvature of the same shape as that of the natural stream lines entering a sharp edged orifice then the vena contracta will occur at the exit from the orifice with

so that

 

Figure 5 Bell Mouthed Orifice

There will of course still be a loss of head due to friction, however, empirical results indicate that the resulting Coefficient of Velocity Cv is typically 0975.

Thus

Now

and if then

Therefore the energy loss in a bell mouthed orifice is typically 1 - 095 Ho = 005 Ho, i.e. some 5% of the head is lost at a bell mouthed entry. The comparative figure for a sharp edged orifice with Cd = 06 is that

or a 64% loss of head.

 

3.6 Borda Mouthpiece

Figure 6 Borda Mouthpiece Running Free

 

Consider a Borda mouthpiece of cross sectional area a in the base of a tank, then as for any orifice.

The volume flowrate of water discharged will be given by :

and the mass flowrate will be

The force which produces this flow is

Now applying Newtons second law

Force = Rate of change of momentum

and substituting for

hence

Thus the Coefficient of Contraction for a Borda mouthpiece operating in this condition where the effluent stream is running free, i.e. does not touch the sides of the mouthpiece, is one half

and the value of the Coefficient of Discharge will be only slightly smaller since Cv @ 097 to 098.

A further observation on the discharge from a Borda mouthpiece which is running free is that the flow is laminar, the appearance of the stream will be clear and there will be no splashing where the stream meets a normal surface.

3.7 Borda Mouthpiece Running Full

With a normal Borda mouthpiece where the water in the vessel is initially quite steady, the issuing jet, after becoming parallel, does not come into contact with the sides of the orifice. If, however, the water in the tank is agitated sufficiently, the jet after contracting to the vena contracta will again expand to fill the orifice. This is termed as running full. The jet of water will be observed to be turbulent, silvery in appearance and splashing will occur when it meets a solid surface.

 

Figure 7 Borda Mouthpiece Running Full

Now in applying Bernoulli's Theorem to a Borda mouthpiece which is running full there will be a loss of head due to expansion as the flow expands back from the vena contracta to the full diameter.

expansion loss

but

hence expansion loss

Bernoulli's equation therefore becomes

and with

and with the equation reduces to

Thus

The discharge when running full is given by

whereas the discharge when running free was found in paragraph 36 to be

Reverting to the basic equation for an orifice

and comparing with

then Cd for a Borda mouthpiece running free is

and for a Borda mouthpiece running full

=

then Cd for a Borda mouthpiece running full is

The pressure at the vena contracta may be found by applying Bernoulli's equation between the plane of the vena contracta, C, and the exit plane, 0.

expansion loss

From the previous analysis Vc = 2Vo if the mouthpiece is horizontal Zc = Zo and the expansion loss is

Therefore substituting

and re-arranging

But

and if the mouthpiece is discharging to atmosphere, then Po = 0, hence

or

The pressure at the vena contracta is therefore less than atmospheric by an amount equal to the incident head Ho. This reduction in pressure therefore explains the increase in quantity of water discharged by a full running Borda mouthpiece compared with a free running Borda mouthpiece. Since in the case of a free running Borda it is discharging against atmospheric pressure whilst in the full running Borda it is discharging against a reduced pressure of Pa - Ho.

3.8 Square And Triangular Orifices

Provided the orifice dimensions are small in comparison with the head then the results of the small orifice analysis in paragraph 2.2 apply. Thus

For the square orifice with length of side d then a = d2.

For the equilateral triangular orifice with side d then a = Ö 3 d = 0866 d2

3.9 Form Of The Jet

When a jet of water issues from a circular orifice mounted in the base of a vessel the water accelerates under normal gravitational force. As the velocity increases the cross sectional area decreases and eventually the surface tension will cause the jet to break up into droplets.

With orifices of other cross sections such as square or triangular orifices the jet undergoes a continuous oscillatory change in its section. This is again caused by surface tension which tries to pull the jet into a circular cross section. The inertia of the water transverse to the jet causes the correction of form to be overdone so that vertices become bases and bases become vertices as illustrated below.

 

Figure 8 Jet Cross Sections

With horizontal discharge the velocities in the jet increase with depth which causes the jet to become narrower in depth as the trajectory of the faster moving particles at the bottom of the jet intersect the trajectory of the slower particles at the top of the jet. Collisions between particles causes the jet to widen in the lateral horizontal direction.

3.10 Orifice With External Tube

Unlike a Borda mouthpiece which has a tube extending into the vessel an orifice with external tube does not exhibit the possibility for free running. The flow always expands after the vena contracta to fill the tube.

Figure 9 Orifice with External Tube

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4. EXPERIMENTS

4.1 List Of Experiments

Cussons P6223 Elementary Orifice Set and Cussons P6224 Advanced Orifice Set allow the following experiments to be performed.

P6223 Elementary Orifices

Experiment 1 Flow Through a Circular Orifice
Experiment 2 Trajectory of a Horizontal Jet
Experiment 3 Time to Empty a Vessel

P6224 Advanced Orifice Set

Experiment 4 Bellmouth Orifice
Experiment 5 Borda Mouthpiece
Experiment 6 Square and Triangular Orifice and the Shape of the Jet

Cussons P6103 Constant Head Inlet Tank is an essential accessory for all of these experiments. P6107 Hook Gauge and Scale is necessary for experiment 2.

4.2 Setting Up The Orifice Experiment

a)

If the Hook Gauge and Scale P6107 are to be used to measure the trajectory of horizontal jets then place the two positioning rails on the worktop of the Hydraulics Bench engaging them onto the locating pegs. Ensure that the engraved rail is placed closest to the front of the Hydraulics Bench with the engraved side uppermost.

b)

Position the Constant Head Inlet Tank P6103 onto the worktop of the Hydraulics Bench (over the Hook Gauge positioning rails, if fitted) at the left hand side engaging two of the feet of the Inlet Tank onto the locating pegs. If the orifices are to be fitted into the base of the inlet tank then the left hand support feet of the inlet tank should engage with the locating pegs so that the orifice can discharge downwards into the weir channel. If the orifice is to be fitted into the side of the inlet tank then it should be moved to the left so that the right hand support feet engage with the locating pegs.

c) 

Remove the hexagonal (37mm across flats) bush and adaptor from the side of the inlet tank. Fit the required orifice into either the screwed hole in the base or in the side and plug the unusued hole using the blanking plug provided.

d)

If using either the triangular or square orifices in the side of the inlet tank, use the orientation tool to twist the orifice into the required position.

e)

Connect the hydraulics bench flexible delivery tube to the connection provided on the rear of the inlet tank base. Insert the flexible overflow take off pipe, which is connected to the boss on the front of the inlet tank, into the overflow pipe of the volumetric measuring tank.

f)

 Remove or refit the overflow extention tube (screwed) in the inlet head tank to obtain a nominal head of 250mm or 500mm above the side orifice.

4.3 Operation

a)

Setting the Overflow   Switch on the pump and control the flow rate by either adjusting the hydraulics bench delivery valve or by adjusting the pump speed. The flow should be adjusted carefully to produce a small but constant overflow and then fine adjusted to give 250 or 500mm head as required.

b)

Flow Measurement   The discharge from the orifice may be measured using the standard weir or much more accurately by using the volumetric measuring tank and taking the time required to collect a quantity of water. The quantity should be chosen so that the time to collect the quantity is at least 120 seconds to obtain a sufficiently accurate result. Because of the action of the overflow it is not possible to use the rotameter for this experiment.

c)

Measurement of Jet Trajectory   Use the Hook Gauge to measure the trajectory of the jet ensuring that the Hook Gauge is correctly assembled as shown in figure 7 of Part 1. The angle scale should be mounted so that the long scale showing a discharging orifice logo is facing the front of the hydraulics bench. The hook gauge is used concave upwards, i.e. È shaped for measurement of the jet trajectory away from the orifice and concave downwards Ç for measurement of the jet close to the orifice. In both cases the cross wire should be aligned in the centre of the jet and the reading read from the top level of the pointer.

d)

Measurement of Head   The scale attached to the side of the inlet tank has its zero level with the centre line of the side outlet boss. The face of the bottom outlet is 38mm below the centre line of the side outlet. When the bottom outlet is used, a fixed increment should be added to the scale reading of 26mm (38 - 12) for all orifices except the Borda mouthpiece where the increment is 19mm (38 - 12 - 7).

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EXPERIMENT 1 FLOW THROUGH A CIRCULAR ORIFICE

Aim   To investigate the discharge characteristics of circular orifices subjected to a constant head.

Equipment Preparation   Prepare the equipment following the general experimental method detailed in section 42.

Vessel P6103 Constant Head Inlet Tank.

Orifice P6223 3, 5 and 8mm circular orifices.

Experimental Procedure

  1. Fit the 3mm diameter orifice into the side of the inlet head tank. Remove the overflow extension pipe. Start the pump and set up an inlet head of 25cm. Measure the flow rate using the volumetric measuring tank.

  2. Replace the overflow extension pipe and set up an inlet head of 50cm. Measure the flow rate.

  3. Remove the orifice and refit it into the base of the inlet tank and refit the blanking plug into the side of the tank. Repeat the readings with the inlet head tank levels of 25 and 50cm, which are now equivalent to a head above the orifice of 276 and 526cm.

  4. Repeat the procedure using the 5mm and 8mm orifices.

Results and Analysis

  1. Record the results on a copy of the result sheet for discharge characteristics.
  2. For each result calculate the flowrate.
  3. Plot a graph of square root of the head against the flow rate for each orifice diameter, the results should lie on a straight line passing through the origin to confirm that

Measure the slope of each graph and calculate the coefficient of discharge for each orifice from

 

EXPERIMENT 1 RESULTS SHEET - FLOW THROUGH A CIRCULAR ORIFICE

 

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EXPERIMENT 2 TRAJECTORY OF A HORIZONTAL JET

Aim   To investigate the trajectory of a horizontal jet issuing from an orifice and hence determine the coefficient of velocity for the orifice.

Equipment Preparation   Prepare the equipment to the following specification following the general method detailed in Section 42.

Tank P6103 Constant Head Inlet Tank.

Orifices Any circular orifice fitted into the side of the Inlet Tank.

Accessories P6107 Hook Gauge and Scale.

Experimental Procedure

  1. Fit the 3mm diameter orifice into the side of the inlet head tank. Remove the overflow extension pipe. Start the pump and set up an inlet head of 250mm.
  2. Measure the trajectory of the jet using the hook gauge by positioning the cross wire through the centre of the jet. Record the horizontal and vertical distances.
  3. Replace the overflow extension tube and establish an inlet head of 500mm. Measure the trajectory of the jet.
  4. Repeat the experiment using the 5mm and 8mm diameter orifices.

Results and Analysis

  1. Draw a graph of y against x to represent the trajectory.
  2. Draw a graph of
against x and draw the best straight line through the points to represent the results. Measure the slope of the line and hence calculate the coefficient of velocity from :

Notes

  1. To achieve reasonably accurate results it is necessary for the Hydraulics Bench working surface to be level. Considerable care and judgement needs to be exercised in positioning the horizontal wire in the centre of the jet. It is possible to judge the correctness of the measurement by observing that the wire divides the jet equally into two streams, the setting should be observed from more than one position.
  2. If the straight line is drawn through the origin then the results will be measured relative to the orifice whereas if the best slope is drawn, then the intercept of the line with the x axis will indicate the position of the vena contracta.

 

 

EXPERIMENT 2 RESULTS SHEET - TRAJECTORY OF HORIZONTAL JET

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EXPERIMENT 3 TIME TO EMPTY A VESSEL

Aim   To investigate the time required to empty a vessel through an orifice and hence determine the coefficient of discharge for the orifice.

Equipment Preparation   Prepare the equipment to the following specification.

Vessel P6103 Constant Head Inlet Tank with overflow extension tube fitted.

Orifice 3mm orifice from the P6223 Elementary Set.

Experimental Procedure

  1. Start the pump and open the bench regulating valve sufficiently to just maintain a constant head of water above the 50cm scale.
  2. Close the bench regulating valve as quickly as possible.
  3. Measure the time taken for the water level to fall in the inlet tank from 45cm to 30cm, 20cm and 10cm. Repeat this procedure twice more and record an average of the readings.
  4. Repeat the test using the 5mm and 8mm diameter orifices.
  5. If required, repeat the test with the orifices installed in the base of the tank.

Results and Analysis

1. Record the results on a copy of the results sheet.
2. Plot a graph of the discharge time against the difference between the square roots of the initial and final heads. Draw a straight line from the origin through the results for each orifice to confirm that

Measure the slope of the lines and calculate the coefficient of discharge for each orifice from

 3. Compare the results for the coefficients of discharge obtained from experiment 1 (which used a constant head test and volumetric flow measurement)  with the results from this experiment.

     

EXPERIMENT 3 RESULTS SHEET - TIME TO EMPTY A VESSEL

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EXPERIMENT 4 BELL MOUTHED ORIFICE

Aim   To investigate the discharge through a bell mouthed orifice.

Equipment Preparation   Prepare the equipment to the following specification.

Vessel P6103 Constant Head Inlet Tank with overflow extension tube removed.

Orifice P6224 bell mouthed orifice and P6223 8mm diameter orifice.

Experimental Procedure

  1. Start the pump and set up an inlet head of 25cm. Measure the flow rate using the volumetric measuring tank.
  2. Replace the overflow extension pipe and set up an inlet head of 50cm. Measure the flow rate.
  3. Remove the orifice and refit it into the base of the inlet tank and refit the blanking plug into the side of the tank. Repeat the readings with the inlet head tank levels of 25 and 50cm, which are now equivalent to a head above the orifice of 276 and 526cm.
  4. Repeat the procedure using the 8mm circular orifice or use the results from experiment 1.

Results and Analysis

  1. Record the results on a copy of the results sheet.
  2. Calculate the flowrate for each result.
  3. Plot a graph of the square root of the head against the flow rate for both the bell mouthed and the square edges orifices. Measure the slope of each graph and calculate the coefficient of discharge for each orifice from

 

EXPERIMENT 4 RESULTS SHEET - BELL MOUTHED ORIFICE

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EXPERIMENT 5 BORDA MOUTHPIECE

Aim   To investigate the discharge through a borda mouthpiece.

Equipment Preparation   Prepare the equipment to the following specification.

Vessel P6173 Constant Head Inlet Tank with overflow extension tube removed.

Orifice P6224 Borda Mouthpiece and P8223 8mm diameter orifice. Treat the Borda mouthpiece with a water repellant before screwing it into the inlet tank.

Experimental Procedure

  1. Start the pump and set up an inlet head of 25cm. Measure the flow rate using the volumetric measuring tank.
  2. Replace the overflow extension pipe and set up an inlet head of 50cm. Measure the flow rate.
  3. Remove the orifice and refit it into the base of the inlet tank and refit the blanking plug into the side of the tank. Repeat the readings with the inlet head tank levels of 25 and 50cm, which are now equivalent to a head above the orifice of 276 and 526cm to which must be subtracted 07cm to allow for the length of the protruding pipe of the Borda mouthpiece. Measure the jet diameter.
  4. Repeat the procedure but this time with the Borda mouthpiece running full. This condition can be promoted by agitating the water close to the mouthpiece intake with a suitable rod. If necessary, clean the water repellant from the mouthpiece.
  5. Repeat the procedure using the 8mm circular orifice or use the results from experiment 1.

Results and Analysis

  1. Record the results on a copy of the results sheet.
  2. Calculate the flowrate for each result.
  3. Plot a graph of the square root of the head against the flow rate for both conditions of the Borda mouthpiece and the square edged orifices. Measure the slope of each graph and calculate the coefficient of discharge for each orifice from

 

EXPERIMENT 5 RESULTS SHEET - BORDA MOUTHPIECE

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EXPERIMENT 6 SQUARE AND TRIANGULAR ORIFICES

Aim   To investigate the discharge through a square and triangular jet.

Equipment Preparation   Prepare the equipment to the following specification.

Vessel P6103 Constant Head Inlet Tank with overflow extension tube removed.

Orifice P6224 7mm square and 10mm equilateral triangular orifices.

P6223 Circular Orifice.

Experimental Procedure

  1. Fit the square orifice into the side of the inlet head tank. Use the orientation tool to align the orifice off or on centres as required of the
  2. c

    jet. If required use the calipers to make measurement of the jet slope.

  3. Replace the square orifice with the triangular orifice and repeat the measurement and observations. Replace the triangular orifice with the 8mm circular orifice and carefully observe the cross sectional shape of the jet. Use the calipers to carefully measure the diameter of the jet in two planes.

Results and Analysis

  1. Calculate the coefficient of discharge for each orifice.
  2. Comment on the shape of the jet.

 

EXPERIMENT 6 RESULTS SHEET - SQUARE AND TRIANGULAR ORIFICES

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