A. SUMMARY:

The topic investigated in this experiment was the venturi Flow Rig. This is very significant in chemical

engineering because the venturi meter offers the best means of measuring the flowrate of fluids out of all

the other means. The major objectives of this lab was to; compare the behaviour of real and ideal fluids

using a venture meter, to record the head loss profile through a venturi meter at different flowrates and to

estimate the coefficient of discharge of a venturi meter.

In the course of the experiment, the following equipments were used; a Perspex venture meter having

eleven (11) manometric tubes to measure the static head of the fluid, a water supply tank, a pump, a level

control device, a stop-clock to take readings of the time it took the water to discharge and a lever system to

measure the weight of water discharge.

From the experiment, it was found out that the Bernoulli's theorem only applies to ideal fluids. That is

fluids that are incompressible (constant density), irrotational (smooth flow or laminar flow), non-viscous

(fluid without internal friction) and finally a steady flow (one with constant velocity at each point. It was

also found out that the head loss across different sections of the Venturi meter were minimal when

compared to that involved in other flow-measuring device such as an orifice. From the plots of various

graphs at different flowrates, it was also found out that the static head, the velocity head and the total head

are directly related (direct proportionally) to the flowrates. Also there is a direct relationship between

pressure drop and flowrate.

One other important finding from the experiment was that the coefficient of discharge of a venturi was

higher due to the fact that the head loss is small. Therefore, it can be inferred that the smaller the head loss,

the greater the coefficient of discharge when measuring the flow of fluids. It was found out also that the

theoretical flow of fluid through venture is slightly higher than the actual value. Also the coefficient of

discharge for the venture was estimated to be 0.95 ± 0.03.

From the experiment, it can be concluded that the ventrui meter is suitable for both compressible and

incompressible fluids. This is true because when it comes to measuring the flow of compressible fluids or

real fluids, the mass flow rate for a compressible fluid can increase with increased upstream pressure,

which will increase the density of the fluid through the constriction even though the velocity will remain

constant (Holland, 1986).

Finally, it can also be concluded that for higher flow rates where power loss could become significant; the

venturi meter could be very suitable.

Page 1 of 20

B. INTRODUCTION:

There were three major objectives of this experiment. These are; to compare the behaviour of real and ideal

fluids using a venture meter, to record the head loss profile through a venturi meter at different flowrates

and to estimate the coefficient of discharge of a venturi meter.

Venturi meters are used to accurately measure fluid flow in a pipe for both compressible and

incompressible fluid. This is true because the head loss when using a venture meter is minimal when

compared to other devices for measuring fluid flow. For an incompressible fluid, its behaviour may be

explained or predicted by Bernoulli’s equation. This can be done by considering it in terms of the total

head. The total head can be calculated as:

pρg

+ v2

2 g + z = H (1)

Where:

pρg

is the static or pressure head (m)

v2

2 g is the velocity head (m)

z is the elevation head (m)

H is the total head (m)

For a horizontal venture (that is one without height), the equation becomes:

pρg

+ v2

2 g = H

(2)

This means that for an incompressible fluid, the pressure head or static head plus the velocity head is

constant or equal to the total head.

It is noteworthy to mention here that the total head (H) should have remained constant at all points along

the tube, but due to the effect of friction, some head losses were experienced. Though, these head losses

were minimal in this case since we are using a venturi meter.

Page 2 of 20

If the flow rate (Q) was measured, then from the continuity equation, the velocity of at each position along

the duct can be calculated.

The continuity equation is:

Q=A1V 1 = A2 V 2 (3)

Where:

Q is the flow rate (m3/s)

A1is the area at the inlet section of the venturi (m2)

V 1 is the velocity at the inlet section of the venturi (m/s)

A2 is the area at the throat of the venturi (m2)

V 2 is the velocity at the throat of the venturi (m/s)

Note: the above equation can be extended to other sections of the tube

The above equation can be transposed for the velocity at each section as:

V n = Qan

(4)

Where:

V n is the velocity at each section of the tube (m/s)

Q is the flow rate (m3/s)

an is the area at each section of the tube (m2)

By determining the velocities at each sections of the duct together with the peizometric heads, the

behaviour of real fluids (compressible fluids) can be compared with that of ideal fluids (incompressible

fluids) as predicted by Bernoulli. For the latter (that is for compressible fluids), the density would not be

constant. It will vary from the density ρ1 at the venturi inlet to the density ρ2 at the venturi throat. (Douglas,

Gasiorek and Swaffield 2001). As mentioned earlier, for a compressible fluid the mass flow rate would

increase with increased upstream pressure, which will increase the density of the fluid through the

Page 3 of 20

constriction (though the velocity will remain constant). The reverse is the case for an incompressible fluid

or ideal fluid.

By measuring the pressure drop across the venturi meter, we can also calculate the value of the coefficient

of discharge ( Cd ).

This can be expressed in terms of the flow rate (Q) as:

Q=Cd*a t * √¿¿) (5)

Where:

Q is the flowrate (m3/s)

Cd is the coefficient of discharge (dimensionless)

a t is the area of the throat of the venturi (m2)

∆ p is the pressure drop across the venturi (Nm-2)

∆ p is the density of water (100 kgm-3)

m is the ratio of the area of the throat of the venturi (a t ¿ to the area at the approach duct a p

Thus, m=a t

ap (6)

Where:

a t is the area of the throat of the venturi (m2)

a p is the area at the approach duct (m2)

The coefficient of discharge is introduced to account for the fact that the theoretical flow of a fluid is

slightly greater than the actual or measured flow. Equation (5) above can be transposed for Cd as:

Cd= Q

at∗√ 2∆ pρ (1−m2)

(7)

Note: all the parameters are as explained above in the previous equations (5) and (6).

Page 4 of 20

Having known the coefficient of discharge Cd, the theoretical flow can also be estimated and compared

with the actual flow using:

Qactual=¿Cd¿ * Qtheoritical (8)

Where:

Qactual is the actual or measured flow (m3/s)

Cd isthe discharge coefficient (dimensionless)

Qtheoritical is the theoretical flow (m3/s)

The theoretical values of the static heads can also be compared with its experimental static heads by

calculating the theoretical heads as:

hn= h1 - [vn

2

2 g -

v12

2 g] (9)

Where

hnis the theoretical static head at various sections of the tube (m)

h1 is the static head at the inlet of the tube (m)

Vn is the velocity at various sections of the tube (m/s)

V1 is the velocity at the inlet of the tube (m/s)

One important relationship that can be shown when using a venturi meter to measure fluid flow is the direct

relationship between the pressure drop and the flowrate. This can be shown by calculating the pressure

drop across the venturi and then plotting a graph of the square-root of the pressure drop against the

flowrate. The pressure drop can be calculated as:

∆ p=ρg ∆ h (10)

Page 5 of 20

Where:

∆ p is the pressure drop across the venturi (Nm-2)

ρ is the density of water = 1000 kgm3

∆ h is the head loss across the venturi (m)

The energy loss can also be calculated in terms of the power consumption as:

Power consumption=Q∗∆ P (11)

Where: Q is flowrate (m3/s)

∆ P is the pressure drop across the venturi (Nm-2)

C. EXPERIMENTAL EQUIPMENT:

The main experimental equipment used in this experiment is a Perspex venturi meter having eleven

manometric tubes to measure the head loss across the venturi. Other equipments include; a lever system to

measure the weight of the water discharged, a stopwatch to measure the time needed for a certain volume

of water to be supplied; in order to then calculate the actual rate of flow. Other equipments are; water

supply tank, a pump and a level control device.

A Venturi Meter consists of a short converging conical tube leading to a cylindrical portion, called the

throat, of smaller diameter of that of the pipeline, which is followed by a diverging section in which the

diameter increases again to that of the main pipeline (Douglas, Gasiorek and Swaffield 2001). The

geometry of the venturi meter is designed to reduce head losses to a minimum. This is accomplished by

providing a relatively streamlined contraction (which eliminates separation ahead of the throat) and a very

gradual expansion downstream of the throat (which eliminates separation in the decelerating portion of the

device. Thus most of the head losses that occurs in a well-designed venturi meter is due to friction losses

along the walls rather than losses associated with separated flows ( Munson, Young and Okiishi 2002). The

diagram of a venturi meter used in this experiment can be seen in figure 1 below.

Page 6 of 20

Taken from http://www.cs.cdu.edu.au/homepages/jmitroy/eng243/VenturiMeter.pdf

The venturi tube has a converging conical inlet, a cylindrical throat, and a diverging recovery cone.   It has

no projections into the fluid, no sharp corners, and no sudden changes in contour. In Figure 3  showing

the venturi tube, the inlet section decreases the area of the fluid stream, causing the velocity to increase and

the head loss to decrease.  The low pressure is measured in the centre of the cylindrical throat since the

pressure  will  be  at  its  lowest  value,  and  neither  the  pressure  nor  the  velocity  is  changing. The

recovery cone allows for the recovery of head such that total head loss is minimal only about 10%.

The working principle of the venturi meter can be explained using a schematic diagram of the venturi

metre on the next page:

Page 7 of 20

Figure 1: Diagram of a venturi meter

manometric tubes

Venturi tube

Taken from www.icaen.uiowa.edu/~fluids/Posting/Home/EFD/EFD.../lab2_lecture.ppt

For ideal fluids, the velocity V1 and V2 are uniform and steady at the points A1 and A2. For real fluids, this

would not be the case. The pressure would fall where the liquid velocity increases in the constriction, thus

the greater the velocity change the greater the pressure drop. As the fluid moves from point 2 to point 3, the

velocity is decreased and the pressure is increased.  The slow rate of change creates a small energy loss;

most of the energy is changed back to pressure.  It is this relatively small loss that allows the venturi

meter’s use in low pressure pipes.  This small loss is also what makes the use of the Venturi meter

attractive for flow measurements. (Watson 1998).

There is a gain in kinetic energy resulting from the increased linear velocity in the throat. This is been

balanced by the decrease of pressure in the throat. This reduction in pressure which occurs when the fluid

flows through the throat is called the venturi effect (Coulson & Richardson, 1999).

D. EXPERIMENTAL PROCEDURE AND OBSERVATIONS:

The procedures undertaken in this experiment was in two (2) stages. First, the pump was switched on, and

the rotameter adjusted to ensure that water flows through the venturi. During this process, it was ensured

that no air was induced into the system via the throat of the venturi.

After establishing a steady flow, the static heads at each manometric tube were measured. Also with the

weigh bench, the weights of water discharged by the system per unit time were measured. This together

with the volume of water discharged was then used to calculate the flowrate (Q). With the dimensions of

the essential parts of the venturi meter that were given in a table behind the venturi tube, the velocity and

the velocity heads were also calculated.

Page 8 of 20

Figure 2: Schematic diagram of a venturi meter

Using the head difference between the manometer tube 1 h1 (approach duct) and the manometer tube in the

venturi throat position ht, the pressure drop across the venturi for the different flow rates were also

calculated. The experiment was repeated for two other flowrates. For full list of measurements taken and

for calculated data, see table 1-6. For graphs showing relationship between different parameters (both

experimental and calculated), see figures 3-6

E. RESULTS AND CALCULATIONS:

The results (both experimental and calculated results) obtained in the experiment are represented in

table 1-4 below. The graphs of static head h, the velocity head hv and the total head H across the

venture at each flowrate are shown in figure 3-5.

Tube 1 (m)

Tube 2 (m)

Tube 3 (m)

Tube 4 (m)

Tube 5 (m)

Tube 6 (m)

Tube 7 (m)

Tube 8 (m)

Tube 9 (m)

Tube 10 (m)

Tube 11 (m)

Time (s)

Flowrate (m3/s)

0.269 0.254 0.156 0.002 0.019 0.108 0.157 0.186 0.205 0.22 0.228

12.24 0.00049

0.253 0.242 0.165 0.049 0.06 0.127 0.165 0.188 0.203 0.214 0.22

13.95 0.00043

0.24 0.23 0.174 0.09 0.097 0.146 0.173 0.19 0.2 0.209 0.21216.6

3 0.00036Static heads gotten from the experiment

Tube 1 (m)

Tube 2 (m)

Tube 3 (m)

Tube 4 (m)

Tube 5 (m)

Tube 6 (m)

Tube 7 (m)

Tube 8 (m)

Tube 9 (m)

Tube 10 (m)

Tube 11 (m) Flowrate

0.04314 0.06858 0.17256 0.30345 0.24893 0.17069 0.12088 0.08747 0.06508 0.04895 0.04314 0.00049

0.03347 0.05308 0.13390 0.23365 0.19202 0.13225 0.09298 0.06747 0.05001 0.03773 0.03347 0.00043

0.02359 0.03686 0.09298 0.16347 0.13390 0.09161 0.06515 0.04702 0.03515 0.02645 0.02359 0.00036

Veocity heads (Vh) across the venturi meter calculated from equation (1)

Page 9 of 20

Table 2: Calculated Results for the velocity heads (Vh) across the venturi meter

Tube 1 (m/s)

Tube 2 (m/s)

Tube 3 (m/s)

Tube 4 (m/s)

Tube 5 (m/s)

Tube 6 (m/s)

Tube 7 (m/s)

Tube 8

(m/s)Tube 9 (m/s)

Tube 10

(m/s)

Tube 11

(m/s)Flowrate

(m3/s)

Table 3: Calculated Results for the total heads (H) across the venturi meter

Tube 1 (m)

Tube 2 (m)

Tube 3 (m)

Tube 4 (m)

Tube 5 (m)

Tube 6 (m)

Tube 7 (m)

Tube 8 (m)

Tube 9 (m)

Tube 10 (m)

Tube 11 (m)

Flowrate (m3/s)

0.312140.3225

8 0.32856 0.30545 0.267930.2786

9 0.27788 0.273470.2700

8 0.26895 0.27114 0.00049

0.286470.2950

8 0.29890 0.28265 0.252020.2592

5 0.25798 0.255470.2530

1 0.25173 0.25347 0.00043

0.263590.2668

6 0.26698 0.25347 0.230900.2376

1 0.23815 0.237020.2351

5 0.23545 0.23559 0.00036

0.92 1.16 1.84 2.44 2.21 1.83 1.54 1.31 1.13 0.98 0.92 0.000490.81 1.02 1.62 2.14 1.94 1.61 1.35 1.15 0.99 0.86 0.81 0.000430.68 0.85 1.35 1.79 1.62 1.34 1.13 0.96 0.83 0.72 0.68 0.00036

Table 5: Calculated Results for Theoretical values of hn

Tube 1 (m)

Tube 2 (m)

Tube 3 (m)

Tube 4 (m)

Tube 5 (m)

Tube 6 (m)

Tube 7 (m)

Tube 8 (m)

Tube 9 (m)

Tube 10 (m)

Tube 11 (m)

Flowrate (m3/s)

0.269 0.244 0.139 0.009 0.024 0.141 0.191 0.225 0.24 0.263 0.269 0.000490.253 0.233 0.153 0.053 0.095 0.116 0.166 0.219 0.237 0.249 0.253 0.00043

0.24 0.227 0.171 0.103 0.131 0.172 0.199 0.217 0.229 0.232 0.24 0.00036Theoretical values of hn was calculated using equation (9)

Table 6: Calculated Results Pressure drop and square-root of pressure drop across the venturi for different

flowrates.

Pressure drop (∆p) Flowrate (m3/s) √ P

(Nm-2) (m3/s) Nm-21510.74 0.00049 38.868238961137.96 0.00043 33.73366271824.04 0.00036 28.70609691

Pressure drop calculated using equation (10)

Page 10 of 20

Table 4: Calculated Results for Velocity ( Vn) across the venturi meter

0 2 4 6 8 10 120

0.05

0.1

0.15

0.2

0.25

0.3

Q1Q2Q3

Tube no. (m)

Stati

c hea

ds, h

(m)

Figure 3: Graph of Static heads against Tube no. for different flowrates Q1, Q2 and Q3

Page 11 of 20

0 2 4 6 8 10 120.00000

0.05000

0.10000

0.15000

0.20000

0.25000

0.30000

0.35000

Q1Q2Q3

Tube no.

Velo

city

head

, Vh

Figure 4: Graph of velocity head hv against Tube no. for different flowrates Q1, Q2 and Q3

Page 12 of 20

0 2 4 6 8 10 120.00000

0.05000

0.10000

0.15000

0.20000

0.25000

0.30000

0.35000

Q1Q2Q3

Tube no.

Tota

l Hea

d, H

(m)

Figure 5: Graph of total head, H against Tube no. for different flowrates Q1, Q2 and Q3

Page 13 of 20

0 2 4 6 8 10 120

0.05

0.1

0.15

0.2

0.25

0.3

Q1 hn experimentalQ2 hn experimentalQ3 hn experimentalQ1 hn theoreticalQ2 hn theoreticalQ3 hn theoretical

Tube no.

hn

exp

erim

enta

l / th

eore

tical

(m)

Figure 5: Graph of hn against tube no. for different flowrates Q1, Q2 and Q3 comparing experimental

and theoretical hn

Page 14 of 20

0.00034 0.00036 0.00038 0.00040 0.00042 0.00044 0.00046 0.00048 0.000500

5

10

15

20

25

30

35

40

45

Flowrate, Q (m3/s)

Figure 6: Graph of √ p against flowrate showing the relationship between pressure drop, ∆ P and flowrate,

Q.

Page 15 of 20

SAMPLE CALCULATIONS:

(A)Sample calculation for velocity across each section of the venturi

Using equation (4): V n = Qan

Velocity across the first section; Where Q = 0.00049 m3/s, A1 = 530.9mm2 = 0.0005309m2

Therefore, V 1 = 0.00046

0.0005309 = 0.92 m/s

(B) Sample calculation for total head across the venturi

Using equation (2): p

ρg + v2

2 g = H

Total head across the first section of the venturi;

Where = p

ρg = static head h = 0.269m, v2

2 g = 0.922

2∗9.81 = 0.04314m

Therefore, total head (H) across first section of the venturi = 0.269m + 0.04314m = 0.31214m

(C)Sample calculation for velocity heads (hv) across the venturi

From equation (1) v2

2 g = velocity head

Velocity head across the first section of the venture; Where v2 = 0.922 = 0.8464 m2/s2

2 * g = 2 * 9.81 = 19.62 m/s2

Therefore, Velocity head across the first section of the venturi = 0.846419.62

= 0.04314 m

(D)Sample calculation for theoretical values of hn across each section of the venturi

Using equation (9): hn= h1 - [vn

2

2 g -

v12

2 g]

Theoretical hn across the second section of the venture; Where v2

2

2∗g = 1.162

19.62 = 0.06858m

v12

2 g = 0.922

19.62 = 0.043139m and h1 = 0.269m

Therefore theoretical hn across the first section of the venture = 0.269 – [0.06858 – 0.043139]

= 0.2435 m

Page 16 of 20

(E) Sample calculation for pressure drop (∆p) the venturi for flowrates Q1, Q2 and Q3

Using equation (10): ∆ p=ρg ∆ h

For flowrate Q1:

Where: ρ=1000 kgm3, g=9.81m/ s2

∆ h=h1 - ht = (0.156 m – 0.002) = 0.154 m

Therefore , ∆ p = 1000 * 9.81 * 0.154 = 1510.74 Pa

For flowrate Q2:

Where ρ=1000 kgm3, g=9.81m/ s2

∆ h=h1 - ht = (0.165 m – 0.049m) = 0.116 m

Therefore , ∆ p = 1000 * 9.81 * 0.154 = 1137.96 Pa

For flowrate Q3:

Where ρ=1000 kgm3, g=9.81m/ s2

∆ h=h1 - ht = (0.174 m – 0.09) = 0.084 m

Therefore , ∆ p = 1000 * 9.81 * 0.084 = 824.04 Pa

(F) Sample calculation for estimation of the energy loss

Using equation (11): Power consumption=Q∗∆ P

Where total pressure loss (∆ P ¿ = (1510.74 + 1137.96 + 824.04) Pa = 3472.74 Pa

Also total flowrate (Q) = Q1 + Q2 + Q3 = (0.00049 + 0.00043 + 0.00036) m3/s = 0.00128 m3/s

Therefore, Power consumption = 0.00128 * 3472.74 = 4.45W

(G)Sample calculation for the coefficient of discharge( Cd ):

From equation (7): Q=Cd*a t * √¿¿)

Transposing the above equation for Cd:

Cd= Q

at∗√ 2∆ pρ (1−m2)

Page 17 of 20

m = at /ap = 2.011∗10−4

2.659∗10−4 = 0.7563

Therefore, m2 = (0.7563)2 = 0.5719

=

0.00128

2.659∗10−4∗√ 2∗3472.741000(1−0.5719)

= 0.95

Therefore, the coefficient of discharge ( Cd ) = 0.95

This is in line with the theoretical value of Cd ( 0.98 ) gotten from textbooks.

(H)Sample calculation for the theoretical flow:

Using equation (8): Qactual=¿Cd¿ * Qtheoritical

Therefore, Qtheoretical = Qactual

Cd = 1.28∗10−3

0.95 = 1.35*10-3 m3/s

F. DISCUSSION OF RESULTS:

The graph in figure 3 shows the relationship between the static heads and the flowrates at each

section of the venture meter. It shows that, the higher the flowrate, the higher the static head,

except for the manometric tubes with smaller area where the head loss was small because of the

small cross-sectional area (for example tube 2).

The graph in figure 4 shows the relationship velocity heads and flowrates across the venturi.

This shows that the higher the flowrate, the higher the velocity head. As it can be seen from

figure 4, the highest flowrate correspond to the highest velocity head while the lowest flowrate

corresponds to the lowest velocity head.

The graph in figure 4 shows that the higher the flowrate, the higher the total head. As such, at

flowrate (Q1), we had higher total head (H) followed by flowrate (Q2) and lastly flowrate (Q3)

Also from the graph in figure 5, it shows that the theoretical values of hn at different flowrates is

slightly higher than the experimental values, except for h1 were it was assumed that the

theoretical values were the same with the experimental values.

Page 18 of 20

From the graph of the square-root of pressure drop √ p against flowrate, Q in figure 6, it can be

inferred that there is a direct relationship between the square-root of the pressure difference;

hence the graph is a straight line graph. This means that if there is a tenfold increase in the

flowrate, there must be a one hundred increase in the pressure difference.

From the calculation of the theoretical flow, it can be seen that the theoretical flow is slightly

higher than the actual or measured flow. This is in line with the theory. It is because of this

slight difference that a coefficient of discharge was introduced.

It is noteworthy that this coefficient of discharge varies with the rate of flow. This can be

explained by equation (6) and (7).

From table 2, it shows that there is a linear increase in velocity at the throat of the venturi. This

increase leads to a gain in kinetic energy and at the same time, this increase is been balanced by

a decrease of pressure in the throat. This explains why the velocities at tube 4 (section D of the

venturi meter) which is the throat were all higher than the velocities at other sections of the

venturi meter. The velocities at tube 4 were 2.44 m/s, 2.14 m/s and 1.79 m/s at different

flowrates respectively. Other velocities were below these ones.

G. CONCLUSIONS:

From the experiment, the following conclusions can be drawn.

(i) The venturi meter is indeed the most accurate method of measuring fluid flow both

incompressible and compressible fluid because the head losses recorded in using it is

minimal compared to other measuring devices.

(ii) Contrary to what we know from the idea of an ideal fluid, where the velocity is constant

at each point, in the case of the venture, the velocity is high at the throat, though this is

balanced by a reduction in the pressure at the throat too due to venturi effect.

(iii) The theoretical discharge was slightly higher than the actual or measured discharge. This

slight difference was accounted for by the coefficient of discharge.

(iv) The head loss, the velocity head and the total head all depends on the rate of flow. The

higher the rate of flow, the higher the head loss, the higher the velocity head and the

higher the total head too.

(v) There exist a direct relationship between the pressure drop across the venturi and the

flowrate. This can be seen when the square-root of the pressure drop, √ P was plotted

against the flowrate, Q. This gave a straight line graph.

Page 19 of 20

References:

Coulson, J.M. and Richardson, J.F. (1999) Coulson & Richardson’s Chemical Engineering. 6th ed. Oxford: Elsevier Science, Volume 1, Chapter 5.

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

Chapter 6 & 17.

Holland, F. (1986) Fluid Flow for Chemical and Process Engineers. 2nd ed. Oxford: Butterworth-

Heinemann.

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.

Mitroy, J. Lab note on Engineering Mechanics (2009) Charles Darwin University [online] Available from:

http://www.cs.cdu.edu.au/homepages/jmitroy/eng243/VenturiMeter.pdf [ accessed 07 March 2011 ].

Watson, K.L. (1998) Foundation Science for Engineers. 2nd ed. London : Macmillan Press Ltd, Chapter 20.

http://www.icaen.uiowa.edu/~fluids/Posting/Home/EFD/EFD.../lab2_lecture.ppt [ accessed 07 March 2011 ]

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