Introduction :
Bernoulli's law states that if a non-viscous fluid is flowing along a pipe of varying cross section,
then the pressure is lower at constrictions where the velocity is higher, and the pressure is
higher where the pipe opens out and the fluid stagnate. Many people find this situation
paradoxical when they first encounter it (higher velocity, lower pressure). This is expressed with
the following equation:
p
v2

 z  h *  Constant
 g 2g
(1.8)
Where,
p
= Fluid static pressure at the cross section
ρ
= Density of the flowing fluid
g
= Acceleration due to gravity
v
= Mean velocity of fluid flow at the cross section
z
= Elevation head of the center at the cross section with respect to a datum
h* =Total (stagnation) head
The venturi meter consists of a venturi tube and differential pressure gauge. The venturi tube
has a converging portion, a throat and a diverging portion as shown in the figure below. The
function of the converging portion is to increase the velocity of the fluid and lower its static
pressure.
A pressure difference between inlet and throat is thus developed, which pressure
difference is correlated with the rate of discharge. The diverging cone serves to change the
area of the stream back to the entrance area and convert velocity head into pressure head.
In metering practice, this non-ideality is accounted by insertion of an experimentally determined
discharge coefficient, Cd that is termed as the coefficient of discharge. With Z 1 = Z2 in this
apparatus, the discharge coefficient is determined as follow:
Cd 
Qa
Qi
Discharge coefficient, Cd usually lies in the range between 0.9 and 0.99.
Objectives :
1.
To determine the discharge coefficient of the venturi meter
(1.20)
2.
To measure flow rate with venturi meter
3.
To demonstrate Bernoulli’s Theorem
Procedure :
Part 1: Discharge Coefficient Determination
1. The General Start-up Procedures was performed.
2. The hypodermic tube from the test section was withdrawed.
3. Discharge valve to the maximum measurable flow rate of the venturi was adjusted.
4. After the level stabilizes, the water flow rate using volumetric method was measured and
the manometers reading was record.
5. Step 4 with at least three decreasing flow rates was repeated by regulating the venturi
discharge valve.
6. The actual flow rate, Qa from the volumetric flow measurement method was obtained.
7. The ideal flow rate, Qi from the head difference between h1 and h3 using Equation 1.18
was calculated.
The chart of “Qa vs Qi” was plotted and finally obtain the discharge coefficient was obtained,
Cd which is the slope (c)
Part 2: Flow Rate Measurement with Venturi Meter
1. The General Start-up Procedures were performed.
2. The hypodermic tube from the test section was withdrawed.
3. Discharge valve to the maximum measurable flow rate of the venturi was adjusted.
4. After the level stabilizes, the water flow rate using volumetric method was measured and
the manometers reading was record.
5. Step 4 with at least three decreasing flow rates was repeated by regulating the venturi
discharge valve.
6. The venturi meter flow rate of each data were calculated by applying the discharge
coefficient obtained.
7. The volumetric flow rate with venturi meter flow rate were compared.
8. Data Analysis:
Throat Diameter, D3 (mm)
16.0
Inlet Diameter, D3 (mm)
26.0
Throat Area, At (m2)
2.011 x 10-4
Inlet Area, Ai (m2)
5.309 x 10-4
g (m/s2)
9.81
ρ (kg/m3)
1000
Part 3: Bernoulli’s Theorem Demonstration
1. The General Start-up Procedures were performed.
2. All manometer tubings are properly connected to the corresponding pressure taps and are
air-bubble free were checked.
3. Discharge valve to the maximum measurable flow rate of the venturi was adjusted.
4. After the level stabilizes, the water flow rate using volumetric method was measured.
5. The hypodermic tube (total head measuring) connected to manometer #G was slided gently,
so that its end reaches the cross section of the Venturi tube at #A. Wait for some time and
the readings from manometer #G and #A were noted down. The reading shown by
manometer #G is the sum of the static head and velocity heads, i.e. the total (or stagnation)
head (h*), because the hypodermic tube is held against the flow of fluid forcing it to a stop
(zero velocity). The reading in manometer #A measures just the pressure head (hi)
because it is connected to the Venturi tube pressure tap, which does not obstruct the flow,
thus measuring the flow static pressure.
6. Step 5 for other cross sections (#B, #C, #D, #E and #F) was repeated.
7. Step 3 to 6 with three other decreasing flow rates were repeated by regulating the venturi
discharge valve.
8. The velocity, ViB was calculated using the Bernoulli’s equation where;
Vi  2  g  (h8  hi )
9. The velocity, ViC was calculated using the continuity equation where
Vi_Con
=
Qav / Ai
10. The difference between two calculated velocities were determined.
Result :
Part 1: Discharge Coefficient Determination
Qav
hA
hB
hC
hD
hE
hF
h A - hC
Qi
(LPM)
(mm)
(mm)
(mm)
(mm)
(mm)
(mm)
(m)
(LPM)
12.15
131
126
90
113
118
123
0.041
11.69
11.93
138
133
99
120
124
130
0.039
11.40
11.72
152
147
115
136
139
145
0.037
11.11
11.11
173
168
136
157
160
166
0.037
11.11
10.22
209
201
178
195
198
203
0.031
10.17
Part 2: Flow Rate Measurement with Venturi Meter
Qav
hA
hB
hC
hD
hE
hF
(LPM)
(mm)
(mm)
(mm)
(mm)
(mm)
(mm)
12.15
131
126
90
113
118
123
11.93
138
133
99
120
124
130
11.72
152
147
115
136
139
145
11.11
173
168
136
157
160
166
10.22
209
201
178
195
198
203
Qav(LPM)
hA - hC (m)
Qi(LPM)
Calculated Flow Rate (LPM)
Error(%)
12.15
0.0041
11.69
12.04
0.91
11.93
0.0039
11.40
11.75
1.51
11.72
0.0037
11.11
11.45
2.30
11.11
0.0037
11.11
11.45
3.06
10.22
0.0031
10.17
10.48
2.54
Part 3: Bernoulli’s Theorem Demonstration
Cross
Section
Using Continuity
Using Bernoulli equation
equation
Difference
h*=hG
hi
ViB =
Ai =
ViC =
ViB-ViC
(mm)
(mm)
√[2*g*(h* - hi )]
π Di2 / 4
Qav / Ai
(m/s)
(m/s)
(m2)
(m/s)
#
A
150
131
0.61
0.00053
0.38
0.23
B
159
126
0.80
0.00037
0.55
0.25
C
160
90
1.17
0.00020
1.01
0.16
D
160
113
0.96
0.00031
0.65
0.31
E
159
118
0.90
0.00038
0.53
0.37
F
155
123
0.79
0.00053
0.38
0.41
Calculation :
Part 1: Discharge Coefficient Determination
Qi (LPM)
  A 2 
Qi  A2V2  A2 1   2  
  A1  
1 / 2
  p1  p 2

 Z1  Z 2 
2 g 

  
1000*60*{0.0082 * π[1-(0.008/0.013) 4]-0.5(2*9.81*0.041) 0.5}=11.69
1000*60*{0.0082 * π[1-(0.008/0.013) 4]-0.5(2*9.81*0.039) 0.5}=11.40
1000*60*{0.0082 * π[1-(0.008/0.013) 4]-0.5(2*9.81*0.037) 0.5}=11.11
1000*60*{0.0082 * π[1-(0.008/0.013) 4]-0.5(2*9.81*0.037) 0.5}=11.11
1/ 2
1000*60*{0.0082 * π[1-(0.008/0.013) 4]-0.5(2*9.81*0.031) 0.5}=10.17
According to the graph,
Cd=1.0303
Part 2: Flow Rate Measurement with Venturi Meter
Calculated Flow Rate (LPM)
Error(%)
1.0303*11.69=12.04
| [(12.04-12.15)/12.15] |*100% =0.91
1.0303*11.40=11.75
| [(11.75-11.93)/11.93] |*100% =1.51
1.0303*11.11=11.45
| [(11.45-11.72)/11.72] |*100% =2.30
1.0303*11.11=11.45
| [(11.45-11.11)/11.11] |*100% =3.06
1.0303*10.17=10.48
| [(10.48-10.22)/10.22] |*100% =2.54
Part 3: Bernoulli’s Theorem Demonstration
ViB =
√[2*g*(h*
π
- hi )]
Ai =
ViC =
ViB-ViC
Di2
Qav / Ai
(m/s)
/4
(m2)
(m/s)
(m/s)
√[2*9.81*(150-131 )]=0.61
π
/ 4=0.00053
0.00020/0.00053=0.38
0.61-0.38=0.23
√[2*9.81*(159-126 )]=0.80
π 21.62/4=0.00037
0.00020/0.00037=0.54
0.80-0.54=0.26
√[2*9.81*(160-90 )]=1.17
262
π
162
/ 4=0.00020
0.00020/0.00020=1.00
1.17-1.00=0.17
√[2*9.81*(160-113 )]=0.96
π
202
/ 4=0.00031
0.00020/0.00031=0.65
0.96-0.65=0.31
√[2*9.81*(159-118 )]=0.90
π
222
/ 4=0.00038
0.00020/0.00038=0.53
0.90-0.53=0.37
√[2*9.81*(155-123 )]=0.79
π
262
/ 4=0.00053
0.00020/0.00053=0.38
0.79-0.38=0.41
Discussion :
Part 1 :
The Qi was calculated by the equation:
  A 2 
Qi  A2V2  A2 1   2  
  A1  
1 / 2
  p1  p 2

 Z1  Z 2 
2 g 
  

1/ 2
According to the Qav from the data, use Qav and Qi to plot the graph and then find the slope
which is the discharge coefficient, Cd. In Bernoulli's law, the pressure of pipe would become
smaller when the diameter decrease, which is shown good in the all data. According to the
theory, discharge coefficient, Cd usually lies in the range between 0.9 and 0.99. However, the
calculation of Cd in part 1, Cd=1.0303, which is larger than 0.99. Here are some reasons why
the error generated: 1. The error come from the leaking of the machine, in the experiment, not
only the tube, but also the pump was leaking. 2.The error of manometers reading and the
measure of flow rate Qav., according to the data of the result, there is a data that different Qav
have the same hA - hC. which is impossible in the real life.
3.There still had some bubbles in
the plastic transfer tube effect the manometers reading.
The way to reduce those error are : Check the function of device carefully before the experiment
and make sure the air bubbles are escape from the pipe. Testing five times of the time then use
the average time to measure the flow rate Qav.
Part 2 :
Use the Cd which we find in part 1 and the formula :
Cd 
Qa
Qi
to calculate the calculated
flow rate. Then use the equation :error = | (Qa-Qav)/Qa |*100% to find the error. According to
the data, some of the calculated flow rate are larger than Qav. , but if the calculated flow rate
were calculated by using the ideally discharge coefficient, the Qav would become larger than
the calculated flow rate. So the error of the calculated flow rate was come form the error of the
Cd calculation.
Part 3 :
Use the Bernoulli equation : ViB =√[2*g*(h* - hi )] and the data we collect to calculate the flow
rate ViB. After that, use the Continuity equation : Ai = π Di2 / 4 and ViC = Qav / Ai to calculate
the flow rate ViC. Finally, compare two flow rate.In Bernoulli's law, the velocity of flow rate would
increase when the diameter decrease, which is shown well in the data in the part 3. The two
flow rate should be same with each theoretically. However, after the calculation, all of the ViB
were larger than ViC. In Bernoulli equation part, the error can only come from the manometers
reading of hi and h*. In Continuity equation part, the error is come from the calculation of Qav.
The error of Qav is due to the reading and the time measure. Use the average time to calculate
Qav can decrease the amount of error. Activity the error of the Continuity equation part should
be less than the Bernoulli equation part if the Qav is calculated by the average time, the Qav
would become almost same with the ideally flow rate. However there still have those error in
the manometers reading of hi and h*, which make the Bernoulli equation part more inaccurate.
Conclusion :
In Bernoulli's law, the pressure is lower at constrictions where the velocity is higher, and the
pressure is higher where the pipe opens out and the fluid stagnate. In the pipe, the velocity is
increasing when the diameter is decreasing, the velocity become higher when the diameter
become lower. According to the result, the Bernoulli equation is proven.
Reference :
Applied Fluid Mechanics 5th Edition, Robert L. Mott, Prentice-Hall
Elementary Fluid Mechanics 7th Edition, Robert L. Street, Gary Z. Watters, John
K. Vennard,
John Wiley & Sons Inc.
Fluid mechanics 4th Edition, Reynold C. Binder
Fluid Mechanics with applications, Anthony Esposito, Prentice-Hall International Inc.