Experiment 1 FLUID STATICS

Fluid Mechanics Department – Faculty of Civil Engineering – HCMUT

Assoc. Prof. Le Van Duc

1

Experiment 1 FLUID STATICS

I. PURPOSE

It is aimed to help students understand the basic equation of hydrostatic and know how to apply

it in several issues related to the incompressible fluid at rest.

II. THEORY

The status of static fluid, incompressible, is described by basic equation:

constp

z =

+ (1.1)

Z is the elevation of any point in a static fluid environment having constant specific weight γ,

and p is pressure at that point, the value of

+p

z also known as hydraulic head with the unit of

meter. According to this equation, we have some applications as follows:

1- Isobars

Isobaric surface is a surface on which the pressure is equal.

The image of a specific isobar, that we can observe, is the free surface of fluid. When the pressure at

the points on the free surface are equal, the elevation of these points have to be equal. This means that

the free surface is a horizontal surface.

2- Fluid Manometers

Applying the basic equation (1.1) to calculate the pressure at any point in a static fluid volume.

If you know the elevation at a point being z0, along with pressure p0, and if we know the elevation of

aother point, z (see Figure 1.1), then the pressure at this point will be:

( ) hpzzpp +=−+= 000 (1.2)

In which plane 0-0, a datum (plane) is used to compute the elevation. By measuring the height h, we

can determine the pressure p.

3- Determination of the specific weight of a fluid

If we know p0, p, z0 and z, from (1.2) we can obtain the specific weight of the fluid:

zz

pp

−

−=

0

0 (1.3)

It should be reminded that basic equation (1.1) is set without the influence of capillary. In the

case of liquid inside a small tube with diameter d < 3mm, the influence of capillary is significant, so

0 0

z

z0

p

p0

h

Figure 1.1



2

the equation (1.1) is no longer accurate. If the liquid is water, alcohol, oil ... the liquid levels in small

diameter pipe is higher than in large diameter pipe, however, if the liquid is mercury, in contrast, the

fluid level in the pipe with smaller diameter is lower than that in larger diameter pipe.

III. EXPERIMENTAL EQUIPMENT

T

Đ

Figure 1.2

1 21 22 3 4 5 7 8 9 10

11

12

23 6

Hydrostatic Experiment Set:

Manometer set consists of a group of glass tubes numbered from 1 to 10 mounted on a table,

along with elevation scales having its accuracy up to millimetre as shown in Figure 1.2. The diameter

of the tubes di = 5mm (except the tubes in group 2 having their diameters ≤ 3mm). In these tubes:

• The pair of tubes 1-3 is used as a liquid manometer to measure the gas pressure in the closed

Tank T.

• Group 2 including 3 tubes: 21, 22 and 23 with diameters of 1mm, 2mm and 3mm, respectively,

is used to observe the capillary phenomena.

• 3 U-tube pairs 4-5, 6-7 and 8-9 containing the liquids whose specific weights are required to be

measured.

• Tubes 10 and 3 are used to observe the isobar. Pressure on the surface of fluid in the tubes 1, 4,

6 and 8 is the closed gas pressure in Tank T. The elevation of fluid surface in the tube is denoted

as Li.

• The part of equipment to create pressure includes a fixed Tank T containing closed air, and a

movable open-air water-tank (Tank D) which is mounted on a pulley 12. Thanking on the crank

11, we can change the elevation of the Tank D, resulting the air pressure in Tank T having

different pressures.

IV. THE EXPERIMENTAL PROCESS

1- Check the elevation datum of the scales (on the level scales in mm) whether they lie on the

same horizontal plane or not by reading the water level in the tubes 3 and 10. These two water levels

have to be horizontal.

2- Using the crank 11 to move the Tank D upward, such that the water surface in Tank D is higher

than the water surface in Tank T ( zD- zT ) about 15 20cm.

− To measure the liquid levels in the tubes from No. 1 to No. 10, and record them in a Table

(including the tubes in Group 2).

− Using the crank 11 to lower the Tank D to the average elevation (zD - zT = 5 7cm) and

low elevation (zD- zT = -15 -20cm). Proceed to measure as mentioned above, and record

these measuring results in the Table (in the laboratory report)

Scale in mm



3

V. GUIDANCE ON THE COMPUTATION AND REPORT:

1- To compute the pressure of closed air in Tank T:

From (1.2), the pressure of the air, in Tank T can be computed as follows:

( )13OHaT LLpp2

−+= (1.4a)

The specific weight of water in tanks, OH2 is defined depending on the temperature of the

environment. If using pa = 0, we can compute the gage presssure of air in Tank T as follows:

( )13OHT LLp2d

−= (1.4b)

According to three locations of Tank D, we can compute three values of pressure of air in Tank T.

2- To compute the specific weight of fluids in the U-Tubes:

Considering the fluid with its specific weight, , being at rest in the tubes i and i + 1 of the U-

Tube. Pressure of air in Tank T is also computed as follows:

( )i1iaT LLpp −+= +

Comparing this equation with the equation (1.4a), the formula to compute the specific weight

of fluid in these U-Tubes can be computed as follows:

( )( )i1i

13OH

LL

LL2 −

−=

+

(1.5)

3- To compute the error in the method of pressure measurement:

Utilizing the theory on the error, from the formula (1.4b), the relative error, p from the

pressure measurement can be computed as follows:

( ) ( ) 13

13

OH

OH

13OH

13OH

T

Tp

LL

LL

LL

LL

p

p

2

2

2

2

−

++

=

−

−=

=

Therefore, relative error from the gage pressure measurement is the addition of two terms:

- Relative error from the computation of the specific weight of water:

OHOHOH 222=

- Relative error from the readings on the level scales:

13

13L

LL

LL13 −

+=

−

Therefore: 132 LOHp −

+= (1.6)

In general, we use OH2 = 0,12% and L1 = L3 = 0,5mm

4- To compute the error in definition of specific weight, of fluid in the U-Tubes

From formula (1.5), we can compute the relative error in the measurement of specific weight

in the U-tubes as follows:

i1i

i1i

13

13

OH

OH

LL

LL

LL

LL

2

2

−

++

−

++

=

=

+

+

Or, iiLLOHp −+−

++=1132

(1.7)

Similarly, OH2 = 0,12% and L1 = L3 = Li = Li+1 = 0,5mm



4

EXPERIMENTAL RESULT RECORD SHEET

Student name: .................................................... ID + GROUP No: ...................................................

Experimental date: ................................. Report submit date: ............................................................

Experiment 1 FLUID STATICS

Barometric pressure and temperature of open air during the experiment time are:

Pa = ........................................ mmHg; t0 = ........................................ 0C

The specific weight of water is: (using the appendix in lecture note)

H2O = ................................................. N/m3

I. EXPERIMENTAL RESULTS AND OBSERVATION:

1. According to the three elevations of Tank D in comparison with Tank T, please record the

measured values of 9 tubes, amd Group Tube 2 in the following Table 1.

TABLE 1a. Measurement results (in unit of cm)

L1 L3 L4 L5 L6 L7 L8 L9 L10 NOTES

1

2

3

TABLE 1b. Measurement results of Group Tube 2 (in unit of cm)

L21 L22 L23 NOTES

1

2

3

II. COMPUTATION AND EXPERIMENTAL RESULT PRESETATION.

1. In the fluid static experiment set, which water levels of tubes, or tanks are in the same horizontal

plane ? Why ?

.............................................................................................................................................................

.............................................................................................................................................................

.............................................................................................................................................................

.............................................................................................................................................................

.............................................................................................................................................................

.............................................................................................................................................................

.............................................................................................................................................................



5

2. In the fluid static experiment set, which water levels of tubes are not followed the law of fluid

static (not in the same horizontal plane) ? Why ?

...............................................................................................................................................................

...............................................................................................................................................................

...............................................................................................................................................................

...............................................................................................................................................................

...............................................................................................................................................................

...............................................................................................................................................................

...............................................................................................................................................................

3. Compute the absolute and gage pressure of air in Tank T and these relative errors in three

measurement cases. Write them in Table 2.

4. Compute the specific weight of three fluids in U-Tubes 4 - 5, 6 - 7, and 8 - 9 and their relative

errors in three measurement cases. Write them in Table 2.

TABLE 2. COMPUTATION RESULTS

Please give your comments on:

a) The pressure in closed air in Tank T

...............................................................................................................................................................

...............................................................................................................................................................

...............................................................................................................................................................

...............................................................................................................................................................

...............................................................................................................................................................

b) The variation of water surface elevations between two tanks (T and D)

...............................................................................................................................................................

...............................................................................................................................................................

...............................................................................................................................................................

...............................................................................................................................................................

...............................................................................................................................................................

pt pd 4-5 6-7 8-9 4-5 6-7 8-9

103N/m2 % 103N/m3 %

1

2

3



6

Experiment 2 THE WORK AND ENERGY EQUATION

I. EQUIPMENT

The experimental apparatus (see Figure 2.1) includes

- Water from Tank 6 is pumped into Tank 1, flows into main channel (3) passing through valve

2 for adjustment of discharge of flow.

- Glass main rectangular channel (3) is horizontal, having the bottom width B= 78mm.

- A rising step (4) having the longitudinal cross section is a form of isosceles trapezoid; the

inclined side faces makes a 45o with the horizontal plane. And the height of rising step (a) is

33,1 mm

- The water level after step is changed by valve 5, located at the end of the channel. The water

then flows into the tank 6, through the rectangular sharp-crested weir.

- Elevation gauge 7 is mounted on glass channel 3 to measure channel bottom elevation and

water level in glass channel.

II. EXPERIMENTAL PROCESS

1. Determine the position of sections on the glass channel with the points from 1 to 6 in the order

from the upstream to the downstream of rising step as shown in Figure 3.4, where the midpoint

of the step corresponds to the 40cm-index on the glass channel, the sectional position in turn

corresponds to the glass channel mark as shown in the figure. The distance between sections

is as follows:

Figure 2.1

Figure 3.3



7

L1-2 = L2-3 = L4-5 = L5-6 = 20 cm; L3-4 = 3.7cm

2. Measure the elevations, Zdi of the channel bottom and step, corresponding to sections from 1

to 6. Record these results in Table 1 (in the experimental report).

3. Use valve 5 to adjust the flow and water level in the channel so that the water level at the

downstream of step is higher than the water level on the step (see the available line in the glass

channel).

4. Wait for the flow in the channel stable, using Vernier and point gauge to measure the values

of free water surface elevations Zi at sections 1 to 6. Write the results in Table 1 (in the

experimental report) [Case 1]

5. Keep discharge of flow constant, adjust other water levels at downstream by valve 5 (this time

the downstream water level is lower than the water level on the step, watch the black line as

a mark on the channel). Wait for the water level in the glass not to fluctuate, repeat the

experimental steps in Section 4. Write the results in Table 1 (in the experimental report) [Case

2]

6. Using your phone to take the picture on the water surface along the rising step for two cases

in (4) and (5).

Table 1: Measured experimental data Unit (cm)

Section 1 2 3 4 5 6 Notes

Bottom Level

Water surface (Case 1)

Water surface (Case 2)

III. GUIDE ON COMPUTATION AND REPORT

1. Computation of velocity

Mean velocity at section I can be computed as follows:

Vi = Q/Ai (2.1)

(Where, Q = 0.3 lit/s)

Where, Ai, cross sectional area at section i; Ai = B.hi

Figure 3.4



8

Where,

width of channel, B = 78mm;

The height of water level computed from channel bottom: hi = ǀZđi – Ziǀ ,

where Zđi and Zi are elevations of bottom level and water surface level, respectively, at section i

(Figure 3.3).

Notice:

- Check the whole system whether the system safety to be operated or not.

- Turn on the electric box to start the pump

- When the water is pumped into the Tank 1, observe the water level in the glass channel, such

that the water level is not higher than the channel side, to avoid the water spill out.

2. Computation of velocity head

Velocity head at section i can be computed using the following formula:

hvi = 𝑉𝑖2

2𝑔 (2.2)

3. Study on energy loss

Energy loss between two cross sections i and j is determined by using the Bernoulli equation (2.3)

between these two sections.

Where, Zi and Zj are the water surface levels at section i and j, respectively. 𝑝𝑖

𝛾 =

𝑝𝑗

𝛾 = 0. And the

value α = 1, and velocity head is computed using Formula (2.2).

Apply (2.3) to compute energy loss of flow from section 1-1 to 2-2 (denoted as hf1-2); from 2-2 to 3-

3 (denoted as hf2-3); from 3-3 to 4-4 (denoted as hf3-4); from 4-4 to 5-5 (denoted as hf4-5); from 5-5

to 6-6 (denoted as hf5-6). The computations are applied for both case of water surface levels.

4. Study on the variation of water surface elevation

If the flow energy loss is omitted from section 1 to any section i (i = 2 to i= 6), and if the flow velocity

distribution in these sections is considered to be uniform, (2.3) becomes

Select the channel bottom as a datum (in this experiment, the bottom channel keeps horizontal),

then we can write:

Zi = hi if section i before or after rising step;

Zi = hi +a , if section i on the rising step

(2.4)

(2.5)

(2.3)



9

Replace Vi from (2.2), Zi from (2.3) into (2.4), we have:

- If section i before and after rising step, then

- If section i on the rising step, then

In the equations (2.6) and (2.7), if the value on the left-hand side is known, we can solve for hi from

the cubic equation. We can solve these equations by using the trial and error method, or iteration

method to find out the unknowns hi (Students can be solve cubic equations directly to find out

unknowns).

Notice:

It is assumed that, the water surface level at downstream rising step is higher than the height of rising

step. Using trial and error method, a guess initial value of hi is given, and substituted into RHS of

Equation (2.6) or (2.7). The computed values of RHS is then obtained, and compared with the given

value on LHS.

If this computed value of RHS is greater than value of LHS, then decreasing guess value hi, and

repeating the computation and comparation (it should choose hi such that: h1 – a < hi < h1). Else if

this computed value of RHS is less than value of LHS, then increasing instead of decreasing.

5. Drawing the water surface profile in channel

Draw on the same graph: channel bottom, rising step and water surface calculated from (2.6) and

measured. Give your conclusions (draw only for one case of downstream water level higher than the

water level above the rising step)

6. Analysis and evaluation

about the Give your global analysis and evaluation on the flow through the broad-crested weir and

the flow succeeding this weir and the energy dissipation at its downstream.

(2.6)

(2.7)



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Experiment 3 FRICTION LOSS IN PIPE

I. OBJECTIVES OF EXPERIMENT

- To investigate the variation of friction head along a circular pipe with the mean flow velocity in the

pipe.

- To investigate the friction factor against Reynolds number and roughness.

II. EQUIPMENT SET – UP

A pipe of 10.64cm diameter is supplied by water a centrifugal pump. Five test sections with

interval of 3 m are connected to a bank of pressurized manometer tubes. Water from the pipe flow

into the concrete channel, and at the end of channel a vee-notch, is installed to measure the flow

rate in the channel, this flow rate is equal to the flow rate in the pipe. Water level over the vee-

notch is measured by a point gauge mounted on a small tank which is opened to the channel.

The flow rate over the vee-notch is calculated by formula as follow:

2/5

0.2215

8hgCtgQ D

=

(1)

Where: = 90o

CD = 0.58

g = 9.81 m/s2

ho = ZCR – z

Where z is water level, ZCR is the elevation of the crest of Vee-notch, ZCR = 27.8 cm

The flow rate over the Vee-notch is regulated by a control valve of pump, and an ampere meter

mounted on an electric box will show the current intensity of motor corresponding to the flow rate in

the pipe. The difference of pressure between the test sections in the pipe are measured by reading the

water level in the water level in the tubes of manometer.



11

III. SUMMARY OF THEORY

Considering flow at two sections i,j in a pipe, Bernouilli’s equation may be written as:

ijj

jj

i

ii hZP

g

VZ

P

g

V+++=++

22

22

(2)

Where: Vi, Vj : velocity at section i, j respectively

Pi, Pj : pressure at section i, j respectively

Zi, Zj : elevation at of water surface section i, j

hi,j : friction loss from section i to section j

For this apparatus, Zi = Zj, Vi = Vj, hence

ji

ji

ji hPP

h ,, =−

=

(3)

hi,j : the difference of manometer reading at section i and section j.

On the other hand, the friction factor can be determined by Darcy’s formula :

g

V

D

Lfh ji

2

2

, = (4)

Where : f : friction factor

L : the distance between section i and j

D : diameter of pipe

V : velocity of pipe

The friction coefficient depends upon the Reynolds number of flow and upon the ratio e/D, the relative

roughness of the pipe.

=

D

eRFf e , (5)

Where : Re : Reynolds’ number

Re = VD/ ( : Kinematic viscosity coefficient)

e : size of roughness

D

e: relative roughness

For a given pipe, D

e is a constant

IV. PROCEDURE

1. Open the control valve (1) to discharge water into pumping chamber. When the pumping chamber

is full of water, close control valve (1).

2. Open the valve at the end of the pipe (2).

3. Turn on power to start pump and use the control valve to regulate the flow rate in the pipe

according to the current intensity shown in ampere meter.

4. For 3 large flow rates corresponding to current 21A ≤ I ≤ 22A :

- Closing the valves on the tubes 3,4 of manometer.

- Reading the water level in the tubes 1 and 2, and measuring the water level before the

Vee-notch in the channel by point gauge.

5. For 5 flow rates corresponding to current I ≤ 20.5A, repeat the same procedure above.

The data table may be arranged as follows:



12

Table 1

No I Manometer reading (mm) Water

Ampere (1) (2) (3) (4) Level (z) (cm)

1 23.0

2 22.5

3 22.0

1 21.5

2 21.0

3 20.5

4 20.0

5 19.5

V. REPORT

▪ For 3 large flow rates

1. Compute the flow rates in the pipe from the measured data in table 1.

2. Compute the head loss between section (1) and (2). And then plot the head loss against flow rates.

Conclude the relationship between the head loss and flow rates.

▪ For 5 small flow rates

3. Determine 7 small flow rates by using the measured data in Table 1.

4. Compute the head loss between sections 2-1, 3-1 and 4-1. And then plot the graph of the friction

loss against the distances. Give the conclusions about the relationship between the friction loss

and the distances.

▪ For 8 flow rates (3 large flow rates and 5 small flow rates)

5. Determine the friction factors f by using the Darcy’s formula. Plot the curve of log (f) against log

(Re) and give the conclusion about the regime of the flow in the pipe.



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Experiment 4 MEASUREMENT OF VOLUMETRIC FLOW RATE

I. OBJECTIVES OF EXPERIMENT

Comparison of flow measurement devices in a duct:

− Orifice plat.

− Venturi nozzle.

II. EQUIPMENT SET - UP

The fan inlet is a duct 149 mm diameter (1) provided with pressure tapings whereby the static

pressure may be measured simultaneously at each of 4 sections. All four pressure tapings are

connected to a bank of pressurised manometer tubes (4). Two flow measurement devices are:

- 65mm orifice plat (2)

- 65mm (d) – 149mm (D) diameter ventury nozzle (3).

III. SUMMARY OF THEORY

The volume flow rate at the orifice plate and venturi nozzle in the pipe is determined by

formula as follows

Where: ppdC

Q /241 4

−=

(1)

Q: volumetric flow rate.

C: discharge coefficient.

p: pressure difference from inlet to throat. The manometer containing liquid of density 1 is

used to indicate p, the pressure difference may be expressed in terms of the manometric head

differential h by:

p = (1 –)gh (2)

: flow density. = 1.226 kg/m3

1: water density 1 = 1000 kg/m3

: diameter ratio = d/D.

: expandability factor. The expandability factory is also detailed in the code and allows for

the effects of density change in gas flows where a high pressure reduction occurs. For liquid flows

and gas flows with moderate variation in pressure at the meter, ≈ 1.00.



14

The discharge coefficients of the orifice plat and the venturi nozzle can be determined by empirical

formula. For the orifice plate.

Where:

Re: Reynolds number

(4)

U: upstream pipe velocity.

Q: discharge in pipe

D: diameter of pipe

: dynamic viscosity

When determining Q from p, it is necessary to estimate a value C initially as Re cannot be

calculated until Q is known. From an initial estimate of C (example C = 1), Q can be calculated and

thus Re found. The value of C can then be corrected and new values of Q and Re cure calculated.

For the venturi nozzle.

(5)

IV. PROCEDURE

Check water level in the tank, it should be at the level of half of tank. Open the valve at the outlet

of fan and turn on the motor.

Take the readings of 4 manometer levels, and the reading values at two electric pressure measurer

(give the different values of gage pressure between two sections in KPa).

In order to decrease or increase the volume flow rate, the valve is closed or opened partly according

to the rotating velocity of the motor decrease or increase.

Repeat for three valve settings (three rotating velocities: 800 rpm, 600 rpm and 400 rpm), and

write the readings of manometer levels and the values of electronic pressure measurer into Table 1

Table 1 Measured experimental data

Rotating velocity (rpm) 800 600 400

Tube

N0

Cross

Section

Manometer Levels (mm)

Exp.1 Exp.2 Exp.3

1 1

2 2

3 3

4 4

Electronic

pressure

measurer

1-2

3-4

D

QUD

4Re ==

5.4196.09858.0 −=C

(3)



15

V. REPORT

1. Derive the formula (1) in case of C = 1.

2. Derive the formula (2)

3. Determine the volumetric flow rater in three experiments by using orifice flat.

4. Determine the volumetric flow rater in three experiments by using venturi nozzle.

5. Compare the computed results between the using of orifice plat and venturi nozzle, give

the conclusions.



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APPENDIX 1: VERNIER AND POINT GAUGES

I. STRUCTURE:

The point gauge and Vernier are used

in the laboratory to measure the

altitude (or depth) as well as the

coordinates of the points (see Figure

A1). It is composed of the following

sections:

- Scale: is a cylindrical steel bar

carved in millimeters. Its end is

attached with a small needle to touch

the measuring point.

- Vernier: is used to determine the

measure of the scale. The Vernier is

engraved, the number of lines

divided equals to 1mm of the

measure of scale. These scales in the

laboratory has the accuracy up to 1 /

20mm (corresponding to the Vernier

having 20 bars). The measure of the

scale is determined according to the

position of the line in the Vernier.

- Adjustment crew: is a wheel,

having the responsibility in sliding

scale up and down so that the needle

tip can touch the surface to be

measured.

- The support frame: is responsible

for keeping the Vernier fixed.

II. USAGE:

1. Use the adjustment crew (wheel), to slide the scale up and down so that the tip of the needle

contacts the surface of the water to be measured. (Note that the water surface must be stable

at the time measuring).

2. Read the water depth value on the scale, right at the "0" line of Vernier. This "0" line is usually

in the middle of any line on the scale, the actual value will fall in the middle.

3. Determine the decimal part of 1mm: find a line on the Vernier that coincides with one line on

the scale. The remaining decimal part of 1mm is the number of overlapping lines on the

Vernier multiplying with 1/20mm.

Example: in Figure A.1, number of main scale is 78mm (determined by reading on the main

scale corresponding with Zero of Vernier) and the decimal is 0.75mm (15/20) (based on the

value of a line on Vernier, here is 15, coinciding with some line on the main scale).

Figure A-1

Experiment 1 FLUID STATICS

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