## Bernoulli’s Theorem

Bernoulli’s Theorem

LABORATORY MANUAL for MECE2860U-Fluid Mechanics

Experiment # 3

Bernoulli’s Theorem

Demonstration Apparatus

LABORATORY MANUAL for MECE2860U-Fluid Mechanics

2

Experiment # 3: Bernoulli’s Theorem Demonstration Apparatus

1.1 Objective

The objective of this experiment is to investigate Bernoulli’s law, perform measurements along a venturi tube

and determine the flow rate factor (K).

1.2 Introduction and Theoretical Background

Bernoulli’s Equation is a very important integral form of the equation of fluid motion. It is one of the most

commonly used equations in fluid mechanics. The Bernoulli equation is named in honor of Daniel Bernoulli

(1700-1782). Many phenomena regarding the flow of liquids and gases can be analyzed by simply using the

Bernoulli equation. However, due to its simplicity, the Bernoulli equation may not provide an accurate

enough answer for many situations, but it is a good place to start. It can certainly provide a first estimate of

parameter values. Modifications to the Bernoulli equation to incorporate viscous losses, compressibility, and

unsteady behavior can be found in other (more complex) calculations in the textbook and Ref. [1]. When

viscous effects are incorporated, the resulting equation is called the “energy equation”.

This experiment utilizes a Venturi tube (as a flow-area varying device) and a Prandtl tube-manometer set up

(as a flow measurement device) to demonstrate some of the key concepts of the Bernoulli’s Equation.

The Bernoulli’s Equation is a description of the momentum of steady, incompressible, irrotational, and

frictionless flow (Figure 1).

Figure 1. Steady, incompressible, irritation and frictionless flow

A general form of Bernoulli’s Equation can be expressed as:

LABORATORY MANUAL for MECE2860U-Fluid Mechanics

p + V + gh = p + V + gh2 = const

2

1 2 2

2

1 1 2

1

2

1 ? ? ? ? (1)

where p is static pressure, ? is density, V is velocity, g is gravity constant, and h is the height with respect to

the reference level (i.e. sea level). The subscripts 1 and 2 denote the stream wise locations of the flow.

Equation (1) can be interpreted as: the total energy (sum of static pressure pstat = p , dynamic pressure

2

2

pdyn = 1 ?V and body force pbdy = ?gh ) of a fluid body flowing along the streamline always remains

constant. For gas, since the density is low, the body force is practically insignificant. Thus, the term

pbdy = ?gh can be ignored and Equation (1) can be simplified to:

2 V const

V p 1 2

p 1 2

2 2

2

1 + ? 1 = + ? = (2)

This expression is also termed the total pressure pt:

2

2

pt = pstat + pdyn = p + 1 ?V (3)

Proper use of the Bernoulli equation requires close attention to the assumptions used in its derivation. To use it

correctly we must constantly remember the basic assumptions used in its derivation: (a) viscous effects are

assumed negligible, (b) the flow is assumed to be steady, (c) the flow is assumed to be incompressible, (d) the

equation is applicable along a streamline. In the derivation of Equation (1), we assume that the flow takes

place in a plane (the x–z plane). In general, this equation is valid for both planar and nonplanar (threedimensional)

flows, provided it is applied along the streamline.

The Bernoulli equation is used to analyze fluid flow along a streamline from a location 1 to a location 2.

Most liquids meet the incompressible assumption and many gases can even be treated as incompressible if

their density varies only slightly from 1 to 2. The steady flow requirement is usually not too hard to achieve

for situations typically analyzed by the Bernoulli equation. Steady flow means that the flow rate (i.e.

discharge) does not vary with time. The inviscid fluid requirement implies that the fluid has no viscosity. All

fluids have viscosity; however, viscous effects are minimized if travel distances are small. Flow along a 100

km river has significant viscous effects (friction between the channel material and the flowing fluid), but

viscous effects along just a 10 m reach of that channel where a sluice gate occurs would be minimal. Sluice

gates are typically analyzed with the Bernoulli equation

1.3 Equipment

The schematic layout and photos of the Bernouilli’s theorem demonstration on apparatus are shown in Figures

2-4, respectively.

Body

force

Dynamic

pressure

Static

pressure

LABORATORY MANUAL for MECE2860U-Fluid Mechanics

4

Figure 2. Layout of Bernoulli’s theorem demonstration apparatus

Figure 3. A photo of Bernoulli’s theorem demonstration apparatus

1. Assembly board

2. Water pressure gage

3. Discharge pipe

4. Outlet ball cock

5. Venturi tube with

six measurement

points

6. Compression gland

7. Probe for measuring

overall pressure (can

be moved axially)

8. Hose connection,

water supply

9. Ball cock at water

inlet

10. 6-fold water

pressure gage

(pressure

distribution in

venture tube)

LABORATORY MANUAL for MECE2860U-Fluid Mechanics

5

Single water pressure gage

Venturi tube with six measurement points

Figure 4. Various photos of the main components of Bernoulli’s theorem demonstration apparatus

Water pressure gage

LABORATORY MANUAL for MECE2860U-Fluid Mechanics

6

1.4 Operating Instructions and Procedure

The following procedure should be followed during the experiments by taking into account Figure 5.

• Arrange the experimental setup on the on the gravimetric hydraulic bench such that the discharge routes

the water into the channel.

• Make hose connection between the gravimetric hydraulic bench and unit.

• Open discharge of the gravimetric hydraulic bench.

• Set cap nut (1) of probe compression gland such that slight resistance is felt on moving probe

• Open inlet and outlet ball cock.

• Close drain valve (2) at bottom of single water pressure gauge.

• Switch on pump and slowly open main cock of the gravimetric hydraulic bench.

• Open vent valves (3) on water pressure gauges

• Close outlet cock carefully until pressure gauges are flushed.

• Regulate water level in pressure gauges by simultaneously setting inlet and outlet cock such that neither

upper nor lower range limit (45) is overshot or undershot.

• Record pressures at all measurement points. Then move overall pressure probe to corresponding

measurement level and note down overall pressure.

• Determine volumetric flow rate. To do so, use stopwatch to establish time t required for raising the level in

the volumetric tank of the gravimetric hydraulic bench from 20 to 30 liters.

Note that the experimental setup should be arranged absolutely plane to avoid falsification of measurement

results (use of spirit level recommended). For taking pressure measurements, the volumetric tank of the

gravimetric hydraulic bench must be empty and the outlet cock open, as otherwise the delivery head of the

pump will change as the water level in the volumetric tank increases. This results in fluctuating pressure

conditions. A constant pump delivery pressure is important with low flow rates to prevent biasing of the

measurement results.

The zero of the single pressure gauge is 80 mm below that of the 6-fold pressure gauge. Allowance is to be

made for this fact when reading the pressure level and performing calculations. Both ball cocks must be reset

whenever the flow changes to ensure that the measured pressures are within the display ranges.

Figure 5. Experimental procedure steps

LABORATORY MANUAL for MECE2860U-Fluid Mechanics

1.5 Calculations

For steady, inviscid, incompressible flow the total energy remains constant along a streamline. The concept of

“head” was introduced by dividing each term in equation (1) by the specific weight, ?=?g, to give the

Bernoulli equation in the following form.

z H

2g

p V2

+ + =

?

(=constant on a streamline) (4)

Each of the terms in this equation has the units of length (feet or meters) and represents a certain type of head.

The Bernoulli equation states that the sum of the pressure head, the velocity head, and the elevation head is

constant along a streamline. This constant is called the total head, H.

Assuming that z1=z2, Equation (4) can be written as follows (Figure 6):

+ =

2g

V

h

2

1

1 L

2

2

2 h

2g

V

h + + (5)

where h1 and h2 are the pressure heads at cross-sectional areas A1 and A2, respectively, while hL is the

pressure loss head.

Figure 6. Cross-sectional areas of the venturi

To conserve mass, the inflow rate must equal the outflow rate. If the inlet is designated as (1) and the outlet as

(2), it follows that

m?? 1 = m?? 2 (6)

Thus, conservation of mass require

?1A1V1 = ?2A2V2 (7a)

If the density remains constant, then ?1=?2 and the above becomes the continuity equation for incompressible

flow

A1V1 = A2V2 or Q1 = Q2 (7b)

Condition 1 Condition 2

LABORATORY MANUAL for MECE2860U-Fluid Mechanics

1.5.1 Velocity profile in the venturi tube

The venturi tube used has 6 measurement points. Table 1 shows the standardized reference velocity Vsr, while

the measurement points along the venturi are illustrated in Figure 7. This parameter is derived from the

geometry of the venturi tube and is given by the relation

i

1

SR A

A

V = (8)

Table 1. Standardized reference velocities

Figure 7. Measurement points along the venturi

Calculate the theoretical velocity values (Vcal) at the 6 measuring points of the venturi tube by multiplying the

reference velocity values with a starting value.

cal SR 1 V = V V (9)

with the starting value for calculating the theoretical velocity at a constant flow rate

1

1 A

V = Q (10)

The dynamic pressure head is calculated as:

h dyn = h t -80 mm – hstat (11)

where 80 mm is subtracted, as there is a zero-point difference of 80 mm between the pressure gages.

Point i

Inside diameter

Di (mm)

Ai

(m210-4)

Reference

velocity

VSR (-)

1 28.4 6.33 1.00

2 22.5 3.97 1.59

3 14.0 1.54 4.11

4 17.2 2.32 2.72

5 24.2 4.60 1.37

6 28.4 6.33 1.00

LABORATORY MANUAL for MECE2860U-Fluid Mechanics

The measured velocity (Vmeas) is calculated from the dynamic pressure as follows:

V (m / s) 2 p (Pa) / (kg / m3 )

meas = ? dyn ? (12a)

or

V (m / s) 2g(m / s )h dyn (mWC)

2

meas = (12b)

Plot the measured and calculated velocity profile along the venture tube at a recorded volumetric flow rate.

1.5.2 Pressure distribution along the venturi tube

Plot the values for hdyn, hstat and ht (mmWC) along the venture tube using the values measured and obtained

from Equation (11).

1.5.3 Determination of flow rate factor

A venturi tube can be used for flow rate measurements. In comparison with orifice or nozzle, there is a far

more smaller pressure loss during measurements of flow rate. The pressure loss ?p between largest and

smallest diameter of the tube is used as measure for the flow rate (Figure 8):

1 3 Q K p – = ? (13a)

Figure 8. Measurement points for the pressure loss (?p1-3)

The flow rate factor K is generally made available for the user by the manufacturer of a venturi tube. If the

flow rate factor is unknown, it can be determined from the pressure loss ?p1-3 as follows:

( )

p (bar)

K1/ s bar Q(l / s)

? 1-3

= (13b)

Table 2 shows the pressure loss for various flow rates as well as the flow rate factor K. Read off the pressure

loss from the six–tube manometer in mm water column (mmWC) and set in the equation as bar. The flow rate

can be used with unit l/s (liter/s).

LABORATORY MANUAL for MECE2860U-Fluid Mechanics

10

Table 2. Pressure loss for various flow rates and flow rate factors

Q=0.275 l/s

Q=0.256 l/s Q=0.166 l/s

Measurement

points

?p

(mmWC)

K

)

s bar )

( l

?p

(mmWC) )

s bar )

( l

?p

(mmWC) )

s bar )

( l

1-3 160 2.1 143 2.1 65 2.1

1 mmWC (or mmH2O) = 0.0980665 mbar ˜ 0.1 mbar = 0.0001 bar = 10 Pa

1.6 Worksheet for Experimental Data

*1 mmWC (or mmH2O) = 0.0980665 mbar ˜ 0.1 mbar = 0.0001 bar = 10 Pa

Volume of water in the tank at the beginning of the experiment (V1)= l

Volume of water in the tank after t=60 s (V2) = l

?V = V2 – V1= – = l t = 60 s Q = V/t = /60 = l/s

Measurement points

(See Figure 7)

h1

(mmWC)

h2

(mmWC)

h3

(mmWC)

h4

(mmWC)

h5

(mmWC)

h6

(mmWC)

hstat (mmWC)

measured

ht (mmWC)

measured

hdyn (mmWC)

Using Equation (11):

h dyn = h t -80 mm – hstat

Vmeas (m/s)

Using

Equation (12b):

2g(m/ s )h (mWC)

V (m / s)

dyn

2

meas =

Vcal

Using Equations (9)-(10) and

Table 1:

V1 = Q / A1 ; cal SR 1 V = V V

?p1-3 (See Figure 8) mmWC bar*

Flow rate factor (K)

Using Equation (13b):

( )

p (bar)

K1/ s bar Q(l / s)

? 1-3

=

s bar

l

bar

(l / s)

p (bar)

K Q(l / s)

1 3

= =

?

=

–

LABORATORY MANUAL for MECE2860U-Fluid Mechanics

11

Nomenclature

A : Cross-sectional area (m2)

g : Gravitational acceleration (m/s2)

h : Height with respect to the reference level (m)

hL : Pressure head loss (mWC)

K : Flow rate factor (1/ s bar )

m?? : Mass flow rate (kg/s)

pdyn : Dynamic pressure (bar)

pstat : Static pressure (bar)

pt : Total pressure (bar)

Q : Volumetric flow rate (m3/s)

V : Volume (m3)

Vcal : Calculated velocity (m/s)

Vmeas : Measured velocity (m/s)

VSR : Reference velocity (m/s)

z : Elevation head (mWC)

?p1-3 : Pressure loss (bar)

? : Density (kg/m3)

References

1. Bernoulli Equation Calculator with Applications. Available at: http://www.lmnoeng.com/Flow/bernoulli.

htm.

2. Equipment for Engineering Education, Instruction and Operation Manuals, Gunt Hamburg Germany,

1998.

3. Munson, B. A., Young, D. F., and Okiishi, T. H. Fundamentals of Fluid Mechanics. 4th Edition, Wiley,

New York, 2002.