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.

find the cost of your paper