ENT 319 THERMOFLUID LABORATORY 1 1 LAB MANUAL BERNOULLI’S THEOREM DEMONSTRATION 1.0 EQUIPMENT / APPARATUS Please familiarize with the unit before operating the unit. The unit consists of the followings: a) Venturi The venturi meter is made of transparent acrylic with the following specifications: Throat diameter : 16 mm Upstream Diameter : 26 mm Designed Flow Rate : 20 LPM b) Manometer There are eight manometer tubes; each length 320 mm, for static pressure and total head measuring along the venturi meter. The manometer tubes are connected to an air bleed screw for air release as well as tubes pressurization. c) Baseboard The baseboard is epoxy coated and designed with 4 height adjustable stands to level the venturi meter. d) Discharge valve One discharge valve is installed at the venturi discharge section for flow rate control. e) Connections Hose Connections are installed at both inlet and outlet. f) Hydraulic Bench Sump tank : 120 litres Volumetric tank : 50 litres Centrifugal pump : 0.6 kW, 60 LPM
Transcript
ENT 319 THERMOFLUID LABORATORY 1
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LAB MANUAL
BERNOULLI’S THEOREM DEMONSTRATION
1.0 EQUIPMENT / APPARATUS Please familiarize with the unit before operating the unit. The unit consists of the followings:
a) Venturi
The venturi meter is made of transparent acrylic with the following specifications: Throat diameter : 16 mm
Upstream Diameter : 26 mm Designed Flow Rate : 20 LPM
b) Manometer
There are eight manometer tubes; each length 320 mm, for static pressure and
total head measuring along the venturi meter. The manometer tubes are connected to an air bleed screw for air release as well as tubes pressurization.
c) Baseboard
The baseboard is epoxy coated and designed with 4 height adjustable stands to level the venturi meter.
d) Discharge valve
One discharge valve is installed at the venturi discharge section for flow rate control.
e) Connections
Hose Connections are installed at both inlet and outlet.
f) Hydraulic Bench
Sump tank : 120 litres Volumetric tank : 50 litres
Centrifugal pump : 0.6 kW, 60 LPM
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Figure 1: Parts Identification Diagram
1. Manometer Tubes 6. Flow Control Valve
2. Test Section 7. Gland Nut
3. Water Inlet 8. Hypodermic Probe
4. Unions 9. Adjustable Feet
5. Air Bleed Screw
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2.0 INTRODUCTION AND THEORY Bernoulli’s Theorem Demonstration (Model: FM 24) apparatus consists of a classical Venturi made of clear acrylic. A series of wall tappings allow measurement of the static pressure
distribution along the converging duct, while a total head tube is provided to traverse along the centre line of the test section. These tappings are connected to a manometer bank incorporating a manifold with air bleed valve. Pressurization of the manometers is facilitated by a hand pump.
The unit is mounted on a base board which is to be placed on top of the Hydraulic Bench (Model: FM110). This base board has four adjustable feet to level the apparatus. The main test section is an accurately machined acrylic venturi of varying circular cross section. It is
provided with a number of side hole pressure tappings, which are connected to the manometer tubes on the rig. These tappings allow the measurement of static pressure head simultaneously at each of 6 sections. The tapping positions and the test section diameters
are shown in Appendix A. The test section incorporates two unions, one at either end, to facilitate reversal for convergent or divergent testing as illustrated in Figure 1 and Figure 2.
Figure 2: Front View of Bernoulli’s Theorem Demonstration Unit (Model: FM24)
Figure 3: Top View of Bernoulli’s Theorem Demonstration Unit (Model: FM24)
A hypodermic tube, the total pressure head probe, is provided which may be positioned to read the total pressure head at any section of the duct. This total pressure head probe may be moved after slacking the gland nut; this nut should be re-tightened by hand after adjustment. An additional tapping is provided to facilitate setting up. All eight pressure tapings are connected to a bank of pressurized manometer tubes. Pressurization of the manometers is facilitated by connecting any hand pump to the inlet valve on the manometer manifold.
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The unit is connected to the hydraulic bench using flexible hoses. The hoses and the
connections are equipped with rapid action couplings. The flexible hose attached to the outlet pipe which should be directed to the volumetric measuring tank on the hydraulics bench. A flow control valve is incorporated downstream of the test section. Flow rate and pressure in the apparatus may be varied independently by adjustment of the flow control valve and the bench supply control valve.
2.1 THEORY 2.1.1 Derivation Using Streamline Coordin
Euler’s equation for steady flow along a streamline is
s
zV
s
zg
s
p
1 (2.1)
If a fluid particle moves a distance, ds, along a streamline,
dpdss
p
(the change in pressure) (2.2)
dzdss
z
(the change in elevation) (2.3)
dVdss
V
(the change in speed) (2.4)
Thus, after multiplying Equation 2.1 by ds,
VdVgdz
dp
(2.5)
Integration of this equation gives:
tconsgzVdp
tan2
2
(2.6)
The relation between pressure and density must be applied in this equation. For the special case of incompressible flow, ρ = constant, and Equation 2.6 becomes the Bernoulli’s Equation.
2.1.2 Bernoulli’s Law 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:
tconshz
g
v
g
ptan
2
*2
(2.8)
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 terms on the left-hand-side of the above equation represent the pressure head (h), velocity head (hv ), and elevation head (z), respectively. The sum of these terms is
known as the total head (h*). According to the Bernoulli’s theorem of fluid flow through
a pipe, the total head h*
at any cross section is constant. In a real flow due to friction and other imperfections, as well as measurement uncertainties, the results will deviate from the theoretical ones.
In our experimental setup, the centerline of all the cross sections we are considering lie on the same horizontal plane (which we may choose as the datum, z = 0, and thus, all the
‘z’ values are zeros so that the above equation reduces to:
tconshg
v
g
ptan
2
*2
(2.9)
This represents the total head at a cross section.
For the experiments, the pressure head is denoted as hi and the total head as h*i,
where i represents the cross sections at different tapping points.
2.1.3 Static, Stagnation and Dynamic Pressures The pressure, p, which we have used in deriving the Bernoulli’s equation, Equation 2.7, is the thermodynamic pressure; it is commonly called the static pressure. The static pressure is that pressure which would be measured by an instrument moving with the flow. However, such a measurement is rather difficult to make in a practical situation.
As we know, there was no pressure variation normal to straight streamlines. This fact
makes it possible to measure the static pressure in a flowing fluid using a wall pressure tapping, placed in a region where the flow streamlines are straight, as shown in Figure 4 (a). The pressure tap is a small hole, drilled carefully in the wall, with its axis
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perpendicular to the surface. If the hole is perpendicular to the duct wall and free
from burrs, accurate measurement of static pressure can be made by connecting the tap to a suitable pressure measuring instrument.
Figure 4: Measurement of Static Pressure
In a fluid stream far from a wall, or where streamlines are curved, accurate static pressure measurements can be made by careful use of a static pressure probe, shown in
Figure 4 (b). Such probes must be designed so that the measuring holes are place correctly with respect to the probe tip and stem to avoid erroneous results. In use, the measuring section must be aligned with the local flow direction.
Static pressure probes or any variety of forms are available commercially in sizes as small as 1.5 mm (1/16 in.) in diameter. The stagnation pressure is obtained when a flowing fluid is decelerated to zero speed by a frictionless process. In incompressible flow, the Bernoulli Equation can be used to relate changes in speed and pressure along a streamline for such a process. Neglecting elevation differences, Equation 2.7 becomes
tconsg
v
g
ptan
2
2
(2.10)
If the static pressure is p at a point in the flow where the speed is v, then the stagnation pressure, Po, where the stagnation speed, Vo, is zero, may be computed from
22
22v
g
pv
g
p oo
(2.11)
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Therefore,
2
2
1vppo (2.12)
Equation 2.12 is a mathematical statement of stagnation pressure, valid for incompressible flow. The term ½ ρV² generally is the dynamic pressure. Solving the dynamic pressure gives:
ppv o 2
2
1 (2.13)
Or
)(2 ppv o (2.14)
or
)(2 hhgv o (2.15)
Thus, if the stagnation pressure and the static pressure could be measured at a point, Equation 2.14 would give the local flow speed.
Figure 5: Measurement of Stagnation Pressure
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Figure 6: Simultaneous Measurement of Stagnation and Static Pressures
Stagnation pressure is measured in the laboratory using a probe with a hole that
faces directly upstream as shown in Figure 5. Such a probe is called a stagnation pressure probe (hypodermic probe) or Pitot (pronounced pea-toe) tube. Again, the measuring section must be aligned with the local flow direction.
We have seen that static pressure at a point can be measured with a static pressure tap or probe (Figure 4). If we know the stagnation pressure at the same point, then the flow speed could be computed from Equation 2.14. Two possible experimental setups are shown in Figure 6.
In Figure 6(a), the static pressure corresponding to point A is read from the wall
static pressure tap. The stagnation pressure is measured directly at A by the total head tube, as shown. (The stem of the total head tube is placed downstream from the measurement location to minimize disturbance of the local flow)
Two probes often are combined, as in the Pitot-static tube shown in Figure 6(b). The inner tube is used to measure the stagnation pressure at point B, while the static pressure at C is sensed using the tapping on the wall. In flow fields where the static pressure variatio in the streamwise direction is small, the Pitot-static tube may be used to infer the speed at point B in the flow by assuming pB =pC and using Equation 2.14. (Note
that when pB ≠ pC, this procedure will give erroneous results)
Remember that the Bernoulli equation applies only for incompressible flow (Mach number, M ≤ 0.3).
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2.1.4 Venturi Meter 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.
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APPENDIX: DATA & RESULTS FLOWRATE 1
Liter Time (min) Flow Rate (liter/min) Flow Rate (m3/s)
20
Cross Section
Using Bernoulli equation
Using Continuity equation
Percentage Error
# h*=hG
(mm)
hi
(mm)
ViB (m/s)
Ai (m2)
ViC
(m/s)
(%)
A B C D E F
FLOWRATE 2
Liter Time (min) Flow Rate (liter/min) Flow Rate (m3/s)
20
Cross Section
Using Bernoulli equation
Using Continuity equation
Percentage Error
# h*=hG
(mm)
hi
(mm)
ViB (m/s)
Ai (m2)
ViC
(m/s)
(%)
A B C D E F
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FLOWRATE 3
Liter Time (min) Flow Rate (liter/min) Flow Rate (m3/s)
20
Cross Section
Using Bernoulli equation
Using Continuity equation
Percentage Error
# h*=hG
(mm)
hi
(mm)
ViB (m/s)
Ai (m2)
ViC
(m/s)
(%)
A B C D E F
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LABORATORY REPORT
LAB 1 : BERNOULLI’S THEOREM
NAME : ______________________________________ MATRIX NUMBER : ______________________________________ PROGRAM : ______________________________________ GROUP NUMBER : ______________________________________ DATE OF EXPERIMENT : ______________________________________ DUE : ____________________________ COMMENTS : _______________________________________ _______________________________________ _______________________________________ MARKS :
