2183-261 Engineering Mechanical Laboratory
Part 1) Problem Definition: Flow Measurements
1.
Introduction
1.1. Applications (General)
Many buildings and plants have pipelines for conveying fluid mediums. When either the quantity or the quality
of flow in pipe are necessary to be known. Among many parameters, flow rate is an important parameter informing
the physics of the flow. Although, flow rate can be measured by many equipments, herein, a venturi is employed.
1.2. Background and Motivation (for the specific problem to be studied)
A venturi is a differential-pressure type flowmeter. Due to its curvy shape, fluid changes speed when flowing
through different cross-sectional areas to maintain constant mass flow rate at steady state inside a venturi. This
results in change in pressure along the flow direction, corresponding to change in flow speed. If one can relate flow
rate to the fluid pressure along a venturi, one can measure flow rate indirectly from pressure difference in a venturi.
The relation stated above is derived and shown in Appendix A.
1.3. Definition of The Experiment
1.3.1.
Problem Statement / Objective
1.
1.3.2.
To calibrate a venturi with a direct method and obtain a discharge coefficient (Cd) for improving the
theoretical formula
Experimental Conditions
First, since the theoretical relation is derived at steady state (see detail in Appendix A), assure this condition
before measuring data. Second, the datum points should be broad enough for a certain range of Re, which the rig
can do. Third (optional), inlet flow condition may be controlled as much as possible.
1.
Justification / Approach
1.1. Experimental Principle
1.
2.
Conservation of mass
Conservation of energy of a flow in a straight pipe
1.2. Experimental Details
1.2.1.
Experimental Setup (See also figure 1)
At any steady flow rate, pressure difference between two ports is measured as a function of the measured flow
rate. This is achieved through: Set the flow to circulate in the experiment set by turning on the pump set. Adjust
flow rate to a certain speed by using the valve behind the venturi. Measure flow rate by the balance. Measure
pressure difference on the venturi by multi-tube manometer. Record the data. Then change the flow rate and record
more data. Finally figure out the relation between flow rate and pressure difference.
1.2.2.
Measurement and Instrumentation: (See details in Appendix B.)
1.
2.
3.
Measuring tank: accumulating water at an end of the pipe line.
Manometer: using hydro-static pressure.
Venturi: using the relation between speed difference and pressure difference .
2.
Experimental Results
2.1. Graphical Presentation of Results
1.
Plot flow rate (Q) VS square root of head difference ( H )
2.
Calculate (non) linear regression of the plot in item 1 (optional)
2.2. Experimental Results
3.
1.
How does the trend of the graph in 2.1 look like; a linear line, 2 linear lines, parabolic line, or else?
2.
What does it happen with the graph in 2.1 when flow rate is high and low?
3.
Does the graph in 2.1 intercept the origin (0,0)? Can it be explained with any theories?
4.
How much is the Cd obtained from this experiment?
5.
How is the obtained Cd? (lower or higher when compared with a reference, discuss)
6.
Can we improve the equation (A9) to make it more accurate for predicting flow rate? (optional)
Conclusions
1.
What is the theoretical relation between flow rate and pressure different?
2.
How accurate is the relation obtained in the conclusion 1?
3.
How much is the Cd obtained from this experiment?
4.
What is the uncertainty of the Cd? (optional)
(a) Schematic diagram
(b) Venturi
Module
28mm
14mm
Multi-tube
manometer
21o
10o
Venturi
Orifice
Pitot tube
Flow
direction
Controller
0.00
Measuring tank
Supply tank
Supply tank
Pump
Hydraulic bench
Figure 1: (a) Schematic diagram of Flow
Through a Venturi, consists of supply tank, pump,
venturi, manometer and measuring tank. The
pump forces water (from supply tank) to circulate
along the pipe work, through the venturi, the
measuring tank and return back to the supply tank
at last. Flow rate can be controlled by the
controller of the pump. Pressure difference
between the inlet and the minimum crosssectional area of the venturi is obtained by the
manometer. Meanwhile the measuring tank is
used for measuring amount of water accumulated
in tank when the bottom valve of the tank is in
close condition. (b) The dimension of the venturi.
2103-390 Mechanical Engineering Experimentation and Laboratory I
Part 2) Appendix: Flow Through A Venturi
Contents:
1. Appendix A: Theoretical relation between flow rate and pressure difference of flow through a venturi.
2. Appendix B: Principal concepts of some important apparatuses.
3. Appendix C: Derivation of energy equation in pipe flow.
1. Appendix A: Theoretical relation between flow rate and pressure difference of flow through a venturi.
v1
v2
1
2
Figure A1: Simple diagram of flow through a venturi, showing the location and velocity of 2 important sections (i.e.
the inlet and the smallest cross-sectional area of the venturi) for illustrating the derivation of the relation between
flow rate and pressure difference these 2 specified points. [Hidden lines stand for considered cross-sections and
Arrows stand velocity normal to the nearest cross-section.]
Assumptions
1. Fluid is incompressible
2. Steady state
3. Venturi aligns on horizon
4.
5.
6.
Frictionless
Uniform velocity profile
Constant temperature
We start to derive the relation between flow rate and pressure difference on a venturi from 2 basic equations, i.e.
Continuity equation and Energy equation in pipe flow. At first, Continuity equation is manipulated. The concept of
this equation that mass flow rate across section area 1 is equal to that of section area 2 in the case that no
accumulated fluid in the portion between them. So, we can readily write down the Continuity equation as follows.
(A1)
1 A1v1  2 A2 v2
However, the fluid in this experiment is incompressible, so
below.
A1v1  A2v2
v1 
1   2  
= constant and (A1) becomes simpler as
A2
v2
A1
The next equation is the Energy equation of a flow in a straight pipe,
1
1




2
2
 P1  1u1  2 1 1v1  1 gz1  Q   P2  2 u2  2  2 2 v2  2 gz2  Q   gH lT Q .




(A2)
(A3)
The derivation of Eq. (A3) is shown in Appendix C.
Next the assumption 1 and 6 are employed, we thus can rearrange terms in (A3) and obtain:
 P P

1v12   2 v22  2  2 1  g  z2  z1   gH lT 
 

(A4)
Since the venturi aligns on horizon as stated in assumption 3,
 P2  P1
1v12   2 v22  2 


z1  z 2 and (A4) will reduce to be:

 gH lT 

(A5)
Substitute (A2) into (A5), we get:

 P P

A2  2
  2  1 22  v2  2  1 2  gH lT 
A1 
 


v2 
 P P

2  1 2  gH lT 



2



  2  1  A2  

 A1  

(A6)
Multiply both sides of (A6) with A2 to calculate volumetric flow rate.
A2 v2  A2
Q  A2
 P P

2  1 2  gH lT 
 

2



  2  1  A2  
A

 1  

 P P

2  1 2  gH lT 
 

2



  2  1  A2  
A

 1  

(A7)
Generally, the pressure difference read from manometer is in unit of water height. For convenience, we will rewrite
(A7) in the form of water height. That is:
Q  A2
Q  A2
  gH1   gH 2

2
 gH lT 



2



  2  1  A2  

 A1  

2 g  H1  H 2  H lT 
2


 
  2  1  A2  

 A1  

2g
 H1  H 2  H lT 
2

 A2  
  2  1 
 
A1  




If we apply assumption 4 and 5, Eq. (A8) will yield
Q  A2
Q  A2
2g
2


 
1  1 A2  

 A1  

 H1  H 2 
Where Q = volumetric flow rate
A = cross-sectional area
H = water height in manometer
g = gravity acceleration
subscription 1 = inlet section
subscription 2 = minimum area section
(A8)
(A9)
2. Appendix B: Principal concepts of some important apparatuses.
Balance:
Figure B1: Schematic diagram of a
measuring tank, expressing how it works.
The reservoir tank receives water inflow
from above and drains water via the
bottom control valve. If the control valve is
in close position, water will be trapped in
the tank, lead to balance between the
weight of water in the tank and the
reference weight suspended at the opposite
side of the beam.
A measuring tank is a tank as shown in figure B1. It is used for measuring the amount of water trapped in the
reservoir tank. The concept is that when the valve at the bottom of the tank is closed, the water will be trapped in
the tank and we will be able to know the accumulated water volume by reading the scale of the left hand side.
To utilize the measuring tank, a stop watch is needed. And the procedure is as shown in figure B1. First, close
the control valve below the tank. Second, wait until the amount of water trapped in the tank reaches a certain value.
Third, stop the stop watch. The fraction between the accumulated water volume and the recorded time interval from
stop watch is the flow rate of water. (Do not forget to re-open the control valve to drain water out of the measuring
tank after use).
Manometer:
extended scale
Upright manometer
Inclined manometer
Figure B2: Showing the typical shape of an upright manometer and an inclined manometer. Also, the figure shows
extended scale on inclined manometer, which gives more resolution of pressure measuring by inclined manometer.
Manometer is a transparent tube containing some amount of fluid. One end of the tube is jointed to the point to
measure. And another end is opened to atmosphere or jointed to a reference point. Manometer is used for measuring
pressure difference between both ends of the tube. So, when one end of the manometer is opened to atmosphere, it
gives us gauge pressure of another end. The fluid inside manometer can be selected to be any fluid, which does not
react with other parts. Usually lead, water and oil are widely used in manometer. Generally, the unit of pressure
difference is displayed in the unit of fluid height, e.g. water meter, lead inch. There are mainly two types of
manometer, as shown in Figure B2, upright and inclined manometer. Upright manometer is made in simpler shape
and easy to manage, while inclined manometer gives more resolution of measuring scale. This is because the scale
on inclined manometer will be extended longer than that of upright manometer, results in more accurate reading.
Venturi:
A venturi is physically a short pipe whose cross-sectional area varies gradually from large to small, then turns
large again (mostly to the same size before reduction: see figure B3 for illustration). Therefore, a venturi is
separated into 2 portions, converging and diverging portion according to its shape. We call the portion that crosssectional area reduce from large to small as converging portion, in contrast, the portion that cross-sectional area
turns from small to large is called as diverging portion. The importance of venturi is due to its varying crosssectional area. Fluid changes its speed when flowing through different cross-sectional areas to maintain constant
mass flow rate at steady state. This results in different pressure of flowing fluid. If one can relate mass flow rate to
the fluid pressure along a venturi, the venturi can be applied as a measuring device that is useful in industrial sphere.
high speed
low pressure
slow speed
high pressure
slow speed
high pressure
converging
portion
diverging
portion
Figure B3: Typical shape of a venturi consists of 2 portions, converging and diverging portion. Converging portion
is defined on the portion that cross-sectional area reduces. And diverging portion is defined on the portion that
cross-sectional area raises. The vector arrows show symbolically velocity at inlet, minimum cross-sectional area
and outlet, respectively. The speed varies from slow to high and slow, depending on the size of cross-sectional area.
Moreover, It theoretically shows that pressure is high at the point of slow speed and vice versa.
3. Appendix C: Derivation of energy equation in pipe flow.
Qheat &Wcv
Qheat &Wcv
Qheat &Wcv
CV
CV
CV
influx
pAin
pAin
pAout
(b)
effflux
Lout
Lin
Lin
(a)
influx
(c)
Figure C1: Schematic diagram of the energy conservation for a control volume
If our system is closed in a control volume as shown in figure C1 (a), the first law of Thermodynamics will say
that
ECV  Qheat  Wcv .
If there is an energy influx as shown in figure C1 (b), the influx will not only add an amount of energy to the control
volume but it will also give some amount of work to the control volume, that is






ECV  Qheat  Wcv 
 Qheat  Wcv
 Qheat  Wcv
 Qheat  Wcv
 Qheat  Wcv
 Qheat  Wcv
Ain
Ain
Ain
Ain
Ain
Ain
in vin ein dA  Win
in vin ein dA  Fin  Lin t
in vin ein dA  Pin Ain  Lin t
in vin ein dA  Pin Ain Lin t
in vin ein dA  PinQin
in vin ein dA  min Pin in
Identical to influx in figure C1 (b), if there is an energy efflux as shown in figure C1 (c), the efflux will subtract
both some amount of energy and work from the control volume. Consequently, the energy equation turns to be
ECV  Qheat  Wcv 

Ain
in vin ein dA  min Pin in 

Aout
out vout eout dA  mout Pout out
Or
ECV 

Aout
out vout eout dA  mout Pout out 

Ain
in vin ein dA  min Pin in  Qheat  Wcv .
(C1)
If we want to consider energy balance of a flow in pipe, we will draw a control volume as shown with the dash
lines in figure C2. The obtained control volume is thus a cylinder with diameter and length are D and L,
respectively. There are only 2 opened surfaces, the inlet is the surface 1 on the left and the outlet is the surface 2 on
the right. According to Eq. (C1), we may write the change of energy in the control volume as follows.
ECV 

A2
2v2e2 dA  m2 P2 2 

A1
1v1e1dA  m1P1 1  Qheat  Wcv
(C2)
p1
p2
L
w
r x v1
v2
D
Figure C2: A control volume of a flow in pipe (shown with the dash line)
The work rate ( Wcv ) is the work rate that the fluid has to give to surroundings, in the case it is the effort that the
fluid has to overcome the wall friction (  w ), that is
Wcv  Fw  v   w Awv .
(C3)
And we can relate the term  w AwV  with the pressure difference across the inlet and the outlet as
 w Awv   P1  P2  Apipe v   P1  P2  Q
(C4)
With Eq. (C3) and (C4), we can rewrite Eq. (C2) as below.
ECV 

A2
2v2e2 dA  m2 P2 2 

A1
1v1e1dA  m1P1 1  Qheat   P1  P2  Q
(C5)
Equation (C5) is a general form of energy equation of flow in pipe. But when there is no heat flux, the equation can
be reduced to be
ECV 

A2
2v2e2 dA  m2 P2 2 

A1
1v1e1dA  m1P1 1    P1  P2  Q .
(C6)
Moreover, if we apply the equation with steady flow, Eq. (C6) will be simpler as below.

A2
2v2e2 dA  m2 P2 2 
m1P1 1 

A1

A1
1v1e1dA  m1P1 1    P1  P2  Q
1v1e1dA  m2 P2 2 

A2
2v2e2 dA   P1  P2  Q
(C7)
The specific energy ( e ) is the summation of internal, kinetic and potential energies ( u  12 v 2  gz ). Therefore Eq.
(C7) will be
m1P1 1 

A1


1v1 u1  12 v12  gz1 dA  m2 P2 2 

A2


2v2 u2  12 v22  gz2 dA   P1  P2  Q
If specific internal energy and altitude are assumed to be constant across a cross sectional area, this will yield
m1P1 1  1v1 A1u1 

1
A1 2
1v13dA  1v1 A1 gz1  m2 P2 2  2v2 A2u2 

1
A2 2
2v23dA  2v2 A2 gz2   P1  P2  Q
m1P1 1  1Qu1 

1v13dA  1Qgz1  m2 P2 2  2Qu2 
1
A1 2
1QP1 1  1Qu1  12

A1

1
A2 2
1v13dA  1Qgz1  2QP2 2  2Qu2  12
2v23dA  2Qgz2   P1  P2  Q

A2
2v23dA  2Qgz2   P1  P2  Q
1
1




2
2
 P1  1u1  2 1 1v1  1 gz1  Q   P2  2 u2  2  2 2 v2  2 gz2  Q   gH lT Q .




(C8)
The kinetic energy coefficient (  ) is defined as


 v3 A
Apipe
 v 3 Apipe
or


 v3 A
Apipe
 v 2Q
(C9)
When a function of a velocity profile is uniform ( v  v ; similar to velocity profile at the inlet of figure C),  will
be equal to unity. But if the function of a velocity profile is parabolic ( v  0.25umax  D  2r  D  2r  D 2 , similar
to velocity profile at the outlet of figure C),  will be equal to 2.
A quick question: What is the value of  if the velocity profile is linear ( v  vmax  D  2r  D )?