Experiment 1: Flow through a Convergent – Divergent Duct Objective of Experiment
To demonstrate the application of the Bernoulli equation as applied to the flow in a convergent – divergent duct. Introduction
Bernoulli’s principle can be applied to various types of fluid flow resulting in what is loosely denoted as Bernoulli equation. In fact, there are different forms of Bernoulli equation for different types of flow. The simple form of Bernoulli equation is valid for incompressible flow as well as compressible flow. Bernoulli’s Principle states that the pressure of a fluid decreases as the speed of the fluid increases.! "ithin the same fluid, high#speed flow is associated with low pressure. This phenomenon applied to the flow in a convergent#divergent duct. $luid %water& may be considered to be incompressible and flow through convergent# divergent duct with a different set of flow rate. 'ecord the pressure head in the manifold in order to demonstrate the application of the Bernoulli equation. The fluid can be either is gas or liquid and there are few assumptions to be made which are $luid flows smoothly $luid flows without eddies of fluid swirled in the venture duct $luid flows everywhere through the pipe Incompressible fluid has the same density in any position.
• • • •
The Bernoulli equation is given by the following relationship( P * ρ * g
+
v*
)
) g
+
z *
=
P ) ρ ) g
+
v)
)
) g
+
z )
=
H
"here, ρ * + ρ )
density of water
g
gravitational acceleration
v
*
velocity at cross section *
v)
velocity at cross section )
z
*
H
+ z )
elevation total pressure head
"ater may be considered to be incompressible,
ρ * ρ ) ρ
$or a hori-ontal convergent#divergent duct,
z ) z
The velocity of the flow
v
*
z
*
+ v) can be calculated from the cross sectional area + the
volume flow rate . The volume flow rate A*
π
=
0
)
d * + A)
π
=
0
d )
v
*
/* v) /)
)
/ssuming no frictional losses in the ideal case, 1 remains constant. P * ρ * g
+
v*
)
) g
+
z *
=
P ) ρ ) g
+
v)
)
) g
+
z )
=
H o
The Ideal pressure head is obtained from P ) ρ ) g
+
z )
=
H o
−
v)
)
) g
%Ideal height of manometer column&
1owever because of frictional losses, the actual 1 drops along the venturi duct. Therefore the /ctual pressure head is less than the Ideal Pressure 1ead.
rocedure
*. The water outlet tube is directed to the sump of the apparatus. ). The water pump is switched on. 2. The main input water flow valve is 3ept fully open. The bypass water valve is ad4usted to control the volume flow rate. 0. The water flow rate is ad4usted to the ma5imum possible flow rate by closing the bypass valve. The flow is allowed to stabili-e and to remove all air bubbles in the system. 6. The air pump provided is fit to the right end of the manifold on top of the manometers. 7. The pump is used to control the base line level of the manometer readings by increasing or decreasing he pressure in the manifold. The base line need not fall e5actly at the -ero line because the velocity calculations used only the pressure difference. 8. The pressure in the manifold can be increased by pumping action of the air pump. 9. The pressure in the manifold can be decreased by pressing on the needle valve on the right hand side of the manifold to release the air in the manifold.
:. The volume flow rate reading and all the manometer readings are ta3en .The readings are entered into the spreadsheet in the computer provided for the purpose. *;. The bypass valve is ad4usted to decrease the flow rate to obtain the different sets of manometer readings for flow rates of 7.6,7.;,6.;,0.; and 2.; gallons
Calculation:
*& $low rate of water, 2gal
=
*;;;kg < m
+ g :.9*m
2
=levation, >
2;mm
?iameter
)0mm
/rea, /
π 0
D )
0.6)2:=#0m ) Q
@elocity, v
A
;.;;;*9:) 0.6)20 × *;
;.0)m
)
) g
=
( ;.0))
)
) × :.9*
9.:mm ;.;;9:m
$or ideal cases( 1; ))*.6mm
P ) ρ ) g
=
H o
−
v)
)
) g
−
z )
))*.6mm – 9.:mm – 2;mm *9).7mm $or actual cases( 1 )*9.:mm
P ) ρ ) g
=
H −
v)
)
) g
−
z )
)*9.:mm#9.:mm#2;mm *9;mm Percentage error( *9).7 − *9; *9).7
)&
× *;;A =
*.0)A
$low rate of water, 0gal
−0
;.;;;)6)2m2
=
*;;;kg < m
+ g :.9*m
2
=levation, >
2;mm
?iameter
)0mm
/rea, /
π 0
D )
0.6)2:=#0m) Q
@elocity, v
A
;.;;;)6)2 0.6)20 × *;
;.67m
)
) g
=
( ;.67)
)
;.;*6:m
) × :.9*
$or ideal cases( 1; )6*.6mm
P ) ρ ) g
=
H o
−
v)
)
) g
−
z )
)6*.6mm – *6.:mm – 2;mm );6.7mm
$or actual cases( 1 )6;.:mm
P ) ρ ) g
=
H −
v)
)
) g
−
z )
)6;.9#*6.:#2; );0.:mm Percentage error( );6.7 − );0.: );6.7
3)
× *;;A =
;.20A
$low rate of water, 6gal
−0
ρ
=
*;;;kg < m
2
:.9*m
+g
=levation, >
2;mm
?iameter
)0mm
/rea, /
π 0
D )
0.6)2:=#0m) Q
@elocity, v
A
;.;;;2*60 0.6)20 × *;
−0
;.8;m
)
) g
=
( ;.8;)
)
) × :.9*
)0.9mm
$or ideal cases( 1; )89.;mm
P ) ρ ) g
=
H o
−
v)
)
) g
−
z )
)89mm–)0.9mm – 2;mm ))2.)mm
$or actual cases( 1 )8:.9mm P ) ρ ) g
=
H
−
v)
)
) g
−
z )
)8:.9#)0.9#2; ))6.;mm
Percentage error( ))2.) − ))6.; ))2.)
4)
× *;;A =
;.9;7A
Flow rate of water, Q
= 6gal/min = 0.0003785m3/s
ρ
=
*;;;kg < m
= 9.81m/s
&g
2
!le"ation, #
= 30mm
$iameter
= 4mm
%rea, %
=
π 0
D )
= 4.539!4m Q
'elo(it, " =
=
A
;.;;;2896 0.6)20 × *;
−0
= 0.84m/s v)
)
=
) g
( ;.90)
)
= 0.0357 m
) × :.9*
For i*eal (ases+ 0 = 310.9mm P ) ρ ) g
=
H o
−
v)
)
) g
−
z )
= 310.935.730 = 45.mm
For a(t-al (ases+ = 310.7mm P ) ρ ) g
=
H −
v)
)
) g
−
z )
= 310.735.730 = 45.0mm er(entage error+ )06.) − )06 )06.)
5)
× *;;A =
;.;9A
Flow rate of water, Q
ρ
=
*;;;kg < m
2
&g
= 6.5gal/min =0.0004100m3/s
= 9.81m/s
!le"ation, #
= 30mm
$iameter
= 4mm
%rea, %
=
π 0
D )
= 4.539!4m Q
'elo(it, " =
=
A
;.;;;0*;; 0.6)20 × *;
−0
= 0.91m/s v)
)
) g
=
( ;.:*)
)
= 0.0419m
) × :.9*
For i*eal (ases+ 0 = 35.4mm P ) ρ ) g
=
H o
−
v)
)
) g
−
z )
= 35.441.930 = 53.5mm
For a(t-al (ases+ = 36.9mm P ) ρ ) g
=
H
−
v)
)
) g
−
z )
= 36.941.930 = 55.0mm
er(entage error+ )62.6 − )66 )62.6
× *;;A =
;.6:A
Di!cu!!ion
$rom the calculation has shown, the actual pressure head is differ than ideal pressure head this is due to the frictional losses occurred when the water flow through the convergent# divergent duct. There are two types of frictional losses occurred in the duct where the first friction is called the s3in#friction which is due to the roughness in the inner part of the where
the fluid comes in the contact of the pipe material. The second one is called form friction which is due to the obstructions present in the line flow and it may be due to a band and control valve or anything which changes the course of motion of the flowing fluid. $urthermore, the fluid is assumed without any eddies of fluid swirled in the venture duct however, this will only happen in the ideal condition and ideal apparatus only and in actual case there are frictional losses and eddies of fluid swirled in the venture duct and caused the results that we obtained is different. astly, the paralla5 error is also one of the causes that caused actual cases result different with the e5pectation results. This phenomenon occurred when we recorded the reading from the manifolds .
f te on"ergent$i"ergent *-(t is in(line* -2war*s, te (orres2on*ing total ea* , will sta (onstant an* te 2ress-re ea* will increase at the upstream due to the difference in height e ow will (ontin-e -ntil te
2ress-re ea* at -2stream e-als te ele"ation ea* at *ownstream t-s "elo(it ea* will *e(rease -ntil ero wen te -i* (omes to rest. is is ase* on te teor of erno-lli e-ation were te "elo(it of -i* in(reases wen te 2ress-re *e(rease* an* "i(e "ersa. Coreover, the velocity of the fluid will get affected by the cross sectional area . $rom the graphs has shown it is obvious that the pressure head will decrease from wide area duct to narrow area duct. This phenomenon is due to the velocity increase when the water reaches the narrow area than decrease again when it pass through the wide area duct. where the velocity increases when the fluid flows through a small cross sectional area and vice versa. $ew precautions must be ta3en in this e5periment. $irst of all, the readings must be ta3en carefully to prevent paralla5 error and our eyes must be parallel with the measurement so that accurate date can be obtained. /ll the bubbles are ma3e sure to be removed when ad4usting the water flow rate.
Conclu!ion The results shown that the actual pressure head is lower than ideal pressure head this is due to the few assumptions that have been made and we 3now that some frictional losses and human errors such as paralla5 error occurred in the e5periment. The velocity of the water is low when the pressure is high, and vice versa. $urthermore, the si-e of the cross sectional area will also affected the velocity of the fluid. astly I should conclude that this e5periment was demonstrated the application of the Bernoulli equation as applied to the flow in a convergent# divergent duct.
"eference http(<
