FLUID MECHANICS LABORATORY LABORATORY EXPERIMENT 1
FLOW MEASUREMENT USING VENTURI METER
1.0
INTRODUCTION
Venturi nturi meter meter is used used to measur measuree the fluid flow rate rate by reduci reducing ng the cross sectio sectional nal area in the flow path, path, genera generatin ting g a pressu pressure re differe difference. nce.
It makes use of
Bernoulli’s Principle, a restatement of the law of Conservation of nergy as applied to the fluid flow. In this e!periment, the Cussons P"##$ Venturi %eter is used in con&unction with the Inlet Inlet 'ead 'ead (ank ank P")*+ P")*+ and the P")* P")* Variable riable 'ead 'ead -utlet -utlet (ank. (ank. (he P")*" P")*" manometer board is reuired for pressure measurement. (he /eedback P")*0 may be used instead of the inlet head tank to increase the flow range. (he venturi, which is manufactured from transparent acrylic material, follows the classic #)1 2 )*1 convergent2divergent design which forms the basis of most engineering standards for venturi flow meters. (he P"##$ complies with the British 3tandard B3)*# for flow measurement. (he dimensions of o f the Venturi Venturi %eter are shown in /igure ).). (he upstr upstream eam and and thro throat at pres pressu sure re tappi tapping ngss are are used used for for flow flow meas measur urem ement ent whil whilst st the the downstream downstream tappings tappings allows an assessment assessment of the pressure pressure recovery to be made. (he throat diameter is )*mm and the upstream and downstream pipe diameters are both #)mm.
/igure).) 4imensions of Cussons P"##$ Venturi enturi meter.
1.1
THEORIES AND EXPLANATION EXPLANATION
D1
D2
P1
P2
/igure).# 4iameters and differential pressure across venturi meter at section ) 5 #. 6efer to /igure ).#, from consideration of continuity between the mouth of the venture at section ) and the throat at section #7
Q = A)V )
= A#V # 8).)9
and on the introducing the diameter ratio : ; 4# < 4), then
A# A)
= β # =
V ) V # 8).#9
=pplying Bernoulli’s theorem to the venture meter between section) and section#, neglecting losses and assuming the venturi is installed hori>ontally
P )
ρ g
+
V )# # g
=
P #
ρ g
+
V ## # g
8).+9
6earranging,
P ) − P #
ρ g
= H =
V ##
− V )#
ρ g 8).9
and solving for V#, V # =
#8 P ) − P # 9
V )# ρ ) − # V #
# g H
=
) − β , 8).09
β
D2 =
D1
normally , β
8)."9
0.5
=
(he volumetric flow rate is then given by
Q = A#V #
= A#
# g H )
− β , 8).$9
(he actual discharge will be less than this due to losses causing the velocity through the throat to be less than that predicted by Bernoulli’s (heorem, therefore it is necessary to introduce an e!perimentally determined coefficient of discharge Cd. (he actual discharge will then be given by7
Q = C d A#
# g H )
− β , 8).?9
where
A 2
=
π 2 D2 4
8).@9
(he coefficient of discharge varies with both the 6eynolds number and area ratio. (ypically values for a machined venturi meter are between *.@$0 and *.@@0.
(he pressure loss across the venture meter is less than the pressure difference measured between the mouth and the throat due to the pressure recovery which occurs in the divergence as the kinetic energy is reduced.
1.2
OBJECTIVES
). (o calculate the coefficient of discharge from e!perimental data for a venturi meter. #. (o investigate the measurement of volumetric flowrate using a venturi meter.
1.3
EQUIPMENT PREPARATION
Inlet
P")*+ Constant 'ead Inlet (ank with overflow pipe e!tension fitted.
1.4
(est 3ection
Cussons P"##$ Venturi meter
-utlet
P")* Variable 'ead -utlet (ank
%anometer
(wo of the single manometer tubes.
EXPERIMENTAL PROCEDURE
).
3et up the apparatus as per instructions in -peration Chapter.
#.
3tart the pump and establish a water flow through the test section.
+.
nsure that any air bubbles are bled from the manometer tubes.
.
Ae!t, raise the water flow rate until ) m+
0.
ait until the water level in the inlet and outlet of manometer stabili>ed.
".
6ecord the reading for the inlet and outlet.
$.
6epeat step +2" for flow rate of *.@, *.?, *.$ until *.) m+
1.
RESULTS SHEET
4iameter of venturi mouth
; DDDDDD mm
4iameter of venturi throat
; DDDDDD mm
M%,"*'&'$
F!"#$%&' (L)*+,-
)
I,!'& (*# + =verage
#+ ## #) #* )@ )? )$ )" )0 ) )+ )# )) OBSERVATIONS
M%,"*'&'$ O/&!'&
)
#
(*+ =verage
S%*!'
T+*'
(L-
(-
1.
RESULTS AND ANALYSIS
). 6ecord the results on a copy of the results sheet. #. Calculate volumetric flow rate for each result. +. Plot graph of flow rate against the suare root of the head and draw the best straight line from the origin through the results. Calculate the slope and determine the coefficient of discharge for the venturi meter.
Euantities of water collected, E 8F9 (ime to collect water, t 8s9 Volume flow rate, E 8F
FLUID MECHANICS LABORATORY EXPERIMENT 2
FRICTION LOSSES IN STRAIGHT PIPES
2.0
INTRODUCTION
(he pressure loss along a pipe is caused by friction and changes in velocity or direction of flow. In order to find the friction loss in pipes, the fluid friction measurements apparatus are specially designed to allow the detailed study of the fluid friction head losses which occur, when an incompressible fluid flows through pipes, bends, valves and other pipe flow metering devices. /riction head losses in straight pipes of different si>es can be investigated over a range of 6eynolds Aumber. (he friction factors depend on the 6eynolds number of the flow and the pipe relative roughness. Cussons P"##* Faminar /low =pparatus used in this e!periment consists of a tubular test section of + mm internal bore and 0*?mm long, including a )+ mm bell nose entry, which is supported inside a protective outer #0 mm tube and is terminated at each end in bushed unions. Besides that, two test sections of Cussons P"##) Fosses In Pipes and /ittings =pparatus are used in this e!periment, i.e. the $ mm nominal bore pipe and the )* mm nominal bore pipe with two static pressure tappings +"* mm apart, each " mm long. It is intended that the test section should be mounted between the P")*+ Constant 'ead Inlet (ank and theP")* Variable 'ead -utlet (ank. (he P")*" %anometer Board is used to measure the head loss across the tubular test section.
2.1
THEORIES AND EXPLANATION 2.1.1
F!"# +, +'
If fluid flows down a pipe at low velocities it is found that individual fluid particles follow flow paths which are parallel but those particles nearer the centre of the pipe move faster than those near the wall. (his type of flow is known as laminar flow or stream line flow. =t much higher velocities it is found that secondary irregular motions are superimposed on the movement of the particles and a significant amount of mi!ing takes place, the flow is said to be turbulent. -sbourne 6eynolds investigated these two
different types of flow and concluded that the parameters which were involved in the flow characteristics were7
G V 4
(he density of the fluid (he velocity of the fluid Internal diameter of pipe (he absolute viscosity of the fluid
Hg m2+ m s2) m As m2#
6eynolds was able to show that the character of the flow could be described with the aid of a dimensionless parameter, which is now known as 6e ynolds number,
ρ VD µ 6e ;
8#.)9
/luid motion was found to be laminar for the values of 6e below #*** and turbulent for value 6e greater than ***. 4ifferent laws of fluid resistance apply to laminar and turbulent flows. /or laminar flow it is found that the pressure drop or head loss is proportional to velocity and that this can be represented by Poiseuille’s euation for the hydraulic gradient
i
=
h f L
=
+# µ V
ρ gD #
8#.#9
/or turbulent flow the relationship between head loss and velocity is e!ponential hf J Vn
8#.+9
and although there is no simple euation for turbulent flow it is accepted engineering practice to use an empirical relationship for the hydraulic gradient which is attributed to 4arcy and eisbach
h f L i;
=
, fV
#
D # g
8#.9
where f is an e!ponentially determined friction factor which varies with both 6eynolds number and the internal roughness of the pipe. = different friction factor may be used which is four times larger than the friction factor in the 4arcy2eisbach formula.
2.1.2
N'#&", L%# "5 V+"+&6
hen a layer of fluid is moved laterally relative to an ad&acent layer, a force is set up within the fluid which is in opposition to the shearing action. (his internal resistance known as the absolute viscosity of the fluid is caused by molecular adhesion and acts along the common boundary of the fluid layers. In the 3I system absolute viscosity is defined as the force in Aewton which would produce unit velocity in a plate of unit area at unit distance from a parallel stationary plate, that is
µ = −σ
∂V ∂ y
8#.09
(he unit of viscosity is in the cgs system is the KpoiseL . = measure of the Mfluidity’ of a substance is the kinematics viscosity whch is defined as72
KinematicsViscosity
=
absolute viscosity density
8#."9
µ ρ i.e. N ;
2.1.3
8#.$9
L%*+,%$ F!"# +, % C+$/!%$ P+'
Consider the flow of fluid in a concentric in a circular pipe as shown in figure. Fet the pressure drop due to fluid friction over a pipe length OF be OP.
/ig #.)
3treamtube in a
Circular Pipe
(he force differential pressure on tube in the direction of the
e!erted by the the fluid contained in the stream flow is given by OP d# <
8#.?9 -pposing this force is a shear force created by the viscous resistance to flow which is proportional to the shear stress and the wetted area of the streamtube Q dOF 8#.@9 /or dynamic euilibrium these two forces must balance OP d# < ; Q dOF 8#.)*9 ∆ P d ⋅ σ = ∆ L , 8#.))9 σ * =t the wall of the pipe where d ; 4 the shear stress is ∆ P d σ * = ⋅ ∆ L , 8#.)#9
δ P < δ L =nd substitute for
back into the euation gives
σ σ * = = cons tan t d
D
8#.)+9
/rom which it follows that the shear stress varies linearly from >ero at the centre to a ma!imum at the pipe wall. (he shear stress is related to velocity by Aewton’s law of viscosity
σ = − µ
δ V δ r
8#.)9
uating these two e!pressions for shear stress
− µ
δ V ∆ P d = ⋅ δ r ∆ L , 8#.)09
6eplacing d by #r
δ V =
r δ r ∆ P # µ
∆ L 8#.)"9
Investigating from the pipe centre 8r ; *9 to the pipe wall 8r ; 6 and V ; *9 yields R # − r # ∆ P V = , µ ∆ L 8#.)$9 (he velocity distribution is therefore parabolic with ma!imum velocity at the centre of the pipe R # ∆ P V ma! = , µ ∆ L D # ∆ P = )" µ ∆ L 8#.)?9 =nd the mean velocity is half of the ma!imum velocity V mean
D # ∆ P = +# µ ∆ L 8#.)@9
6earranging,
∆ P =
+# µ LV mean D # 8#.#*9
!pressing the pressure loss as a head due to friction, 'f over the pipe length l7 +# µ LV mean ∆ f = ρ gD # 8#.#)9
h f L (he head loss per unit length of pipe symbol Mi’, is then given by
i=
h f L
=
which is known as the hydraulic gradient,
+# µ V mean
ρ gD #
8#.##9
which is known as Poiseuilles euation for laminar flow. Aote that V is now taken to signify the mean velocity.
2.1.4
T/$7/!',& F!"# +, C+$/!%$ P+'
(he velocity distribution of turbulent flow across pipe is more uniform than the parabolic velocity distribution of laminar flow. Consider a section of pipe length OF over which the pressure drop is OP, as shown in figure #.#.
/ig #.# (urbulent /low in Circular Pipe
(he forces acting on the cylinder of fluid are the pressure forces producing the flow and the opposing shear forces caused by frictional resistance at the wall.
σ ο π LD = ∆ P π D # < ,
σ ο =
8#.#+9
∆ PD L, 8#.#9
Aow accepting that the shear stress is proportional to the suare of the mean velocity Q J V#
8#.#09
σ = KV # then,
8#.#"9
where H is a constant.
uating these two e!pressions for shear stress
σ =
∆ P D .
L
,
= KV #
∆ P =
, KLV
8#.#$9
#
D 8#.#?9
!pressing the pressure loss as a head loss due to friction, hf
, KLV
#
ρ gD hf ; # , fL V D # g
;
8#.#@9
# K
ρ
where f ;
is the 4arcy friction factor.
(he alternative definition of friction factor is often shown as f’ 8f dash9 and the head loss euation is then written as
f R L V # D # g hf ;
2.1.
R'6,"!8 ,/*7'$
8#.+*9
hen 6eynolds plotted the results of his investigation of how the energy head loss varied with the velocity of flow, he obtained two distinct regions separated by a transition >one. In the laminar region the hydraulic gradient is directly proportional to the mean velocity. In the turbulent flow region the hydraulic gradient is proportional to the mean velocity raised to some power n value of n being influenced b the roughness of the pipe wall. i J V).$ i J V# i J V).$ to #
/or smooth pipe /or very rough pipe In the transition region
2.1.F$+&+", 5%&"$
(he head loss due to friction for both laminar and turbulent flow can be presicted by the 4arcy eisbatch euation
i=
+# µ V
ρ gD # 8#.+)9
By multiplying top and bottom by V and rearranging i=
, × )" µ V #
i
VD ρ # g
=
", µ V
#
VD ρ # g
or i
=
, f V
8#.+#9
#
i=
D # g
f R V # D # g
or f = where
)" µ
VD ρ
=
)"
8#.++9 f R =
Re or
", µ
VD ρ
=
",
Re 8#.+9
Care has to be used due to these two different definitions of the friction factor, which are both in eually common use, and therefore in choosing the appropriate relationship between the friction factor and the head loss. hen using graph of friction
factor against 6eynolds number always check the relationship for laminar flow as a mean of distinguishing between the two. f =
)"
h f =
Re
If
, fLV
#
D # g
then use f R =
",
8#.+09 f R LV #
h f =
Re
If
D # g
then use
8#.+"9
/or turbulent flow the friction factor is a function of 6eynolds number, the
ε < D relative roughness of the pipe wall
. /or highly turbulent flows the friction factor
became independent of the 6eynolds number in a flow regime known as fullydeveloped turbulent flo! (he most widely accepted data for friction factors for use with the 4arley eisback formula is that produced by Professor F./.%oody. 3election of pipe si>e for a pipe to carry a given flow rate, which is a very common e!ercise, is made easier if the relationship between the head loss and pipe diameter is known for specific case of constant flow rate. /or a given flow rate, the mean velocity In the pipe is given by7 Q=
π D # ,
•
V =
V
,Q
π D #
hence
8#.+$9
3ubstituting for V into Poiseuille’s euation for laminar flow
i
=
h f L
=
, fV
#
D # g
#
,Q = D # g π D , f
)
#
D , hence i "
=nd using the 4arcy2eisbach euation for turbulent flow
8#.+?9
i
=
h f L
=
, fV
#
D # g
#
,Q = D # g π D , f
)
#
D 0 hence i "
8#.+@9
(he head loss is therefore inversely proportional to the diameter of the pipe raise to the forth power for laminar flow and inversely proportional to the fifth power for turbulent flow.
2.2
OBJECTIVES
). (o investigate the pressure loss due to friction in a pipe. #. (o compare the relationship between the friction factor and 6eynolds number with empirical data.
2.3
EQUIPMENT PREPARATION
Inlet
Initially P")*+ Constant 'ead Inlet (ank with overflow pipe e!tension fitted.
(est 3ection
P"##) Fosses in Pipe $ mm and )* mm test section.
-utlet
P")* Variable 'ead -utlet (ank.
%anometer
(wo of the single manometer tubes.
=ssembly
nsure the bell mouthed entry end of the P"##* test section is at the left hand end and that is correctly inserted into the inlet
tank. nsure that the P"##) $ mm bore test section is installed the correct way round with the conical inlet at the left hand end.
2.4
).
EXPERIMENTAL PROCEDURE
3tart the pump and establish a water flow through the test section. 6aise the swivel tube of the outlet tank so that it is close to the vertical.
#.
=d&ust the bench regulating valve 8or pump speed9 to provide a small overflow from the inlet tank and overflow pipe. nsure that any air bubbles are bled from the manometer tubes.
+.
3et up a serial of flow conditions with differential heads starting at #0mm in steps of #0mm up to )0*mm and thereafter in steps 0*mm up to a ma!imum of 0**mm. =t each condition carefully measure the flow rate using the volumetric tank and a stop watch.
.
%easure the water temperature.
0.
6eport the test with the other test sections.
2.
RESULTS SHEET
). (est 3ection 4iameter ater (emperature
..mm C
Constant 'ead Inlet (ank .mm Variable 'ead -utlet (ank Euantity of water Collected, E 8Fitres9 (ime to Collect ater, t 8sec9 Inlet 'ead, ') 8mm9 -utlet 'ead '# 8mm9
#. (est 3ection 4iameter ..mm ater (emperature
C
Constant 'ead Inlet (ank .mm Variable 'ead -utlet (ank Euantity of water Collected, E 8Fitres9 (ime to Collect ater, t 8sec9 Inlet 'ead, ') 8mm9 -utlet 'ead '# 8mm9
OBSERVATIONS 2.
RESULTS AND ANALYSIS
).
6ecord the results on a copy of the results sheet.
#.
4etermine the water density and viscosity from =nne! ) of Part ) of the manual.
+.
/or each result calculate the mean velocity and hence the 6eynolds number and friction factor S# .
.
Plot a graph of logT hf against logT V, draw a straight line through the results and measure its slope to e!press the relationship between hf and V in the from hf J Vn .
0.
-n a photocopy of the graph on pages + U )0 plot the points of friction factor against 6eynolds number.
".
/rom the graph of fraction factor against 6eynolds number on page + 2)0 determine the empirical friction factor S’ using the 6eynolds number for each result and assuming a pipe roughness of *.**)0mm. ). (est 3ection 4iameter7mm ater (emperature7.. C
4ensity7.. kg
Euantity of ater Collected, E 8litre9 (ime to Collect ater, t 8sec9 Volume /low 6ate, E 8litres
4ensity7.. kg
Foge hf /riction /actor, f’ Foge f’
APPENDIX
/igure #.+ (he 3tanton 4iagram<%oody Chart
