 Open Access
 Total Downloads : 350
 Authors : Muralidhar Lakkanna, Yashwanth Nagamallappa, R Nagaraja
 Paper ID : IJERTV2IS110084
 Volume & Issue : Volume 02, Issue 11 (November 2013)
 Published (First Online): 30102013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Implementation of Conservation principles for Runner conduit in Plastic Injection Mould Design
Muralidhar Lakkanna1, Yashwanth Nagamallappa 2, R Nagaraja3
Government Tool room & Training Centre, Bangalore, Karnataka, India
Abstract
Conservation principles are used to represent all physical transformations occurring in the universe, accordingly are also adopted to design runner conduit for thermoplastic melt injection. Conservation principles for thermoplastic melt injection through runner conduit are implemented by considering cylindrical coordinates system relevant to its geometrical configuration for deriving governing equations. While the continuity equation ensures volumetric conservation of thermoplastic melt, the momentum equation represents equilibrium of forces on thermoplastic melt injection through runner conduit. During an injection moulding cycle heat and work done energy transformations are balanced by implementing first and second law of thermodynamics. Thermoplastic melt state change through the runner conduit for a particular cycle is appreciated by heat conduction equation. Traditionally inertia and entropy contribution is neglected to skip rigorousness, nevertheless they continue to prevail. Especially in very frequently used noncircular runner cross section conduits, their influence is highly significant. Hence the current endeavour attempts to computationally model, continuousness, equilibrium, energy balance and phase transformative runner design criteria by implementing conservation principles.
Keywords: Runner Design, Plastic Injection Mould, Conservation Principles

Introduction
Designing moulds necessitate fundamental insight into physical injection moulding behaviour of chosen melt relative to desired component characteristics as well as complementary classical principles of mechanics. Off mould design runner system is the focussed subject of investigation. Runner conduit is one
of the integral and influential elements of injection mould design to process plastic parts [1]. Specific spatial region defining runner conduit is considered in isolation from everything else as a welldefined control surface or system boundary [2]. Nonnewtonian thermoplastic melt is then injected through runner system conduit to contrive component impression space. Since runner conduit involves continuous melt injection through its inlet and exit boundary surfaces, it will be an open system often referred here afterwards as control volume [3]. Here the runner conduit is considered as a Euclidian flow field and thus thermoplastic melt properties are defined as spatiotemporal functions [4]. Hence here we consider an elemental volume of thermoplastic melt inside runner conduit and conservation laws are implemented by differential approach [5].
The condition of thermoplastic melt at any instant of time is typically referred as its state and expressed as a function of melt characteristics [6], which are basically a nonnewtonian. Though it experiences the runner conduit for fraction of cycle, it directly influences the ideal feed system design. Consequent to injection action, exclusive melt state metrics and their extents are used to design effective runner parameters by implementing mass, momentum and energy principles [7]. Thus its control volume formulation is critically important to represent overall balance [8].
The mass conservation principle implements continuity of thermoplastic melt injection rate. The momentum conservation principle appreciates forces acting on thermoplastic melt over the entire runner conduit [9]. Similarly energy conservation principle apprehends energy change of thermoplastic melt throughout the process. Here two simultaneous modes of energy transformations are recognised, injection work and solidification heat transfer. The primary objective of mould designer is to realize maximum injection power (rate of injection work transfer) from runner conduit design criteria. The energy conservation principle articulates energy is conserved during a
process but doesnt hint anything about whether the
process but doesnt hint anything about whether the
Nomenclature
m Mass Kg
V Volume m3
P
Pressure
Kgf / m2
T
Temperature
K
k
Thermal conductivity
W/m
A
Crosssection area
m2
U
Ur
Linear velocity
Velocity in radial direction
m / s
m / s
U
Velocity in tangential direction
m / s
U Velocity in arbitrary direction m / s
a Acceleration
m / s2
M Linear momentum Kg m / s

Angular momentum Kg m / s

Moment of inertia
Kg m2
e Specific total energy KJ / Kg
u Specific internal energy KJ / Kg
T Resultant torque N m
M Resultant moment N m
F Resultant force
N / m2
Fr Force acting in radial direction
F Force acting in tangential direction
F Force acting in arbitrary direction
N / m2 N / m2 N / m2
Q Rate of heat transfer KW
q Rate of heat transfer per unit mass KW
Wv Rate of work done by viscous forces KW
Wp Rate of work done by pressure forces KW
dS Entropy change KJ / Kg
Cv Specific heat at constant volume KJ / KgK
Cp Specific heat at constant pressure KJ / KgK
n Unit normal vector
r Position vector
Greek symbols
Density
Specific volume
Kg / m3
m3 / Kg
Angular acceleration
m/ s2
Surface force
N / m2
Shear stress
Viscous dissipation function
N / m2
Angular acceleration
m/ s2
Surface force
N / m2
Shear stress
Viscous dissipation function
N / m2
Angular Velocity m / s
energy exchange is reversible or not. However in most circumstances, melt injection through runner conduit for a particular cycle is an irreversible process. So an additional quantity called entropy is required to balance energy transformations. To gain better understanding of realizable energy transformations the concept of entropy is implemented [10].
Hence fundamental conservation of injection mechanics from mass, momentum and energy perspective is essential to realize specific relationships among chosen thermoplastic, desired component and available machine. All these relationships along with processing objectives facilitate designing an idealistic runner system.


Conservation laws
Plastic injection phenomena through runner conduit can be explained by using conservation laws. The conservation equations fundamentally seek to tell us how three important field variables are distributed in space and time. According to conservation laws any measurable property of the system does not change as the system evolves/ undergoes a change. Thus during
any process the quantities of the system will be
figure 1, subtending an angle d at the centre. Since thereare no sources or sinks in the conduit, we can conveniently postulate that the mass of representative melt element does not change in position and time i.e.,
mass is conserved spatiotemporally. If U is melt velocity then Ur is velocity in radial direction, U is velocity in tangential direction and U in arbitrary
injection direction respectively. Here is linear
arbitrary path function in the YZ plane on machine.

Conservation of Mass
The continuity equation comes from the basic principle that matter can neither be created nor be destroyed. This principle is then applied to a small volume of thermoplastic melt under injection resulting in equation representing continuity.
Lemma
Melt state change at any instant within runner conduit is always equal to influx and efflux melt state change rates across the runner conduit inlet and outlet respectively
conserved. This concept of conservation can be applied
dV AU AU 0
(1)
on thermoplastic melt diffusing through runner conduit of injection mould to determine the properties of thermoplastic melt throughout runner conduit.
Let us begin by considering a three dimensional
rcv t
Proof
out in
runner conduit of a thermoplastic injection mould in cylindrical coordinates system having a representative melt element between r and r dr as shown in
Consider injection in arbitrary direction, Mass of melt entering/unit time
Figure 1: Three dimensional runner conduit in cylindrical coordinates [5]
= density Ã— velocity in direction Ã— inlet cross section area
(rUr ) ( U ) r ( U ) 0
t r
= U rddr
U ( U ) 1 ( U ) ( U )
Similarly mass of melt leaving/unit time
r r 0
(4)
U
t r
r r
U d rd dr
Now by substituting vector gradient operator,
U U 1 U U
.U = r r
(5)
Thus, total change of mass in direction/unit time
r r r
= mass entering – mass leaving
U
U
rdrd d
Hence equation reduces (4) to
. U 0
(6)
t
U
dV
The above equation is called continuity equation.
The above equation describes that the thermoplastic melt injection is compressible and melt is conserved
Correspondingly,
Total change of mass in direction/unit time
r
r
1 U dV
Total change of mass in rdirection/unit time
Ur Ur dV
r r
Therefore net change of melt mass in element/unit time is
Ur Ur dV 1 U dV U dV
through the runner conduit [11], during complete injection moulding cycle.

Conservation of Momentum
We hereby develop governing equations for runner conduit design considering dynamic forces consequent to thermoplastic melt injection. According to Newtonian mechanics thermoplastic melt injection through runner conduit should obey Newtons Second Law of Motion for Conservation of Momentum. The implied momentum could be linear or angular and its
corresponding actions are force F and moment
Mrespectively.
r r
r
U U 1 U
U U 1 U
U
r r dV
r r r

Conservation of Linear Momentum Lemma
(rU ) U
(rU ) U
U
r r dV
(2)
At any moment resultant force acting on
thermoplastic melt is proportional to its instantaneous
r
Instantaneous mass of melt within the runner conduit element = Density Ã— Volume
injection momentum change rates in that direction and momentum change rates across the runner conduit inlet and outlet
mU
= dV
So rate of change of melt in element
(dV)
t
i.e., F
We have,
mU mU
t
t
out in
(7)
Since runner element volume dV is stationary and invariable with respect to time or independent of time
change it becomes dv
mU
t
UdV
t rcv
out
and
t
Hence by applying conservation law we get,
mU mU UUdA
out in in
out
dV (rUr ) U r
U dV 0
(3)
F t rcv UdV UUdA
(8)
t r
in
The term UUdA represents momentum change rates through mass transfer across runner conduit inlet and outlet.
Proof
The element is so small that volume integral simply reduces to a derivative term
Therefore body forces can be neglected for runner conduit design.
Consider stresses in rdirection,
Resultant force acting at runner conduit entrance (boundary)/unit time rr n.rdd
Since mass is entering into the runner control volume, unit normal vector to exit boundary n 1
i.e., UdV UdV
(9)
Therefore resultant force acting at runner conduit
t rcv t
entrance/unit time rr rd d
Consider rate of momentum change in rdirection
Resultant force acting at runner conduit exit/unit time
Rate of momentum change at inlet of runner conduit
rr
= Ur Urd d
rr
r dr n. r dr d d
Rate of momentum change at exit of runner conduit
Ur U
Unit normal vector to entrance boundary, n = 1 Thus Resultant force
= U U
r
r dr r drd d
=

rr dr
r drd d
rd d
rr r rr
Rate of momentum change in rdirection
U U
= rr rr rdrd d
U U r dr n.r drd d U Urd d
r r
r r r
= rr rr dV
r r
Ur U Ur UdV = r. Ur UdV
r r r
So that the net resultant force in rdirection is given by
dF ( rr ) 1 ( r ) ( r ) rr dV
Similarly in and directions
r,resultant
r r
r
Rate of momentum change in – direction
Similarly in other directions,
1 U U
1
2
dV
dF
r
r dV
r
,resultant
r r
r
Rate of momentum change in – direction
r
1
r
U U
dF,resultant
dV
r r
r
dV
Therefore net momentum change rate through mass
Net resultant surface force due to injection/melt convection exerts stress on runner conduit boundary.
P
transfer
r. Ur U
1 U U
U U
ij
rr
r
r
P
r
dV (10)
P
r r
r
Forces acting on thermoplastic melt
The forces acting on melt mass are body forces and surface forces. The body forces are mainly due to gravity and their contribution in momentum conservation would be very less because plastic melt within the runner conduit is a small fractio of shot volume (approximately 50 times), melt injection forces are predominantly larger (more than 100 times) than
inertial forces and melt density itself being very less.
Splitting this into pressure and viscous stresses, we can rewrite it as
dV r r r r r
dV r r r r r
dFr P 1 (r rr ) 1 ( r ) ( r )
Similarly in other directions,
r r r
r r r
dV r r2 r r r
dV r r2 r r r
dF 1 P 1 (r2 ) 1 ( ) ( )
dF P 1 (r r ) 1 ( ) ( )
dV r r r
Hence net resultant forces is given by
dF P 1 P P
This can be expanded as follows Momentum in rdirection
dV
r r
P 1 (r )
1 ( )
( r )
resultant
rr r
1 (r rr ) 1 ( r ) ( r )
r r r r r
r r r r
U U U U U U 2
r U
r r U
r
1 (r2 )
1 ( )
( )
t
r r r
r
r r r
r2 r r r
Momentum in direction
1 (r r )
1 ( )
( )
1 P 1 (r2 ) 1 ( )
( )

r

r r
r r r
r
r2
r r r
(11) = (U ) U
U U U U
U Ur U
That is,
t r r r r
dF
P dF
(12)
Momentum in direction
dV dV
resultant
viscous
P 1 (r r ) 1 ( ) ( )
dF .
r r r
dV
viscous
dF
U
U
U U U U
U
P . (13)
t
r r r
dV resultant
The resultant force is thus the sum of the pressure gradient vector and the divergence of the viscous stress tensor.
According to conservation law,
Substituting (9), (10) & (13) in equation (8) we get
Since the contribution of body forces is negligible, the rate of change of momentum of thermoplastic melt inside runner conduit is completely depended on surface forces which is a combination of pressure and viscous forces
U r. Ur U 1 U U U U
P .


Conservation of Angular Momentum
t
r r
(14)
Equation on the right side can be split into,
U r. Ur U 1 U U U U
Lemma
Resultant torque acting on thermoplastic melt at any instant is equal to instantaneous angular momentum change within the conduit and angular
momentum change rates at runner conduit inlet & exit
t r r
H
U
. U dU
i.e.,
T
H H
t
t
out in
(17)
t dt
Where, H Angular momentum.
We know that, from continuity equation number (6)
Also we have
. U 0
H
r U
t
Hence,
t t rcv dV
out
U r. Ur U 1 U U U U dU
H H r UUdS
out in in
t r r dt
i.e., dF dU (15)
The vector cross product r Urepresents angular momentum per unit mass. Where r is position vector
dV
dV
resultant dt
Hence equation becomes,
from origin or fixed central axis and U is linear
dU velocity. Thus we can designate rUdV as
P .
dt
(16)
angular momentum acting on representative melt
The above equation is also called Cauchy equation
element. So the right hand side of equation (17) becomes
From equation (6) we have
.U 0
t
H H H
t out in
out
(18)
Thus above equation (22) reduces to
dr U
t
t
rcv r UdV r UUdS
in
. r dt
(23)
Since body torque is negligible as stated in section
2.2.1 only surface torque acting on runner conduit accounts to net moment. Since contribution of pressure forces on angular momentum is negligible, the surface torque will mainly due to viscous torques. So left hand
side of equation (17) becomes,
This is called angular momentum equation
The above equation describes that rate of change of angular momentum of the thermoplastic melt inside the runner conduit is proportional to viscous forces which causes the angular motion. Angular momentum being
negligible in circular or axissymmetric runner conduits
T rcs
r dS
(19)
has a significant influence particularly on parametric distribution in noncircular cross section runner
Substituting equation (18) and (19) in (17), we get
conduits.
out
t
t
rcs r dS rcv r UdV r UUdS
in
For surface integral,
r r n. r n. r
(20)

Conservation of Energy
Here we seek to adopt Joules energy principle to mathematically express conservation on the basis of
By using Gauss Divergence Theorem, the left hand side of equation is converted from surface integral to
volume integral
rcs r dS rcs n. rdS rcv . rdV
out
First of law of thermodynamics by explicitly quantifying various energy entities and balancing them in accordance with the conservation notion. So in runner design we recognise melt heat, injection
momentum work and polymers internal energy as
Similarly,
r UUdS
in
different forms of energy. In physical mould design sense, heat is thought of an energy exchange by melt in
becomes .r UUdV
rcv
Thus equation (20) becomes,
runner conduit to the surrounding indeformable mould runner insert.
t
t
. rdV
r UdV .r UUdV

First Law of Thermodynamics
rcv
rcv
rcv
Lemma
t
t
. r r U .r UU
But we have
r U
(21)
The algebraic sum of thermoplastic melt heat transfer rates across runner conduit inlet and exit plus rate of injection force acting on it throughout runner
conduit is proportional to rate of internal energy
t .r UU
change within runner conduit and rate of internal
dr U
energy changes across the runner conduit inlet and
r U
.U
exit
dt t
dr U
dr U
Hence equation (21) becomes,
Q Wv Wp t rcv edV rcs e U.ndA (24)
r U .U . r
(22)
Here e is total energy and it is sum of internal and
dt t
kinetic energy.
U2 Therefore net rate of energy change
i.e., e= u+
2
r. U U
r
r
U
= 1
dV
(28)
The product
U.ndA represents melt change rate at
r r
runner conduit inlet and exit
Here rate of work done by pressure forces Wp
p
W = PU.ndA = P U.ndA
Hence the equation (24) becomes
(25)
Net rate of heat change of thermoplastic melt in runner conduit
To evaluate net heat change Q , we neglect radiation
and consider only heat conduction through the
P
thermoplastic melt.
Q Wv t edV e U.ndA
(26)
According to Fourier law of heat conduction,
Rate of heat change per unit area is proportional to
This is very convenient form of energy equation since pressure work is now combined with energy of thermoplastic melt leaving at runner conduit outlet; we
gradience of temperature
q kT
Where k= thermal conductivity
(29)
no longer have to deal with pressure work.
Q W edV m P e m P e
(27)
Consider rate of heat change through radial direction Rate of heat change at conduit inlet
v t
Proof
out in
qr rd d
Consider rate of energy leaving at runner conduit exit by mass transfer in rdirection,
Rate of heat change at conduit outlet
q qr dr r drd d
r r
Let = p e
Rate of energy change at runner conduit inlet
Ur rd d
By subtracting inlet term with outlet term, we obtain heat change in that direction
Hence rate of heat change in radial direction
Rate of energy change at runner conduit outlet
q rd d q qr dr r drd d
r r r
U
Ur
r dr r drd d
q drd d
qr
rdrd d
r
r r
Therefore rate of energy change in rdirection
rqr
U
r dV
= U r dr r drd d U rd d
r r r
Similarly in other directions,
U
r. U
Rate of heat change in tangential direction
= Ur
r dV =
r dV
1 q dV
r r
r
r
1
1
Similarly in " " and " " directions Rate of energy change in " " direction
U
dV
Rate of heat change in longitudinal direction
q dV
r
Therefore net rate of heat change on thermoplastic melt
rqr 1 q q
Rate of energy change in – direction
r r
dV
U
dV
Q .qdV
From above, rate of heat change is proportional to runner conduit volume. Thus Introducing Fourier law
But the right hand side of the equation can be expanded as,
of heat conduction, we have net rate of heat conducted
e
r. Ur
1 U
U
Q .kTdV
(30)
t r r
Rate of work done on thermoplastic melt due to viscous forces
e U
t
e 1 U
r r r
e U
e
Net rate of work done by viscous forces on thermoplastic melt in rdirection is given by
U U 1 U
U U 1 U
U
P r r
r r r
r U U
rr r 1 ( r Ur )
r r drd d
r. U U U
r r
e 1 r 1
t r
r r
Similarly in and – directions
r r U
1 U
U
U
P U
1 P U
P
rdrd d
r r
r
r r
Further,
r r U
1 U
U
e
r. Ur
1 U
U
rdrd d
r r
t r r
Therefore, net rate of work done by surface forces is given by
de P.U e . U U.P
dt t
r rr Ur 1 ( U ) r Ur
r r
r r
But . U 0
t
t
Hence right hand side of equation reduces to
r U 1 U U
r
rdrd d
e
r. Ur
1 U
U
r r
t
r r
r r U
1 U
U de
(33)
P.U U.P
dt
r r
.U. rdrd d
.U. dV
(31)
Substituting (33) in equation (32), we get
dt
dt
.kT .U. de .PU
U2
Hence from conservation law, Substituting (28), (30) &
d u 2
(31) in equation (27), we get
.kTdV .U. dV
.kT .U. .PU
dt
dt
From equation of mechanical energy,
(34)
e
r. Ur
1 U
U
U2
dV
d
t r r
2
dt
dt
.PU P.U .U. : U
(35)
.kTdV .U. dV
Subtracting equation (35) from equation (34) we get,
e
r. Ur
1 U
U
(32)
du
dV
.kT P.U : U
(36)
t r r dt
The above equation describes that rate of change of
. q and irreversible 1 q.T effects of heat
T T2
internal energy of the thermoplastic melt inside runner
conduit is proportional to amount of energy added to
transfer.
the thermoplastic melt via heat and work done.
Hence equation (41) becomes,
dS . q 1 q.T
(42)

Second Law of Thermodynamics
Lemma
Rate of injective work done on thermoplastic melt across the runner conduit inlet and outlet is proportional to thermoplastic melt energy change rates within the runner conduit and rate of entropy change across the runner conduit inlet and outlet
Entropy change (dS) gives us,
TdS de Pd
dt T T2 T
This is called Equation of Entropy
The above equation describes that rate of change of entropy during the injection moulding cycle is proportional to energy added to the thermoplastic melt during that cycle.


Volumetric Heat Absorption
Thermoplastic melt state change for a particular
Substituting specific volume 1
TdS de P d
2
we get,
(37)
cycle through the runner conduit of an injection mould is considered. Accounting to runner conduit configuration heat transfer is dominant in radial direction compared to that of heat transfer in the
Since injection moulding is a dynamic process, we now differentiate above equation (37) for a small representative interval to get,
dS de P d
direction of injection which is very less and hence neglected. This provides us with an advantage of considering heat conduction in radial direction alone.
T
dt dt 2 dt
(38)
runner volume element with heat transfer taking place
Equation of internal energy is given by,
de .q P .U
dt
where = Viscous dissipation function
in rdirection alone. Since the runner geometry is best explained by cylindrical coordinate system, the same is taken to derive heat conduction equation.
Lemma
de
dt
.q
P.U
(39)
Rate of volumetric thermal energy absorbed across thermoplastic melt core and conduit interface wall is proportional to difference between net rate of
Equation (6) can also be written as,
d .U dt
Substituting (39) & (40) in equation (38) we get,
T dS .q
(40)
heat conducted radially across runner conduit cross section and rate of internal energy change throughout an injection cycle
Proof
dt
By rearranging we get,
The rate of volumetric heat absorbed across thermoplastic melt core and conduit wall interface is
dS
dt
.q
T T
(41)
given by
qvh rdr
(43)
Using vector calculus the term
.q T
is split as
Let the rate of heat conducted into differential runner
element in rdirection

.q . q q .T to separate reversible Q k T r
T T T2
r r
Let the rate of heat conducted away from differential
Where, Ttrans
is the glass transition temperature which is
runner element in rdirection
almost a linear function of pressure [13].
Q
Ttrans b5 b6P
(51)
Q Q r dr
r
r
rdr
Hence governing equations are:
Continuity equation
Hence governing equations are:
Continuity equation
r
t
Linear momentum equation
t
Linear momentum equation
. U 0
. U 0
Thus net rate of heat conducted into differential runner element in rdirection
Q Q
1 kr T rdr
(44)
r rdr
r r
r
dt
Angular momentum equation
dt
Angular momentum equation
P . dU
P . dU
The rate of internal energy change of thermoplastic melt is given by
C
T rdr
v
v
t
(45)
dt
First law of thermodynamics
dt
First law of thermodynamics
Hence from equations (43), (44) and (45) we have
.r d r U
.r d r U
du .kT P.U : U
du .kT P.U : U
q rdr 1 kr T rdr C T rdr
vh r r
r
v t
dt
Second law of thermodynamics
dt
Second law of thermodynamics
dS . q 1 q.T
dS . q 1 q.T
Neglecting the volume terms and expanding the above terms, the equation becomes
dt
dt
T
T
T2
T2
T
T
2T k T T
Volumetric heat absorption
Volumetric heat absorption
T
T
2
2
C
C
k
k
T T
T T
t
t
qvh k r2 r
r Cv t
(46)
The above equation represents the melt state change during a particular injection moulding cycle through the runner conduit.




Equation of State (Tait Equation)
Thermoplastic melt specific volume streaming through runner conduit at any instant is represented as a function of its pressure P (t) and temperature T (t). This equation is popularly known as Tait equation [12].
qvh
k r2 r r
v
qvh
k r2 r r
v

Conclusion
The authors have earlier proposed spatiotemporal conservation principle to design runner system for a plastic injection mould [14]. Accordingly governing equations for thermoplastic melt inside runner conduit
T, P
T Cln 1 P
T, P
(47)
throughout an injection moulding cycle has been
0 1 BT t
obtained after implementing conservation principles.
Where C=0.0894 is a universal constant.
Further set of equations tabled above are then used to
0 T is given by,
T b m b m T b , if T > T
(48)
derive a computational model to design the ideal runner conduit size for feeding thermoplastic melt relative to
specific combination of injection moulding machine
0 1 2 5 trans,
Where, subscripts (m) represent molten state of the polymer.
B (T) is given by
available, characteristics of thermoplastic material chosen and desired features of component being moulded.
BT b m exp b m T b , if T > T
(49)
3 4 5
trans
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For thermoplastic melt streaming through runner conduit under general injection moulding processing
conditions,

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t T, P 0
(50)
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F. M. White, Fluid Mechanics, 6th ed., Tata McGraw Hill, 2008, p. 227

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