COMPUTATIONAL MODEL TO DESIGN CIRCULAR RUNNER CONDUIT FOR PLASTIC INJECTION MOULD

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IJRET: International Journal of Research in Engineering and Technology
eISSN: 2319-1163 | pISSN: 2321-7308
COMPUTATIONAL MODEL TO DESIGN CIRCULAR RUNNER
CONDUIT FOR PLASTIC INJECTION MOULD
Muralidhar Lakkanna*, Yashwanth Nagamallappa, R Nagaraja
PG & PD Studies, Government Tool room & Training Centre, Bangalore, Karnataka, India
* Corresponding author Tel +91 8105899334, Email: [email protected]
Abstract
Analytical solution quest for viscoelastic shear thinning fluid flow through circular conduit is a matter of great prominence as it
directly evolvesmost efficient criteria to investigate various responses of independent parameters. Envisaging this facet present
endeavour attempts to develop a computational model for designing runner conduit lateral dimension in a plastic injection mould
through which thermoplastic melt gets injected. At outset injection phenomenon is represented by governing equations on the basis of
mass, momentum and energy conservation principles [1]. Embracing Hagen Poiseuille flow problem analogous to runner conduit
injection the manuscript uniquely imposes runner conduit inlet and outlet boundary conditions along with relative to appropriate
assumptions; governing equations evolve a computation model as criteria for designing. To overwhelm Non-Newtonian's abstruse
Weissenberg-Rabinowitsch correction factor has been adopted byaccommodating thermoplastic melt behaviour towards the final
stage of derivation. The resulting final computational model is believed to express runner conduit dimensions as a function of
available type of injection moulding machine specifications, characteristics of thermoplastic melt and required features of component
being moulded. Later the equation so obtained has been verified by using dimensional analysis method.
Keywords: Computational modelling, Runner conduit design, Plastic injection mould, Hagen-Poiseuille flow
---------------------------------------------------------------------***---------------------------------------------------------------------
1. INTRODUCTION
The physical process of thermoplastic melt injection through
runner conduit in a typical plastic injection mould is
Mathematical models for polymer processingisby and large
represented by a set of expressions, which insights adept
deterministic (as are the processes)typically transport based
acquaintance of in-situ physical phenomena that occurs within
unsteady
(cyclic
process)
and
distributed
parameter.
actualprocessing[3].
Mathematical
modelling
involves
Particularly complex thermoplastic melt injection mould
assembling sets of various mathematical equations, which
system was broken into clearly defined subsystems for
originates from engineering fundamentals, such as the
modelling. Runner dimension plays a vital role in the
material, energy and momentum balance equations [4]. Herein
idealization of an injection mould and hence deriving a
representative
mathematical
equations
attempts
to
computational model to determine runner dimension was of
computationally model the interrelations that govern actual
immense need. There have been a fairly large number of
processing situate[5]. More complex the mathematical model,
Newtonian apprehensions for which a closed form analytical
the more accurately it mimics the actual process [6]. Towards
solution
are
prevailing.
However,
for
non-Newtonian
obtaining an analytical solution we must first simplify the
apprehension fluids such as thermoplastic melts exact
balance equations, although the resulting equations are
solutions are rare. In general, non-Newtonian melt injection
fundamental, rigorous,nonlinear, collective, complex and
behaviours are more complicated and subtle compared to
difficult to solve [2]. Therefore, the resulting equations are
Newtonian fluid circumstances[2].Developing a model for
sufficiently simplified by considering appropriate assumptions
complex process like thermoplastic melt (which is non-
that correspond to those the actual processing interrelations
Newtonian highly viscoelastic, shinning type fluid) injection
between variables and parameters. These assumptions are
through runner conduit (circular conduit) requires a clear
geometric simplifications, initial conditions and physical
objective definition. Hereto the sole objective of this
assumptions, such as isothermal systems, isotropic materials
derivation is to obtain a runner diameter design criteria as a
as well as material models, such as Newtonian, elastic, visco-
function of injection moulding machine specifications used for
elastic, shear thinning, or others [6]. Finally boundary
the purpose of injection, type of thermoplastic melt being
conditions like velocity and temperature profiles are applied to
injected and features of the component being moulded as
simplifying the resulting equations completely [7]. Further
known parameters.
meticulous rearranging of the functions leads us to a complete
computational model that enables design engineers to
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418

IJRET: International Journal of Research in Engineering and Technology
eISSN: 2319-1163 | pISSN: 2321-7308
confidently design superior moulds at bonus costs thus
U

U


U
U


U


U U

r

idealizing the process from mould design perspective [8].

+ U
+
+ U
+

r

t

r

r


r

(6)
2. GOVERNING EQUATIONS
2
1 P


1 (r ) 1





( ) ( )


-

r
r
r
= -
+
+
+
+

2
The phenomenon ofactual melt injection through the runner
r
r
r

r


r

conduit
is
represented
by
governing
equations
that
discriminately appreciate compressibility, unsteadiness and
U

U

U U

U





non-Newtonian factors. Hence are highly rigorous, nonlinear,

+ U
+
+ U

r




comprehensive, complex and difficult to solve. Few explicit
t
r
r

(7)
assumptions are made herein are:
P

1 (r )



1 (
)

( )
r

(a) Thermoplastic viscosity remain consistent
= -
+
+
+

r
r

r


(b)Body forces are neglected when compared to that of
viscous forces
(c) Thermoplastic melt thermal conductivity is considered
The Newtonian constitutive relations are
constant.
U
2
ur
r
= - 2
-
.
U
(8)
rr
( )
2.1 Equation of Continuity (Mass Balance)


r
3

U
U
1 ( U )
(U )
r
( r )
+
+
+
+

= 0
1 U

U
2
ur
t

r
r

r


(1)

r


= - 2
+
- ( .
U)
r
(9)
r
3


ur
+.(U) = 0
(2)
t

U
2 ur


= - 2
- ( .
U)


(10)
3
ur
ur
ur

Substituting .( U) = .U + .U , in equation (2),
U 1 U


r



= = -


+

r
ur
ur


r
r
+

r
r r
(11)

.
U + .U = 0
(3)
t
U

U

r
d
ur




= = -
+

+ .U = 0
r
r

r


(12)

dt


d(log )
ur

U
1 U


+



.
U = 0
= = -
+



(13)
dt

r

ur
d (log )
ur
1
1 U

U



.
U = -
(4)
.
U =
(rU +
+
r )
dt
r r

r

(14)
Substituting Newtonian constitutive relation Eqn. (8) to (14) in
equations of motion Eqn. (5) to (7) we get,
2.2 Equations of Motion
2
U

U

U
U

U

U

r
r
r
r

+ U
+
+ U
-


r

t

r

r


r
2
U

U

U
U

U

U

r
r
r
r

+ U
+
+ U
-




ur
r
P
U
2
2 U

1 U

U

t

r

r


r
r
r

r
= -
+
2
-
.
U +

-
-

(5)
r

r

r

3

r
r

r
r
P

1 (r ) 1 ( ) ( )

r
rr
r
= -
+
+
+
-
1 1 U

U

U




U

U


r
r

r

r
r

r

r
+

+
-
+

+

r r
r

r

r

(15)
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IJRET: International Journal of Research in Engineering and Technology
eISSN: 2319-1163 | pISSN: 2321-7308
U

We can apprehend from Eqn. (19) and (21) that entropy is

U
U

U
U


U U

r

+ U
+
+ U
+



r
already present quantitatively in energy equation Eqn. (19)
t
r

r


r

itself.
1 P
1 2 U
U
2
ur
= -
+

+ 2 -
.
U
r
r r
r
3

3. SOLUTION FOR GOVERNING EQUATIONS
2 1 U
U
U


1 U

U
U
r
r
+

+
-
+

+
-
Geometrical conditions
r r
r

r
r
r
r

r
a)
Analogous to pipe flow transverse velocity components



could be considered almost zero (geometrical constraint)
1 U
U
+

+


i.e., U = U = 0 ,
accordingly
transverse
pressure
r

r
P
P
(16)
gradience would also be zero. Hence
=
= 0



U

r

U U U
U






+ U
+
+ U

b)
Since runner cross section is axis-symmetric profile,
r
t
r

r


tangential gradience could be considered zero i.e.,
P


U 2 ur U

U

r
= -
+


2
-
.
U +
+

=

0


3

r
r


c)
Only lateral gradience of temperature is considered
1


1 U U



U

U


r
+



+
+

+

because radial gradience far exceeds than other two
r r

r

r

directionsi.e.,
(17)
T

T

T


0,
0




r

2.3 Energy Balance Equation
d u
ur
t
ur

+
Upon substituting above (a) to (c), governing equations reduce
P (.U) = .
(kT) + : U
(18)
dt
to,
d (

log )
U
= -
From thermodynamic relation we have,
dt

(22)
d u = C , du = C dT
U

U




P

4 U



U


v
v

dT

+ U
= -
+

+
r


t



3
(23)

r
r

r

Hence Eqn. (18) simplifies as
2
2


T
U





1
T
4
U
3
U
C
+ P
=
kr
+
+



(24)
dT
ur
t
ur
v

t


r r

r 3

4
r

C
+ P .U = . kT + : U
(19)


v
( ) ( )
dt
Substituting Eqn. (22) in Eqn. (23) and Eqn. (24) we get
dT
ur
1
T
1
T

T


C
+ P .
U =
kr
+
k
+
k

v
( )
2
dt
r r

r
r







U

d


- U


(log )
2
2
2



U


t
dt
1 U

U
U



r

r


+
+
+


(25)
r

r


r



P

4 d
U





= -
-

(log )+
r

2
2





3
dt
r
r
r
1 U

U
1 U

1 1 U






U
r
+ 2

+
-
+
+
+

2
r

r
r


2

r




T

d

2

C
- P
log
v
(
)
1 U


U

ur

1
+
t

dt

+
-

(

.
U)2
r
2

r
3

(26)


2
2


1
T
4
d



=

kr
+ -
(log)
3
U
+



(20)
r r

r
3 dt

4
r



Equation of Entropy
3.1 Hagen-Poiscuille Velocity Profile
dS
t
ur
T
= .(kT) + : U
(21)
Thermoplastic
melt
transportation
studies
are
critical
dt
fordesigninglateral dimension of runner conduits as well as
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IJRET: International Journal of Research in Engineering and Technology
eISSN: 2319-1163 | pISSN: 2321-7308
injectionbehaviours [9]. Cognising axial velocity of non-


4 d

Newtonian, shear thinning thermoplastic melt injection can

(log )



conversely enable conduit dimension determination, because




2
3
dt
2
r =
+ R


(28)
Axial velocity component is a function of conduit radius
1 P
d




(


log )
P

-
through which melt is being injected [10]. Since thermoplastic


4
dt

t



melt injection through circular runner conduit occurs at creep
level Reynolds number, the flow is fully developed and
laminar. Hence well-established incompressible laminar flow
Similarly substitute Eqn.(27) in Eqn.(26) we get,
Hagen-Poiscuille velocity profile can be considered analogous
to represent velocity for thermoplastic melt injection through
T

d
1
T

C
= P
log +
kr

v
(
)
circular conduits. Parabolic velocity profile through circular
t

dt
r r

r

conduits varies from core to wall in such a way that core
2
2




-

velocity would be maximum while almost zero at the rigid
4
d
+ - (
)
3
1 P
log
+

( 2 2
R - r )


stationary wall. Accordingly velocity profile would be,
3
dt

4 r
4



1
- P

U
2
=
( 2 2
R - r )
T

d
1
T
4 d

4
(27)
C
= P
log +
kr
+
log
v
(
)
(
)
t

dt
r r

r
3 dt

2
Although Eqn. (27) neglects compressibility factor, herein we
2
r P

+
retain it right from equation of continuity to appreciate actual


4
expandability and compressibility phenomenon through each
cycle typically involved in injection moulding. Substituting
Eqn. (27) in Eqn. (25) we get,
2


4
d

(
)
d

( ) 1
T

T

log
P
log
kr
C

+
+
-
v






2
3
dt
dt
r r
r
t
r = -



1
- P

2





1
P

(
1
- P

d

2
2
R - r ) -
( 2 2
R - r ) (log )





t
4
4
dt


4

(29)
P

4 d

= -
-

( )


1
- P


log
+
r

( 2 2
R - r )

3 dt

r
r
r
4

Now consideringHagen-Poiscuille temperature profiles for
thermoplastic melt injection through circular conduit
( 2 2
- ) ( 2 2
R
r
R - r
P
) P d
-


+

(

log )

4
t


4

dt

T
- T =
(U )2
(30)
max
w
max
4k
P

4 d
= -
-

( ) P

log
+


3

dt



Eqn. 30featuresmaximum temperature at conduit core with an
almost constant streaming gradience, while in actual injection


consequent to concurrent cooling melt temperature reduces
4 d (


non-linearly, despite cooling melt streams keep moving ahead.
log )



(




Hence temperature profile is appropriately modified as,
2
2
- )
3
dt
R
r
= -

1 P
d




( ) P

log

-



4
dt

t



T - T =
(U)2
(30a)
w
4k


4 d (

-

1 P
log )



Where U =
( 2 2
R - r )




4
2
2
3
dt
r - R =

1 P
d

T

= wall temperature


( ) P

log

-


w
4
dt

t



Substituting Eqn. (27) in Eqn. (30a) we get,
1
- P


T =

(R -r ) 2
2
2
+ Tw
4k 4

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2




2
4
2 2
2


1
P



(r - R r
4
d
1
) P

T =


R - r
+ T
(log )


+


2

(
)2
2
2
w
4k 16









3
dt
r r

16









C

( 2 2
R - r


d

v
)2
2
(
P

2
2
R - r )2
2

+P (log ) -


P








T =

+ T
dt
64 k
t




w
2
64k

(31)

r = -
2
1 P


Now substituting Eqn. (31) in Eqn. (29), we get
4






2


( 3
2
4r - 2R r

4
d
1
)
2
2




( 2 2
R - r
4
d
1
)2
2
P
d

(log) +

+ P
(log)


( )
P


log
+
kr

+ Tw
3
dt

r r

r

64k



3 dt

r
16

dt








C ( 2 2
R - r




v
)2
2
P
(


2
2


R - r

d
)2
2
-






( )
P


+P
log
- C
+ T




64k
t



v
w


dt
t
64k








2




r = -
2
r = -
2

2
1
P
1 P





4
4


C ( 2 2
R - r )2
2
2



v
P
4 d



(




-

log )




As wall temperature variation throughout the cycle is very
64k
t

3 dt





nominal, it can be considered to be almost constant. Thus

( 2 2
2r - R

d
)
2
applying the condition, equation reduces to,

( )
P

P
log

-
-




dt
8


2
r =
2
2



(32)
4 d
2



( ) 1 r P

log
+

1
P


-




3 dt

r r 64 r ( R
r )2
2
2
)






4



C ( 2 2
R - r



d

v
)2
2
+
( )
P
P
log
-






dt
64k
t




Thus equating Eqn. (28) and (32) we get,




2
r = -
2
1 P



4 d




(log )



4




2
3
dt
R +
=


1 P d




( ) P
log

-




2
2



4
dt
t

4 d
( ) 1 r P

log
+


2

( 2 2
R - r )( 2
- r)
2


2
2
2

3 dt

r r 64




C R - r



v (
)
2


P
4
d









- (log )

64k
t


3
dt









C ( 2 2
R - r





d

v
)2
2
+
( )
P
P
log
-






( 2 2
2r - R

d
)
2


P
dt
64k
t



P

(log )

-
-







2



r = -
dt
8


2
1 P

2
1 P




4
4
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Rearranging the above equation
Rearranging the above equation


2
2

C
R
P
P d
P

( 2 2
R - r )2
2
2



v
P
4 d




(log)-



(


-

log )



64k
t


3
dt





4
dt
t






3
2

4 P
d

4 d
P


( 2 2

2r - R

d
)
2
-

(log) + (log)






( )
P

P
log

-
-









3
dt
3
dt
t


dt
8







2
2

P
d

d
P


2
R =
1 P


P
-

(log) +P (log)





dt

dt
t


4



=

3
2

2
2







R P
d
R P
P

4 d
-


(log)+

(







log )




8 dt
8 t




3
dt


-







2



1 P
d
4
d
P




-


(log)






(

)
P
log

-





4

3 dt



dt

t




2
Substituting the condition at the boundary, that is at wall r=R,
2


R
P
P d




(log)
P
-



the above equation simplifies into
4 dt

t





3
2
2
2
2







R
P
d
R
P
P

( 2R

4
d
d
)
2
-
+


(log)-

( ) -
( )
P

log
P
log

-












8
dt
8
t

3 dt

dt
8




2
3
2








1 P


4
P d
4
d
P


-

(log) + (log)


4










3
dt
3
dt
t

2
R


=



2




P
d

d
P


4 d
= P
-

(log) +
(log)

P



(


log )









dt
dt
t


3
dt





-


2




1 P
d
4
d
P


-


(log)





(



)
P
log


-



4
3 dt

dt

t





r=R
3
2






2

1
P
d
1
P
P





-


( 2R


4
d
d
)
2
(log )
-





( ) - ( )
P

log
P
log


-






4
dt
4
t
3 dt

dt
8



2


R



3
2

1 P d

1 P
P

2



+



(log )-
P
d









( ) P
4 d
-

-

( ) P

log
log



8 dt
8 t
dt

t
3 dt

2
R =
3
2





2

4
P
d
4
d
P
1 P
P
d
-

(log ) + (log )



( )

P


log
-




3 dt

3 dt

t

4 dt

t










2



P
d
= -
d
P

P

(log ) + P (log )



dt

dt
t








2
r=R
4 d
-


( ) P


log



3 dt


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IJRET: International Journal of Research in Engineering and Technology
eISSN: 2319-1163 | pISSN: 2321-7308
3
2


shear rate, viscosity becomes obvious as it is the ratio of shear
3 P
d

3 P
P

2
R


(log )-





stress at the wall to true shear rate at the wall of the capillary.
8 dt
8 t






3
2


4 P
d


R
-
=

( ) 4 d
+

( ) P

log
log




(36)

3 dt

3 dt

t

&R


2
P
d
= -


( )

d
+
( ) P

P
log
P
log


Accordingly viscosity for axisymmetric flow in terms of wall

dt

dt
t




shear stress and apparent shear rate is
2
4 d

-

( ) P

log





1
-
3n +1
3 dt


R
=


&

(37)
4n
a
3
2
4 P
d

-

( ) 4 d
+

( ) P

log
log



d ln

3 dt

3 dt

t

Where n= power law index/shear thinning index
R
n =


d ln &

2
P
d
a

( )

d
+
( ) P


-P
log
P
log




For n=1 the true and apparent viscosity values are identical.
dt

dt
t





For shear-thinning (Pseudo- plastic) n<1, this means that for
2
4 d
aqueous thermoplastic melts, true shear rate would always be
-


( ) P


log



3 dt


greater than apparent shear rate [12]. Thus Eqn (37) would
2
R =
(33)
2
now be,
3 P
P
d




( ) P

log
-


8 dt

t

4n
=
(38)
0


+
From Tait Equation we know that,
3n 1



dT
dP

Where
= Apparent viscosity
0

-b b exp b b - b T
+


3
4
( 4 5 4 )
dT
dt
dt
0.0894b
+ 0.0894


4

dt

(b exp b b - b T + P


Thus
adopting
Weissenberg-Rabinowitsch
correction
in
3
( 4 5 4 ) )






equation (33) the non-Newtonian behaviour of polymer melt is


1


+ 0.0894ln b + 0.0894b b - T


accommodated.
3
4
b
( 5
) dT
2


-


b + b T - b b

0
- .0894ln b exp


-b T - b + P dt
1
2
2
5
d
( 3
4 (
5 )
)
(


log ) =
dt
1 + 0.0894ln b + 0.0894b b - T

4 (
5
)
3


4. MATHEMATICAL VALIDATION OF THE
0
- .0894ln

(b exp-b T -b + P
3
4 (
5 )
)
RUNNER
EQUATION
USING
DIMENSIONAL
(34)
ANALYSIS
3.2 Weissenberg-Rabinowitsch Correction
Dimension of all physical parameters being a unique
combination of basic constituting physicalquantification can
Since thermoplastic melt is non-Newtonian type, herein
be expressed in terms of the fundamental dimensions (or base
inequality is implicit inabove Newtonian constitutive relations,
dimensions) M, L, and T - also these form 3-dimensional
reasoning
wallshear
rate
differencefor
non-Newtonian
vector space.Itmandates strategic relevance to choice of data
constitutive relations. Further to equate true non-newtonian
and
isadoptedherein
to
deduce
the
credibility
of
viscosity
is
obtained
from
Weissenberg-Rabinowitsch
derivedequations and thereon computations. Its most basic
correction [11]. Accordingly correct shear rate at the wall for a
benefit beinghomogeneity i.e., only commensurable equations
non-Newtonian thermoplastic could now be calculated from
could be substantiated by havingidentical dimensions on either
below equation,
sides. Accordingly dimensional analysis (also referred as Unit
Factor Method) is adapted to characterising Eqn. (33),
1
d ln &

a
&
= & 3 +

R
a 4

d ln
(35)

kg
R

1
-3
1
=
= M L
T = k =
3
m
The term in square brackets is the Weissenberg-Rabinowitsch
(WR) correction, by correcting apparent shear rate to true
__________________________________________________________________________________________
Volume: 02 Issue: 11 | Nov-2013, Available @ http://www.ijret.org
424

IJRET: International Journal of Research in Engineering and Technology
eISSN: 2319-1163 | pISSN: 2321-7308
N
kg
CONCLUSIONS
1
-1 -2
P =
=
= M L T
2
2
m
m - s
This manuscript features a step by step derivation towards a
3
3
m
-
m
1
3
-1 3 -1
computational model for determining runner dimension of a
b =
= M L
b =
= M L
1
2
kg
kg - k
plastic injection mould. On the basis of governing equations,
Weissenberg-Rabinowitsch
correction
for
non-Newtonian
nature of thermoplastic melt Eqn. (33) was derived as a
N
kg
1
-1 -2
b =
=
= M L T
function of thermoplastic melt properties such as viscosity and
3
2
2
m
m - s
density, injection moulding machinespecificationssuch as
maximum injection pressure and nozzle tip temperature as
1
-1
1
1
well as temporal parameter that feature the mould impression
b =
=
b = k =
dt = s = T
4
5
k
in totality owing to processing dynamics. Ultimately we
believe Eqn. (33)computational model would offer a definite
d (
-
value of runner dimension that might still diverge from perfect
log )
-
N s
kg
1
1
1
-1 -1
= T , = m = L and =
=
= M L T
2
or ideal design owing to computational rigour.
dt
m
m - s
2
2
2
REFERENCES
Consider L.H.S R = m = L
[1]
M. Lakkanna, Y. Nagamalappa and Nagaraja R (6 Nov
C o n sid e r R .H .S
2013) "Implementation of Conservation principles for

k g


k g

Runner
conduit
in
Plastic
Injection
Mould

2


2
k g
1
k g
1
1

m - s
m - s
-
x
x
+
x
x x

Design"International Journal of Engineering Research
3
2
m

- s
m
s
m


- s s
s
m





and Technology, vol. 2, no. 11, pp. 88-99




[2]
Z. Tadmor and C. G. Gogos (2006)Principles of

k g


k g

Polymer Processing, 2nd ed., New Jersey: WILEY

2


2
k g
1
k g
1
1

m - s
m - s
-
x
x
+
x x x

[3]
E.
Mitsoulis(2010)"Computational
Polymer
2
2
2
m - s
m
s
m - s
s
s
m





Processing", Modelling and simulation in polymers, A.




I. Purushotham,D.Gujrati, Ed., Wiley-VCH Verlag
2

k g
k g





GmbH
&
Co.
KGaA,
Weinheim,
Germany.

2
1

m - s
m - s

-
x
x

doi: 10.1002/9783527630257.ch4

k g
m - s
m






[4]
R.
Patani,
I.
Coccorullo,
V.
Speranza
and
G.
3

m




Titomanlio(12
September
2005)"Modelling
of
2

k g





morphology evolution in the injection moulding process
m - s


m - s
x
2

k g
1
k g
1




x
x -
x
of thermoplastic polymers", Progress in Polymer
k g
2
2
2
2

m


m - s
s
m
- s
s




Science, pp. 1185-1222




[5]
M. Lakkanna, R. Kadoli and Mohan Kumar G C
Upon simplifying
(2013)"Governing Equations to Inject Thermoplastic
2
2
2
2
kg
kg
kg
kg

Melt Through Runner conduit in a plastic injection
-
+
-
+

mould",
Proceedings
of
National
Conference
on
3
6
3
6
3
6
3
6
m - s m - s m - s m - s
Innovations in Mechanical Engineering, Madanapalle
2
kg

-
[6]
T AOsswald and J. P. Hernandez-Ortiz (2006), Polymer


3
6
m - s
processing
modelling
and
simulation,
Hanser
Publishers. Germany
2
kg



[7]
Mahmood, S. Parveen, A. Ara and N. Khan(15 January
5
6
m - s
2009) "Exact analytic solutions for the unsteady flow of
a non-Newtonian fluid between two cylinders with
2
kg
fractional
derivative
model"
Communications
in
Nonlinear Science and Numerical Simulation pp. 3309-
3
6
m - s
2
2
=
= m = L
3319
2
kg
[8]
L. l. Grange, G. Greyvenstein, W de Kock and J. Meyer
5
6
m - s
(1993), "A numerical model for solving polymer melt
flow", R&D Journal, Vol. 9, Issue 2, pp. 12-17
Since LHS = RHS, Runner equation has been verified
[9]
S.
Kumar
and
S.
Kumar
(8
March
2009)"A
dimensionally.
Mathematical
Model
for
Newtonian
and
Non-
Newtonian Flow" Indian Journal of Biomechanics, pp.
191-195
__________________________________________________________________________________________
Volume: 02 Issue: 11 | Nov-2013, Available @ http://www.ijret.org
425

IJRET: International Journal of Research in Engineering and Technology
eISSN: 2319-1163 | pISSN: 2321-7308
[10]
F. Pinho and J. Whitelaw (May 1990)"Flow of Non-
NOMENCLATURE:
Newtonian fluids through pipe" Journal of Non-
Newtonian Fluid Mechanics, Vol 25, pp. 129-144
m
Mass
Kg
[11]
P C Beaupre (2002)"A Comparison of the axisymmetric
V
Volume
3
m
& planar elongation viscosities of a polymer" , MSc
P
Pressure
2
Mech.
Engg.
Thesis,
Michigan
Technological
Kgf / m
University, USA
T
Temperature
K
[12]
N. Martins (2009), "Rheological investigation of iron
T
Wall Temperature
K
based feedstock for the metal injection moulding
w
process," Dubendorf, Switzerland
K
Thermal conductivity
W/m
A
Cross-section area
2
m
BIOGRAPHIES
R
Runner radius
m
Muralidhar
Lakkanna
B.E(Mech)'96,
ur
Linear velocity
m / s
Post-Dipl-Tool
Design'99,
M.Mktg
U
uur
Mngt'03, M.Tech(Tool Engg)'04, M.Phil
U
Velocity in radial direction
m / s
r
(ToolroomMngt)'05, PG-SQC'09 Since 16
uuur
U
Velocity in tangential direction
m / s
years actively engaged in tool, die and

uuur
mould
manufacturing
has
designed,
U
Velocity in arbitrary direction
m / s

developed & commissioned more than
a
Acceleration
2
5000 high value tooling projects. Research
m / s
uur
interests are high performance tools, dies & moulds, plastic
M
Linear momentum
Kg - m / s
injection mould design, plastic injection mould mechanics,
ur
H
Angular momentum
Kg - m / s
computational plastic injection dynamics, etc., Currently
National Tool, Die & Mould Making Consultant, Regd at
I
Moment of inertia
2
Kg - m
Ministry of MSME, Government of India and Tool, Die &
e
Specific total energy
KJ / Kg
Mould Technology Innovation & Management (TIMEIS)
Expert, Regd at Department of Science & Technology,
u
Specific internal energy
KJ / Kg
Government of India
T
Resultant torque
N - m
[email protected]
M
Resultant moment
N - m
Yashwanth Nagamallappa B.E (Mech)

Resultant force
2
'11 Presently studying M.Tech in Tool
F
N / m
Engineering at Government Toolroom&
F
Force acting in radial direction
2
N / m
Training
Centre,
Bangalore
Possess
r
2
research interests in injection mould
F
Force acting in tangential

N / m
design arena
direction
2
[email protected]
F
Force acting in arbitrary
N / m
direction
R Nagaraja B.E (Mech)'88,M.Tech(Tool
Q
&
Rate of heat transfer
KW
Engg)'92 Since 21 years training at PG
q
Rate of heat transfer per unit
KW
level in tool design arena like plastic
mass
moulds, advance moulding techniques,
W
&
Rate of work done by viscous
KW
press tools, component materials, die
v
forces
casting, jigs & fixtures, etc. Currently in-
W
&
Rate of work done by pressure
KW
charge principal for PG&PD studies at
p
forces
GT&TC, Government of Karnataka, Bangalore
dS
Entropy change
KJ / Kg
[email protected]
C
Specific heat at constant
KJ / KgK
v
volume
C
Specific heat at constant
KJ / KgK
p
pressure
r
n
Unit normal vector
r
r
Position vector
n
Shear thinning index
__________________________________________________________________________________________
Volume: 02 Issue: 11 | Nov-2013, Available @ http://www.ijret.org
426

IJRET: International Journal of Research in Engineering and Technology
eISSN: 2319-1163 | pISSN: 2321-7308
GREEK SYMBOLS

Density
3
Kg / m

Specific volume
3
m / Kg
ur

Angular Velocity
m / s

Angular acceleration
2
m / s
&

True shear rate
1/ s
R
&

Apparent shear rate
1/ s
a

Surface force
2
N / m

Shear stress
2
N / m

True Viscosity
2
N - s / m

Apparent viscosity
2
-
0
N s / m

Viscous dissipation function
__________________________________________________________________________________________
Volume: 02 Issue: 11 | Nov-2013, Available @ http://www.ijret.org
427