fluid flow and heat transfer characteristics across an internally heated finned … · ·...
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J. Energy Power Sources Vol. 1, No. 6, 2014, pp. 296-303 Received: August 31, 2014, Published: December 30, 2014
Journal of Energy and Power Sources
www.ethanpublishing.com
Fluid Flow and Heat Transfer Characteristics across an
Internally Heated Finned Duct
Sofiane Touahri and Toufik Boufendi
Energy Physics Laboratory, Physics Dept., Constantine1 University, Constantine, Algeria
Corresponding author: Toufik Boufendi ([email protected])
Abstract: We numerically study the fluid flow and the heat transfer characteristics in a horizontal pipe equipped by longitudinal and transversal fins attached on its internal surface used in several areas of thermal sciences. The longitudinal fins number is: Two vertical, four and eight, while the transversal fins number is eight. The considered heights fins are 0.12 and 0.24 cm. The convection in the fluid domain is conjugated to conduction in the pipe and in the fins thickness. The physical properties of the fluid are thermal dependant and the heat losses to the ambient are considered. The model equations are numerically solved by a finite volume method with a second order discretization. As expected for the longitudinal fins, the axial Nusselt number increase with increasing of number and height of fins. For Gr = 5.1 × 105, the Nusselt number without fins is equal to 13.144. The introduce of longitudinal fins gives a Nusselt number equal to 16.83, 22.52 and 32.06 for two vertical, four and eight fins respectively. The participation of fins located in the lower part of the tube on the improvement of heat transfer is higher than the upper fins. The longitudinal fins participate directly on increase the heat transfer; this is justified by the large local Nusselt number along the fins interface. This participation is moderate in the case of transverse fins, these latter are used to mix the fluid for increase the local Nusselt number in the axial sections following the transversal fins. Keywords: Conjugate heat transfer, mixed convection, internal fins, numerical simulation.
Nomenclature:
D Pipe diameter, (m)
Gr* Modified Grashof number, Gr* = gβGDi
5
Ksν2
G* Non-dimensional heat generation= Ks
Re0 Pr0
H Height of the fin, (m)
h Heat transfer coefficient, (W/m2K)
K Thermal conductivity, (W/mK)
K* Non-dimensional thermal conductivity= K
K0
L* Non-dimensional pipe length=
L
Di
Nu(θ,z) Local Nusselt number = K K0⁄
Nu Nusselt number (dimensionless)
P Pressure, (Pa)
P* Non-dimensional pressure = (P-P0) ρ0V02
Pr Prandtl number= ν α⁄
q Heat flux, (W/m2)
R Radial coordinate, (m)
Re Reynolds number = V0Di ν0⁄
r* Non-dimensional radial coordinate= r Di⁄
T Temperature, (°C)
T* Non-dimensional temperature= (T-T0) (GDi2 Ks )
V Velocity, (m/s)
V* Non-dimensional velocity= V V0⁄
Y+ Dimensionless wall distance
Z Axial coordinate, (m)
Z* Non-dimensional axial coordinate = z Di⁄
1. Introduction
Finned tubes are often used in many engineering
sectors for extend the contact surface between the tube
wall and the fluid and improve the heat transfer; the
researchers have studied the problem of optimizing the
shape and geometry of attached fins in order to increase
heat transfer effectiveness. Most of the studies
performed on this optimization consider longitudinal
fins which have symmetrical lateral profiles; this
assumption simplifies the treatment of the problem
with regard to the boundary conditions and gives
Fluid Flow and Heat Transfer Characteristics across an Internally Heated Finned Duct
297
symmetrical results concerning velocity and
temperature profiles. Many investigations, both
experimental and numerical, have been conducted for
different kinds of internally finned tubes. An analytical
model for fully developed turbulent air flow in
internally finned tubes and annuli was presented by [1].
In this study, the longitudinal fins are attached in the
inner wall, and the constant heat flux, is the thermal
boundary condition at the inner surface. The results
concern the heat transfer and the pressure drop
coefficients. A combined numerical and experimental
study of plate-fin and tube heat exchangers was
examined by [2]. The detailed numerical results of
pressure drop and heat transfer coefficient are
presented. A numerical study on hydrodynamically
fully developed, thermally developing flow inside
circular tubes with internal longitudinal fins having
tapered lateral profiles was conducted by [3]. The
results showed significant heat transfer enhancement
with the inclusion of internal fins. Water and engine oil
were assumed as fluids in their numerical studies, and
they concluded water to be a better coolant as
compared to engine oil. In Ref. [4], a numerical study
of thermally developing flow in an elliptical duct with
four longitudinal internal fins of zero thickness is
considered. A control volume based on finite
difference technique was used and an optimum value of
the local Nusselt number was obtained as a function of
the fin height. Ref. [5] performs an experimental
analysis of heat transfer in an internally finned tube, the
experimental results were compared with results from
the smooth channel tube, and a significant
improvement in heat transfer was observed for
internally finned cases. Similar studies were also
examined numerically and experimentally by [6-10].
In this work, we studied numerically the heat
transfer by mixed convection in horizontal pipe
equipped by longitudinal and transversal attached fins
on its internal wall. The mixed convection is conjugate
to thermal conduction in the pipe and fins walls. The
physical properties of the fluid are thermo- dependent
and the heat losses with the external environment are
considered. The objective of our work is study the
enhancement gives to the heat transfer by using
different shape of fins.
2. The Geometry and Mathematical Model
Fig. 1 illustrates the problem geometry. We consider
a long horizontal pipe having a length L = 1 m, an
outside diameter D0 = 1 cm and an inside diameter Di =
0.96 cm, this later is equipped by longitudinal and
transversal attached fins on its internal surface. The
longitudinal fins are fixed at (θ = 0, π/4, π/2, 3π/4, π,
5π/4, 3π/2 and 7π/4) while the transversal fins are
attached at eight axial stations from z* = 9.4404 to z* =
82.3594. The pipe and fins are made of Inconel having
a thermal conductivity Ks = 20 W m-1 K-1. An electric
current passing along the pipe (in the solid thickness)
produced a heat generation by Joulean effect. This heat
is transferred to distilled water flow in the pipe. At the
entrance the flow is of Poiseuille type with an average
axial velocity equal to 7.2 × 10-2 m s-1 and a constant
temperature of 15 °C. The density is a linear function of
temperature and the Boussinesq approximation is
adopted. The physical principles involved in this
problem are well modeled by the following non
dimensional conservation partial differential equations
with their initial and boundaries conditions.
(a)
(b)
Fig. 1 Pipe with (a) longitudinal fin; (b) transversal fin.
*eD*
iD
*eD
*iD
Fluid Flow and Heat Transfer Characteristics across an Internally Heated Finned Duct
298
2.1 Modeling Equations
At 0* =t , 0**** ==== TVVV zr θ (1)
At 0* >t
Mass conservation equation:
( ) 011
*
**
***
**=
∂∂+
∂∂+
∂∂
z
VV
rVr
rrz
r θθ (2)
Radial momentum conservation equation:
( ) ( )
( )( ) ( ) ( )
∂∂+−
∂∂+
∂∂
++∂∂−=−
∂∂
+∂∂+
∂∂+
∂∂ ∗
***
**
***
**0
*20
*0
*
*
*
2***
*
***
*****
*
11
Re
1
)(cosRe
11
zrrrr
rz
rrrr
zrrr
rr
TGr
r
P
r
VVV
z
VVr
VVrrrt
V
τττθ
τ
θ
θ
θθθ
θ
θ
(3)
Angular momentum conservation equation:
( ) ( )( )
( ) ( ) ( )
∂∂+
∂∂+
∂∂
+−∂∂−=+
∂∂
+∂∂+
∂∂+
∂∂
∗
∗∗
**
**
*2**2*
0
*20
*0
**
****
*
**
******
11
Re
1
sinRe
1
11
zr
rz
r
zrr
rr
TGrP
rr
VVVV
z
VVr
VVrrrt
V
θθθθ
θθ
θθθθ
ττθ
τ
θθ
θ
(4)
Axial momentum conservation equation:
( ) ( )( )
( ) ( ) ( )
∂∂+
∂∂+
∂∂
+∂∂−=
∂∂
+∂∂+
∂∂+
∂∂
**
**
****
0
*
***
*
***
******
*
11
Re
1
11
zzzrz
zz
zzrz
zrr
rr
z
PVV
z
VVr
VVrrrt
V
ττθ
τ
θ
θ
θ
(5)
Energy Conservation Equation
( ) ( )( )
( ) ( ) ( )
∂∂+
∂∂+
∂∂
−=∂∂
+∂∂+
∂∂+
∂∂
∗
∗
zr
z
r
qz
qr
qrrrPrRe
GTVz
TVr
TVrrrt
T
**
***
**00
****
***
*****
*
111
11
θ
θ
θ
θ
(6)
where ( )
=fluidthein0
solidtheinPrRe/ 00*
* SKG
The viscous stress tensor components are:
*
*** 2
r
Vrrr ∂
∂= μτ ,
∂∂
+
∂∂==
θμττ θ
θθ
*
**
*
***** 1 r
rrV
rr
V
rr
+
∂∂
=*
**
*** 1
2r
VV
rr
θμτ θ
θθ ,
∂∂
+∂∂
==θ
μττ θθθ
*
**
**** 1 z
zzV
rz
V (7)
2∗
∗∗∗
∂
∂=z
Vzzz μτ ,
∂∂+
∂∂==
*
*
**
**** 1
z
V
rr
V rzzrrz μττ
The heat fluxes are:
*
***
*
*
*
**
*
*** and,
z
TKq
T
r
Kq
r
TKq Zr ∂
∂−=∂∂−=
∂∂−=
θθ (8)
2.2 The Boundaries Conditions
The previous differential equations are solved with
the following boundaries conditions:
(1) At the pipe entrance: z* = 0
• In the fluid domain: 5.0≤≤0 *r and πθ 2≤≤0
)4-1(2,0 2***** rVTVV zr ==== θ (9)
• In the solid domain: 5208.0≤≤5.0 *r and πθ 2≤≤0
0**** ==== TVVV zr θ (10)
(2) At the pipe exit: z* = 104.17
• In the fluid domain: 5.0≤≤0 *r and πθ 2≤≤0
0*
**
**
*
*
*
*
*
=
∂∂
∂∂=
∂∂=
∂∂=
∂∂
z
TK
zz
V
z
V
z
V zr θ (11)
• In the solid domain: 5208.0≤≤5.0 *r and
πθ 2≤≤0
0*
**
**** =
∂∂
∂∂===
z
TK
zVVV zr θ (12)
(3) At the outer wall of the pipe: r* = 0.5208
( )
+=∂∂−
===*
0*
**
*** 0
TK
Dhh
r
TK
VVV
icr
zr θ (13)
( )( )∞2∞
2 TTTThr ++= εσ (14)
The emissivity of the outer wall ε is arbitrarily
chosen to 0.9 while hc is derived from the correlation of
[11] valid for all Pr and for Rayleigh numbers in the
range 10−6 ≤ Ra ≤ 109.
[ ]
( )( )2
27/816/96/1 Pr/559.01/387.06.0
/
++=
=
air
airic
Ra
KDhNu
(15)
with
( )[ ]airairair
airair
oo PrDTzRTg
Ra αννα
θβ/,
-,, 3∞ == (16)
In Eq. (16) the thermophysical properties of the air
ambient are evaluated at the local film temperature
given as: [ ] 2),,( ∞TzRTT ofilm += θ .
In our calculations we have considered the solid as a
fluid with a dynamic viscosity equal to 1030. This very
Fluid Flow and Heat Transfer Characteristics across an Internally Heated Finned Duct
299
large viscosity within the solid domain ensures that the
velocity of this part remains null and consequently the
heat transfer is only by conduction deducted from Eq.
(6).
2.3 The Nusselt Numbers
At the cylindrical solid-fluid interface, the local
Nusselt number is defined as:
( )
∂∂== =
)(-),,5.0(
),(),(
****
5.0***
0
**
*
zTzT
rTK
K
DzhzNu
b
ri
θθ
θ (17)
The axial Nusselt number for the cylindrical
interface is:
∫2
0
** ),(2
1)(
π
θθπ
dzNuzNu = (18)
Finally, we can define an average Nusselt number
for the whole cylindrical solid-fluid interface:
( )( ) ∫∫2
0
17.104
0
**),(17.1042
1π
θθπ
ddzzNuNuA = (19)
At the longitudinal fin interface, the local Nusselt
number is defined as
( )( )
∂∂==
=
)(),,(
),(),(
*****
***
0
****
zTzrT
TrK
K
DzrhzrNu
mfin
h fin
θ
θ θθ (20)
The axial Nusselt number for the longitudinal fin is:
( ) ( )*
*****
***
*
****
*
∫
∫*
**
*
**
)(),,(
1
),(1
)(
drzTzrT
TrK
H
drzrNuH
zNu
i
i
fin
i
i
R
HR mfin
R
HR
∂∂=
=
=
θ
θ θθ (21)
The average Nusselt number for the longitudinal fin
interface is defined as:
∫*
0
***
)(1
L
A dzzNuL
Nu =
(22)
At the transversal fin interface, the local Nusselt
number is
( )( )
∂∂==
=
)(),,(
),(),(
*****
**
0
**
zTzrT
zTK
K
DrhrNu
mfin
zzh fin
θθθ (23)
The axial Nusselt number for the transversal fin
interface is defined as:
( )( )( )
∫∫2
0
******
**
**
*
**
**
)(),,(2
1)(
π
θθπ
i
i
fin
R
HR mfin
zzddr
zTzrT
zTK
HzNu
∂∂=
= (24)
3. Numerical Resolution
For the numerical solution of modeling equations,
we used the finite volume method well described by
[12]; the using of this method involves the
discretization of the physical domain into a discrete
domain constituted of finite volumes where the
modeling equations are discretized in a typical volume.
We used a temporal discretization with a truncation
error of ∆t*2 order. The convective and nonlinear
terms have been discretized according to the
Adams-Bashforth numerical scheme, with a truncation
error of ∆t*2 order, the diffusive and pressure terms
are implicit. Regarding the spatial discretization, we
used the central differences pattern with a truncation
error of ∆r* 2, ∆θ 2 and ∆z* 2
order. So our
spatio-temporal discretization is second order. The
mesh used contains 52 × 88 × 162 points in the radial,
angular and axial directions. The considered time step
is ∆t*=5 × 10-4and the time marching is pursued until
the steady state is reached. The steady state is
controlled by the satisfaction of the global mass and
energy balances as well as the leveling off of the time
evolution of the hydrodynamic and thermal fields.
The accuracy of the results of our numerical code has
been tested by the comparison of our results with those
of other researchers. A comparison with the results of
[13] who studied the non conjugate and the conjugate
mixed convection heat transfer in a pipe with constant
physical properties of the fluid. Some of their results
concern the simultaneously developing heat transfer
and fluid flow in a uniformly heated inclined pipe
( 40=α ).The controlling parameters of the problem
are: Re = 500, Pr = 7.0, Gr = 104 and 106, 90=iDL ,
583.0=io DR , 700 =KKs . The used grid is 40 × 36
× 182 in the r*, θ and z* directions, respectively. We
reproduced the results of the cited reference with the
Fluid Flow and Heat Transfer Characteristics across an Internally Heated Finned Duct
300
Fig. 2 Axial evolution of the circumferentially mean axial Nusselt number; A comparison with the result of [13].
first order calculus code concerning the conjugate and
non conjugate mixed convection. In Fig. 2, we
illustrate the axial evolution of the circumferentially
averaged Nusselt number. It is seen that there is a good
agreement between our results and theirs.
4. Results and Discussion
4.1 Development of the Secondary Flow
All the results presented in this paper were
calculated for Reynolds number, Re = 606.85, and the
Prandtl number, Pr = 8.082, while the Grashof number
is equal to 5.1 × 105. The obtained flow for the studied
cases is characterized by a main flow along the axial
direction and a secondary flow influenced by the
density variation with temperature, which occurs in the
plane (r* − θ), this flows are presented for eight fins in
the longitudinal case and four fins in the transversal
case. In the reference case (forced convection) the
transverse motion is nonexistent; the only flow is in
axial direction. In the presence of volumetric heating in
the pipe and fins wall, a transverse flow exists and
explained as follows: The hot fluid moves along the
hot wall from the bottom of the outer tube (θ=π)
upwards (θ=0) and moves down from the top to the
bottom along the centre of tube. The vertical plane
passing through the angles (θ=0) and (θ=π) is a plane
of symmetry. The transverse flow in the (r*, θ) plane is
represented by counter rotating cells; the cells number
is proportional to longitudinal fins number.
Fig. 3 The secondary flow vectors at the pipe exit for longitudinal fins.
Fig. 4 The secondary flow vectors at the fourth transversal fin section.
In Fig. 3, we present the secondary flow vectors at the
pipe exit for the case of longitudinal fins. For the case
of transversal fins, the position of fins is only in
selected axial sections. Far from these sections, the
secondary motion is similar to that of simple
cylindrical pipe. The Fig. 4, illustrates the secondary
flow vectors at the fourth transversal fin section (z* =
40.6914).
4.2 The Axial Flow Development
At the entrance, the axial flow is axisymmetric with
the maximum velocity in the center of the pipe. In the
presence of volumetric heating in the pipe and the fins
walls, the configuration of the axial flow completely
changes because the secondary flow causes an angular
Z*=104.17
Z*=40.6914
0,000 0,003 0,006 0,009 0,012 0,015 0,018 0,021 0,0240
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
30
35
40
45
50Ouzzane and Galanis [13]: Gr =104 non conjugate case
Gr =106 non conjugate case
Gr =104 conjugate case
Our results: Gr =104 non conjugate case
Gr =106 non conjugate case
Gr =104 conjugate case Nu
z*/(RePr)
Fluid Flow and Heat Transfer Characteristics across an Internally Heated Finned Duct
301
Fig. 5 Axial velocity profiles at the pipe exit for longitudinal fins.
Fig. 6 Axial velocity profiles at the fourth transversal fin section.
variation which has a direct influence on the
distribution of axisymmetric axial flow. In Fig. 5, the
axial flow is presented at the exit of the pipe for the
case of longitudinal fins (height H* = 0.1875). In Fig. 6,
the axial flow is presented at the fourth transversal fin
section (z* = 40.6914). Through these figures, it is clear
that the axial velocity is null in the fins walls.
4.3 Development of the Temperature Field
In the presence of volumetric heating, a transverse
with flow exists and thus changes the axisymmetric
distribution of fluid and gives it an angular variation,
this variation explained as follows: The hot fluid near from
Fig. 7 The isotherms at the pipe exit for longitudinal fins.
Fig. 8 The isotherms at the fourth transversal fin section.
the pipe and fins walls moves upwards under the
buoyancy force effect, the relatively cold fluid
descends down in de centre of the pipe. A permanent
generation of heat in the pipe and the fins walls
imposes a continuous increase of the temperature of the
fluid up to the exit of the pipe.
The obtained results show that at given section, the
maximum fluid temperature T* is all the time located at
r* = 0.5 and θ = 0 (top of solid-fluid interface), because
the hot fluid is driven by the secondary motion towards
the top of the pipe. The isotherms are presented at the
exit of the pipe for the case of longitudinal fins having
height H* equal to 0.1875 in Fig. 7 and at the fourth
transversal fin section (z* = 40.6914) in Fig. 8.
Z*=104.17
2.3672.2682.1702.0711.9731.8741.7751.6771.5781.4791.3811.2821.1841.0850.9860.8880.7890.6900.5920.4930.3950.2960.1970.0990.000
Z*=40.6914
3.1002.9712.8422.7132.5832.4542.3252.1962.0671.9381.8081.6791.5501.4211.2921.1631.0330.9040.7750.6460.5170.3880.2580.1290.000
Z*=104.17
0.1710.1640.1570.1500.1430.1350.1280.1210.1140.1070.1000.0930.0860.0780.0710.0640.0570.0500.0430.0360.0290.0210.0140.0070.000
Z*=40.6914
0.1700.1630.1560.1490.1420.1350.1280.1210.1140.1060.0990.0920.0850.0780.0710.0640.0570.0500.0430.0350.0280.0210.0140.0070.000
Fluid Flow and Heat Transfer Characteristics across an Internally Heated Finned Duct
302
4.4 The Heat Transfer
The phenomenon of heat transfer has been
characterized in terms of Nusselt numbers calculated at
the inner wall of the pipe, obtained by Eq. (17) and
those calculated at the fins wall, obtained by Eq. (20)
and Eq. (23). The variation of local Nusselt number at
cylindrical interface is presented in Fig. 9 for the case
of longitudinal fins and in Fig. 10 for the case of
transversal fins. The local Nusselt number of
longitudinal fins placed in the right side of the pipe at
(θ = 0, π/4, π/2, 3π/4, π) is shown in Fig. 11. The local
Fig. 9 The local Nusselt number variation at cylindrical interface for longitudinal fins case.
Fig. 10 The local Nusselt number variation at cylindrical interface for transversal fins case.
Nusselt number takes a maximum value equal to 63.56
on the fin placed at (θ = 3π/4), z* = 104.17 and r* =
0.3841. The comparison of axial Nusselt numbers
between finned tube and smooth tube is shown in Fig.
12. Quantitatively, there is a large increase in the axial
Nusselt when the number of fins is increased. At the
exit of the pipe, the axial Nusselt number is equal to
15.79, 23.47, 31.34 and 47.31 for the cases: Smooth tube,
(a)
(b)
(c)
(d)
48.73046.70044.66942.63940.60838.57836.54734.51732.48730.45628.42626.39524.36522.33520.30418.27416.24314.21312.18210.152
8.1226.0914.0612.0300.000
48.83046.79544.76142.72640.69238.65736.62334.58832.55330.51928.48426.45024.41522.38020.34618.31116.27714.24212.20810.1738.1386.1044.0692.0350.000
6 3 . 5 6
5 8 . 2 6
5 2 . 9 6
4 7 . 6 7
4 2 . 3 7
3 7 . 0 7
3 1 . 7 8
2 6 . 4 8
2 1 . 1 8
1 5 . 8 9
1 0 . 5 9
5 . 2 9
0 . 0 0
6 3 . 5 6
5 8 . 2 6
5 2 . 9 6
4 7 . 6 7
4 2 . 3 7
3 7 . 0 7
3 1 . 7 8
2 6 . 4 8
2 1 . 1 8
1 5 . 8 9
1 0 . 5 9
5 . 2 9
0 . 0 0
6 3 . 5 6
5 8 . 2 6
5 2 . 9 6
4 7 . 6 7
4 2 . 3 7
3 7 . 0 7
3 1 . 7 8
2 6 . 4 8
2 1 . 1 8
1 5 . 8 9
1 0 . 5 9
5 . 2 9
0 . 0 0
6 3 . 5 6
5 8 . 2 6
5 2 . 9 6
4 7 . 6 7
4 2 . 3 7
3 7 . 0 7
3 1 . 7 8
2 6 . 4 8
2 1 . 1 8
1 5 . 8 9
1 0 . 5 9
5 . 2 9
0 . 0 0
Fluid Flow and Heat Transfer Characteristics across an Internally Heated Finned Duct
303
(e)
Fig. 11 The local Nusselt number variation at longitudinal fins interface: (a) fin placed at (θ = 0); (b) at (θ = π/4); (c) at (θ = π/2); (d) at (θ = 3π/4); (e) fin placed at (θ = π).
0 10 20 30 40 50 60 70 80 90 100
10
15
20
25
30
35
40
45
500 10 20 30 40 50 60 70 80 90 100
10
15
20
25
30
35
40
45
50
Axi
al N
usse
lt n
umbe
r
Z*
smooth tube
2 fins H*=0.125 4 fins H*=0.125 8 fins H*=0.125
Fig. 12 Variation of the axial Nusselt number for the different longitudinal fins studied cases.
two vertical fins, four fins and eight fins respectively.
The average Nusselt numbers for these cases are
13.144, 16.830, 22.523 and 32.066.
5. Conclusions
This study considers the numerical simulation of the
three dimensional mixed convection heat transfer in a
horizontal pipe equipped by internal, longitudinal and
transversal fins. The pipe and fins are heated by an
electrical intensity passing through its small thickness.
The obtained results show that the longitudinal fins
participate directly in improving the heat transfer; this
is justified by the high local Nusselt number at the
interface of longitudinal fins. By against, the transverse
fins participate in an indirect way in improving the heat
transfer their location facing the flow allowed to
rearrange the structure of the fluid for each passage
through the fins, which is used to mix the fluid and to
increase the heat transfer to the cylindrical interfaces.
The number and fins height are also important factors
in improving the heat transfer.
References
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6 3 . 5 6
5 8 . 2 6
5 2 . 9 6
4 7 . 6 7
4 2 . 3 7
3 7 . 0 7
3 1 . 7 8
2 6 . 4 8
2 1 . 1 8
1 5 . 8 9
1 0 . 5 9
5 . 2 9
0 . 0 0